Found problems: 24
2016 IberoAmerican, 4
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 the $2$ main diagonals and the ones parallel to those ones.
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.
2016 IberoAmerican, 1
Find all prime numbers $p,q,r,k$ such that $pq+qr+rp = 12k+1$
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:
\[
45 < x_{1000} < 45.1.
\]
2016 Chile TST IMO, 2
There are 2016 points near a line such that the distance from each point to the line is less than 1 cm, and the distance between any two points is always greater than 2 cm. Prove that there exist two points whose distance is at least 17 meters.
2017 Iberoamerican, 4
Let $ABC$ be an acute triangle with $AC > AB$ and $O$ its circumcenter. Let $D$ be a point on segment $BC$ such that $O$ lies inside triangle $ADC$ and $\angle DAO + \angle ADB = \angle ADC$. Let $P$ and $Q$ be the circumcenters of triangles $ABD$ and $ACD$ respectively, and let $M$ be the intersection of lines $BP$ and $CQ$. Show that lines $AM, PQ$ and $BC$ are concurrent.
[i]Pablo Jaén, Panama[/i]
2023 Chile TST Ibero., 4
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 \(BP = CQ\).
2016 IberoAmerican, 3
Let $ABC$ be an acute triangle and $\Gamma$ its circumcircle. The lines tangent to $\Gamma$ through $B$ and $C$ meet at $P$. Let $M$ be a point on the arc $AC$ that does not contain $B$ such that $M \neq A$ and $M \neq C$, and $K$ be the point where the lines $BC$ and $AM$ meet. Let $R$ be the point symmetrical to $P$ with respect to the line $AM$ and $Q$ the point of intersection of lines $RA$ and $PM$. Let $J$ be the midpoint of $BC$ and $L$ be the intersection point of the line $PJ$ and the line through $A$ parallel to $PR$. Prove that $L, J, A, Q,$ and $K$ all lie on a circle.
2017 Iberoamerican, 2
Let $ABC$ be an acute angled triangle and $\Gamma$ its circumcircle. Led $D$ be a point on segment $BC$, different from $B$ and $C$, and let $M$ be the midpoint of $AD$. The line perpendicular to $AB$ that passes through $D$ intersects $AB$ in $E$ and $\Gamma$ in $F$, with point $D$ between $E$ and $F$. Lines $FC$ and $EM$ intersect at point $X$. If $\angle DAE = \angle AFE$, show that line $AX$ is tangent to $\Gamma$.
2020 IberoAmerican, 5
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(xf(x-y))+yf(x)=x+y+f(x^2),$$ for all real numbers $x$ and $y.$
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:
\[
45 < x_{1000} < 45.1.
\]
2017 Iberoamerican, 3
Consider the configurations of integers
$a_{1,1}$
$a_{2,1} \quad a_{2,2}$
$a_{3,1} \quad a_{3,2} \quad a_{3,3}$
$\dots \quad \dots \quad \dots$
$a_{2017,1} \quad a_{2017,2} \quad a_{2017,3} \quad \dots \quad a_{2017,2017}$
Where $a_{i,j} = a_{i+1,j} + a_{i+1,j+1}$ for all $i,j$ such that $1 \leq j \leq i \leq 2016$.
Determine the maximum amount of odd integers that such configuration can contain.
2016 Chile TST IMO, 4
Let \( f \) and \( g \) be two nonzero polynomials with integer coefficients such that \( \deg(f) > \deg(g) \). Suppose that for infinitely many prime numbers \( p \), the polynomial \( pf + g \) has a rational root. Prove that \( f \) has a rational root.
Clarification: A rational root of a polynomial \( f \) is a number \( q \in \mathbb{Q} \) such that \( f(q) = 0 \).
2014 Chile TST Ibero, 2
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:
\[
\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)}.
\]
2020 IberoAmerican, 2
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 $m\ge T_m$.
[i]Proposed by Nicolás De la Hoz [/i]
2013 IberoAmerican, 3
Let $A = \{1,...,n\}$ with $n \textgreater 5$. Prove that one can find $B$ a finite set of positive integers such that $A$ is a subset of $B$ and
$\displaystyle\sum_{x \in B} x^2 = \displaystyle\prod_{x \in B} x$
2016 IberoAmerican, 2
Find all positive real numbers $(x,y,z)$ such that:
$$x = \frac{1}{y^2+y-1}$$
$$y = \frac{1}{z^2+z-1}$$
$$z = \frac{1}{x^2+x-1}$$
2017 Iberoamerican, 5
Given a positive integer $n$, all of its positive integer divisors are written on a board. Two players $A$ and $B$ play the following game:
Each turn, each player colors one of these divisors either red or blue. They may choose whichever color they wish, but they may only color numbers that have not been colored before. The game ends once every number has been colored. $A$ wins if the product of all of the red numbers is a perfect square, or if no number has been colored red, $B$ wins otherwise. If $A$ goes first, determine who has a winning strategy for each $n$.
2017 Iberoamerican, 6
Let $n > 2$ be an even positive integer and let $a_1 < a_2 < \dots < a_n$ be real numbers such that $a_{k + 1} - a_k \leq 1$ for each $1 \leq k \leq n - 1$. Let $A$ be the set of ordered pairs $(i, j)$ with $1 \leq i < j \leq n$ such that $j - i$ is even, and let $B$ the set of ordered pairs $(i, j)$ with $1 \leq i < j \leq n$ such that $j - i$ is odd. Show that
$$\prod_{(i, j) \in A} (a_j - a_i) > \prod_{(i, j) \in B} (a_j - a_i)$$
2023 Chile TST Ibero., 4
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 \(BP = CQ\).
2014 Chile TST Ibero, 2
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:
\[
\frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)}.
\]
2017 Iberoamerican, 1
For every positive integer $n$ let $S(n)$ be the sum of its digits. We say $n$ has a property $P$ if all terms in the infinite secuence $n, S(n), S(S(n)),...$ are even numbers, and we say $n$ has a property $I$ if all terms in this secuence are odd. Show that for, $1 \le n \le 2017$ there are more $n$ that have property $I$ than those who have $P$.
2019 IberoAmerican, 4
Let $ABCD$ be a trapezoid with $AB\parallel CD$ and inscribed in a circumference $\Gamma$. Let $P$ and $Q$ be two points on segment $AB$ ($A$, $P$, $Q$, $B$ appear in that order and are distinct) such that $AP=QB$. Let $E$ and $F$ be the second intersection points of lines $CP$ and $CQ$ with $\Gamma$, respectively. Lines $AB$ and $EF$ intersect at $G$. Prove that line $DG$ is tangent to $\Gamma$.
2016 Chile TST IMO, 3
A set \( A \) of integers is said to be \textit{admissible} if it satisfies the property:
\[
\text{If } x, y \in A, \text{ then } x^2 + kxy + y^2 \in A \text{ for all } k \in \mathbb{Z}.
\]
Determine all pairs \( (m, n) \) of nonzero integers such that the only admissible set containing both \( m \) and \( n \) is the set of all integers.