Olympiad problems

2009 All-Russian Olympiad, 7

Tags: combinatorics

We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). Then what is the maximum number of times which the rook jumped over the fence?

1969 Spain Mathematical Olympiad, 5

Tags: geometry , convex polygon , similarity , combinatorial geometry , tiles

Show that a convex polygon with more than four sides cannot be decomposed into two others, both similar to the first (directly or inversely), by means of a single rectilinear cut. Reasonably specify which are the quadrilaterals and triangles that admit a decomposition of this type.

2011 Belarus Team Selection Test, 1

Tags: number theory , digit , sum , divisible

Let $g(n)$ be the number of all $n$-digit natural numbers each consisting only of digits $0,1,2,3$ (but not nessesarily all of them) such that the sum of no two neighbouring digits equals $2$. Determine whether $g(2010)$ and $g(2011)$ are divisible by $11$. I.Kozlov

2009 Sharygin Geometry Olympiad, 2

Tags: geometry , trapezoid

Given quadrilateral $ABCD$. Its sidelines$ AB$ and $CD$ intersect in point $K$. It's diagonals intersect in point $L$. It is known that line $KL$ pass through the centroid of $ABCD$. Prove that $ABCD$ is trapezoid. (F.Nilov)

2005 Slovenia National Olympiad, Problem 2

Tags: number theory , prime number

For which prime numbers $p$ and $q$ is $(p+1)^q$ a perfect square?

2018 Kazakhstan National Olympiad, 6

Tags: inequalities , geometric inequality , geometry

Inside of convex quadrilateral $ABCD$ found a point $M$ such that $\angle AMB=\angle ADM+\angle BCM$ and $\angle AMD=\angle ABM+\angle DCM$.Prove that $$AM\cdot CM+BM\cdot DM\ge \sqrt{AB\cdot BC\cdot CD\cdot DA}.$$

2015 China Team Selection Test, 3

Tags: number theory , function

For all natural numbers $n$, define $f(n) = \tau (n!) - \tau ((n-1)!)$, where $\tau(a)$ denotes the number of positive divisors of $a$. Prove that there exist infinitely many composite $n$, such that for all naturals $m < n$, we have $f(m) < f(n)$.

2022 Tuymaada Olympiad, 8

Tags: sine , geometry , median

In an acute triangle $\triangle ABC$ the points $C_m, A_m, B_m$ are the midpoints of $AB, BC, CA$ respectively. Inside the triangle $\triangle ABC$ a point $P$ is chosen so that $\angle PCB = \angle B_mBC$ and $\angle PAB = \angle ABB_m.$ A line passing through $P$ and perpendicular to $AC$ meets the median $BB_m$ at $E.$ Prove that $E$ lies on the circumcircle of the triangle $\triangle A_mB_mC_m.$ [i](K. Ivanov )[/i]

2018 Federal Competition For Advanced Students, P2, 1

Tags: algebra , functional equation

Let $a \ne 0$ be a real number. Find all functions $f : R_{>0}\to R_{>0}$ with $$f(f(x) + y) = ax + \frac{1}{f\left(\frac{1}{y}\right)}$$ for all $x, y \in R_{>0}$. [i](Proposed by Walther Janous)[/i]

2023 Bulgarian Autumn Math Competition, 11.4

Tags: combinatorics

Let $G$ be a complete bipartite graph with partition sets $A$ and $B$ of sizes $km$ and $kn$, respectively. The edges of $G$ are colored in $k$ colors. Prove that there exists a monochromatic connected component with at least $m+n$ vertices (which means that there exists a color and a set of vertices, such that between any two of them, there is a path consisting of edges only in that color).

2022 Federal Competition For Advanced Students, P1, 4

Tags: prime number , number theory , diophantine equation , diophantine

Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

2001 Turkey Team Selection Test, 2

Tags: geometric transformation , reflection , geometry

A circle touches to diameter $AB$ of a unit circle with center $O$ at $T$ where $OT>1$. These circles intersect at two different points $C$ and $D$. The circle through $O$, $D$, and $C$ meet the line $AB$ at $P$ different from $O$. Show that \[|PA|\cdot |PB| = \dfrac {|PT|^2}{|OT|^2}.\]

2011 China Western Mathematical Olympiad, 2

Tags: trigonometry , inequalities

Let $a,b,c > 0$, prove that \[\frac{(a-b)^2}{(c+a)(c+b)} + \frac{(b-c)^2}{(a+b)(a+c)} + \frac{(c-a)^2}{(b+c)(b+a)} \geq \frac{(a-b)^2}{a^2+b^2+c^2}\]

2000 Italy TST, 2

Tags: linear algebra , matrix , symmetry , angle bisector , geometry

Let $ ABC$ be an isosceles right triangle and $M$ be the midpoint of its hypotenuse $AB$. Points $D$ and $E$ are taken on the legs $AC$ and $BC$ respectively such that $AD=2DC$ and $BE=2EC$. Lines $AE$ and $DM$ intersect at $F$. Show that $FC$ bisects the $\angle DFE$.

2022 Germany Team Selection Test, 2

Tags: number theory , divisibility , cubefree

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

Kvant 2023, M2737

Tags: number theory , combinatorics

All the divisors of a) $8\cdot 10^6$ and b) $360^{10}$ are written on a board. At a move, we can take two numbers, neither of which is divisible by the other, and replace them with their greatest common divisor and lowest common multiple. At some point, we will no longer be able to perform new operations. How many different numbers will be on the board at this moment? [i]Proposed by V. Bragin[/i]

1987 AIME Problems, 4

Tags: geometry , analytic geometry , inequalities

Find the area of the region enclosed by the graph of $|x-60|+|y|=|x/4|.$

2010 National Olympiad First Round, 30

Tags: floor function , modular arithmetic

If $N=\lfloor \frac{2}{5} \rfloor + \lfloor \frac{2^2}{5} \rfloor +\dots \lfloor \frac{2^{2009}}{5} \rfloor$, what is the remainder when $2^{2010}$ is divided by $N$? $ \textbf{(A)}\ 5034 \qquad\textbf{(B)}\ 5032 \qquad\textbf{(C)}\ 5031 \qquad\textbf{(D)}\ 5028 \qquad\textbf{(E)}\ 5024 $

2023 All-Russian Olympiad, 1

Tags: rotation , circumcircle , geometry

Sidelines of an acute-angled triangle $T$ are colored in red, green, and blue. These lines were rotated about the circumcenter of $T$ clockwise by $120^\circ$ (we assume that the line has the same color after rotation). Prove that three points of pairs of lines of the same color are the vertices of a triangle which is congruent to $T$.

1978 IMO Longlists, 12

Tags: algebra

The equation $x^3 + ax^2 + bx + c = 0$ has three (not necessarily distinct) real roots $t, u, v$. For which $a, b, c$ do the numbers $t^3, u^3, v^3$ satisfy the equation $x^3 + a^3x^2 + b^3x + c^3 = 0$?

1987 India National Olympiad, 7

Tags: geometry , ratio , algebra

Construct the $ \triangle ABC$, given $ h_a$, $ h_b$ (the altitudes from $ A$ and $ B$) and $ m_a$, the median from the vertex $ A$.

1957 Moscow Mathematical Olympiad, 349

Tags: geometry , rectangle table , table , sum , combinatorics

For any column and any row in a rectangular numerical table, the product of the sum of the numbers in a column by the sum of the numbers in a row is equal to the number at the intersection of the column and the row. Prove that either the sum of all the numbers in the table is equal to $1$ or all the numbers are equal to $0$.

2014-2015 SDML (High School), 4

Tags:

Evaluate $$1+\frac{1+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}{2+\frac{1+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}{2+\frac{1+\frac{1+\cdots}{2+\cdots}}{2+\frac{1+\cdots}{2+\cdots}}}}.$$ $\text{(A) }\frac{\sqrt{3}}{2}\qquad\text{(B) }\frac{1+\sqrt{5}}{2}\qquad\text{(C) }\frac{2+\sqrt{3}}{2}\qquad\text{(D) }\frac{3+\sqrt{5}}{2}\qquad\text{(E) }\frac{3+\sqrt{13}}{2}$

2002 AMC 8, 2

Tags:

How many different combinations of $5$ bills and $2$ bills can be used to make a total of $17$? Order does not matter in this problem. $ \text{(A)}\ 2\qquad\text{(B)}\ 3\qquad\text{(C)}\ 4\qquad\text{(D)}\ 5\qquad\text{(E)}\ 6 $

1999 Baltic Way, 2

Tags: geometry , 3d geometry , number theory

Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.