Olympiad problems

2021 CMIMC Integration Bee, 3

Tags:

$$\int_{0}^{\frac{\pi}{2}}\sin^2(x)\sin(2x)\,dx$$ [i]Proposed by Connor Gordon[/i]

2011 Singapore MO Open, 4

Find all polynomials $P(x)$ with real coefficients such that \[P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.\]

1986 IMO Longlists, 66

Tags: triangle inequality , combinatorics , inequalities

One hundred red points and one hundred blue points are chosen in the plane, no three of them lying on a line. Show that these points can be connected pairwise, red ones with blue ones, by disjoint line segments.

2010 Iran MO (3rd Round), 7

Tags: function , abstract algebra , algebra

[b]interesting function[/b] $S$ is a set with $n$ elements and $P(S)$ is the set of all subsets of $S$ and $f : P(S) \rightarrow \mathbb N$ is a function with these properties: for every subset $A$ of $S$ we have $f(A)=f(S-A)$. for every two subsets of $S$ like $A$ and $B$ we have $max(f(A),f(B))\ge f(A\cup B)$ prove that number of natural numbers like $x$ such that there exists $A\subseteq S$ and $f(A)=x$ is less than $n$. time allowed for this question was 1 hours and 30 minutes.

2018 Brazil National Olympiad, 2

Tags:

We say that a quadruple $(A,B,C,D)$ is [i]dobarulho[/i] when $A,B,C$ are non-zero algorisms and $D$ is a positive integer such that: $1.$ $A \leq 8$ $2.$ $D>1$ $3.$ $D$ divides the six numbers $\overline{ABC}$, $\overline{BCA}$, $\overline{CAB}$, $\overline{(A+1)CB}$, $\overline{CB(A+1)}$, $\overline{B(A+1)C}$. Find all such quadruples.

2018 Czech-Polish-Slovak Match, 3

Tags: combinatorics

There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns. [i]Proposed by Peter Novotný, Slovakia[/i]

2021 Balkan MO Shortlist, N4

Tags: shortlist , number theory

Can every positive rational number $q$ be written as $$\frac{a^{2021} + b^{2023}}{c^{2022} + d^{2024}},$$ where $a, b, c, d$ are all positive integers? [i]Proposed by Dominic Yeo, UK[/i]

2005 Alexandru Myller, 3

Tags: inequalities , algebra

Let be three positive real numbers $ a,b,c $ whose sum is $ 1. $ Prove: $$ 0\le\sum_{\text{cyc}} \log_a\frac{(abc)^a}{a^2+b^2+c^2} $$

2018 IMO Shortlist, A5

Tags: function , algebra

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

2015 Math Prize for Girls Problems, 9

Tags:

Say that a rational number is [i]special[/i] if its decimal expansion is of the form $0.\overline{abcdef}$, where $a$, $b$, $c$, $d$, $e$, and $f$ are digits (possibly equal) that include each of the digits $2$, $0$, $1$, and $5$ at least once (in some order). How many special rational numbers are there?

2020 AIME Problems, 2

Tags: geometric sequence , logarithm

There is a unique positive real number $x$ such that the three numbers $\log_8(2x),\log_4x,$ and $\log_2x,$ in that order, form a geometric progression with positive common ratio. The number $x$ can be written as $\tfrac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2024 Harvard-MIT Mathematics Tournament, 27

Tags: guts

A deck of $100$ cards is labeled $1,2,\ldots,100$ from top to bottom. The top two cards are drawn; one of them is discarded at random, and the other is inserted back at the bottom of the deck. This process is repeated until only one card remains in the deck. Compute the expected value of the label of the remaining card.

2005 Hong kong National Olympiad, 2

Tags: triangle inequality , geometry , inequalities

Suppose there are $4n$ line segments of unit length inside a circle of radius $n$. Furthermore, a straight line $L$ is given. Prove that there exists a straight line $L'$ that is either parallel or perpendicular to $L$ and that $L'$ cuts at least two of the given line segments.

2006 Tournament of Towns, 4

Tags: geometry , inradius , hexagon , tangent

A circle of radius $R$ is inscribed into an acute triangle. Three tangents to the circle split the triangle into three right angle triangles and a hexagon that has perimeter $Q$. Find the sum of diameters of circles inscribed into the three right triangles. (6)

2022 Latvia Baltic Way TST, P9

Tags: geometry , cyclic quadrilateral , circumcircle

Let $ABCD$ be a cyclic quadrilateral inscribed in circle $\Omega$. Let the lines $AB$ and $CD$ intersect at $P$, and the lines $AD$ and $BC$ intersect at $Q$. Let then the circumcircle of the triangle $\triangle APQ$ intersect $\Omega$ at $R \neq A$. Prove that the line $CR$ goes through the midpoint of the segment $PQ$.

1997 Denmark MO - Mohr Contest, 2

Tags: square , area , geometry

Two squares, both with side length $1$, are arranged so that one has one vertex in the center of the other. Determine the area of the gray area. [img]https://1.bp.blogspot.com/-xt3pe0rp1SI/XzcGLgEw1EI/AAAAAAAAMYM/vFKxvvVuLvAJ5FO_yX315X3Fg_iFaK2fACLcBGAsYHQ/s0/1997%2BMohr%2Bp2.png[/img]

2002 India Regional Mathematical Olympiad, 4

Tags:

Suppose the integers $1,2,\ldots 10$ are split into two disjoint collections $a_1,a_2, \ldots a_5$ and $b_1 , \ldots b_5$ such that $a_1 <a _2 < a_3 <a_4 <a _5 , b_1 < b_2 < b_3 < b_4 < b_5$ (i) Show that the larger number in any pair $\{ a_j, b_j \}$ , $1 \leq j \leq 5$ is at least $6$. (ii) Show that $\sum_{i=1} ^{5} | a_i - b_i|$ = 25 for every such partition.

2013 Dutch IMO TST, 4

Tags: combinatorics , combinatorial geometry , rectangle , board

Let $n \ge 3$ be an integer, and consider a $n \times n$-board, divided into $n^2$ unit squares. For all $m \ge 1$, arbitrarily many $1\times m$-rectangles (type I) and arbitrarily many $m\times 1$-rectangles (type II) are available. We cover the board with $N$ such rectangles, without overlaps, and such that every rectangle lies entirely inside the board. We require that the number of type I rectangles used is equal to the number of type II rectangles used.(Note that a $1 \times 1$-rectangle has both types.) What is the minimal value of $N$ for which this is possible?

PEN H Problems, 76

Tags: diophantine equation

Find all pairs $(m,n)$ of integers that satisfy the equation \[(m-n)^{2}=\frac{4mn}{m+n-1}.\]

2023 Bulgarian Spring Mathematical Competition, 9.3

Tags: number theory

Given a prime $p$, find $\gcd(\binom{2^pp}{1},\binom{2^pp}{3},\ldots, \binom{2^pp}{2^pp-1}) $.

2008 All-Russian Olympiad, 7

Tags: number theory

For which integers $ n>1$ do there exist natural numbers $ b_1,b_2,\ldots,b_n$ not all equal such that the number $ (b_1\plus{}k)(b_2\plus{}k)\cdots(b_n\plus{}k)$ is a power of an integer for each natural number $ k$? (The exponents may depend on $ k$, but must be greater than $ 1$)

2013 Sharygin Geometry Olympiad, 21

Tags: power of a point , reflection , angle chasing , similar triangles , geometry

Chords $BC$ and $DE$ of circle $\omega$ meet at point $A$. The line through $D$ parallel to $BC$ meets $\omega$ again at $F$, and $FA$ meets $\omega$ again at $T$. Let $M = ET \cap BC$ and let $N$ be the reflection of $A$ over $M$. Show that $(DEN)$ passes through the midpoint of $BC$.

2022 USEMO, 1

Tags: perimeter , combinatorics , grid

A [i]stick[/i] is defined as a $1 \times k$ or $k\times 1$ rectangle for any integer $k \ge 1$. We wish to partition the cells of a $2022 \times 2022$ chessboard into $m$ non-overlapping sticks, such that any two of these $m$ sticks share at most one unit of perimeter. Determine the smallest $m$ for which this is possible. [i]Holden Mui[/i]

2017 International Zhautykov Olympiad, 3

Tags: 3d geometry , tetrahedron , inequalities , geometry

Let $ABCD$ be the regular tetrahedron, and $M, N$ points in space. Prove that: $AM \cdot AN + BM \cdot BN + CM \cdot CN \geq DM \cdot DN$

2010 Today's Calculation Of Integral, 555

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

For $ \frac {1}{e} < t < 1$, find the minimum value of $ \int_0^1 |xe^{ \minus{} x} \minus{} tx|dx$.