Olympiad problems

2014 Contests, 3

Tags: combinatorics

There are $n$ students sitting on a round table. You collect all of $ n $ name tags and give them back arbitrarily. Each student gets one of $n$ name tags. Now $n$ students repeat following operation: The students who have their own name tags exit the table. The other students give their name tags to the student who is sitting right to him. Find the number of ways giving name tags such that there exist a student who don't exit the table after 4 operations.

2001 AIME Problems, 1

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Find the sum of all positive two-digit integers that are divisible by each of their digits.

LMT Guts Rounds, 2020 F1

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Find the remainder when $2020!$ is divided by $2020^2.$ [i]Proposed by Kevin Zhao[/i]

1995 Spain Mathematical Olympiad, 6

Tags: geometry , equilateral triangle

Let $C$ be a variable interior point of a fixed segment $AB$. Equilateral triangles $ACB' $ and $CBA'$ are constructed on the same side and $ABC' $ on the other side of the line $AB$. (a) Prove that the lines $AA' ,BB'$ , and $CC'$ meet at some point $P$. (b) Find the locus of $P$ as $C$ varies. (c) Prove that the centers $A'' ,B'' ,C''$ of the three triangles form an equilateral triangle. (d) Prove that $A'' ,B'',C''$ , and $P$ lie on a circle.

The Golden Digits 2024, P1

Tags: function , algebra , functional equation , number theory

Find all functions $f:\mathbb{Z}_{>0}\rightarrow\mathbb{Z}_{>0}$ with the following properties: 1) For every natural number $n\geq 3$, $\gcd(f(n),n)\neq 1$. 2) For every natural number $n\geq 3$, there exists $i_n\in\mathbb{Z}_{>0}$, $1\leq i_n\leq n-1$, such that $f(n)=f(i_n)+f(n-i_n)$. [i]Proposed by Pavel Ciurea[/i]

2022 China National Olympiad, 1

Tags: geometry

Let $a$ and $b$ be two positive real numbers, and $AB$ a segment of length $a$ on a plane. Let $C,D$ be two variable points on the plane such that $ABCD$ is a non-degenerate convex quadrilateral with $BC=CD=b$ and $DA=a$. It is easy to see that there is a circle tangent to all four sides of the quadrilateral $ABCD$. Find the precise locus of the point $I$.

2020 AMC 8 -, 19

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A number is called [i]flippy[/i] if its digits alternate between two distinct digits. For example, $2020$ and $37373$ are flippy, but $3883$ and $123123$ are not. How many five-digit flippy numbers are divisible by $15$? $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 8$

2013 NIMO Problems, 7

Tags: logarithm

For each integer $k\ge2$, the decimal expansions of the numbers $1024,1024^2,\dots,1024^k$ are concatenated, in that order, to obtain a number $X_k$. (For example, $X_2 = 10241048576$.) If \[ \frac{X_n}{1024^n} \] is an odd integer, find the smallest possible value of $n$, where $n\ge2$ is an integer. [i]Proposed by Evan Chen[/i]

2019 BMT Spring, 7

Tags: algebra , factorization , sum of cubes

Let $ r_1 $, $ r_2 $, $ r_3 $ be the (possibly complex) roots of the polynomial $ x^3 + ax^2 + bx + \dfrac{4}{3} $. How many pairs of integers $ a $, $ b $ exist such that $ r_1^3 + r_2^3 + r_3^3 = 0 $?

2020 Balkan MO Shortlist, G3

Tags: geometry , circumcircle , perimeter

Let $ABC$ be a triangle. On the sides $BC$, $CA$, $AB$ of the triangle, construct outwardly three squares with centres $O_a$, $O_b$, $O_c$ respectively. Let $\omega$ be the circumcircle of $\vartriangle O_aO_bO_c$. Given that $A$ lies on $\omega$, prove that the centre of $\omega$ lies on the perimeter of $\vartriangle ABC$. [i]Sam Bealing, United Kingdom[/i]

2023 All-Russian Olympiad Regional Round, 11.8

Tags: geometry , perpendicular bisector

Given is a triangle $ABC$ with circumcenter $O$. Points $D, E$ are chosen on the angle bisector of $\angle ABC$ such that $EA=EB, DB=DC$. If $P, Q$ are the circumcenters of $(AOE), (COD)$, prove that either the line $PQ$ coincides with $AC$ or $PQCA$ is cyclic.

2012 Iran Team Selection Test, 1

Tags: inequalities

For positive reals $a,b$ and $c$ with $ab+bc+ca=1$, show that \[\sqrt{3}({\sqrt{a}+\sqrt{b}+\sqrt{c})\le \frac{a\sqrt{a}}{bc}+\frac{b\sqrt{b}}{ca}+\frac{c\sqrt{c}}{ab}.}\] [i]Proposed by Morteza Saghafian[/i]

2017 Thailand TSTST, 1

Tags: midpoint , geometry , incenter , concurrency

In $\vartriangle ABC, D, E, F$ are the midpoints of $AB, BC, CA$ respectively. Denote by $O_A, O_B, O_C$ the incenters of $\vartriangle ADF, \vartriangle BED, \vartriangle CFE$ respectively. Prove that $O_AE, O_BF, O_CD$ are concurrent.

2011 IMO, 3

Tags: function , algebra , functional inequality

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies \[f(x + y) \leq yf(x) + f(f(x))\] for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$. [i]Proposed by Igor Voronovich, Belarus[/i]

2014 May Olympiad, 3

Tags: combinatorics

There are nine boxes. In the first there is $1$ stone, in the second there are $2$ stones, in the third there are $3$ stones, and thus continuing, in the eighth there are $8$ stones and in the ninth there are $9$ stones. The allowed operation is to remove the same number of stones from two different boxes and place them in a third box. The goal is that all stones are in a single box. Describe how to do it with the minimum number of operations allowed. Explain why it is impossible to achieve it with fewer operations.

2024 India National Olympiad, 2

Tags: combinatorics

All the squares of a $2024 \times 2024$ board are coloured white. In one move, Mohit can select one row or column whose every square is white, choose exactly $1000$ squares in that row or column, and colour all of them red. Find maximum number of squares Mohit can colour in a finite number of moves. $\quad$ [i]Proposed[/i] by Pranjal Srivastava

2004 Estonia Team Selection Test, 5

Tags: number theory , lcm , power of 2 , divisor

Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.

2000 AMC 10, 21

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If all alligators are ferocious creatures and some creepy crawlers are alligators, which statement(s) [b]must[/b] be true? I. All alligators are creepy crawlers. II. Some ferocious creatures are creepy crawlers. III. Some alligators are not creepy crawlers. $\text{(A)}\ \text{I only}\qquad\text{(B)}\ \text{II only}\qquad\text{(C)}\ \text{III only}\qquad\text{(D)}\ \text{II and III only}\qquad\text{(E)}\ \text{None must be true}$

1994 AMC 8, 17

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Pauline Bunyan can shovel snow at the rate of $20$ cubic yards for the first hour, $19$ cubic yards for the second, $18$ for the third, etc., always shoveling one cubic yard less per hour than the previous hour. If her driveway is $4$ yards wide, $10$ yards long, and covered with snow $3$ yards deep, then the number of hours it will take her to shovel it clean is closest to $\text{(A)}\ 4 \qquad \text{(B)}\ 5 \qquad \text{(C)}\ 6 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 12$

2004 Romania Team Selection Test, 13

Tags: euler , modular arithmetic , inequalities , function , number theory , relatively prime

Let $m\geq 2$ be an integer. A positive integer $n$ has the property that for any positive integer $a$ coprime with $n$, we have $a^m - 1\equiv 0 \pmod n$. Prove that $n \leq 4m(2^m-1)$. Created by Harazi, modified by Marian Andronache.

2022 Baltic Way, 15

Tags: geometry

Let $\Omega$ be a circle, and $B, C$ are two fixed points on $\Omega$. Given a third point $A$ on $\Omega$, let $X$ and $Y$ denote the feet of the altitudes from $B$ and $C$, respectively, in the triangle $ABC$. Prove that there exists a fixed circle $\Gamma$ such that $XY$ is tangent to $\Gamma$ regardless of the choice of the point $A$.

2013 Harvard-MIT Mathematics Tournament, 5

Tags: hmmt

Rahul has ten cards face-down, which consist of five distinct pairs of matching cards. During each move of his game, Rahul chooses one card to turn face-up, looks at it, and then chooses another to turn face-up and looks at it. If the two face-up cards match, the game ends. If not, Rahul flips both cards face-down and keeps repeating this process. Initially, Rahul doesn't know which cards are which. Assuming that he has perfect memory, find the smallest number of moves after which he can guarantee that the game has ended.

MOAA Gunga Bowls, 2021.21

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King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andrew Wen[/i]

2022 Tuymaada Olympiad, 3

Tags: geometry

Bisectors of a right triangle $\triangle ABC$ with right angle $B$ meet at point $I.$ The perpendicular to $IC$ drawn from $B$ meets the line $IA$ at $D;$ the perpendicular to $IA$ drawn from $B$ meets the line $IC$ at $E.$ Prove that the circumcenter of the triangle $\triangle IDE$ lies on the line $AC.$ [i](A. Kuznetsov )[/i]

2023 AMC 10, 14

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A number is chosen at random from among the first $100$ positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by $11$? $\textbf{(A)}~\frac{4}{100}\qquad\textbf{(B)}~\frac{9}{200} \qquad \textbf{(C)}~\frac{1}{20} \qquad\textbf{(D)}~\frac{11}{200}\qquad\textbf{(E)}~\frac{3}{50}$