Olympiad problems

2016 Sharygin Geometry Olympiad, 7

Diagonals of a quadrilateral $ABCD$ are equal and meet at point $O$. The perpendicular bisectors to segments $AB$ and $CD$ meet at point $P$, and the perpendicular bisectors to $BC$ and $AD$ meet at point $Q$. Find angle $\angle POQ$. by A.Zaslavsky

I Soros Olympiad 1994-95 (Rus + Ukr), 10.1

Tags: algebra , number theory , functional

The function $f: Z \to Z$ satisfies the following conditions: 1) $f(f(n))=n$ for all integers $n$ 2) $f(f(n+2)+2) = n$ for all integers $n$ 3) $f(0)=1$. Find the value of $f(1995)$ and $f(-1994)$.

2006 International Zhautykov Olympiad, 2

Tags: geometry , parallelogram , analytic geometry , function , trigonometry , trapezoid , ratio

Let $ ABC$ be a triangle and $ K$ and $ L$ be two points on $ (AB)$, $ (AC)$ such that $ BK \equal{} CL$ and let $ P \equal{} CK\cap BL$. Let the parallel through $ P$ to the interior angle bisector of $ \angle BAC$ intersect $ AC$ in $ M$. Prove that $ CM \equal{} AB$.

2005 Today's Calculation Of Integral, 88

Tags: calculus , integration , function , calculus computations

A function $f(x)$ satisfies $\begin{cases} f(x)=-f''(x)-(4x-2)f'(x)\\ f(0)=a,\ f(1)=b \end{cases}$ Evaluate $\int_0^1 f(x)(x^2-x)\ dx.$

2002 All-Russian Olympiad Regional Round, 8.1

Tags: combinatorics , number theory

Is it possible to fill all the cells of the table $9 \times 2002$ with natural numbers so that the sum of the numbers in any column and the sum of the numbers in any string would be prime numbers?

2006 ISI B.Math Entrance Exam, 6

Tags: calculus , number theory theorems

You are standing at the edge of a river which is $1$ km wide. You have to go to your camp on the opposite bank . The distance to the camp from the point on the opposite bank directly across you is $1$ km . You can swim at $2$ km/hr and walk at $3$ km-hr . What is the shortest time you will take to reach your camp?(Ignore the speed of the river and assume that the river banks are straight and parallel).

2002 Tuymaada Olympiad, 2

Tags: geometry , ratio , equilateral , equilateral triangle

Points on the sides $ BC $, $ CA $ and $ AB $ of the triangle $ ABC $ are respectively $ A_1 $, $ B_1 $ and $ C_1 $ such that $ AC_1: C_1B = BA_1: A_1C = CB_1: B_1A = 2: 1 $. Prove that if triangle $ A_1B_1C_1 $ is equilateral, then triangle $ ABC $ is also equilateral.

2021 Korea Junior Math Olympiad, 2

Tags: modular arithmetic , sequence , integer sequence , number theory

Let $\{a_n\}$ be a sequence of integers satisfying the following conditions. [list] [*] $a_1=2021^{2021}$ [*] $0 \le a_k < k$ for all integers $k \ge 2$ [*] $a_1-a_2+a_3-a_4+ \cdots + (-1)^{k+1}a_k$ is multiple of $k$ for all positive integers $k$. [/list] Determine the $2021^{2022}$th term of the sequence $\{a_n\}$.

2022 Pan-African, 1

Tags: geometry , circumcircle

Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$, and $AB$ its shortest side. Let $H$ be the orthocenter of $ABC$. Let $\Gamma$ be the circle with center $B$ and radius $BA$. Let $D$ be the second point where the line $CA$ meets $\Gamma$. Let $E$ be the second point where $\Gamma$ meets the circumcircle of the triangle $BCD$. Let $F$ be the intersection point of the lines $DE$ and $BH$. Prove that the line $BD$ is tangent to the circumcircle of the triangle $DFH$.

2014 Serbia National Math Olympiad, 1

Tags: function , functional equation

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x$, $y \in \mathbb{R}$ hold: $$f(xf(y)-yf(x))=f(xy)-xy$$ [i]Proposed by Dusan Djukic[/i]

2002 All-Russian Olympiad Regional Round, 9.4

Tags: combinatorics , combinatorial geometry , rectangle , geometry

Located on the plane $\left[ \frac43 n \right]$ rectangles with sides parallel to the coordinate axes. It is known that any rectangle intersects at least n rectangles. Prove that exists a rectangle that intersects all rectangles.

2010 ELMO Shortlist, 2

Tags: modular arithmetic , algebra , difference of squares , special factorizations , number theory

Given a prime $p$, show that \[\left(1+p\sum_{k=1}^{p-1}k^{-1}\right)^2 \equiv 1-p^2\sum_{k=1}^{p-1}k^{-2} \pmod{p^4}.\] [i]Timothy Chu.[/i]

2024 Philippine Math Olympiad, P1

Tags: algebra , functional equation

Let $f:\mathbb{Z}^2\rightarrow\mathbb{Z}$ be a function satisfying \[f(x+1,y)+f(x,y+1)+1=f(x,y)+f(x+1,y+1)\] for all integers $x$ and $y$. Can it happen that $|f(x,y)|\leq 2024$ for all $x,y\in\mathbb{Z}$?

1995 IberoAmerican, 1

Tags: number theory

Find all the possible values of the sum of the digits of all the perfect squares. [Commented by djimenez] [b]Comment: [/b]I would rewrite it as follows: Let $f: \mathbb{N}\rightarrow \mathbb{N}$ such that $f(n)$ is the sum of all the digits of the number $n^2$. Find the image of $f$ (where, by image it is understood the set of all $x$ such that exists an $n$ with $f(n)=x$).

2014 Contests, 1

Tags: algebra , inequality

Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds. [i]UK - David Monk[/i]

2022 Iranian Geometry Olympiad, 4

Tags: geometry , angle bisector

Let $AD$ be the internal angle bisector of triangle $ABC$. The incircles of triangles $ABC$ and $ACD$ touch each other externally. Prove that $\angle ABC > 120^{\circ}$. (Recall that the incircle of a triangle is a circle inside the triangle that is tangent to its three sides.) [i]Proposed by Volodymyr Brayman (Ukraine)[/i]

1998 Croatia National Olympiad, Problem 1

Tags: complex number , algebra , equation

Solve the equation $2z^3-(5+6i)z^2+9iz+1-3i=0$ over $\mathbb C$ given that one of the solutions is real.

2018 MIG, 7

Tags:

The accuracy of a mystic's prediction is related to the volume of his or her crystal ball by the equation $P = 1 - (0.999)^V$, where $P$ is the probability the mystic's prediction is correct and $V$ is the volume of his ball in cubic centimeters. These crystal balls are sold wth radii that are a whole number of centimeters. What is the radius (in centimeters) of the smallest ball that gives the mystic over a $99\%$ chance to make an accurate prediction.

2014 Turkey Team Selection Test, 2

Tags: function , symmetry , algebra , functional equation

Find all $f$ functions from real numbers to itself such that for all real numbers $x,y$ the equation \[f(f(y)+x^2+1)+2x=y+(f(x+1))^2\] holds.

2012 Polish MO Finals, 5

Tags: tangent , triangle , circumcircle , geometry

Point $O$ is a center of circumcircle of acute triangle $ABC$, bisector of angle $BAC$ cuts side $BC$ in point $D$. Let $M$ be a point such that, $MC \perp BC$ and $MA \perp AD$. Lines $BM$ and $OA$ intersect in point $P$. Show that circle of center in point $P$ passing through a point $A$ is tangent to line $BC$.

2010 JBMO Shortlist, 2

Tags: circumcircle , parallelogram , perpendicular bisector , geometry

Let $ABC$ be acute-angled triangle . A circle $\omega_1(O_1,R_1)$ passes through points $B$ and $C$ and meets the sides $AB$ and $AC$ at points $D$ and $E$ ,respectively . Let $\omega_2(O_2,R_2)$ be the circumcircle of triangle $ADE$ . Prove that $O_1O_2$ is equal to the circumradius of triangle $ABC$ .

1999 Romania National Olympiad, 2

Tags: number theory , prime

Let $a, b, c$ be non zero integers,$ a\ne c$ such that $$\frac{a}{c}=\frac{a^2+b^2}{c^2+b^2}$$ Prove that $a^2 +b^2 +c^2$ cannot be a prime number.

2010 Iran MO (3rd Round), 4

Tags: function , number theory

sppose that $\sigma_k:\mathbb N \longrightarrow \mathbb R$ is a function such that $\sigma_k(n)=\sum_{d|n}d^k$. $\rho_k:\mathbb N \longrightarrow \mathbb R$ is a function such that $\rho_k\ast \sigma_k=\delta$. find a formula for $\rho_k$.($\frac{100}{6}$ points)

2018 Iran Team Selection Test, 4

Tags: number theory

Call a positive integer "useful but not optimized " (!), if it can be written as a sum of distinct powers of $3$ and powers of $5$. Prove that there exist infinitely many positive integers which they are not "useful but not optimized". (e.g. $37=(3^0+3^1+3^3)+(5^0+5^1)$ is a " useful but not optimized" number) [i]Proposed by Mohsen Jamali[/i]

2019 Romanian Master of Mathematics Shortlist, G1

Tags: geometry , circumcircle , circumcenter , median

Let $BM$ be a median in an acute-angled triangle $ABC$. A point $K$ is chosen on the line through $C$ tangent to the circumcircle of $\vartriangle BMC$ so that $\angle KBC = 90^\circ$. The segments $AK$ and $BM$ meet at $J$. Prove that the circumcenter of $\triangle BJK$ lies on the line $AC$. Aleksandr Kuznetsov, Russia