Olympiad problems

2018 Belarusian National Olympiad, 11.1

Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality $$ f(f(x))=x^2f(x)+ax^2 $$ for all real $x$.

2010 Tuymaada Olympiad, 3

Tags: geometry , circumcircle , geometric transformation , reflection , cyclic quadrilateral , geometry unsolved

In a cyclic quadrilateral $ABCD$, the extensions of sides $AB$ and $CD$ meet at point $P$, and the extensions of sides $AD$ and $BC$ meet at point $Q$. Prove that the distance between the orthocenters of triangles $APD$ and $AQB$ is equal to the distance between the orthocenters of triangles $CQD$ and $BPC$.

2024 CIIM, 6

Tags: series , fractional part , Convergence , ana , real analysis

Given a real number $x$, define the series \[ S(x) = \sum_{n=1}^{\infty} \{n! \cdot x\}, \] where $\{s\} = s - \lfloor s \rfloor$ is the fractional part of the number $s$. Determine if there exists an irrational number $x$ for which the series $S(x)$ converges.

2018 Moldova Team Selection Test, 10

Tags: inequalities

The positive real numbers $a,b, c,d$ satisfy the equality $ \frac {1}{a+1} + \frac {1}{b+1} + \frac {1}{c+1} + \frac{ 1}{d+1} = 3 $ . Prove the inequality $\sqrt [3]{abc} + \sqrt [3]{bcd} + \sqrt [3]{cda} + \sqrt [3]{dab} \le \frac {4}{3} $.

2021 Yasinsky Geometry Olympiad, 4

Tags: geometry , angle bisector , orthocenter

In triangle $ABC$, the point $H$ is the orthocenter. A circle centered at point $H$ and with radius $AH$ intersects the lines $AB$ and $AC$ at points $E$ and $D$, respectively. The point $X$ is the symmetric of the point $A$ with respect to the line $BC$ . Prove that $XH$ is the bisector of the angle $DXE$. (Matthew of Kursk)

1983 Czech and Slovak Olympiad III A, 5

Tags: Inequality , algebra , national olympiad , inequalities

Find all pair $(x,y)$ of positive integers satisfying $$\left|\frac{x}{y}-\sqrt2\right|<\frac{1}{y^3}.$$

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.

1979 IMO Longlists, 18

Tags: number theory proposed , number theory

Show that for no integers $a \geq 1, n \geq 1$ is the sum \[1+\frac{1}{1+a}+\frac{1}{1+2a}+\cdots+\frac{1}{1+na}\] an integer.

2003 Putnam, 3

Tags: Putnam , number theory , least common multiple , floor function , prime factorization , college contests

Show that for each positive integer n, \[n!=\prod_{i=1}^n \; \text{lcm} \; \{1, 2, \ldots, \left\lfloor\frac{n}{i} \right\rfloor\}\] (Here lcm denotes the least common multiple, and $\lfloor x\rfloor$ denotes the greatest integer $\le x$.)

2003 AMC 10, 24

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Sally has ﬁve red cards numbered $ 1$ through $ 5$ and four blue cards numbered $ 3$ through $ 6$. She stacks the cards so that the colors alternate and so that the number on each red card divides evenly into the number on each neighboring blue card. What is the sum of the numbers on the middle three cards? $ \textbf{(A)}\ 8 \qquad \textbf{(B)}\ 9 \qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 11 \qquad \textbf{(E)}\ 12$

1985 Balkan MO, 3

Tags: analytic geometry , combinatorics proposed , combinatorics

Let $S$ be the set of all positive integers of the form $19a+85b$, where $a,b$ are arbitrary positive integers. On the real axis, the points of $S$ are colored in red and the remaining integer numbers are colored in green. Find, with proof, whether or not there exists a point $A$ on the real axis such that any two points with integer coordinates which are symmetrical with respect to $A$ have necessarily distinct colors.

2016 Dutch BxMO TST, 4

Tags: combinatorics , group theory

The Facebook group Olympiad training has at least five members. There is a certain integer $k$ with following property: [i]for each $k$-tuple of members there is at least one member of this $k$-tuple friends with each of the other $k - 1$.[/i] (Friendship is mutual: if $A$ is friends with $B$, then also $B$ is friends with $A$.) (a) Suppose $k = 4$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members? (b) Suppose $k = 5$. Can you say with certainty that the Facebook group has a member that is friends with each of the other members?

2017 CMIMC Computer Science, 1

Tags: 2017 , computer science

What is the minimum number of times you have to take your pencil off the paper to draw the following figure (the dots are for decoration)? You must lift your pencil off the paper after you're done, and this is included in the number of times you take your pencil off the paper. You're not allowed to draw over an edge twice. [center][img]http://i.imgur.com/CBGmPmv.png[/img][/center]

2019 Switzerland - Final Round, 5

Tags: combinatorics

A group of children is sitting around a round table . At first, each child has an even number of candies. Each turn, each child gives half of his candies to the child sitting at his right. If, after a turn, a child has an odd number of candies, the teacher gives him\her an extra candy. Show that after a finite number of rounds all children will have the same number of candies.

1987 Tournament Of Towns, (141) 1

Tags: number theory , Sum of Squares , Perfect Square , odd

Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?

2024 Belarusian National Olympiad, 8.2

Tags: combinatorics , function

Let $S$ be the set of all non-increasing sequences of numbers $a_1 \geq a_2 \geq \ldots \geq a_{101}$ such that $a_i \in \{ 0,1,\ldots ,101 \}$ for all $1 \leq i \leq 101$ For every sequence $s \in S$ let $$f(s)=\lceil \frac{a_1}{2} \rceil+\lfloor \frac{a_2}{2} \rfloor + \lceil \frac{a_3}{2} \rceil + \ldots + \lfloor \frac{a_{100}}{2} \rfloor + \lceil \frac{a_{101}}{2} \rceil$$ where $\lfloor x \rfloor$ is the greatest integer, not exceeding $x$, and $\lceil x \rceil$ is the least integer at least $x$. Prove that the number of sequences $s \in S$ for which $f(s)$ is even is the same, as the number of sequences $s$ for which $f(s)$ is odd [i]M. Zorka[/i]

2004 IMC, 3

Tags: inequalities , triangle inequality , IMC , college contests

Let $D$ be the closed unit disk in the plane, and let $z_1,z_2,\ldots,z_n$ be fixed points in $D$. Prove that there exists a point $z$ in $D$ such that the sum of the distances from $z$ to each of the $n$ points is greater or equal than $n$.

2023 VN Math Olympiad For High School Students, Problem 2

Tags: geometry

Prove that: $3$ symmedians of a triangle are concurrent at a point; the concurrent point is called the [i]Lemoine[/i] point of the given triangle.

2017 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra

Sasha’s computer can do the following two operations: If you load the card with number $a$, it will return that card back and also prints another card with number $a+1$, and if you consecutively load the cards with numbers $a$ and $b$, it will return them back and also prints cards with all the roots of the quadratic trinomial $x^2+ax+b$ (possibly one, two, or none cards.) Initially, Sasha had only one card with number $s$. Is it true that, for any $s> 0$, Sasha can get a card with number $\sqrt{s}$?

2011 India IMO Training Camp, 3

Tags: number theory , greatest common divisor , search , combinatorics unsolved , combinatorics

Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let \[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\] Prove that : $a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b]. $b)$ the number of good subsets of $T$ is [b]odd[/b].

2003 Miklós Schweitzer, 8

Tags: college contests , Miklos Schweitzer , function , series

Let $f_1, f_2, \ldots$ be continuous real functions on the real line. Is it true that if the series $\sum_{n=1}^{\infty} f_n(x)$ is divergent for every $x$, then this holds also true for any typical choice of the signs in the sum (i.e. the set of those $\{ \epsilon _n\}_{n=1}^{\infty} \in \{ +1, -1\}^{\mathbb{N}}$ sequences, for which there series $\sum_{n=1}^{\infty} \epsilon_nf_n(x)$ is convergent at least at one point $x$, forms a subset of first category within the set $\{+1,-1\}^{\mathbb{N}} $)? (translated by L. Erdős)

1966 AMC 12/AHSME, 9

Tags: logarithms

If $x=(\log_82)^{(\log_28)}$, then $\log_3x$ equals: $\text{(A)} \ -3 \qquad \text{(B)} \ -\frac13 \qquad \text{(C)} \ \frac13 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 9$

2020 CCA Math Bonanza, L2.4

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If \[ \sum_{k=1}^{1000}\left( \frac{k+1}{k}+\frac{k}{k+1}\right)=\frac{m}{n} \] for relatively prime positive integers $m,n$, compute $m+n$. [i]2020 CCA Math Bonanza Lightning Round #2.4[/i]

2015 Caucasus Mathematical Olympiad, 3

Tags: combinatorics , combinatorial geometry , tiles , Tiling

The workers laid a floor of size $n\times n$ ($10 <n <20$) with two types of tiles: $2 \times 2$ and $5\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen? (You can’t cut tiles and also put them on top of each other.)

1990 AIME Problems, 4

Tags: quadratics , blogs

Find the positive solution to \[ \frac 1{x^2-10x-29}+\frac1{x^2-10x-45}-\frac 2{x^2-10x-69}=0 \]