Found problems: 132
TNO 2023 Senior, 3
Let \( \triangle ABC \) be an acute triangle with orthocenter \( H \), and let \( M \) be the midpoint of \( BC \). Let \( P \) be the foot of the perpendicular from \( H \) to \( AM \). Prove that \( AM \cdot MP = BM^2 \).
2024 Chile Classification NMO Seniors, 4
Consider a right triangle $\triangle ABC$ with right angle at $A$. Let $CD$ be the bisector of angle $\angle ACB$, where $D$ lies on segment $AB$. The perpendicular line from $B$ to $BC$ intersects $CD$ at $E$. Let $F$ be the reflection of $E$ over $B$, and let $P$ be the intersection of $DF$ with $BC$. Prove that lines $EP$ and $CF$ are perpendicular.
TNO 2008 Senior, 2
The sequence $a_n$ for $n \in \mathbb{N}$ is defined as follows:
\[ a_0 = 6, \quad a_1 = 7, \quad a_{n+2} = 3a_{n+1} - 2a_n \]
Find all values of $n$ such that $n^2 = a_n$.
TNO 2024 Senior, 2
Consider the acute triangle $ABC$. Let $C_1$ and $C_2$ be semicircles with diameters $AB$ and $AC$, respectively, positioned outside triangle $ABC$. The altitude passing through $C$ intersects $C_1$ at $P$, and similarly, $Q$ is the intersection of $C_2$ with the extension of the altitude passing through $B$. Prove that $AP = AQ$.
2024 Chile National Olympiad., 4
Find all pairs \((x, y)\) of real numbers that satisfy the system
\[
(x + 1)(x^2 + 1) = y^3 + 1
\]
\[
(y + 1)(y^2 + 1) = x^3 + 1
\]
TNO 2024 Senior, 4
In a lake, there are 2024 leaves arranged in a row. Two frogs are positioned, one on the first leaf and the other on the second leaf. Every minute, both frogs jump simultaneously. Each time a frog jumps, it decides whether to jump to the next leaf or to the leaf that is three positions ahead. Is it possible for each leaf to be visited exactly once by exactly one of the frogs?
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 National Olympiad., 3
Let \( AD \) and \( BE \) be altitudes of triangle \( \triangle ABC \) that meet at the orthocenter \( H \). The midpoints of segments \( AB \) and \( CH \) are \( X \) and \( Y \), respectively. Prove that the line \( XY \) is perpendicular to line \( DE \).
TNO 2023 Senior, 1
Let \( n \geq 4 \) be an integer. Show that at a party of \( n \) people, it is possible for each person to have greeted exactly three other people if and only if \( n \) is even.
TNO 2008 Senior, 5
Consider the polynomial with real coefficients:
\[ p(x) = a_{2008}x^{2008} + a_{2007}x^{2007} + \dots + a_1x + a_0 \]
and it is given that its coefficients satisfy:
\[ a_i + a_{i+1} = a_{i+2}, \quad i \in \{0,1,2,\dots,2006\} \]
If $p(1) = 2008$ and $p(-1) = 0$, compute $a_{2008} - a_0$.
2024 Chile TST Ibero., 3
Find all natural numbers \( n \) for which it is possible to construct an \( n \times n \) square using only tetrominoes like the one below:
2014 Chile TST Ibero, 1
Consider a function $f: \mathbb{R} \to \mathbb{R}$ satisfying for all $x \in \mathbb{R}$:
\[
f(x+1) = \frac{1}{2} + \sqrt{f(x) - f(x)^2}.
\]
Prove that there exists a $b > 0$ such that $f(x + b) = f(x)$ for all $x \in \mathbb{R}$.
TNO 2008 Senior, 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 $(25,32)$, is it possible to obtain the pair $(82,98)$ 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 $(34,60)$, is it possible to obtain the pair $(2000, 2008)$ 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 $(145,220)$, is it possible to obtain the pair $(363,498)$ after a series of transformations?
TNO 2008 Senior, 6
If a square is drawn externally on each side of a parallelogram, prove that:
(a) The quadrilateral formed by the centers of these squares is also a square.
(b) The diagonals of the new square formed are concurrent with the diagonals of the original parallelogram.
TNO 2023 Junior, 3
The following sequence of letters is written on a board:
\[
\text{TNOTNOTNO...TNOTN}
\]
where the sequence repeats 2024 times.
At each step, one of the following operations can be performed:
1. Take two different adjacent letters and replace them with two copies of the missing letter.
2. Take three consecutive identical letters and remove them.
After a certain number of steps, only two identical letters remain. Determine which letter it is possible to reach.
TNO 2008 Junior, 2
A cube of size $4 \times 4 \times 4$ is divided into 16 equal squares per face, with numbers from 1 to 96 randomly assigned to these squares. An operation consists of taking two squares that share a vertex, summing their numbers, and rewriting this sum in one of the squares while leaving the other blank. After performing several such operations, only one number remains. Prove that regardless of the order of operations, the final remaining number is always the same. Additionally, find this number.
TNO 2008 Junior, 11
(a) Consider a $6 \times 6$ board with two squares removed at diagonally opposite corners. Prove that it is not possible to exactly cover it with $2 \times 1$ dominoes.
(b) Consider a box with dimensions $4 \times 4 \times 4$ from which two $1 \times 1 \times 1$ cubes have been removed at diagonally opposite corners. Is it possible to fill the remaining space exactly with bricks of dimensions $2 \times 1 \times 1$?
TNO 2008 Senior, 9
Let $f: \mathbb{N} \to \mathbb{N}$ be a function that satisfies:
\[
f(1) = 2008,
\]
\[
f(4n^2) = 4f(n^2),
\]
\[
f(4n^2 + 2) = 4f(n^2) + 3,
\]
\[
f(4n(n+1)) = 4f(n(n+1)) + 1,
\]
\[
f(4n(n+1) + 3) = 4f(n(n+1)) + 4.
\]
Determine whether there exists a natural number $m$ such that:
\[
1^2 + 2^2 + \dots + m^2 + f(1^2) + \dots + f(m^2) = 2008m + 251.
\]
2015 Chile TST Ibero, 1
Determine the number of functions $f: \mathbb{N} \to \mathbb{N}$ and $g: \mathbb{N} \to \mathbb{N}$ such that for all $n \in \mathbb{N}$:
\[
f(g(n)) = n + 2015,
\]
\[
g(f(n)) = n^2 + 2015.
\]
2013 Chile TST Ibero, 1
Prove that the equation
\[
x^z + y^z = z^z
\]
has no solutions in postive integers.
TNO 2008 Senior, 10
Let $\triangle ABC$ and a point $D$ on $AC$ such that $BD = DC = 3$. If $AD = 6$ and $\angle ACB = 30^\circ$, calculate $\angle ABD$.
2024 Chile TST Ibero., 1
Determine all integers \( x \) for which the expression \( x^2 + 10x + 160 \) is a perfect square.
TNO 2008 Junior, 12
(a) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $11n$ is twice the sum of the digits of $n$.
(b) Prove that there exist infinitely many natural numbers $n$ such that the sum of the digits of $5n + 1$ is six times the sum of the digits of $n$.
2023 Chile Classification NMO Juniors, 3
The following light grid is given:
\begin{tabular}{cccc}
o & o & o & o \\
o & o & o & o \\
o & o & o & o \\
o & o & o & o
\end{tabular}
where `o` represents a switched-off light and `•` represents a switched-on light. Each time a light is pressed, it toggles its state (on/off) as well as the state of its four adjacent neighbors (left, right, above, below). The bottom edge lights are considered to be immediately above the top edge lights, and the same applies to the lateral edges.The right figure illustrates the effect of pressing a light in a corner.
Pressing a certain combination of lights results in all lights turning on. Prove that all lights must have been pressed at least once.
2024 Chile National Olympiad., 2
On a table, there are many coins and a container with two coins. Vale and Diego play the following game, where Vale starts and then Diego plays, alternating turns. If at the beginning of a turn the container contains \( n \) coins, the player can add a number \( d \) of coins, where \( d \) divides exactly into \( n \) and \( d < n \). The first player to complete at least 2024 coins in the container wins. Prove that there exists a strategy for Vale to win, no matter the decisions made by Diego.