Found problems: 132
2014 Chile TST IMO, 4
Let \( f(n) \) be a polynomial with integer coefficients. Prove that if \( f(-1) \), \( f(0) \), and \( f(1) \) are not divisible by 3, then \( f(n) \neq 0 \) for all integers \( n \).
2013 Chile TST Ibero, 3
The incircle of triangle $\triangle ABC$ touches $AC$ and $BC$ at $E$ and $D$ respectively. The excircle corresponding to $A$ touches the extensions of $BC$ at $A_1$, $CA$ at $B_1$, and $AB$ at $C_1$. Let $DE \cap A_1B_1 = L$. Prove that $L$ belongs to the circumcircle of triangle $\triangle A_1B_1C_1$.
2015 Chile TST Ibero, 3
Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.
TNO 2023 Junior, 1
In the convex quadrilateral $ABCD$, it is given that $\angle BAD = \angle DCB = 90^\circ$, $AB = 7$, $CD = 11$, and that $BC$ and $AD$ are integers greater than 11. Determine the values of $BC$ and $AD$.
TNO 2008 Junior, 9
(a) Is it possible to form a prime number using all the digits 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 exactly once?
(b) Consider the following magic square where the sum of each row, column, and diagonal is the same (in this case, 15):
\[
\begin{array}{ccc}
6 & 7 & 2 \\
1 & 5 & 9 \\
8 & 3 & 4 \\
\end{array}
\]
Is it possible to create a magic square with the same properties using the numbers 11, 12, 13, 14, 15, 16, 17, 18, and 19?
TNO 2008 Senior, 8
Two mathematicians discuss two positive integers. One of them states that the square of the ratio between their product and their sum is exactly one more than this ratio. What is the smaller of these two numbers?
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 \(BP = CQ\).
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 $n \in \mathbb{N}$:
\[
f(g(n)) = n + 2015,
\]
\[
g(f(n)) = n^2 + 2015.
\]
2014 Chile TST IMO, 2
Given \(n, k \in \mathbb{N}\), prove that \((n-1)^2\) divides \(n^k - 1\) if and only if \(n-1 \mid k\).
2024 Chile Junior Math Olympiad, 3
Determine all triples \( (a, b, c) \) of positive integers such that:
\[
a + b + c = abc.
\]
2024 Chile Junior Math Olympiad, 6
In a regular polygon with 100 vertices, 10 vertices are painted blue, and 10 other vertices are painted red.
1. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( R_1 \) is equal to the distance between \( A_2 \) and \( R_2 \).
2. Prove that there exist two distinct blue vertices \( A_1 \) and \( A_2 \), and two distinct red vertices \( R_1 \) and \( R_2 \), such that the distance between \( A_1 \) and \( A_2 \) is equal to the distance between \( R_1 \) and \( R_2 \).
TNO 2008 Senior, 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 $4n + 3$ is equal to the sum of the digits of $n$.
(c) Prove that for any natural number $n$, it is possible to find $n$ consecutive numbers such that none of them is prime.
TNO 2024 Senior, 6
Let $C$ be a circle, and let $A, B, P$ be three points on $C$. Let $L_A$ and $L_B$ be the tangent lines to $C$ passing through $A$ and $B$, respectively. Let $a$ and $b$ be the distances from $P$ to $L_A$ and $L_B$, respectively, and let $c$ be the distance from $P$ to the chord of $C$ determined by $A$ and $B$. Prove that $c^2 = a \cdot b$.
2015 Chile TST Ibero, 3
Prove that in a scalene acute-angled triangle, the orthocenter, the incenter, and the circumcenter are not collinear.
TNO 2008 Junior, 3
Luis' friends played a prank on him in his geometry homework. They erased the entire triangle but left traces equivalent to two sides measuring $a$ and $b$, with $b > a$, and the height $h$ falling on the side measuring $b$, with $h < a$. Help Luis reconstruct the original triangle using only a straightedge and compass. Since Luis' method does not involve measurements, prove that his method results in a triangle longer than its given sides and height.
2023 Chile Classification NMO Seniors, 4
When writing the product of two three-digit numbers, the multiplication sign was omitted, forming a six-digit number. It turns out that the six-digit number is equal to three times the product.
Find the six-digit number.
2024 Chile Classification NMO Seniors, 1
Bus tickets from a transportation company are numbered with six digits, ranging from 000000 to 999999. A ticket is considered "lucky" if the sum of the first three digits equals the sum of the last three digits. For example, ticket 721055 is lucky, whereas 003101 is not.
Determine how many consecutive tickets a person must buy to guarantee obtaining at least one lucky ticket, regardless of the starting ticket number.
2013 Chile TST Ibero, 3
The incircle of triangle $\triangle ABC$ touches $AC$ and $BC$ at $E$ and $D$ respectively. The excircle corresponding to $A$ touches the extensions of $BC$ at $A_1$, $CA$ at $B_1$, and $AB$ at $C_1$. Let $DE \cap A_1B_1 = L$. Prove that $L$ belongs to the circumcircle of triangle $\triangle A_1B_1C_1$.
TNO 2008 Junior, 8
A traffic accident involved three cars: one blue, one green, and one red. Three witnesses spoke to the police and gave the following statements:
**Person 1:** The red car was guilty, and either the green or the blue one was involved.
**Person 2:** Either the green car or the red car was guilty, but not both.
**Person 3:** Only one of the cars was guilty, but it was not the blue one.
The police know that at least one car was guilty and that at least one car was not. However, the police do not know if any of the three witnesses lied.
Which car(s) were responsible for the accident?
TNO 2008 Senior, 11
Each face of a cube is painted with a different color. How many distinct cubes can be created this way? (*Observation: The ways to color the cube are $6!$, since each time a color is used on one face, there is one fewer available for the others. However, this does not determine $6!$ different cubes, since colorings that differ only by rotation should be considered the same.*)
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.
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:
\[
45 < x_{1000} < 45.1.
\]
2014 Chile TST IMO, 1
Given positive real numbers \(a\), \(b\), and \(c\) such that \(a+b+c \leq \frac{3}{2}\), find the minimum of
\[
a+b+c + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.
\]
2023 Chile Classification NMO Juniors, 2
There are 2023 points on the plane. Prove that there exists a circle that contains 2000 points inside it and leaves the remaining 23 outside.
For example, if we had 5 points on the plane, we could find a circle that contains 4 of them inside and leaves 1 outside. Similarly, for 10 points, there exists a circle that contains 7 inside and leaves 3 outside. This reasoning extends to 2023 points, ensuring that such a division is always possible.
2013 Chile TST Ibero, 1
Prove that the equation
\[
x^z + y^z = z^z
\]
has no solutions in postive integers.