This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 132

TNO 2008 Senior, 7
Tags: number theory , algebra , Chile , prime numbers
Find all pairs of prime numbers $p$ and $q$ such that: \[ p(p + q) = q^p+ 1. \]

TNO 2024 Junior, 5
Tags: number theory , Chile
The nine digits from 1 to 9 are to be placed around a circle so that the average of any three consecutive digits is a multiple of 3. Is this possible? Justify your answer.

2024 Chile TST Ibero., 3
Tags: TST , Chile , combinatorics
Find all natural numbers \( n \) for which it is possible to construct an \( n \times n \) square using only tetrominoes like the one below:

2023 Chile TST Ibero., 1
Tags: TST , Chile , number theory
Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers \[ a_n = 4^n c + \frac{4^n - (-1)^n}{5} \] contains at least one perfect square.

2022 Chile TST IMO, 2
Tags: geometry , Chile
Let $ABC$ be an acute-angled triangle with $|AB| \neq |AC|$. Let $D$ be the foot of the altitude from $A$ to $BC$, and let $E$ be the intersection of the bisector of angle $\angle BAC$ with side $BC$. Let $P$ and $Q$ be the intersection points of the circumcircle of triangle $ADE$ with $AC$ and $AB$, respectively. Prove that the lines $AD$, $BP$, and $CQ$ pass through a common point.

TNO 2023 Senior, 4
Tags: combinatorics , Chile
In a country, there are \( n \) cities. Each pair of cities is connected either by a paved road or a dirt road. It is known that there exists a pair of cities such that it is impossible to travel between them using only paved roads. Show that, in this case, it is possible to travel between any two cities using only dirt roads.

2015 Chile TST Ibero, 2
Tags: combinatorics , TST , Chile
In the country of Muilejistan, there exists a network of roads connecting all its cities. The network has the particular property that for any two cities, there is a unique path without backtracking (i.e., a path where the traveler never returns along the same road). The longest possible path between two cities is 600 kilometers. For instance, the path from the city of Mlar to the city of Nlar is 600 kilometers. Similarly, the path from the city of Klar to the city of Glar is also 600 kilometers. 1. If Jalim departs from Mlar towards Nlar at noon and Kalim departs from Klar towards Glar also at noon, both traveling at the same speed, prove that they meet at some point on their journey. 2. If the distance in kilometers between any two cities is an integer, prove that the distance from Glar to Mlar is even.

TNO 2024 Junior, 4
Tags: combinatorics , Chile
Tomás is an avid domino player. One day, while playing with the tiles, he realized he could arrange all the tiles in a single row following the rules, meaning that the number on the right side of each tile matches the number on the left side of the next tile. If the number on the left side of the first tile is 5, what is the number on the right side of the last tile?

TNO 2024 Junior, 3
Tags: combinatorics , Chile
Antonia and Benjamin play the following game: First, Antonia writes an integer from 1 to 2024. Then, Benjamin writes a different integer from 1 to 2024. They alternate turns, each writing a new integer different from the ones previously written, until no more numbers are left. Each time Antonia writes a number, she gains a point for each digit '2' in the number and loses a point for each digit '5'. Benjamin, on the other hand, gains a point for each digit '5' in his number and loses a point for each digit '2'. Who can guarantee victory in this game?

2023 Chile TST IMO, 3
Tags: algebra , Chile , TST
Solve the system of equations in real numbers: \[ \frac{x}{y} + \frac{y}{z} + \frac{z}{x} = \frac{x}{z} + \frac{z}{y} + \frac{y}{x} \] \[ x^2 + y^2 + z^2 = 294 \] \[ x + y + z = 0 \]

2024 Chile TST Ibero., 2
Tags: TST , algebra , Chile
A collection of regular polygons with sides of equal length is said to "fit" if, when arranged around a common vertex, they exactly complete the surrounding area of the point on the plane. For example, a square fits with two octagons. Determine all possible collections of regular polygons that fit.

2025 Chile TST IMO-Cono, 3
Tags: algebra , Chile
Let \( a, b, c, d \) be real numbers such that \( abcd = 1 \), and \[ a + \frac{1}{a} + b + \frac{1}{b} + c + \frac{1}{c} + d + \frac{1}{d} = 0. \] Prove that one of the numbers \( ab, ac \) or \( ad \) is equal to \( -1 \).

2023 Chile TST Ibero., 4
Tags: Chile , geometry , Iberoamerican , TST
Let \(ABC\) be a triangle with \(AB < AC\) and let \(\omega\) be its circumcircle. Let \(M\) denote the midpoint of side \(BC\) and \(N\) the midpoint of arc \(BC\) of \(\omega\) that contains \(A\). The circumcircle of triangle \(AMN\) intersects sides \(AB\) and \(AC\) at points \(P\) and \(Q\), respectively. Prove that \(BP = CQ\).

2023 Chile TST Ibero., 2
Tags: TST , Chile , algebra
Consider a function \( n \mapsto f(n) \) that satisfies the following conditions: \( f(n) \) is an integer for each \( n \). \( f(0) = 1 \). \( f(n+1) > f(n) + f(n-1) + \cdots + f(0) \) for each \( n = 0, 1, 2, \dots \). Determine the smallest possible value of \( f(2023) \).

2023 Chile TST Ibero., 3
Tags: TST , Chile , number theory
Determine the smallest positive integer \( n \) with the following property: for every triple of positive integers \( x, y, z \), with \( x \) dividing \( y^3 \), \( y \) dividing \( z^3 \), and \( z \) dividing \( x^3 \), it also holds that \( (xyz) \) divides \( (x + y + z)^n \).

2023 Chile TST IMO, 4
Tags: combinatorics , Chile , TST
On a \( 10 \times 10 \) chessboard, there are 91 white pawns placed in different squares. Nico picks a white pawn, paints it black, and places it in an empty square, repeating the process until all pawns have been painted. Prove that at some point, there will be two pawns of different colors placed on squares that share a common edge.

2023 Chile TST IMO, 1
Tags: TST , Chile , geometry
Let \( \triangle ABC \) be an equilateral triangle, and let \( M \) be the midpoint of \( BC \). Let \( C_1 \) be the circumcircle of triangle \( \triangle ABC \) and \( C_2 \) the circumcircle of triangle \( \triangle ABM \). Determine the ratio between the areas of the circles \( C_1 \) and \( C_2 \).

2014 Chile TST Ibero, 2
Tags: geometry , TST , Chile , Iberoamerican
Let $\triangle ABC$ be a triangle and points $P, Q, R$ on the sides $AB, BC,$ and $CA$ respectively, such that: \[ \frac{AP}{AB} = \frac{BQ}{BC} = \frac{CR}{CA} = \frac{1}{n} \] for $n \in \mathbb{N}$. The segments $AQ$ and $CP$ intersect at $D$, the segments $BR$ and $AQ$ intersect at $E$, and the segments $BR$ and $CP$ intersect at $F$. Compute the ratio: \[ \frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)}. \]

2024 Chile Junior Math Olympiad, 5
Tags: combinatorics , Chile
You have a collection of at least two tokens where each one has a number less than or equal to 10 written on it. The sum of the numbers on the tokens is \( S \). Find all possible values of \( S \) that guarantee that the tokens can be separated into two groups such that the sum of each group does not exceed 80.

2023 Chile Classification NMO Seniors, 3
Tags: geometry , Chile
In the convex quadrilateral $ABCD$, $M$ is the midpoint of side $AD$, $AD = BD$, lines $CM$ and $AB$ are parallel, and $3\angle LBAC = \angle LACD$. Find the measure of angle $\angle ACB$.

2014 Chile TST IMO, 3
Tags: geometry , Chile , projective geometry
In a triangle \( ABC \), \( D \) is the foot of the altitude from \( C \). Let \( P \in \overline{CD} \). \( Q \) is the intersection of \( \overline{AP} \) and \( \overline{CB} \), and \( R \) is the intersection of \( \overline{BP} \) and \( \overline{CA} \). Prove that \( \angle RDC = \angle QDC \).

TNO 2008 Senior, 4
Tags: geometry , number theory , Chile
Prove that the diagonals of a convex quadrilateral are perpendicular if and only if the sum of the squares of one pair of opposite sides is equal to the sum of the squares of the other pair.

TNO 2024 Senior, 3
Tags: combinatorics , geometry , Chile , combinatorics geometry
In the Cartesian plane, each point with integer coordinates is colored either red, green, or blue. It is possible to form right isosceles triangles ($45^\circ - 90^\circ - 45^\circ$) using colored points as vertices. Prove that regardless of how the coloring is done, there always exists a right isosceles triangle such that all its vertices are either the same color or all different colors.

TNO 2008 Junior, 6
Tags: algebra , Chile
(a) Given $2 + 4 + 6 + \dots + p = 6480$, find $p$. (b) Given $7 + 11 + 15 + \dots + q = 5250$, find $q$. (c) Given $2^2 + 4^2 + 6^2 + \dots + r^2 - 1^2 - 3^2 - 5^2 - \dots - (r-1)^2 = 2485$, compute $r$.

2024 Chile Junior Math Olympiad, 2
Tags: combinatorics , Chile
Emilia and Julieta have a pile of 2024 cards and play the following game: they take turns, and each player removes a number of cards that must be a power of two, i.e., \(1, 2, 4, 8, \dots\). The player who removes the last card wins. Julieta starts the game. Prove that there exists a strategy for Julieta that guarantees her victory, no matter how Emilia plays.