domain listlengths 0 3 | difficulty float64 5 5 | problem stringlengths 48 1.74k | solution stringlengths 3 5.84k | answer stringlengths 1 340 | source stringclasses 34
values |
|---|---|---|---|---|---|
[
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 5 | Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible. |
To solve this problem, we need to find the smallest positive integer \( n \) such that all 73 fractions of the form \(\frac{k}{n+k+20}\) for \( k = 19, 20, \ldots, 91 \) are irreducible. A fraction \(\frac{a}{b}\) is irreducible if and only if \(\gcd(a, b) = 1\).
For the fractions \(\frac{k}{n+k+20}\) to be irreducib... | 95 | cono_sur_olympiad |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 5 | Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that
$a + b + c = 3k + 1$,
$ab + bc + ca = 3k^2 + 2k$.
Slovakia |
To solve this problem, we need to find all triples of positive integers \((a, b, c)\) such that:
1. \(a + b + c = 3k + 1\),
2. \(ab + bc + ca = 3k^2 + 2k\).
First, we observe that the problem is set with symmetric conditions which often suggest that \((a, b, c)\) could take a symmetric form. Therefore, let's assume ... | (k+1, k, k) | czech-polish-slovak matches |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$
if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$. |
Given the equation \( x^2y^2 + xy + 1 = 3y^2 \), we need to determine the largest and smallest values of the fraction \( F = \frac{y-x}{x+4y} \).
### Step 1: Analyze the Given Equation
To simplify the problem, we first explore the given equation:
\[
x^2y^2 + xy + 1 = 3y^2
\]
Rearranging terms, we get:
\[
x^2y^2 + xy... | $0 \leq \frac{y-x}{x+4y} \leq 4$ | czech-polish-slovak matches |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Number Theory -> Congruences"
] | 5 | Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$ |
We are tasked with finding all quadruples of positive integers \((p, q, a, b)\), where \(p\) and \(q\) are prime numbers, \(a > 1\), and they satisfy the equation:
\[
p^a = 1 + 5q^b.
\]
Given the integers involved, we will first check small values for \(p\), and test if they can satisfy the equation.
### Case 1: \(... | (2, 3, 4, 1) \text{ and } (3, 2, 4, 4) | junior_balkan_mo |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 5 | Determine the maximum number of bishops that we can place in a $8 \times 8$ chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop.
Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals ... | To solve this problem, we need to determine the maximum number of bishops that can be placed on an \(8 \times 8\) chessboard such that each bishop is threatened by at most one other bishop. The key here is to understand the attacking capability of bishops on a chessboard.
Bishops move diagonally, which means they cont... | 20 | bero_American |
[
"Mathematics -> Geometry -> Plane Geometry -> Area",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares.
Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$. |
Consider a \(9 \times 7\) rectangle that needs to be tiled using two types of tiles: L-shaped tiles, which cover three unit squares, and \(2 \times 2\) square tiles, which cover four unit squares. We need to find all possible values of \(n\), the number of \(2 \times 2\) square tiles used in such a tiling configuratio... | 0 \text{ and } 3 | junior_balkan_mo |
[
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 5 | For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$.
[i]Romania[/i] |
To find the greatest common divisor (GCD) of the sequence \( A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2} \) for each nonnegative integer \( n \), we begin by examining the expression for \( A_n \):
\[
A_n = 2^{3n} + 3^{6n+2} + 5^{6n+2}.
\]
Our goal is to determine \( \gcd(A_0, A_1, \ldots, A_{1999}) \).
### Step 1: Check Di... | 7 | junior_balkan_mo |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 5 | Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$. |
We are tasked with finding all triples of primes \((p, q, r)\) that satisfy the equation:
\[
3p^4 - 5q^4 - 4r^2 = 26.
\]
We begin by analyzing the equation with respect to the properties of prime numbers.
1. **Testing Small Primes**:
- Since \( p^4 \), \( q^4 \), and \( r^2 \) grow rapidly for primes larger than... | (5, 3, 19) | junior_balkan_mo |
[
"Mathematics -> Algebra -> Algebra -> Sequences and Series",
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 5 | A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\geq1$, we have
$$a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.$$
Calculate the value of $\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}$. |
We start by examining the sequence \((a_n)\) given by the recurrence relations \(a_0 = 1\) and \(a_1 = 2015\), with the following recursive formula for \(n \geq 1\):
\[
a_{n+1} = \frac{n-1}{n+1}a_n - \frac{n-2}{n^2+n}a_{n-1}.
\]
The goal is to evaluate the expression:
\[
S = \frac{a_1}{a_2} - \frac{a_2}{a_3} + \fra... | 3021 | centroamerican |
[
"Mathematics -> Number Theory -> Factorization",
"Mathematics -> Number Theory -> Prime Numbers"
] | 5 | Find all the positive perfect cubes that are not divisible by $10$ such that the number obtained by erasing the last three digits is also a perfect cube. |
To solve the given problem, we need to find all positive perfect cubes that are not divisible by \(10\) and have the property that when the last three digits are erased, the resulting number is also a perfect cube.
1. **Understanding the Cube Condition**: Let \( n^3 \) be a perfect cube such that \( n^3 \equiv 0 \pmo... | 1331 \text{ and } 1728 | jbmo_shortlists |
[
"Mathematics -> Number Theory -> Other",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either strictly greater than $n$. | Let \( n \) be an integer such that \( n \geq 1 \). We need to find the smallest integer \( k \geq 2 \) such that for any set of \( k \) real numbers, there exist at least two numbers, say \( x \) and \( y \), where either \( |x - y| < \frac{1}{n} \) or \( |x - y| > n \).
To solve this problem, we will employ a combin... | n^2 + 2 | math_olympiad_for_the_french_speaking |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | Find all nonnegative integers $a, b, c$ such that
$$\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}.$$ |
Given the equation:
\[
\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}
\]
where \(a\), \(b\), and \(c\) are nonnegative integers, we need to find all possible combinations of \((a, b, c)\) that satisfy this condition.
### Step 1: Analyze the equation
Since \(a\), \(b\), and \(c\) are nonnegative integers, \(\sqrt{a}... | (0, 0, 2014) | nordic |
[
"Mathematics -> Number Theory -> Factorization"
] | 5 | Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$ |
To find the greatest positive integer \( x \) such that \( 23^{6+x} \) divides \( 2000! \), we need to determine how many times the prime factor 23 appears in the prime factorization of \( 2000! \).
The exponent of a prime \( p \) in \( n! \) is given by:
\[
\sum_{k=1}^{\infty} \left\lfloor \frac{n}{p^k} \right\rfloo... | 83 | jbmo_shortlists |
[
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)",
"Mathematics -> Number Theory -> Least Common Multiples (LCM)"
] | 5 | Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$:
\[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =
\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left... |
Given the problem, we seek all functions \( f: \mathbb{N}_{\geq 1} \to \mathbb{N}_{\geq 1} \) such that for all positive integers \( m \) and \( n \), the following holds:
\[
\mathrm{GCD}(f(m), n) + \mathrm{LCM}(m, f(n)) = \mathrm{GCD}(m, f(n)) + \mathrm{LCM}(f(m), n).
\]
To solve this, let's explore the properties ... | f(n) = n | math_olympiad_for_the_french_speaking |
[
"Mathematics -> Number Theory -> Other",
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 5 | For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$. | Given the problem, we want to determine all integers \( n \geq 1 \) such that the sum of the squares of the digits of \( n \), denoted as \( s(n) \), is equal to \( n \).
To begin, let's express \( n \) in terms of its digits. Suppose \( n \) is a \( k \)-digit number given by:
\[
n = d_{k-1} \cdot 10^{k-1} + d_{k-2} ... | 1 | bero_American |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 5 | Find all positive integers $a,b$ for which $a^4+4b^4$ is a prime number. |
To find all positive integers \( a, b \) for which \( a^4 + 4b^4 \) is a prime number, we first analyze the expression:
\[
a^4 + 4b^4
\]
This can be rewritten using the Sophie Germain identity:
\[
a^4 + 4b^4 = (a^2 + 2b^2 - 2ab)(a^2 + 2b^2 + 2ab)
\]
For the expression \( a^4 + 4b^4 \) to be a prime number, it must... | (1, 1) | jbmo_shortlists |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 5 | Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points. |
Consider that there are 2009 distinct points in the plane, and each point is colored either blue or red. The objective is to determine the greatest possible number of blue points under the condition that every blue-centered unit circle contains exactly two red points.
To solve this, we need to maximize the number of ... | 45 | junior_balkan_mo |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | One writes 268 numbers around a circle, such that the sum of 20 consectutive numbers is always equal to 75. The number 3, 4 and 9 are written in positions 17, 83 and 144 respectively. Find the number in position 210. | Given the problem, we have to find the number in position 210 under the constraints provided. We have 268 numbers written in a circle, denoted as \( a_1, a_2, \ldots, a_{268} \), and we know that the sum of any 20 consecutive numbers is 75.
This implies:
\[
a_i + a_{i+1} + \cdots + a_{i+19} = 75
\]
for all \( i \). G... | -1 | pan_african MO |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Number Theory -> Prime Numbers"
] | 5 | Determine all triples $(p, q, r)$ of positive integers, where $p, q$ are also primes, such that $\frac{r^2-5q^2}{p^2-1}=2$. |
To find all triples \((p, q, r)\) of positive integers, where \(p, q\) are also primes, such that:
\[
\frac{r^2 - 5q^2}{p^2 - 1} = 2,
\]
we start by rearranging the equation:
\[
r^2 - 5q^2 = 2(p^2 - 1).
\]
This can be further rewritten as:
\[
r^2 = 5q^2 + 2(p^2 - 1).
\]
Since \(p\) and \(q\) are primes, we will ... | (3, 2, 6) | th_centroamerican_and_caribbean_math_olympiad_havana |
[
"Mathematics -> Number Theory -> Other",
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 5 | It is given that $2^{333}$ is a 101-digit number whose first digit is 1. How many of the numbers $2^k$, $1\le k\le 332$ have first digit 4? |
To determine how many numbers \( 2^k \), for \( 1 \leq k \leq 332 \), have the first digit as 4, we can approach the problem using logarithms to examine the leading digits.
### Step 1: Understanding the Leading Digit
For a number \( 2^k \) to have a first digit of 4, it must satisfy:
\[
4 \times 10^m \leq 2^k < 5 \t... | 32 | ToT |
[
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 5 | Let $ABCDE$ be a convex pentagon such that $AB=AE=CD=1$, $\angle ABC=\angle DEA=90^\circ$ and $BC+DE=1$. Compute the area of the pentagon.
[i]Greece[/i] |
To find the area of the pentagon \(ABCDE\), we will use the given conditions:
1. \(AB = AE = CD = 1\),
2. \(\angle ABC = \angle DEA = 90^\circ\),
3. \(BC + DE = 1\).
We start by placing the pentagon in the coordinate plane to simplify calculations:
- Let \(A\) be at the origin \((0, 0)\).
- Since \(AB = 1\) and \(\... | 1 | junior_balkan_mo |
[
"Mathematics -> Number Theory -> Other"
] | 5 | Several positive integers are written on a blackboard. The sum of any two of them is some power of two (for example, $2, 4, 8,...$). What is the maximal possible number of different integers on the blackboard? |
To determine the maximal possible number of different positive integers on the blackboard, given the condition that the sum of any two of them must be a power of two, we proceed as follows:
First, recall that a power of two can be expressed as \(2^k\) for some integer \(k\). The integers on the blackboard must sum to... | 2 | ToT |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 5 | As shown below, there is a $40\times30$ paper with a filled $10\times5$ rectangle inside of it. We want to cut out the filled rectangle from the paper using four straight cuts. Each straight cut is a straight line that divides the paper into two pieces, and we keep the piece containing the filled rectangle. The goal is... |
We are given a \( 40 \times 30 \) rectangle (the paper) with a filled \( 10 \times 5 \) rectangle inside it. The objective is to cut out the filled rectangle using four straight cuts with the aim of minimizing the total length of the cuts. Each cut divides the remaining paper into two pieces, and we keep the piece con... | 65 | th_igo |
[
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 5 | In the given figure, $ABCD$ is a parallelogram. We know that $\angle D = 60^\circ$, $AD = 2$ and $AB = \sqrt3 + 1$. Point $M$ is the midpoint of $AD$. Segment $CK$ is the angle bisector of $C$. Find the angle $CKB$. |
We are given a parallelogram \(ABCD\) with \(\angle D = 60^\circ\), \(AD = 2\), and \(AB = \sqrt{3} + 1\). Point \(M\) is the midpoint of \(AD\), and segment \(CK\) is the angle bisector of \(\angle C\). We need to find \(\angle CKB\).
### Step 1: Analyzing the Parallelogram Properties
In a parallelogram, opposite si... | 75^\circ | th_igo |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 5 | Find all prime numbers $a,b,c$ and positive integers $k$ satisfying the equation \[a^2+b^2+16c^2 = 9k^2 + 1.\] |
To find all prime numbers \(a, b, c\) and positive integers \(k\) that satisfy the equation
\[
a^2 + b^2 + 16c^2 = 9k^2 + 1,
\]
we proceed as follows:
1. **Modulo Consideration**: Observe the equation modulo 3. We have:
\[
a^2 \equiv 0 \text{ or } 1 \pmod{3}, \quad b^2 \equiv 0 \text{ or } 1 \pmod{3}, \quad 16... | (3, 3, 2, 3), (3, 37, 3, 13), (37, 3, 3, 13), (3, 17, 3, 7), (17, 3, 3, 7) | junior_balkan_mo |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Number Theory -> Diophantine Equations -> Other"
] | 5 | Find all positive integers $a$, $b$, $c$, and $p$, where $p$ is a prime number, such that
$73p^2 + 6 = 9a^2 + 17b^2 + 17c^2$. |
To find all positive integers \( a \), \( b \), \( c \), and \( p \), where \( p \) is a prime number, satisfying the equation:
\[
73p^2 + 6 = 9a^2 + 17b^2 + 17c^2,
\]
we proceed as follows:
### Step 1: Investigate the Equation
The equation is balanced on both sides, with terms involving squares of integers and a ... | (2, 1, 4, 1) \text{ and } (2, 1, 1, 4) | jbmo_shortlist |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 5 | Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square. |
Let \( n \) be an integer such that \( n \geq 1 \), and we want to find all values of \( n \) for which \( n \cdot 2^{n+1} + 1 \) is a perfect square. We set:
\[
n \cdot 2^{n+1} + 1 = k^2
\]
for some integer \( k \). Rearranging gives:
\[
n \cdot 2^{n+1} = k^2 - 1
\]
which can be factored as:
\[
(k - 1)(k + 1) = ... | 3 | junior_balkan_mo |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations",
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | Two isosceles triangles with sidelengths $x,x,a$ and $x,x,b$ ($a \neq b$) have equal areas. Find $x$. |
We are given two isosceles triangles with side lengths \( x, x, a \) and \( x, x, b \), where \( a \neq b \), and they have equal areas. We need to find the value of \( x \).
### Step-by-Step Solution
1. **Area of an Isosceles Triangle:**
For an isosceles triangle with sides \( x, x, a \), the area \( A_1 \) can... | \frac{\sqrt{a^2 + b^2}}{2} | cono_sur_olympiad |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 5 | What is the maximum number of colours that can be used to paint an $8 \times 8$ chessboard so that every square is painted in a single colour, and is adjacent , horizontally, vertically but not diagonally, to at least two other squares of its own colour?
(A Shapovalov) |
To find the maximum number of colors that can be used to paint an \(8 \times 8\) chessboard such that each square is adjacent (horizontally or vertically) to at least two other squares of its own color, we need to carefully analyze and construct a feasible coloring pattern under the given constraints.
### Step-by-Ste... | 16 | ToT |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Let $n$ be a positive integer. A child builds a wall along a line with $n$ identical cubes. He lays the first cube on the line and at each subsequent step, he lays the next cube either on the ground or on the top of another cube, so that it has a common face with the previous one. How many such distinct walls exist? |
To solve this problem, we need to determine how many distinct ways the child can build a wall with \( n \) identical cubes. Each cube can be placed in such a way that it shares a face with the previous cube. This can be done either by placing the new cube on the ground or on top of the previously placed cube.
Let's ... | 2^{n-1} | pan_african MO |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 5 | Find all positive integers $x,y,z$ and $t$ such that $2^x3^y+5^z=7^t$. |
We are tasked with finding all positive integers \( x, y, z, \) and \( t \) such that:
\[
2^x 3^y + 5^z = 7^t.
\]
Given the nature of the equation and that it involves powers of prime numbers, let's analyze the problem:
1. **Examine small values for the exponents**: Begin by trying small values for \( x, y, z, \) a... | (x, y, z, t) = (3, 1, 2, 2) | junior_balkan_mo |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 5 | If $a$ , $b$ are integers and $s=a^3+b^3-60ab(a+b)\geq 2012$ , find the least possible value of $s$. |
Given the inequality \( s = a^3 + b^3 - 60ab(a + b) \geq 2012 \), we are tasked with finding the least possible value of \( s \).
To solve this, we start by rewriting \( s \) in terms of a simplified expression:
\[
s = a^3 + b^3 - 60ab(a + b).
\]
Notice that we can factor and simplify the expression using the identi... | 2015 | jbmo_shortlists |
[
"Mathematics -> Discrete Mathematics -> Algorithms",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Esmeralda has created a special knight to play on quadrilateral boards that are identical to chessboards. If a knight is in a square then it can move to another square by moving 1 square in one direction and 3 squares in a perpendicular direction (which is a diagonal of a $2\times4$ rectangle instead of $2\times3$ like... |
To solve this problem, we will analyze the movement of Esmeralda's special knight on a \(5 \times 5\) board and determine the longest possible path without revisiting any square. The knight moves in an "L" shape spanning across a \(2 \times 4\) rectangle, rather than the traditional \(2 \times 3\) from chess. These mo... | 12 | lusophon_mathematical_olympiad |
[
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 5 | The triangle $ABC$ is isosceles with $AB=AC$, and $\angle{BAC}<60^{\circ}$. The points $D$ and $E$ are chosen on the side $AC$ such that, $EB=ED$, and $\angle{ABD}\equiv\angle{CBE}$. Denote by $O$ the intersection point between the internal bisectors of the angles $\angle{BDC}$ and $\angle{ACB}$. Compute $\angle{COD}$. |
Given triangle \( \triangle ABC \), where \( AB = AC \) and \( \angle BAC < 60^\circ \), points \( D \) and \( E \) are chosen on side \( AC \) such that \( EB = ED \) and \( \angle ABD \equiv \angle CBE \). We are tasked with finding \(\angle COD\), where \( O \) is the intersection of the internal bisectors of \(\an... | 120^\circ | junior_balkan_mo |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 5 | $100$ numbers $1$, $1/2$, $1/3$, $...$, $1/100$ are written on the blackboard. One may delete two arbitrary numbers $a$ and $b$ among them and replace them by the number $a + b + ab$. After $99$ such operations only one number is left. What is this final number?
(D. Fomin, Leningrad) |
Initially, we have $100$ numbers written on the blackboard: \(1, \frac{1}{2}, \frac{1}{3}, \ldots, \frac{1}{100}\). In each step, two numbers \(a\) and \(b\) are selected and replaced with the number \(a + b + ab\). This process is repeated until only one number remains.
To solve this, a useful observation involves r... | 101 | ToT |
[
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 5 | Find all integers $m$ and $n$ such that the fifth power of $m$ minus the fifth power of $n$ is equal to $16mn$. |
We are tasked with finding all integer pairs \((m, n)\) such that:
\[
m^5 - n^5 = 16mn
\]
**Step 1: Algebraic Manipulation**
We begin by rewriting the given equation as:
\[
m^5 - n^5 - 16mn = 0
\]
**Step 2: Factorization**
Using the identity for the difference of powers, we have:
\[
m^5 - n^5 = (m - n)(m^4 + m^... | (m, n) = (0, 0) \text{ and } (m, n) = (-2, 2) | jbmo_shortlist |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 5 | The sequence $\{a_n\}_{n\geq 0}$ of real numbers satisfies the relation:
\[ a_{m+n} + a_{m-n} - m + n -1 = \frac12 (a_{2m} + a_{2n}) \]
for all non-negative integers $m$ and $n$, $m \ge n$. If $a_1 = 3$ find $a_{2004}$. |
We are given the sequence \( \{a_n\}_{n \geq 0} \) which satisfies the relation:
\[
a_{m+n} + a_{m-n} - m + n - 1 = \frac{1}{2} (a_{2m} + a_{2n})
\]
for all non-negative integers \( m \) and \( n \) with \( m \ge n \). We are also given that \( a_1 = 3 \), and we need to find \( a_{2004} \).
We start by plugging sp... | 4018021 | balkan_mo |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Calculus -> Optimization -> Other"
] | 5 | Find the smallest positive real $\alpha$, such that $$\frac{x+y} {2}\geq \alpha\sqrt{xy}+(1 - \alpha)\sqrt{\frac{x^2+y^2}{2}}$$ for all positive reals $x, y$. |
To solve the problem of finding the smallest positive real \(\alpha\) such that
\[
\frac{x+y}{2} \geq \alpha\sqrt{xy} + (1 - \alpha)\sqrt{\frac{x^2 + y^2}{2}}
\]
for all positive reals \(x\) and \(y\), we proceed as follows:
### Step 1: Analyze Special Cases
1. **Case \(x = y\):**
If \(x = y\), then both side... | \frac{1}{2} | baltic_way |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | In a math test, there are easy and hard questions. The easy questions worth 3 points and the hard questions worth D points.\\
If all the questions begin to worth 4 points, the total punctuation of the test increases 16 points.\\
Instead, if we exchange the questions scores, scoring D points for the easy questions and 3... |
Let \( x \) be the number of hard questions and \( 9x \) be the number of easy questions in the test. Let the total number of questions be \( n = x + 9x = 10x \).
Given:
- Easy questions are worth 3 points each.
- Hard questions are worth \( D \) points each.
**Initial Total Points**
The initial total score of the... | 20 | lusophon_mathematical_olympiad |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | Find the roots of the equation $(x-a)(x-b)=(x-c)(x-d)$, if you know that $a+d=b+c=2015$ and $a \ne c$ (numbers $a, b, c, d$ are not given). |
To find the roots of the equation \((x-a)(x-b)=(x-c)(x-d)\), given that \(a + d = b + c = 2015\) and \(a \neq c\), we will simplify the equation and determine the solutions.
### Step 1: Expand Both Sides
Expanding both sides of the equation, we have:
\[
(x-a)(x-b) = x^2 - (a+b)x + ab
\]
\[
(x-c)(x-d) = x^2 - (c+d)x +... | \frac{2015}{2} | caucasus_mathematical_olympiad |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 5 | Let $N$ be a positive integer. A collection of $4N^2$ unit tiles with two segments drawn on them as shown is assembled into a $2N\times2N$ board. Tiles can be rotated.
[asy]size(1.5cm);draw((0,0)--(1,0)--(1,1)--(0,1)--cycle);draw((0,0.5)--(0.5,0),red);draw((0.5,1)--(1,0.5),red);[/asy]
The segments on the tiles def... |
Given a collection of \(4N^2\) unit tiles, each with two segments drawn as shown in the problem, we need to determine how these tiles can be assembled into a \(2N \times 2N\) board to minimize and maximize the number of paths created by the segments. Each tile can be rotated to form different path configurations.
###... | minimum $4N$, maximum $(N+1)^{2}+N^{2}$. | benelux MO |
[
"Mathematics -> Algebra -> Abstract Algebra -> Field Theory"
] | 5 | i will post here my solution. i gave it in the contest as well
the solutions are $f\equiv 0$ and $f(x)=x^{2}+a$, with $a$ real.
obviously $f\equiv 0$ satisfies the equation, so i will choose an $x_{0}$ now such that $f(x_{0})\neq 0$. i first claim that any real number can be written as $f(u)-f(v)$, with $u,v$ reals. d... |
Given the functional equation and conditions provided, we need to find all possible functions \( f \) such that:
1. \( f(f(x) + f(y) + z) = f(f(x) - f(y) - z) + 4f(x)(f(y) + z) \) for all real numbers \( x, y, z \).
We are tasked with showing that the solutions are \( f(x) \equiv 0 \) and \( f(x) = x^2 + a \) for so... | f(x) = 0 \text{ and } f(x) = x^2 + a \text{ for some real constant } a. | balkan_mo |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other",
"Mathematics -> Discrete Mathematics -> Logic"
] | 5 | Determine all functions $f : \mathbb{R}^2 \to\mathbb {R}$ for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever $A,B,C,D$ are the vertices of a square with side-length one. | To determine all functions \( f : \mathbb{R}^2 \to \mathbb{R} \) for which
\[
f(A) + f(B) + f(C) + f(D) = 0,
\]
whenever \( A, B, C, D \) are the vertices of a square with side-length one, consider the following steps:
1. **Translation Invariance**: The property holds for any square, particularly for squares center... | $f(x)=0$ | balkan_mo_shortlist |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Algebra -> Abstract Algebra -> Other"
] | 5 | Let $a$, $b$, $c$ be real numbers such that $a^2+b=c^2$, $b^2+c=a^2$, $c^2+a=b^2$. Find all possible values of $abc$. |
Given the equations:
\[
a^2 + b = c^2,
\]
\[
b^2 + c = a^2,
\]
\[
c^2 + a = b^2,
\]
we are tasked with finding all possible values of \(abc\).
### Step 1: Analyze the System of Equations
Let's add all three equations:
\[
(a^2 + b) + (b^2 + c) + (c^2 + a) = c^2 + a^2 + b^2.
\]
Simplifying the left-hand side, we... | $abc=0$ | caucasus_mathematical_olympiad |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Number Theory -> Prime Numbers"
] | 5 | Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$ |
We are tasked with finding all prime numbers \( p, q, r \) that satisfy the equation:
\[
\frac{p}{q} - \frac{4}{r+1} = 1.
\]
First, we rearrange the equation to find a common denominator:
\[
\frac{p}{q} - \frac{4}{r+1} = 1 \implies \frac{p(r+1) - 4q}{q(r+1)} = 1.
\]
This simplifies to:
\[
p(r+1) - 4q = q(r+1).
\]... | (7, 3, 2), (5, 3, 5), (3, 2, 7) | junior_balkan_mo |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | Petya bought one cake, two cupcakes and three bagels, Apya bought three cakes and a bagel, and Kolya bought six cupcakes. They all paid the same amount of money for purchases. Lena bought two cakes and two bagels. And how many cupcakes could be bought for the same amount spent to her? |
To solve this problem, we need to determine how many cupcakes can be purchased for the same amount that Lena spent, given the prices of each pastry type.
Let's denote the prices:
- The price of one cake as \( c \).
- The price of one cupcake as \( p \).
- The price of one bagel as \( b \).
According to the problem,... | $\frac{13}{4}$ | caucasus_mathematical_olympiad |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 5 | Each of given $100$ numbers was increased by $1$. Then each number was increased by $1$ once more. Given that the fi rst time the sum of the squares of the numbers was not changed find how this sum was changed the second time. |
Let the original 100 numbers be \( a_1, a_2, \ldots, a_{100} \). Initially, the sum of their squares is
\[
S = \sum_{i=1}^{100} a_i^2.
\]
When each number is increased by 1 for the first time, the new numbers are \( a_1 + 1, a_2 + 1, \ldots, a_{100} + 1 \). The new sum of squares is:
\[
S_1 = \sum_{i=1}^{100} (a_i... | 200 | ToT |
[
"Mathematics -> Number Theory -> Factorization",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | In how many ways can we fill the cells of a $4\times4$ grid such that each cell contains exactly one positive integer and the product of the numbers in each row and each column is $2020$? |
To solve the problem, we need to fill the cells of a \(4 \times 4\) grid such that each cell contains exactly one positive integer, and the product of the numbers in each row and each column is 2020. We must determine the number of ways to achieve this configuration.
First, observe that the prime factorization of 202... | 576 | lusophon_mathematical_olympiad |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 5 | Let $a,b,c,d$ be non-negative reals such that $a+b+c+d=4$. Prove the inequality
\[\frac{a}{a^3+8}+\frac{b}{b^3+8}+\frac{c}{c^3+8}+\frac{d}{d^3+8}\le\frac{4}{9}\] |
To prove the inequality for non-negative reals \(a\), \(b\), \(c\), and \(d\) such that \(a + b + c + d = 4\):
\[
\frac{a}{a^3+8} + \frac{b}{b^3+8} + \frac{c}{c^3+8} + \frac{d}{d^3+8} \leq \frac{4}{9},
\]
we utilize Jensen's inequality, which applies to convex functions, combined with AM-GM and other inequality proper... | \frac{a}{a^3 + 8} + \frac{b}{b^3 + 8} + \frac{c}{c^3 + 8} + \frac{d}{d^3 + 8} \leq \frac{4}{9} | baltic_way |
[
"Mathematics -> Number Theory -> Other"
] | 5 | Let $S(n)$ be the sum of the digits of the positive integer $n$. Find all $n$ such that $S(n)(S(n)-1)=n-1$. |
Let \( S(n) \) be the sum of the digits of the positive integer \( n \). We want to find all \( n \) such that:
\[
S(n)(S(n) - 1) = n - 1.
\]
Rearranging the equation gives:
\[
S(n)^2 - S(n) = n - 1 \quad \Rightarrow \quad S(n)^2 - S(n) - n + 1 = 0.
\]
This can be rewritten as:
\[
S(n)^2 - S(n) = n - 1.
\]
Denot... | $n=\{1,13,43,91,157\}$ | cono_sur_olympiad |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations"
] | 5 | Find the smallest value that the expression takes $x^4 + y^4 - x^2y - xy^2$, for positive numbers $x$ and $y$ satisfying $x + y \le 1$. |
We wish to find the minimum value of the expression \( x^4 + y^4 - x^2y - xy^2 \) subject to the constraint \( x + y \leq 1 \) where \( x \) and \( y \) are positive real numbers.
First, consider using the Lagrange multipliers method to incorporate the constraint \( x + y = c \leq 1 \). We define the Lagrangian funct... | -\frac{1}{8} | czech-polish-slovak matches |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Given an $m \times n$ table consisting of $mn$ unit cells. Alice and Bob play the following game: Alice goes first and the one who moves colors one of the empty cells with one of the given three colors. Alice wins if there is a figure, such as the ones below, having three different colors. Otherwise Bob is the winner. ... |
Consider an \(m \times n\) table with \(mn\) unit cells. Alice and Bob play a game where Alice goes first, and each player colors one of the empty cells with one of the given three colors. Alice wins if there exists a figure with three different colors as depicted in the problem. Otherwise, Bob wins. We aim to determi... | \text{ when } m \ge 5, n \ge 4\text{ Alice has a winning strategy, otherwise Bob has } | jbmo_shortlist |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 5 | Yesterday, $n\ge 4$ people sat around a round table. Each participant remembers only who his two neighbours were, but not necessarily which one sat on his left and which one sat on his right. Today, you would like the same people to sit around the same round table so that each participant has the same two neighbours as... |
Given a scenario where \( n \geq 4 \) people sat around a round table, each person remembers only who their two neighbors were without specifying which side each neighbor was on. We are tasked with rearranging these people the same way today, allowing for each participant to have the same pair of neighbors as yesterda... | n - 32q - r - 1 | benelux MO |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Let $n$ be a positive integer and $p$ a fixed prime. We have a deck of $n$ cards, numbered $1,\ 2,\ldots,\ n$ and $p$ boxes for put the cards on them. Determine all posible integers $n$ for which is possible to distribute the cards in the boxes in such a way the sum of the numbers of the cards in each box is the same. |
Given:
- \( n \) is a positive integer,
- \( p \) is a fixed prime number,
- We have a deck of \( n \) cards, numbered \( 1, 2, \ldots, n \),
- We have \( p \) boxes to put the cards into,
- We need to distribute the cards into these boxes such that the sum of the numbers on the cards in each box is the same.
We w... | n=kp^{2} | centroamerican |
[
"Mathematics -> Number Theory -> Factorization"
] | 5 | The set of $\{1,2,3,...,63\}$ was divided into three non-empty disjoint sets $A,B$. Let $a,b,c$ be the product of all numbers in each set $A,B,C$ respectively and finally we have determined the greatest common divisor of these three products. What was the biggest result we could get? |
Given the problem, we need to divide the set \(\{1, 2, 3, \ldots, 63\}\) into three non-empty disjoint sets \(A\), \(B\), and \(C\). Let the product of the numbers in these sets be \(a\), \(b\), and \(c\), respectively. We aim to maximize the greatest common divisor (GCD) of these three products, \(\gcd(a, b, c)\).
F... | $\boxed{2^{19} \cdot 3^{10} \cdot 5^{4} \cdot 3^{3} \cdot 11\cdot 13\cdot 17\cdot 19}$ | czech-polish-slovak matches |
[
"Mathematics -> Algebra -> Algebra -> Polynomial Operations",
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 5 | Let $f(x)=x^2-2$ and let $f^{(n)}(x)$ denote the $n$-th iteration of $f$. Let $H=\{x:f^{(100)}(x)\leq -1\}$. Find the length of $H$ (the sum of the lengths of the intervals of $H$). |
Consider the function \( f(x) = x^2 - 2 \). The \( n \)-th iteration of \( f \), denoted as \( f^{(n)}(x) \), is obtained by applying \( f \) iteratively \( n \) times.
We are tasked with finding the set \( H = \{ x : f^{(100)}(x) \leq -1 \} \), and subsequently, the length of this set, which is the sum of the length... | {\frac{2\left(\sin\frac{\pi}{3\cdot 2^{99}}\right)\left(1-\cos\frac{\pi}{2^{98}}\right)}{\sin \frac{\pi}{2^{99}}}} | problems_from_the_kmal_magazine |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Let $n$ be a positive integer. In how many ways can a $4 \times 4n$ grid be tiled with the following tetromino?
[asy]
size(4cm);
draw((1,0)--(3,0)--(3,1)--(0,1)--(0,0)--(1,0)--(1,2)--(2,2)--(2,0));
[/asy] |
We are tasked with determining the number of ways to tile a \(4 \times 4n\) grid using the \(L\)-shaped tetromino described in the problem. The shape of the \(L\)-shaped tetromino can cover precisely 4 unit squares.
### Step-by-step Analysis
1. **Understand the Requirements**:
- A \(4 \times 4n\) grid contains ... | 2^{n+1} - 2 | cono_sur_olympiad |
[
"Mathematics -> Algebra -> Abstract Algebra -> Group Theory"
] | 5 | Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$\gcd(f(x),y)f(xy)=f(x)f(y)$$ for all positive integers $x, y$. |
Let \( f: \mathbb{N} \to \mathbb{N} \) be a function such that for all positive integers \( x, y \), the following holds:
\[
\gcd(f(x), y)f(xy) = f(x)f(y).
\]
We need to find all functions \( f \) satisfying this condition.
### Step 1: Basic Properties
Start by considering the case \( y = 1 \):
\[
\gcd(f(x), 1)f(... | f(x) = 1 \text{ for all } x \in \mathbb{N}. | nordic |
[
"Mathematics -> Number Theory -> Congruences",
"Mathematics -> Algebra -> Abstract Algebra -> Group Theory"
] | 5 | Let $T_n$ denotes the least natural such that
$$n\mid 1+2+3+\cdots +T_n=\sum_{i=1}^{T_n} i$$
Find all naturals $m$ such that $m\ge T_m$. |
Let \( T_n \) be the least natural number such that
\[
n \mid 1 + 2 + 3 + \cdots + T_n = \sum_{i=1}^{T_n} i.
\]
The formula for the sum of the first \( T_n \) natural numbers is
\[
\sum_{i=1}^{T_n} i = \frac{T_n(T_n + 1)}{2}.
\]
Thus, we need \( n \mid \frac{T_n(T_n + 1)}{2} \).
We aim to find all natural numbe... | {\text{all numbers that are not powers of 2 greater than 1.}} | bero_American |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 5 | Let $a,b,c$ be positive real numbers such that $a+b+c = 3$. Find the minimum value of the expression \[A=\dfrac{2-a^3}a+\dfrac{2-b^3}b+\dfrac{2-c^3}c.\] |
To find the minimum value of the expression
\[
A = \frac{2-a^3}{a} + \frac{2-b^3}{b} + \frac{2-c^3}{c},
\]
given that \( a, b, c \) are positive real numbers and \( a + b + c = 3 \), we proceed as follows:
First, we rewrite the expression:
\[
A = \frac{2}{a} - a^2 + \frac{2}{b} - b^2 + \frac{2}{c} - c^2.
\]
Consi... | 3 | junior_balkan_mo |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions"
] | 5 | Let $L$ be the number formed by $2022$ digits equal to $1$, that is, $L=1111\dots 111$.
Compute the sum of the digits of the number $9L^2+2L$. |
Given a number \( L \) consisting of 2022 digits, all equal to 1, we aim to compute the sum of the digits of the number \( 9L^2 + 2L \).
### Step 1: Express \( L \) Numerically
The number \( L \) can be expressed numerically as a sequence of ones, mathematically expressed as:
\[
L = \underbrace{111\ldots111}_{2022\ \... | 4044 | all_levels |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$. Compute the following expression in terms of $k$:
\[ E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}. \]
[i]Ciprus[/i] |
We start with the given equation:
\[
\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k.
\]
Let's set \( a = \frac{x^2+y^2}{x^2-y^2} \) and \( b = \frac{x^2-y^2}{x^2+y^2} \). Therefore, we have:
\[
a + b = k.
\]
Observe that:
\[
ab = \left(\frac{x^2+y^2}{x^2-y^2}\right) \left(\frac{x^2-y^2}{x^2+y^2}\right) = \... | \frac{k^4 - 8k^2 + 16}{4k(k^2 + 4)} | junior_balkan_mo |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | The digits of a calculator (with the exception of 0) are shown in the form indicated by the figure below, where there is also a button ``+":
[img]6965[/img]
Two players $A$ and $B$ play in the following manner: $A$ turns on the calculator and presses a digit, and then presses the button ``+". $A$ passes the calculator ... |
To determine which player has a winning strategy in the given game, we need to analyze the possible moves and outcomes strategically. The setup on the calculator presents us with a logical and combinatorial game theory problem. The key to solving this is to determine which player can force a win by controlling the gam... | \text{Player } B \text{ has a winning strategy.} | centroamerican |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 5 | Find all non-negative integer solutions of the equation $2^a+p^b=n^{p-1}$, where $p$ is a prime number. |
We are tasked to find all non-negative integer solutions \((a, b, n, p)\) of the equation:
\[
2^a + p^b = n^{p-1}
\]
where \(p\) is a prime number.
Firstly, consider the case when \(b = 0\). This simplifies the equation to:
\[
2^a + 1 = n^{p-1}
\]
Now, we need to find the values of \(a\), \(n\), and \(p\) such th... | （a，b，n，p）=（3，0，3，3） | problems_from_the_kmal_magazine |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other",
"Mathematics -> Number Theory -> Prime Numbers"
] | 5 | Students in the class of Peter practice the addition and multiplication of integer numbers.The teacher writes the numbers from $1$ to $9$ on nine cards, one for each number, and places them in an ballot box. Pedro draws three cards, and must calculate the sum and the product of the three corresponding numbers. Ana and... |
Let's analyze the information provided about Pedro and Ana to solve the problem and find out which numbers Julian removed.
### Pedro's Drawn Numbers
We are told that Pedro picks three consecutive numbers whose product is 5 times their sum. Let \( a \), \( a+1 \), and \( a+2 \) be the consecutive numbers drawn by Ped... | 2, 6, 7 | cono_sur_olympiad |
[
"Mathematics -> Discrete Mathematics -> Combinatorics",
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 5 | We colored the $n^2$ unit squares of an $n\times n$ square lattice such that in each $2\times 2$ square, at least two of the four unit squares have the same color. What is the largest number of colors we could have used? |
To solve the problem, we must determine the largest number of distinct colors that can be used to color an \( n \times n \) square lattice, under the condition that within every \( 2 \times 2 \) sub-square, at least two of the four unit squares share the same color.
### Analysis
1. **Understanding the Conditions:**
... | [\frac{n^2+2n-1}{2}] | problems_from_the_kmal_magazine |
[
"Mathematics -> Number Theory -> Other"
] | 5 | A palindrome is a positive integer which reads in the same way in both directions (for example, $1$, $343$ and $2002$ are palindromes, while $2005$ is not). Is it possible to find $2005$ pairs in the form of $(n, n + 110)$ where both numbers are palindromes? | \section*{Solving Process}
To determine whether it is possible to find \( 2005 \) pairs in the form of \( (n, n+110) \) where both numbers are palindromes, we need to analyze the properties of palindromic numbers and the effect of adding \( 110 \).
A number is a palindrome if it reads the same forwards and backwards.... | \text{Yes} | ToT |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | In what case does the system of equations
$\begin{matrix} x + y + mz = a \\ x + my + z = b \\ mx + y + z = c \end{matrix}$
have a solution? Find conditions under which the unique solution of the above system is an arithmetic progression. |
To determine the conditions under which the given system of equations has a solution and when the solution is an arithmetic progression, we analyze the system:
\[
\begin{cases}
x + y + mz = a \\
x + my + z = b \\
mx + y + z = c
\end{cases}
\]
**Step 1: Determine When the System Has a Solution**
First, let's expr... | \[ y = \frac{1}{m - 1} \cdot \left(b - \frac{S}{m + 2}\right),\ x = \frac{1}{m - 1} \cdot \left(c - \frac{S}{m + 2}\right),z = \frac{1}{m - 1} \cdot \left(a - \frac{S}{m + 2}\right). \] | imo_longlists |
[
"Mathematics -> Number Theory -> Other"
] | 5 | Find all positive integers $n$ such that the decimal representation of $n^2$ consists of odd digits only. |
We are tasked with finding all positive integers \( n \) such that the decimal representation of \( n^2 \) consists only of odd digits. To approach this problem, we need to analyze the properties of squares of integers and the constraints that arise due to having only odd digits.
First, let's consider the possible la... | n \in \{1, 3\} | baltic_way |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | Find all triples $(a,b,c)$ of real numbers such that the following system holds:
$$\begin{cases} a+b+c=\frac{1}{a}+\frac{1}{b}+\frac{1}{c} \\a^2+b^2+c^2=\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}\end{cases}$$ |
To find all triples \((a, b, c)\) of real numbers that satisfy the given system of equations, we start by analyzing the conditions:
1. The first equation is:
\[
a + b + c = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.
\]
This can be rewritten as:
\[
a + b + c = \frac{ab + ac + bc}{abc}.
\]
Multipl... | \boxed{(\pm 1,x,\frac{1}{x})}\text{ and permutations.} | jbmo_shortlist |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities",
"Mathematics -> Algebra -> Prealgebra -> Other",
"Mathematics -> Discrete Mathematics -> Logic"
] | 5 | Find all real solutions $x$ to the equation $\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2$. |
To solve the equation \(\lfloor x^2 - 2x \rfloor + 2\lfloor x \rfloor = \lfloor x \rfloor^2\), we proceed with the following steps:
### Step 1: Understanding the Floor Function
The floor function, \(\lfloor y \rfloor\), represents the greatest integer less than or equal to \(y\). This means that for any real number ... | {x \in \mathbb{Z} \cup \bigcup_{n = 1} ^{\infty} (n, \sqrt{(n-1)^2 + 1} + 1)} | pan_african MO |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Alice and Bob play the following game: starting with the number $2$ written on a blackboard, each player in turn changes the current number $n$ to a number $n + p$, where $p$ is a prime divisor of $n$. Alice goes first and the players alternate in turn. The game is lost by the one who is forced to write a number greate... |
Alice and Bob are playing a game starting with the number \( n = 2 \) on a blackboard. Each player, in turn, changes the current number \( n \) to \( n + p \), where \( p \) is a prime divisor of \( n \). The game is lost by the player who is forced to write a number greater than \( \underbrace{22\ldots2}_{2020\text{ ... | \text{Alice} | jbmo_shortlist |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 5 | Solve in prime numbers the equation $x^y - y^x = xy^2 - 19$. |
To find the solutions of the equation \(x^y - y^x = xy^2 - 19\) in prime numbers, we will begin by analyzing possible small prime candidates, as powers of small primes often have manageable forms that can be verified manually.
**Step 1: Try small primes for \(x\) and \(y\) and verify conditions.**
Since \(x\) and \(... | (2, 3)(2, 7) | balkan_mo |
[
"Mathematics -> Number Theory -> Other",
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 5 | Solve the equation $a^3+b^3+c^3=2001$ in positive integers.
[i]Mircea Becheanu, Romania[/i] |
To solve the equation \( a^3 + b^3 + c^3 = 2001 \) in positive integers \( (a, b, c) \), we can explore possible values for \( a \), \( b \), and \( c \) that satisfy this equation.
First, calculate some bounds for the integers. Assume \( a \leq b \leq c \), which implies \( 3a^3 \le 2001 \). Thus, we find:
\[
a^3 \l... | (a, b, c) = (10, 10, 1), (10, 1, 10), (1, 10, 10) | junior_balkan_mo |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 5 | Fix a triangle $ABC$. We say that triangle $XYZ$ is elegant if $X$ lies on segment $BC$, $Y$ lies on segment $CA$, $Z$ lies on segment $AB$, and $XYZ$ is similar to $ABC$ (i.e., $\angle A=\angle X, \angle B=\angle Y, \angle C=\angle Z $). Of all the elegant triangles, which one has the smallest perimeter? |
To solve the problem, we need to find an elegant triangle \( XYZ \) with the smallest perimeter, where \( XYZ \) is similar to triangle \( ABC \) and the points \( X, Y, Z \) lie on the segments \( BC, CA, AB \) respectively. The condition for similarity implies that the angles of both triangles match: \(\angle A = \a... | \text{the medial triangle} | problems_from_the_kmal_magazine |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | In three piles there are $51, 49$, and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each? |
To determine if it is possible to transform the initial configuration of piles into 105 piles with one stone each, let's first examine the operations allowed and their effects on the total number of piles:
1. **Combining Two Piles**: When any two piles are combined into one pile, the total number of piles decreases b... | \text{No} | ToT |
[
"Mathematics -> Number Theory -> Diophantine Equations -> Other"
] | 5 | Solve the equation
\[ 3^x \minus{} 5^y \equal{} z^2.\]
in positive integers. |
To solve the equation
\[ 3^x - 5^y = z^2 \]
in positive integers, we begin by investigating small values of \( x \) and \( y \) to find solutions.
### Step 1: Test for Small Values
1. **Substitute \( x = 1 \):**
\[
3^1 - 5^y = z^2 \implies 3 - 5^y = z^2.
\]
As \( y \) must be a positive integer, \( 5^y... | (x, y, z) = (2, 1, 2) | balkan_mo |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | An arrangement of chips in the squares of $ n\times n$ table is called [i]sparse[/i] if every $ 2\times 2$ square contains at most 3 chips. Serge put chips in some squares of the table (one in a square) and obtained a sparse arrangement. He noted however that if any chip is moved to any free square then the arrangement... |
Consider an \( n \times n \) table where each square can potentially hold one chip. We define a "sparse" arrangement as an arrangement where every \( 2 \times 2 \) sub-square of the table contains at most 3 chips. The problem conditions further specify that if any chip from this arrangement is moved to any free square... | \text{All odd integers} | tuymaada_olympiad |
[
"Mathematics -> Number Theory -> Other",
"Mathematics -> Algebra -> Other"
] | 5 | Is it possible to arrange the numbers $1^1, 2^2,..., 2008^{2008}$ one after the other, in such a way that the obtained number is a perfect square? (Explain your answer.) |
We are tasked with determining whether it is possible to arrange the numbers \( 1^1, 2^2, \ldots, 2008^{2008} \) in such a way that the resulting sequence forms a perfect square.
To address this question, let's analyze the problem step-by-step.
### Overview and Initial Considerations
1. **Understand the Range of Nu... | \text{No} | jbmo_shortlist |
[
"Mathematics -> Number Theory -> Prime Numbers",
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | Find all square numbers $ S_1$ and $ S_2$ such that $ S_1 \minus{} S_2 \equal{} 1989.$ |
Given the equation \( S_1 - S_2 = 1989 \), where \( S_1 \) and \( S_2 \) are square numbers, we seek to find all such pairs \((S_1, S_2)\). Let \( S_1 = a^2 \) and \( S_2 = b^2 \), where \( a > b \) are integers. Thus, we have:
\[
a^2 - b^2 = 1989.
\]
This can be factored using the difference of squares:
\[
(a - b)... | $ (S_1,S_2)\in \{ (995^2,994^2), (333^2,330^2), (115^2,106^2), (83^2, 70^2), (67^2,50^2), (45^2, 6^2)\}$ | imo_longlists |
[
"Mathematics -> Number Theory -> Other"
] | 5 | Peter picked an arbitrary positive integer, multiplied it by 5, multiplied the result by 5, then multiplied the result by 5 again and so on. Is it true that from some moment all the numbers that Peter obtains contain 5 in their decimal representation? |
To determine whether from some moment all the numbers that Peter obtains contain the digit 5 in their decimal representation, let us examine the operation being performed.
Peter repeatedly multiplies a positive integer by 5. Let's denote the initial integer by \( a_1 \).
1. The sequence generated by Peter is:
\[
... | \text{True} | ToT |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | In Mathcity, there are infinitely many buses and infinitely many stations. The stations are indexed by the powers of $2: 1, 2, 4, 8, 16, ...$ Each bus goes by finitely many stations, and the bus number is the sum of all the stations it goes by. For simplifications, the mayor of Mathcity wishes that the bus numbers form... |
We are given a problem involving buses and stations in Mathcity, where stations are indexed by powers of 2, such as \(1, 2, 4, 8, \ldots\). Each bus visits finitely many stations, and its number is the sum of the indices of these stations. The mayor wants the bus numbers to form an arithmetic progression with the firs... | \text{all odd }r | jbmo_shortlist |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Anna's number is obtained by writing down $20$ consecutive positive integers, one after another in arbitrary order. Bob's number is obtained in the same way, but with $21$ consecutive positive integers. Can they obtain the same number? |
To determine if Anna's and Bob's numbers can be the same, we consider how each number is formed:
Anna's number is created by writing down 20 consecutive positive integers. Let these integers be \(a, a+1, a+2, \ldots, a+19\). When these numbers are concatenated, they form Anna's complete number.
Bob's number is simil... | \text{Not equal} | ToT |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | The workers laid a floor of size $n\times n$ ($10 <n <20$) with two types of tiles: $2 \times 2$ and $5\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen? (You can’t cut tiles and also put them on top of each ... |
To solve this problem, we aim to find all integer values of \( n \) (where \( 10 < n < 20 \)) for which an \( n \times n \) floor can be completely covered using the same number of \( 2 \times 2 \) and \( 5 \times 1 \) tiles. We cannot cut the tiles and they should not overlap.
First, we calculate the total area of t... | 12, 15, 18 | caucasus_mathematical_olympiad |
[
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 5 | Is it possible to cover a $13\times 13$ chessboard with forty-two pieces of dimensions $4\times 1$ such that only the central square of the chessboard remains uncovered? |
To determine if it is possible to cover a \(13 \times 13\) chessboard using forty-two pieces of dimensions \(4 \times 1\) such that only the central square remains uncovered, we begin by analyzing the total area and the constraints given by the problem.
Firstly, the total number of squares on a \(13 \times 13\) chess... | \text{No} | baltic_way |
[
"Mathematics -> Algebra -> Abstract Algebra -> Group Theory"
] | 5 | For an arbitrary positive integer $m$, not divisible by $3$, consider the permutation $x \mapsto 3x \pmod{m}$ on the set $\{ 1,2,\dotsc ,m-1\}$. This permutation can be decomposed into disjointed cycles; for instance, for $m=10$ the cycles are $(1\mapsto 3\to 9,\mapsto 7,\mapsto 1)$, $(2\mapsto 6\mapsto 8\mapsto 4\maps... |
Given a positive integer \( m \), not divisible by 3, we are interested in the permutation \( x \mapsto 3x \pmod{m} \) on the set \(\{ 1, 2, \dotsc, m-1 \}\). The task is to determine for which integers \( m \) the number of cycles in this permutation is odd.
### Understanding the Problem
For a permutation \( \sigma... | m \equiv 2, 5, 7, 10 \pmod{12} | problems_from_the_kmal_magazine |
[
"Mathematics -> Algebra -> Algebra -> Equations and Inequalities"
] | 5 | Solve the following system of equations in integer numbers:
$$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$ |
To solve the given system of equations in integer numbers:
\[
\begin{cases}
x^2 = yz + 1 \\
y^2 = zx + 1 \\
z^2 = xy + 1
\end{cases}
\]
we need to find integer solutions \((x, y, z)\).
### Analysis
First, consider the symmetry of the problem; each equation is structurally similar, suggesting potential symmetry i... | (1, 0, -1) | czech-polish-slovak matches |
[
"Mathematics -> Algebra -> Intermediate Algebra -> Other",
"Mathematics -> Number Theory -> Other"
] | 5 | Let us say that a pair of distinct positive integers is nice if their arithmetic mean and their geometric mean are both integer. Is it true that for each nice pair there is another nice pair with the same arithmetic mean? (The pairs $(a, b)$ and $(b, a)$ are considered to be the same pair.) |
Given a problem that defines a pair of distinct positive integers as "nice" if their arithmetic mean and geometric mean are both integers, we aim to determine if for each nice pair, there exists another nice pair with the same arithmetic mean.
Let's denote the pair of integers as \( (a, b) \) with \( a < b \).
### A... | \text{Yes} | problems_from_the_kvant_magazine |
[
"Mathematics -> Geometry -> Plane Geometry -> Triangulations"
] | 5 | Let $ABC$ be a triangle with $\angle ABC$ obtuse. The [i]$A$-excircle[/i] is a circle in the exterior of $\triangle ABC$ that is tangent to side $BC$ of the triangle and tangent to the extensions of the other two sides. Let $E$, $F$ be the feet of the altitudes from $B$ and $C$ to lines $AC$ and $AB$, respectively. Can... |
To determine whether the line \( EF \) can be tangent to the \( A \)-excircle of \( \triangle ABC \), where \(\angle ABC\) is obtuse, we start by analyzing the geometric properties involved.
### Step 1: Understanding the Geometry
1. **Excircle Properties:** The \( A \)-excircle is a circle in the exterior of \( \tria... | \text{Line } EF \text{ cannot be tangent to the } A\text{-excircle.} | usajmo |
[
"Mathematics -> Algebra -> Algebra -> Algebraic Expressions",
"Mathematics -> Number Theory -> Congruences"
] | 5 | Let $a_0$ be a positive integer and $a_n=5a_{n-1}+4$ for all $n\ge 1$. Can $a_0$ be chosen so that $a_{54}$ is a multiple of $2013$? |
Given the recurrence relation \( a_n = 5a_{n-1} + 4 \) with \( a_0 \) as a positive integer, we aim to determine if there exists a choice of \( a_0 \) such that \( a_{54} \) is a multiple of 2013.
First, let's explore a closed-form expression to represent \( a_n \). Starting with the homogeneous part of the recurrenc... | \text{No} | baltic_way |
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