Olympiad problems

2015 ASDAN Math Tournament, 3

Tags: geometry test

Points $E$ and $F$ are chosen on sides $BC$ and $CD$ respectively of rhombus $ABCD$ such that $AB=AE=AF=EF$, and $FC,DF,BE,EC>0$. Compute the measure of $\angle ABC$.

Today's calculation of integrals, 887

Tags: calculus , integration , trigonometry , derivative , calculus computations , logarithm , function

For the function $f(x)=\int_0^x \frac{dt}{1+t^2}$, answer the questions as follows. Note : Please solve the problems without using directly the formula $\int \frac{1}{1+x^2}\ dx=\tan^{-1}x +C$ for Japanese High School students those who don't study arc sin x, arc cos x, arc tanx. (1) Find $f(\sqrt{3})$ (2) Find $\int_0^{\sqrt{3}} xf(x)\ dx$ (3) Prove that for $x>0$. $f(x)+f\left(\frac{1}{x}\right)$ is constant, then find the value.

2021 Iran Team Selection Test, 5

Tags: sequence , algebra , arithmetic sequence

Call a triple of numbers [b]Nice[/b] if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ [b] Nice[/b] triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios Proposed by [i]Morteza Saghafian[/i]

2013 Saudi Arabia GMO TST, 2

Tags: algebra , inequalities

For positive real numbers $a, b$ and $c$, prove that $$\frac{a^3}{a^2 + ab + b^2} +\frac{b^3}{b^2 + bc + c^2} +\frac{c^3}{ c^2 + ca + a^2} \ge\frac{ a + b + c}{3}$$

2015 IMO Shortlist, N8

Tags: inequalities , number theory , prime factorization , function

For every positive integer $n$ with prime factorization $n = \prod_{i = 1}^{k} p_i^{\alpha_i}$, define \[\mho(n) = \sum_{i: \; p_i > 10^{100}} \alpha_i.\] That is, $\mho(n)$ is the number of prime factors of $n$ greater than $10^{100}$, counted with multiplicity. Find all strictly increasing functions $f: \mathbb{Z} \to \mathbb{Z}$ such that \[\mho(f(a) - f(b)) \le \mho(a - b) \quad \text{for all integers } a \text{ and } b \text{ with } a > b.\] [i]Proposed by Rodrigo Sanches Angelo, Brazil[/i]

1987 Tournament Of Towns, (134) 3

Tags: algebra

We are given two three-litre bottles, one containing $1$ litre of water and the other containing $1$ litre of $2\%$ salt solution . One can pour liquids from one bottle to the other and then mix them to obtain solutions of different concentration . Can one obtain a $1 . 5\%$ solution of salt in the bottle which originally contained water? (S . Fomin, Leningrad),

2019 Czech-Austrian-Polish-Slovak Match, 3

Tags: combinatorics

A dissection of a convex polygon into finitely many triangles by segments is called a [i]trilateration[/i] if no three vertices of the created triangles lie on a single line (vertices of some triangles might lie inside the polygon). We say that a trilateration is [i]good[/i] if its segments can be replaced with one-way arrows in such a way that the arrows along every triangle of the trilateration form a cycle and the arrows along the whole convex polygon also form a cycle. Find all $n\ge 3$ such that the regular $n$-gon has a good trilateration.

2008 Alexandru Myller, 1

Tags: vector geometry , geometry , circumcircle , center of mass

$ O $ is the circumcentre of $ ABC $ and $ A_1\neq A $ is the point on $ AO $ and the circumcircle of $ ABC. $ The centers of mass of $ ABC, A_1BC $ are $ G,G_1, $ respectively, and $ P $ is the intersection of $ AG_1 $ with $ OG. $ Show that $ \frac{PG}{PO}=\frac{2}{3} . $ [i]Gabriel Popa, Paul Georgescu[/i]

2016 Abels Math Contest (Norwegian MO) Final, 2a

Tags: diophantine equation , diophantine , number theory , system of equations

Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\ c + d = ab \end{cases}$ .

1932 Eotvos Mathematical Competition, 3

Tags: trigonometry , geometry , inequalities

Let $\alpha$, $\beta$ and $\gamma$ be the interior angles of an acute triangle. Prove that if $\alpha < \beta < \gamma$ then $$\sin 2\alpha >\ sin 2 \beta > \sin 2\gamma.$$

1966 Vietnam National Olympiad, 1

Tags: algebra , inequalities , maximum value , minimum value

Let $x, y$ and $z$ be nonnegative real numbers satisfying the following conditions: (1) $x + cy \le 36$,(2) $2x+ 3z \le 72$, where $c$ is a given positive number. Prove that if $c \ge 3$ then the maximum of the sum $x + y + z$ is $36$, while if $c < 3$, the maximum of the sum is $24 + \frac{36}{c}$ .

2021 Korea Winter Program Practice Test, 5

Tags: combinatorics , permutation

For positive integers $k$ and $n$, express the number of permutation $P=x_1x_2...x_{2n}$ consisting of $A$ and $B$ that satisfies all three of the following conditions, using $k$ and $n$. $ $ $ $ $(i)$ $A, B$ appear exactly $n$ times respectively in $P$. $ $ $ $ $(ii)$ For each $1\le i\le n$, if we denote the number of $A$ in $x_1,x_2,...,x_i$ as $a_i$ $,$ then $\mid 2a_i -i\mid \le 1$. $ $ $ $ $(iii)$ $AB$ appears exactly $k$ times in $P$. (For example, $AB$ appears 3 times in $ABBABAAB$)

2016 Regional Olympiad of Mexico Southeast, 3

Tags: algebra , polynomial

Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying , for any real $x$ , the identy \[P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})\]

2004 VTRMC, Problem 1

Tags: matrix , linear algebra

Let $I$ denote the $2\times2$ identity matrix $\begin{pmatrix}1&0\\0&1\end{pmatrix}$ and let $$M=\begin{pmatrix}I&A\\B&C\end{pmatrix},\enspace N=\begin{pmatrix}I&B\\A&C\end{pmatrix}$$where $A,B,C$ are arbitrary $2\times2$ matrices which entries in $\mathbb R$, the real numbers. Thus $M$ and $N$ are $4\times4$ matrices with entries in $\mathbb R$. Is it true that $M$ is invertible (i.e. there is a $4\times4$ matrix $X$ such that $MX=XM=I$) implies $N$ is invertible? Justify your answer.

2017 ELMO Shortlist, 4

Tags: combinatorics

nic$\kappa$y is drawing kappas in the cells of a square grid. However, he does not want to draw kappas in three consecutive cells (horizontally, vertically, or diagonally). Find all real numbers $d>0$ such that for every positive integer $n,$ nic$\kappa$y can label at least $dn^2$ cells of an $n\times n$ square. [i]Proposed by Mihir Singhal and Michael Kural[/i]

Kvant 2020, M2617

Tags: combinatorial geometry , combinatorics

The points in the plane are painted in 100 colors. Prove that there are three points of the same color that are the vertices of a triangle of area 1. [i]Proposed by V. Bragin[/i]

2025 Turkey EGMO TST, 6

Tags:

In a chess tournament with 200 participants, 700 matches are arranged such that among any 100 participants, the number of matches played between them is at least $ N $. Determine the maximum possible value of $ N $.

2006 Vietnam Team Selection Test, 1

Tags: circumcircle , trigonometry , parallelogram , ratio , geometry

Given an acute angles triangle $ABC$, and $H$ is its orthocentre. The external bisector of the angle $\angle BHC$ meets the sides $AB$ and $AC$ at the points $D$ and $E$ respectively. The internal bisector of the angle $\angle BAC$ meets the circumcircle of the triangle $ADE$ again at the point $K$. Prove that $HK$ is through the midpoint of the side $BC$.

2022 Bundeswettbewerb Mathematik, 1

Tags: algebra , am-gm

Find all quadrupels $(a, b, c, d)$ of positive real numbers that satisfy the following two equations: \begin{align*} ab + cd &= 8,\\ abcd &= 8 + a + b + c + d. \end{align*}

2004 Mediterranean Mathematics Olympiad, 2

Tags: circumcircle , trigonometry , geometry

In a triangle $ABC$, the altitude from $A$ meets the circumcircle again at $T$ . Let $O$ be the circumcenter. The lines $OA$ and $OT$ intersect the side $BC$ at $Q$ and $M$, respectively. Prove that \[\frac{S_{AQC}}{S_{CMT}} = \biggl( \frac{ \sin B}{\cos C} \biggr)^2 .\]

2014 BAMO, 4

Tags: fibonacci , sequence , divisibility , number theory

Let $F_1, F_2, F_3 \cdots$ be the Fibonacci sequence, the sequence of positive integers satisfying $$F_1 =F_2=1$$ and $$F_{n+2} = F_{n+1} + F_n$$ for all $n \ge 1$. Does there exist an $n \ge 1$ such that $F_n$ is divisible by $2014$? Prove your answer.

2021 Argentina National Olympiad Level 2, 6

Tags: algebra

Decide if it is possible to choose $330$ points in the plane so that among all the distances that are formed between two of them there are at least $1700$ that are equal.

2016 Korea Winter Program Practice Test, 1

Tags: geometry , tangent circle

There is circle $\omega$ and $A, B$ on it. Circle $\gamma_1$ tangent to $\omega$ on $T$ and $AB$ on $D$. Circle $\gamma_2$ tangent to $\omega$ on $S$ and $AB$ on $E$. (like the figure below) Let $AB\cap TS=C$. Prove that $CA=CB$ iff $CD=CE$

2019 Mexico National Olympiad, 5

Tags: number theory , algebra

Let $a > b$ be relatively prime positive integers. A grashopper stands at point $0$ in a number line. Each minute, the grashopper jumps according to the following rules: [list] [*] If the current minute is a multiple of $a$ and not a multiple of $b$, it jumps $a$ units forward. [*] If the current minute is a multiple of $b$ and not a multiple of $a$, it jumps $b$ units backward. [*] If the current minute is both a multiple of $b$ and a multiple of $a$, it jumps $a - b$ units forward. [*] If the current minute is neither a multiple of $a$ nor a multiple of $b$, it doesn't move. [/list] Find all positions on the number line that the grasshopper will eventually reach.

2013 Online Math Open Problems, 18

Tags: floor function , trigonometry , absolute value

Determine the absolute value of the sum \[ \lfloor 2013\sin{0^\circ} \rfloor + \lfloor 2013\sin{1^\circ} \rfloor + \cdots + \lfloor 2013\sin{359^\circ} \rfloor, \] where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. (You may use the fact that $\sin{n^\circ}$ is irrational for positive integers $n$ not divisible by $30$.) [i]Ray Li[/i]