Olympiad problems

1993 National High School Mathematics League, 1

Tags: geometry

In convex quadrilateral $ABCD$, only $D$ is an obtuse angle. Use some line segments to divide it into $n$ obtuse triangles. But on its sides (except $A,B,C,D$ ), there is no vertex of triangles we divided into. Prove that if and only if $n\geq4$, we can divide the convex quadrilateral into such $n$ triangles.

2019 Tuymaada Olympiad, 7

Tags: inequalities , grid

$N$ cells chosen on a rectangular grid. Let $a_i$ is number of chosen cells in $i$-th row, $b_j$ is number of chosen cells in $j$-th column. Prove that $$ \prod_{i} a_i! \cdot \prod_{j} b_j! \leq N! $$

2013 CIIM, Problem 5

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Let $A,B$ be $n\times n$ matrices with complex entries. Show that there exists a matrix $T$ and an invertible matrix $S$ such that \[ B=S(A+T)S^{-1}\ -T \iff \operatorname{tr}(A) = \operatorname{tr}(B) \]

2021 MOAA, 20

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In the interior of square $ABCD$ with side length $1$, a point $P$ is chosen such that the lines $\ell_1, \ell_2$ through $P$ parallel to $AC$ and $BD$, respectively, divide the square into four distinct regions, the smallest of which has area $\mathcal{R}$. The area of the region of all points $P$ for which $\mathcal{R} \geq \tfrac{1}{6}$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ where $\gcd(a,b,d)=1$ and $c$ is not divisible by the square of any prime. Compute $a+b+c+d$. [i]Proposed by Andrew Wen[/i]

2011 Princeton University Math Competition, B3

Tags: algebra

Let $f(x) = x^3-7x^2+16x-10$. As $x$ ranges over all integers, find the sum of distinct prime values taken on by $f(x)$.

2023 4th Memorial "Aleksandar Blazhevski-Cane", P5

Tags: combinatorics , graph theory , double counting , triplets

There are $1000$ students in a school. Every student has exactly $4$ friends. A group of three students $ \left \{A,B,C \right \}$ is said to be a [i]friendly triplet[/i] if any two students in the group are friends. Determine the maximal possible number of friendly triplets. [i]Proposed by Nikola Velov[/i]

2019 Moldova Team Selection Test, 3

Tags: combinatorics , easy

On the table there are written numbers $673, 674, \cdots, 2018, 2019.$ Nibab chooses arbitrarily three numbers $a,b$ and $c$, erases them and writes the number $\frac{\min(a,b,c)}{3}$, then he continues in an analogous way. After Nibab performed this operation $673$ times, on the table remained a single number $k$. Prove that $k\in(0,1).$

2006 Hong kong National Olympiad, 2

Tags: function , modular arithmetic , algebra , logarithm

For a positive integer $k$, let $f_1(k)$ be the square of the sum of the digits of $k$. Define $f_{n+1}$ = $f_1 \circ f_n$ . Evaluate $f_{2007}(2^{2006} )$.

2013 Stanford Mathematics Tournament, 8

Tags: logarithm

According to Moor's Law, the number of shoes in Moor's room doubles every year. In 2013, Moor's room starts out having exactly one pair of shoes. If shoes always come in unique, matching pairs, what is the earliest year when Moor has the ability to wear at least 500 mismatches pairs of shoes? Note that left and right shoes are distinct, and Moor must always wear one of each.

2018 District Olympiad, 4

Tags: superior algebra , finite field

Let $n$ and $q$ be two natural numbers, $n\ge 2$, $q\ge 2$ and $q\not\equiv 1 (\text{mod}\ 4)$ and let $K$ be a finite field which has exactly $q$ elements. Show that for any element $a$ from $K$, there exist $x$ and $y$ in $K$ such that $a = x^{2^n} + y^{2^n}$. (Every finite field is commutative).

2010 Math Prize For Girls Problems, 18

Tags: trigonometry

If $a$ and $b$ are positive integers such that \[ \sqrt{8 + \sqrt{32 + \sqrt{768}}} = a \cos \frac{\pi}{b} \, , \] compute the ordered pair $(a, b)$.

2022 China Team Selection Test, 4

Tags: combinatorics , geometry , combinatorial geometry , lattice points

Find all positive integer $k$ such that one can find a number of triangles in the Cartesian plane, the centroid of each triangle is a lattice point, the union of these triangles is a square of side length $k$ (the sides of the square are not necessarily parallel to the axis, the vertices of the square are not necessarily lattice points), and the intersection of any two triangles is an empty-set, a common point or a common edge.

1957 Moscow Mathematical Olympiad, 360

Tags: combinatorics , combinatorial geometry , geometry

(a) A radio lamp has a $7$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $7$ holes. Is it possible to number the contacts and the holes so that for any insertion at least one contact would match the hole with the same number? (b) A radio lamp has a $20$-contact plug, with the contacts arranged in a circle. The plug is inserted into a socket with $20$ holes. Let the contacts in the plug and the socket be already numbered. Is it always possible to insert the plug so that none of the contacts matches its socket?

2019 Jozsef Wildt International Math Competition, W. 61

Tags: summation , inequalities

If $a$, $b$, $c \in \mathbb{R}$ then$$\sum \limits_{cyc} \sqrt{(c+a)^2b^2+c^2a^2}+\sqrt{5}\left |\sum \limits_{cyc} \sqrt{ab}\right |\geq \sum \limits_{cyc}\sqrt{(ab+2bc+ca)^2+(b+c)^2a^2}$$

Kvant 2024, M2810

Tags: combinatorics

The positive integer $n \geqslant 2$ is given. How many ways can the cells of the $n\times n$ square be colored in four colors so that any two cells with a common side or vertex are colored in different colors? [i] I. Efremov [/i]

2016 Costa Rica - Final Round, LR2

Tags: combinatorics

There are $2016$ participants in the Olcotournament of chess. It is known that in any set of four participants, there is one of them who faced the other three. Prove there is at least $2013$ participants who faced everyone else.

2018 Kyiv Mathematical Festival, 5

Tags: combinatorics , game strategy , geometry

A circle is divided by $2019$ points into equal parts. Two players delete these points in turns. A player wins, if after his turn it is possible to draw a diameter of the circle such that there are no undeleted points on one side of it. Which player has a winning strategy?

2016 Oral Moscow Geometry Olympiad, 4

Tags: geometry , right triangle , excircle , collinear

Let $M$ and $N$ be the midpoints of the hypotenuse $AB$ and the leg $BC$ of a right triangles $ABC$ respectively. The excircle of the triangle $ACM$ touches the side $AM$ at point $Q$, and line $AC$ at point $P$. Prove that points $P, Q$ and $N$ lie on one straight line.

1998 Baltic Way, 11

Tags: trigonometry , geometry , inequalities

If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that \[ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}\] When does equality hold?

2007 Princeton University Math Competition, 10

Tags: geometry , trigonometry , trapezoid

In triangle $ABC$ with $AB \neq AC$, points $N \in CA$, $M \in AB$, $P \in BC$, and $Q \in BC$ are chosen such that $MP \parallel AC$, $NQ \parallel AB$, $\frac{BP}{AB} = \frac{CQ}{AC}$, and $A, M, Q, P, N$ are concyclic. Find $\angle BAC$.

2023 ELMO Shortlist, A2

Tags: algebra , functional equation

Let $\mathbb R_{>0}$ denote the set of positive real numbers. Find all functions $f:\mathbb R_{>0}\to\mathbb R_{>0}$ such that for all positive real numbers $x$ and $y$, \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\] [i]Proposed by Luke Robitaille[/i]

2003 National High School Mathematics League, 11

Tags: geometry , 3d geometry , sphere

Eight spheres with radius of $1$ are put into a circular column. There are two floors, and each sphere is tangent to adjacent four spheres, one of the bottom surfaces, and the flank. Then the height of the circular column is________.

2006 Sharygin Geometry Olympiad, 21

Tags: cevian , area of a triangle , geometric inequality , geometry , area , concurrency

On the sides $AB, BC, CA$ of triangle $ABC$, points $C', A', B'$ are taken. Prove that for the areas of the corresponding triangles, the inequality holds: $$S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}$$ and equality is achieved if and only if the lines $AA', BB', CC'$ intersect at one point.

Kvant 2023, M2739

Tags: geometry

In an acute triangle $ABC$, let $M$ and $N$ be the midpoints of $AB$ and $AC$ and let $BH$ be its altitude from $B$. Its incircle touches $AC$ at $K$ and the line through $K$ parallel to $MH$ meets $MN$ at $P$. Prove that $AMPK$ has an incircle.

2019 CCA Math Bonanza, T3

Tags: trigonometry

What is the sum of all possible values of $\cos\left(2\theta\right)$ if $\cos\left(2\theta\right)=2\cos\left(\theta\right)$ for a real number $\theta$? [i]2019 CCA Math Bonanza Team Round #3[/i]