Olympiad problems

1979 Brazil National Olympiad, 1

Show that if $a < b$ are in the interval $\left[0, \frac{\pi}{2}\right]$ then $a - \sin a < b - \sin b$. Is this true for $a < b$ in the interval $\left[\pi,\frac{3\pi}{2}\right]$?

Brazil L2 Finals (OBM) - geometry, 2019.3

Tags: euclidean geometry , geometry , Brazilian Math Olympiad

Let $ABC$ be an acutangle triangle inscribed in a circle $\Gamma$ of center $O$. Let $D$ be the height of the vertex $A$. Let E and F be points over $\Gamma$ such that $AE = AD = AF$. Let $P$ and $Q$ be the intersection points of the $EF $ with sides $AB$ and $AC$ respectively. Let $X$ be the second intersection point of $\Gamma$ with the circle circumscribed to the triangle $AP Q$. Show that the lines $XD$ and $AO $ meet at a point above sobre

1988 Brazil National Olympiad, 1

Tags: Brazilian Math Olympiad , Brazilian Math Olympiad 1988 , prime numbers , number theory

Find all primes which are sum of two primes and difference of two primes.

2019 Peru Cono Sur TST, P5

Tags: number theory , Brazilian Math Olympiad , Brazilian Math Olympiad 2018 , games

Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations. [b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal. [b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.

1989 Brazil National Olympiad, 2

Tags: Brazilian Math Olympiad 1989 , Brazilian Math Olympiad , number theory , Perfect Squares , Pell equations

Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.

2018 Brazil Undergrad MO, 3

Tags: permutations , Brazilian Math Olympiad , college contests , number theory

How many permutations $a_1, a_2, a_3, a_4$ of $1, 2, 3, 4$ satisfy the condition that for $k = 1, 2, 3,$ the list $a_1,. . . , a_k$ contains a number greater than $k$?

2018 Brazil Undergrad MO, 12

Tags: probability , geometry , Brazilian Math Olympiad , college contests

Let $ABC$ be an equilateral triangle. $A $ point $P$ is chosen at random within this triangle. What is the probability that the sum of the distances from point $P$ to the sides of triangle $ABC$ are measures of the sides of a triangle?

2018 Brazil Undergrad MO, 5

Tags: Sets , algebra , Brazilian Math Olympiad , Brazil , minimum value , minimization

Consider the set $A = \left\{\frac{j}{4}+\frac{100}{j}|j=1,2,3,..\right\} $ What is the smallest number that belongs to the $ A $ set?

2018 Brazil National Olympiad, 2

Tags: number theory , Brazilian Math Olympiad , Brazilian Math Olympiad 2018 , games

Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations. [b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal. [b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.

2018 Brazil National Olympiad, 6

Tags: combinatorial geometry , combinatorics , Brazilian Math Olympiad , Brazilian Math Olympiad 2018 , geometry

Consider $4n$ points in the plane, with no three points collinear. Using these points as vertices, we form $\binom{4n}{3}$ triangles. Show that there exists a point $X$ of the plane that belongs to the interior of at least $2n^3$ of these triangles.

2018 Brazil Undergrad MO, 13

Tags: function , Brazilian Math Olympiad , college contests , algebra

A continuous function $ f: \mathbb {R} \to \mathbb {R} $ satisfies $ f (x) f (f (x)) = 1 $ for every real $ x $ and $ f (2020) = 2019 $ . What is the value of $ f (2018) $?

1979 Brazil National Olympiad, 3

Tags: geometry , Brazilian Math Olympiad , Brazilian Math Olympiad 1979 , orthocenter

The vertex C of the triangle ABC is allowed to vary along a line parallel to AB. Find the locus of the orthocenter.

2017 Brazil National Olympiad, 3.

Tags: geometry , circumscribed quadrilateral , incircle , incenter , Brazilian Math Olympiad , Brazilian Math Olympiad 2017

[b]3.[/b] A quadrilateral $ABCD$ has the incircle $\omega$ and is such that the semi-lines $AB$ and $DC$ intersect at point $P$ and the semi-lines $AD$ and $BC$ intersect at point $Q$. The lines $AC$ and $PQ$ intersect at point $R$. Let $T$ be the point of $\omega$ closest from line $PQ$. Prove that the line $RT$ passes through the incenter of triangle $PQC$.

2019 Brazil National Olympiad, 6

Tags: Brazilian Math Olympiad

In the Cartesian plane, all points with both integer coordinates are painted blue. Blue colon they are said to be [b]mutually visible[/b] if the line segment connecting them has no other blue dots. Prove that There is a set of $ 2019$ blue dots that are mutually visible two by two.

2020 Brazil National Olympiad, 2

Tags: Fibonacci , number theory , Brazilian Math Olympiad , Brazil , Analytic Number Theory

For a positive integer $a$, define $F_1 ^{(a)}=1$, $F_2 ^{(a)}=a$ and for $n>2$, $F_n ^{(a)}=F_{n-1} ^{(a)}+F_{n-2} ^{(a)}$. A positive integer is [i]fibonatic[/i] when it is equal to $F_n ^{(a)}$ for a positive integer $a$ and $n>3$. Prove that there are infintely many not [i]fibonatic[/i] integers.

2003 Brazil National Olympiad, 3

Tags: geometry , rhombus , trigonometry , projective geometry , Brazilian Math Olympiad

$ABCD$ is a rhombus. Take points $E$, $F$, $G$, $H$ on sides $AB$, $BC$, $CD$, $DA$ respectively so that $EF$ and $GH$ are tangent to the incircle of $ABCD$. Show that $EH$ and $FG$ are parallel.

2018 Brazil National Olympiad, 4

Tags: inequalities , algebra , maximum value , Brazilian Math Olympiad , Brazilian Math Olympiad 2018

Esmeralda writes $2n$ real numbers $x_1, x_2, \dots , x_{2n}$, all belonging to the interval $[0, 1]$, around a circle and multiplies all the pairs of numbers neighboring to each other, obtaining, in the counterclockwise direction, the products $p_1 = x_1x_2$, $p_2 = x_2x_3$, $\dots$ , $p_{2n} = x_{2n}x_1$. She adds the products with even indices and subtracts the products with odd indices. What is the maximum possible number Esmeralda can get?

2018 Brazil Undergrad MO, 8

Tags: Brazilian Math Olympiad , college contests , probability

A student will take an exam in which they have to solve three chosen problems by chance of a list of $10$ possible problems. It will be approved if it correctly resolves two problems. Considering that the student can solve five of the problems on the list and not know how to solve others, how likely is he to pass the exam?

2017 Brazil National Olympiad, 5.

Tags: geometry , incenter , orthocenter , Circumcenter , angle bisector , Brazilian Math Olympiad , Brazilian Math Olympiad 2017

[b]5.[/b] In triangle $ABC$, let $r_A$ be the line that passes through the midpoint of $BC$ and is perpendicular to the internal bisector of $\angle{BAC}$. Define $r_B$ and $r_C$ similarly. Let $H$ and $I$ be the orthocenter and incenter of $ABC$, respectively. Suppose that the three lines $r_A$, $r_B$, $r_C$ define a triangle. Prove that the circumcenter of this triangle is the midpoint of $HI$.

2019 Brazil National Olympiad, 4

Tags: euclidean geometry , geometry , Brazilian Math Olympiad

Let $ ABC $ be an acutangle triangle and $ D $ any point on the $ BC $ side. Let $ E $ be the symmetrical of $ D $ in $ AC $ and $ F $ is the symmetrical $ D $ relative to $ AB $. $ A $ straight $ ED $ intersects straight $ AB $ at $ G $, while straight $ F D $ intersects the line $ AC $ in $ H $. Prove that the points $ A, E, F, G$ and $ H $ are on the same circumference.

2018 Brazil Undergrad MO, 14

Tags: modulo , algebra , Brazilian Math Olympiad , college contests

What is the arithmetic mean of all values of the expression $ | a_1-a_2 | + | a_3-a_4 | $ Where $ a_1, a_2, a_3, a_4 $ is a permutation of the elements of the set {$ 1,2,3,4 $}?

2018 Brazil Undergrad MO, 17

Tags: geometry , Brazilian Math Olympiad , college contests

In the figure, a semicircle is folded along the $ AN $ string and intersects the $ MN $ diameter in $ B $. $ MB: BN = 2: 3 $ and $ MN = 10 $ are known to be. If $ AN = x $, what is the value of $ x ^ 2 $?

2021 Brazil National Olympiad, 1

Tags: geometry , Brazilian Math Olympiad , Brazil , convex quadrilateral

Let $ABCD$ be a convex quadrilateral in the plane and let $O_{A}, O_{B}, O_{C}$ and $O_{D}$ be the circumcenters of the triangles $BCD, CDA, DAB$ and $ABC$, respectively. Suppose these four circumcenters are distinct points. Prove that these points are not on a same circle.

1989 Brazil National Olympiad, 4

Tags: Brazilian Math Olympiad 1989 , Brazilian Math Olympiad , combinatorics , games

A game is played by two contestants A and B, each one having ten chips numbered from 1 to 10. The board of game consists of two numbered rows, from 1 to 1492 on the first row and from 1 to 1989 on the second. At the $n$-th turn, $n=1,2,\ldots,10$, A puts his chip numbered $n$ in any empty cell, and B puts his chip numbered $n$ in any empty cell on the row not containing the chip numbered $n$ from A. B wins the game if, after the 10th turn, both rows show the numbers of the chips in the same relative order. Otherwise, A wins. [list=a] [*] Which player has a winning strategy? [*] Suppose now both players has $k$ chips numbered 1 to $k$. Which player has a winning strategy? [*] Suppose further the rows are the set $\mathbb{Q}$ of rationals and the set $\mathbb{Z}$ of integers. Which player has a winning strategy? [/list]

2018 Brazil Undergrad MO, 2

Tags: functional equation , function , Olympiad , Brazilian Math Olympiad , algebra , college contests

Let $ f, g: \mathbb {R} \to \mathbb {R} $ function such that $ f (x + g (y)) = - x + y + 1 $ for each pair of real numbers $ x $ e $ y $. What is the value of $ g (x + f (y) $?