Olympiad problems

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 Junior, 10

Tags: Chile , combinatorics

A jeweler makes necklaces with round stones, four emeralds (green) and four rubies (red), arranged at equal distances from each other. One day, they decide to give away some necklaces. How many necklaces can they give away without the risk of two friends ending up with the same necklace? (*Observation: The necklace is completely symmetrical except for the type of stone, meaning there is not a unique way to form it. Consider this while solving the problem.*)

2024 Chile TST Ibero., 5

Tags: TST , Chile , geometry

Let $\triangle ABC$ be an acute-angled triangle. Let $P$ be the midpoint of $BC$, and $K$ the foot of the altitude from $A$ to side $BC$. Let $D$ be a point on segment $AP$ such that $\angle BDC = 90^\circ$. Let $E$ be the second point of intersection of line $BC$ with the circumcircle of $\triangle ADK$. Let $F$ be the second point of intersection of line $AE$ with the circumcircle of $\triangle ABC$. Prove that $\angle AFD = 90^\circ$.

2025 Chile TST IMO-Cono, 1

Tags: number theory , Chile

Find all triples $ (x, y, z) $ of positive integers that satisfy the equation \[ x + xy + xyz = 31. \]

2024 Chile Junior Math Olympiad, 4

Tags: geometry , Chile

Consider a triangle with sides of length $ a $, $ b $, and $ c $ that satisfy the following conditions: \[ a + b = c + 3 \quad c^2 + 9 = 2ab \] Find the area of the triangle.

TNO 2024 Senior, 5

Tags: combinatorics , Chile

Nine people have attended four different meetings sitting around a circular table. Could they have done so in a way that no two people sat next to each other more than once? Justify your answer.

2024 Chile Junior Math Olympiad, 1

Tags: geometry , Chile , 3D geometry

A plastic ball with a radius of 45 mm has a circular hole made in it. The hole is made to fit a ball with a radius of 35 mm, in such a way that the distance between their centers is 60 mm. Calculate the radius of the hole.

2024 Chile Classification NMO Juniors, 4

Tags: geometry , Chile

Given a square $ABCD$ with a side length of 4 cm and a point $E$ on side $BC$, a square $AEFG$ is constructed with side $AE$, as shown in the figure. It is known that triangle $DFG$ has an area of 1 cm$^2$. Determine the area of square $AEFG$.

TNO 2008 Junior, 4

Tags: Chile , combinatorics

A square cake of uniform height is evenly covered with frosting on the top and all four sides. Find a way to cut the cake into five portions such that: (a) All portions contain the same amount of cake. (b) All portions contain the same amount of cake and frosting.

TNO 2008 Senior, 3

Tags: geometry , Chile

Luis' friends decided to play a prank on him in his geometry homework. They erased most of a triangle and, instead, drew an equivalent triangle with the sum of its three side lengths. Help Luis complete his homework by reconstructing the original triangle using only a straightedge and compass. Since Luis' method involves measurements, prove that his method results in a triangle longer than the sum of its three sides.

TNO 2023 Senior, 5

Tags: algebra , Chile

Find all triples of integers $ (x, y, z) $ such that \[ x - yz = 11 \] \[ xz + y = 13 \]

2016 Chile TST IMO, 4

Tags: TST , Chile , algebra , polynomial , IMO Shortlist , Iberoamerican

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

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: \[ \frac{\text{Area}(\triangle ABC)}{\text{Area}(\triangle DEF)}. \]

2025 Chile TST IMO-Cono, 4

Tags: geometry , Chile

Let $ ABC $ be a triangle with $ AB < AC $. Let $ M $ be the midpoint of $ AC $, and let $ D $ be a point on segment $ AC $ such that $ DB = DC $. Let $ E $ be the point of intersection, different from $ B $, of the circumcircle of triangle $ ABM $ and line $ BD $. Define $ P $ and $ Q $ as the points of intersection of line $ BC $ with $ EM $ and $ AE $, respectively. Prove that $ P $ is the midpoint of $ BQ $.

2025 Chile TST IMO-Cono, 2

Tags: combinatorics , Chile

At a meeting, there are $ N $ people who do not know each other. Prove that it is possible to introduce them in such a way that no three of them have the same number of acquaintances.

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.

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) $.

2024 Chile National Olympiad., 1

Tags: Chile , algebra , Funcion , function

Let $ f(x) = \frac{100^x}{100^x + 10} $. Determine the value of: \[ f\left( \frac{1}{2024} \right) - f\left( \frac{2}{2024} \right) + f\left( \frac{3}{2024} \right) - f\left( \frac{4}{2024} \right) + \ldots - f\left( \frac{2022}{2024} \right) + f\left( \frac{2023}{2024} \right) \]

2023 Chile TST IMO, 5

Tags: geometry , TST , Chile

Let $ \triangle ABC $ be an acute-angled triangle. Let $ D $ and $ E $ be the feet of the altitudes from $ B $ and $ C $, respectively. Let $ E' $ be the reflection of point $ E $ with respect to line $ BD $, which is assumed to lie on the circumcircle of triangle $ \triangle ABC $. Let $ C' $ be the reflection of point $ C $ with respect to line $ BD $. Prove that triangle $ C'AE $ is isosceles and determine the ratio $ AD : DC $.

TNO 2024 Senior, 1

Tags: combinatorics , Chile

Sofía has many boxes where she keeps candies. Every morning, she chooses two of these boxes and places one candy in each. However, each night, a thief selects one box and steals all the candies inside it. Sofía dreams of waking up one day and finding a box with 2024 candies. Prove that Sofía can always fulfill her dream if she has enough boxes.

2024 Chile TST IMO, 1

Tags: TST , Chile , combinatorics

Consider a set of $ n \geq 3 $ points in the plane where no three are collinear. Prove that the points can be labeled as $ P_1, P_2, \dots, P_n $ so that the angles $ \angle P_i P_{i+1} P_{i+2} $ are less than $ 90^\circ $ for all $ i $.

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 $.

2024 Chile TST IMO, 2

Tags: number theory , Chile , TST

Find all natural numbers that have a multiple consisting only of the digit 9.

TNO 2024 Junior, 2

Tags: geometry , Chile

Prove that the area enclosed by three semicircles, tangent at their ends, is equal to the area of the circle whose diameter is $CD$, perpendicular to the diameter $AB$.

TNO 2024 Junior, 6

Tags: number theory , combinatorics , Chile

A box contains 900 cards numbered from 100 to 999. Cards are drawn randomly, one at a time, without replacement, and the sum of their digits is recorded. What is the minimum number of cards that must be drawn to guarantee that at least three of these sums are the same?