Olympiad problems

Kyiv City MO 1984-93 - geometry, 1985.8.3

The longest diagonal of a convex hexagon is $2$. Is there necessarily a side or diagonal in this hexagon whose length does not exceed $1$?

2024/2025 TOURNAMENT OF TOWNS, P2

Tags: combinatorics

Pete puts 100 stones in a row: black one, white one, black one, white one, ..., black one, white one. In a single move either Pete chooses two black stones with only white stones between them, and repaints all these white stones in black, or Pete chooses two white stones with only black stones between them, and repaints all these black stones in white. Can Pete with a sequence of moves described above obtain a row of 50 black stones followed by 50 white stones? Egor Bakaev

2019 Sharygin Geometry Olympiad, 3

Tags: geometry

The rectangle $ABCD$ lies inside a circle. The rays $BA$ and $DA$ meet this circle at points $A_1$ and $A_2$. Let $A_0$ be the midpoint of $A_1A_2$. Points $B_0$, $C_0, D_0$ are defined similarly. Prove that $A_0C_0 = B_0D_0$.

2011 Argentina National Olympiad, 5

Tags: number theory , divide , divisible

Find all integers $n$ such that $1<n<10^6$ and $n^3-1$ is divisible by $10^6 n-1$.

2023 Romania Team Selection Test, P5

Tags: algebra , sequence

Let $(a_n)_{n\geq 1}$ be a sequence of positive real numbers with the property that $$(a_{n+1})^2 + a_na_{n+2} \leq a_n + a_{n+2}$$ for all positive integers $n$. Show that $a_{2022}\leq 1$.

2011 JBMO Shortlist, 2

Tags: geometry

Let $AD,BF$ and ${CE}$ be the altitudes of $\vartriangle ABC$. A line passing through ${D}$ and parallel to ${AB}$intersects the line ${EF}$at the point ${G}$. If ${H}$ is the orthocenter of $\vartriangle ABC$, find the angle ${\angle{CGH}}$.

2024 Princeton University Math Competition, A8

Tags: algebra

Let $[n]$ denote the set of integers $0, 1, \ldots, n-1.$ Let $\omega_n=e^{2\pi i/n}.$ Let $$f(n) = \prod_{\overset{i \in [n]}{\gcd(i,n)=1}} \prod_{\overset{j \in [n]}{\gcd(j,n)=1}} (\omega_n^i - \omega_n^j).$$ Then, $f(2024)=2^{e_1} \cdot 11^{e_2} \cdot 23^{e_3}$ for positive integers $e_1, e_2, e_3.$ Find $e_1+e_2+e_3.$

1995 AMC 12/AHSME, 4

Tags:

If $M$ is $30 \%$ of $Q$, $Q$ is $20 \%$ of $P$, and $N$ is $50 \%$ of $P$, then $\frac{M}{N} =$ $\textbf{(A)}\ \frac{3}{250} \qquad \textbf{(B)}\ \frac{3}{25} \qquad \textbf{(C)}\ 1 \qquad \textbf{(D)}\ \displaystyle \frac{6}{5} \qquad \textbf{(E)}\ \displaystyle \frac{4}{3}$

1997 Belarusian National Olympiad, 1

Tags: number theory

$$Problem 1$$ ;Find all composite numbers $n$ with the following property: For every proper divisor $d$ of $n$ (i.e. $1 < d < n$), it holds that $n-12 \geq d \geq n-20$.

1957 AMC 12/AHSME, 9

Tags:

The value of $ x \minus{} y^{x \minus{} y}$ when $ x \equal{} 2$ and $ y \equal{} \minus{}2$ is: $ \textbf{(A)}\ \minus{}18 \qquad \textbf{(B)}\ \minus{}14\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 256$

2015 ASDAN Math Tournament, 14

Tags: team test

For a given positive integer $m$, the series $$\sum_{k=1,k\neq m}^{\infty}\frac{1}{(k+m)(k-m)}$$ evaluates to $\frac{a}{bm^2}$, where $a$ and $b$ are positive integers. Compute $a+b$.

2017 China Girls Math Olympiad, 4

Tags: analysis , algebra

Partition $\frac1{2002},\frac1{2003},\frac1{2004},\ldots,\frac{1}{2017}$ into two groups. Define $A$ the sum of the numbers in the first group, and $B$ the sum of the numbers in the second group. Find the partition such that $|A-B|$ attains it minimum and explains the reason.

2024 Girls in Mathematics Tournament, 1

Tags: algebra

The nonzero real numbers $a,b,c$ are such that: $a^2-bc= b^2-ac= c^2-ab= a^3+b^3+c^3$. Compute the possible values of $a+b+c$.

2004 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , decimal representation , perfect square

Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$. Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i] [i]Proposed by Laurentiu Panaitopol, Romania[/i]

2002 Manhattan Mathematical Olympiad, 1

Tags:

You are given a rectangular sheet of paper and scissors. Can you cut it into a number of pieces all having the same size and shape of a polygon with five sides? What about polygon with seven sides?

PEN S Problems, 29

Tags: algorithm , modular arithmetic

What is the rightmost nonzero digit of $1000000!$?

2008 AIME Problems, 3

Tags: modular arithmetic , calculus , integration

Ed and Sue bike at equal and constant rates. Similarly, they jog at equal and constant rates, and they swim at equal and constant rates. Ed covers $ 74$ kilometers after biking for $ 2$ hours, jogging for $ 3$ hours, and swimming for $ 4$ hours, while Sue covers $ 91$ kilometers after jogging for $ 2$ hours, swimming for $ 3$ hours, and biking for $ 4$ hours. Their biking, jogging, and swimming rates are all whole numbers of kilometers per hour. Find the sum of the squares of Ed's biking, jogging, and swimming rates.

Kyiv City MO Seniors 2003+ geometry, 2006.11.3

Tags: midpoint , circumcircle , equal segments , geometry

Let $O$ be the center of the circle $\omega$ circumscribed around the acute-angled triangle $\vartriangle ABC$, and $W$ be the midpoint of the arc $BC$ of the circle $\omega$, which does not contain the point $A$, and $H$ be the point of intersection of the heights of the triangle $\vartriangle ABC$. Find the angle $\angle BAC$, if $WO = WH$. (O. Clurman)

2001 National Olympiad First Round, 17

Tags: geometry

Let $ABC$ be a triangle such that midpoints of three altitudes are collinear. If the largest side of triangle is $10$, what is the largest possible area of the triangle? $ \textbf{(A)}\ 20 \qquad\textbf{(B)}\ 25 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 40 \qquad\textbf{(E)}\ 50 $

2020 Greece Junior Math Olympiad, 3

Tags: number theory

Find all positive integers $x$, for which the equation $$a+b+c=xabc$$ has solution in positive integers. Solve the equation for these values of $x$

2021 JHMT HS, 8

Tags: polynomial , complex number , algebra

For complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy $x^3 + cx + p(1 + qi) = 0$. Suppose $c$, $p$, and $q$ were chosen so that all three complex roots $x$ satisfy $\tfrac{5}{6} \leq \tfrac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \tfrac{6}{5}$, where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real part of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.

1949-56 Chisinau City MO, 38

Tags: compare , algebra

Which is more $\log_3 7$ or $\log_{\frac{1}{3}} \frac{1}{7}$ ?

2019 ELMO Shortlist, G6

Tags: geometry

Let $ABC$ be an acute scalene triangle and let $P$ be a point in the plane. For any point $Q\neq A,B,C$, define $T_A$ to be the unique point such that $\triangle T_ABP \sim \triangle T_AQC$ and $\triangle T_ABP, \triangle T_AQC$ are oriented in the same direction (clockwise or counterclockwise). Similarly define $T_B, T_C$. a) Find all $P$ such that there exists a point $Q$ with $T_A,T_B,T_C$ all lying on the circumcircle of $\triangle ABC$. Call such a pair $(P,Q)$ a [i]tasty pair[/i] with respect to $\triangle ABC$. b) Keeping the notations from a), determine if there exists a tasty pair which is also tasty with respect to $\triangle T_AT_BT_C$. [i]Proposed by Vincent Huang[/i]

2014 Contests, 1

Tags: geometry , rhombus

Say that a convex quadrilateral is [i]tasty[/i] if its two diagonals divide the quadrilateral into four nonoverlapping similar triangles. Find all tasty convex quadrilaterals. Justify your answer.

2012 Hanoi Open Mathematics Competitions, 12

Tags: geometry

In an isosceles triangle ABC with the base AB given a point M $\in$ BC: Let O be the center of its circumscribed circle and S be the center of the inscribed circle in ABC and SM // AC: Prove that OM perpendicular BS.