Found problems: 132
2024 Chile TST Ibero., 1
Determine all integers \( x \) for which the expression \( x^2 + 10x + 160 \) is a perfect square.
2024 Chile Classification NMO Juniors, 1
Victor has four types of coins: gold, silver, bronze, and copper. All coins of the same type have the same weight, which is an integer number of grams. Victor performs two weighings:
- He takes 6 gold coins, 13 silver coins, 3 bronze coins, and 7 copper coins, and the total weight on the scale is 162 grams.
- He takes 15 gold coins, 5 silver coins, and 11 bronze coins, and the total weight on the scale is 110 grams.
Determine the weight of each type of coin.
TNO 2008 Junior, 7
A $5 \times 5$ grid is given, called $f_1$:
\[
\begin{array}{ccccc}
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 \\
-1 & 1 & -1 & 1 & -1 \\
1 & 1 & 1 & 1 & 1 \\
1 & 1 & 1 & 1 & 1 \\
\end{array}
\]
A new grid $f_{n+1}$ is constructed where each cell is equal to the product of its neighboring cells in grid $f_n$.
(a) Find the grids $f_6$ and $f_7$.
(b) Find the grids $f_{2008}$ and $f_{2009}$.
(c) Find $f_{2n}$ and $f_{2n+1}$ for any $n \in \mathbb{N}$.
*Note: Neighboring cells are those that share an edge, not just a vertex.*
TNO 2023 Junior, 5
Show that there do not exist five consecutive integers whose sum of squares is itself a perfect square.
2016 Chile TST IMO, 1
An equilateral triangle with side length 20 is subdivided using parallels to its sides into \( 20^2 = 400 \) smaller equilateral triangles of side length 1. Some segments of length 1, which are edges of these small triangles, must be colored red in such a way that no small triangle has all three of its edges colored red. Determine the maximum number of segments of length 1 that can be colored red.
2025 Chile TST IMO-Cono, 5
Let \( u_n \) be the \( n \)-th term of the Fibonacci sequence (where \( u_1 = u_2 = 1 \) and \( u_{n+1} = u_n + u_{n-1} \) for \( n \geq 2 \)). For each prime \( p \), let \( n(p) \) be the smallest integer \( n \) such that \( u_n \) is divisible by \( p \). Find the smallest possible value of \( p - n(p) \).
TNO 2023 Junior, 2
Find all pairs of integers $(x, y)$ such that the number
\[
\frac{x^2 + y^2}{xy}
\]
is an integer.
TNO 2023 Senior, 6
The points inside a circle \( \Gamma \) are painted with \( n \geq 1 \) colors. A color is said to be dense in a circle \( \Omega \) if every circle contained within \( \Omega \) has points of that color in its interior. Prove that there exists at least one color that is dense in some circle contained within \( \Gamma \).
2023 Chile Classification NMO Juniors, 1
There are 10 numbers on a board. The product of any four of them is divisible by 30.
Prove that at least one of the numbers on the board is divisible by 30.
TNO 2023 Junior, 6
Show that for every integer $n \geq 1$, it is possible to express $5^n$ as the sum of two nonzero squares.
2023 Chile Classification NMO Seniors, 1
The function $f(x) = ax + b$ satisfies the following equalities:
\begin{align*}
f(f(f(1))) &= 2023, \\
f(f(f(0))) &= 1996.
\end{align*}
Find the value of $a$.
2023 Chile TST Ibero., 1
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.
2013 Chile TST Ibero, 2
Let $a \in \mathbb{N}$ such that $a + n^2$ can be expressed as the sum of two squares for all $n \in \mathbb{N}$. Prove that $a$ is the square of a natural number.
TNO 2008 Junior, 1
There are three number-transforming machines. We input the pair $(a_1, a_2)$, and the machine returns $(b_1, b_2)$. We denote this transformation as $(a_1, a_2) \to (b_1, b_2)$.
(a) The first machine can perform two transformations:
- $(a, b) \to (a - 1, b - 1)$
- $(a, b) \to (a + 13, b + 5)$
If the input pair is $(5,2)$, is it possible to obtain the pair $(20,22)$ after a series of transformations?
(b) The second machine can perform two transformations:
- $(a, b) \to (a - 1, b - 1)$
- $(a, b) \to (2a, 2b)$
If the input pair is $(15,10)$, is it possible to obtain the pair $(27,23)$ after a series of transformations?
(c) The third machine can perform two transformations:
- $(a, b) \to (a - 2, b + 2)$
- $(a, b) \to (2a - b + 1, 2b - 1 - a)$
If the input pair is $(5,8)$, is it possible to obtain the pair $(13,17)$ after a series of transformations?
TNO 2024 Junior, 1
A group of 6 math students is staying at a mathematical hotel to participate in a math tournament that will take place in the city in the coming days. This group, composed of 3 women and 3 men, was assigned rooms in a specific way by the hotel administration: in separate rooms and alternating between genders, specifically: woman, man, woman, man, woman, man, occupying the last 6 rooms in a corridor numbered from 101 to 110.
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|}
\hline
M & H & M & H & M & H & & & & \\ \hline
110 & 109 & 108 & 107 & 106 & 105 & 104 & 103 & 102 & 101 \\ \hline
\end{tabular}
Against the hotel's rules, the group devised the following game: A valid room exchange occurs when two students in consecutive rooms move to two empty rooms, such that the difference between their new room numbers and their original ones is the same. For example, if the students in rooms 105 and 106 move to rooms 101 and 102, this would be a valid exchange since both numbers decreased by 4 units.
Determine if, following these rules, the students can manage to have rooms 101, 102, and 103 occupied by men and rooms 104, 105, and 106 occupied by women in just 3 valid exchanges.
2024 Chile National Olympiad., 5
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.
TNO 2023 Junior, 4
Find the largest number formed by the digits 1 to 9, without repetition, that is divisible by 18.
2014 Chile TST Ibero, 3
Let $x_0 = 5$ and define the sequence recursively as $x_{n+1} = x_n + \frac{1}{x_n}$. Prove that:
\[
45 < x_{1000} < 45.1.
\]
TNO 2023 Senior, 2
Find all integers \( n > 1 \) such that all prime divisors of \( n^6 - 1 \) divide \( (n^2 - 1)(n^3 - 1) \).
2015 Chile TST Ibero, 4
Let $x, y \in \mathbb{R}^+$. Prove that:
\[
\left( 1 + \frac{1}{x} \right) \left( 1 + \frac{1}{y} \right) \geq \left( 1 + \frac{2}{x + y} \right)^2.
\]
2024 Chile Classification NMO Seniors, 2
Find all real numbers $x$ such that:
\[
2^x + 3^x + 6^x - 4^x - 9^x = 1,
\]
and prove that there are no others.
2024 Chile TST Ibero., 4
Prove that if \( a \), \( b \), and \( c \) are positive real numbers, then the following inequality holds:
\[
\frac{a + 3c}{a + b} + \frac{c + 3a}{b + c} + \frac{4b}{c + a} \geq 6.
\]
2024 Chile National Olympiad., 6
Let \( 133\ldots 33 \) be a number with \( k \geq 2 \) digits, which we assume is prime. Prove that \( k(k + 2) \) is a multiple of 24. (For example, 133...33 is a prime number when \( k = 16\)
2015 Chile TST Ibero, 2
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.
2024 Chile TST IMO, 3
Let $ABC$ be a triangle. Circle $\Gamma$ passes through $A$, meets segments $AB$ and $AC$ again at points $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $BDF$ at $F$ and the tangent to circle $CEG$ at $G$ meet at point $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.