Olympiad problems

1977 IMO Longlists, 55

Through a point $O$ on the diagonal $BD$ of a parallelogram $ABCD$, segments $MN$ parallel to $AB$, and $PQ$ parallel to $AD$, are drawn, with $M$ on $AD$, and $Q$ on $AB$. Prove that diagonals $AO,BP,DN$ (extended if necessary) will be concurrent.

2015 239 Open Mathematical Olympiad, 5

Tags: geometry , 3D geometry

The nodes of a three dimensional unit cube lattice with all three coordinates even are coloured red and blue otherwise. A convex polyhedron with all vertices red is given. Assuming the number of red points on its border is $n$. How many blue vertices can be on its border?

2022 CHMMC Winter (2022-23), 7

Tags: number theory , algebra

Let $N_0$ be the set of all non-negative integers. Let $f : N_0 \times N_0 \to N_0$ be a function such that for all non-negative integers $a,b$: $$f(a,b) = f(b,a),$$ $$f(a,0) = 0,$$ $$f(a+b,b) = f(a,b) +b.$$ Compute $$\sum_{i=0}^{30}\sum_{j=0}^{2^i-1}f(2^i, j)$$

2021 Saint Petersburg Mathematical Olympiad, 2

Tags: number theory

Given are $2021$ prime numbers written in a row. Each number, except for those in the two ends, differs from its two adjacent numbers with $6$ and $12$. Prove that there are at least two equal numbers.

2020 Iranian Our MO, 5

Tags: Arithmetic Progression , number theory

Concider two sequences $x_n=an+b$, $y_n=cn+d$ where $a,b,c,d$ are natural numbers and $gcd(a,b)=gcd(c,d)=1$, prove that there exist infinite $n$ such that $x_n$, $y_n$ are both square-free. [i]Proposed by Siavash Rahimi Shateranloo, Matin Yadollahi[/i] [b]Rated 3[/b]

2004 Iran Team Selection Test, 2

Tags: modular arithmetic , number theory proposed , number theory

Suppose that $ p$ is a prime number. Prove that the equation $ x^2\minus{}py^2\equal{}\minus{}1$ has a solution if and only if $ p\equiv1\pmod 4$.

VI Soros Olympiad 1999 - 2000 (Russia), 11.1

Tags: algebra , polynomial

The game involves two players $A$ and $B$. Player $A$ sets the value of one of the coefficients $a, b$ or $c$ of the polynomial $$x^3 + ax^2 + bx + c.$$ Player $B$ indicates the value of any of the two remaining coefficients . Player $A$ then sets the value of the last coefficients. Is there a strategy for player A such that no matter how player $B$ plays, the equation $$x^3 + ax^2 + bx + c = 0$$ to have three different (real) solutions?

2023 China Team Selection Test, P10

Tags: combinatorics , China TST

The set of nonempty integers $A$ is said to be "elegant" if it is for any $a\in A,$ $1\leq k\leq 2023,$ $$\left| \left\{ b\in A:\left\lfloor\frac b{3^k}\right\rfloor =\left\lfloor\frac a{3^k}\right\rfloor\right\}\right| =2^k.$$ Prove that if the intersection of the integer set $S$ and any "elegant" set is not empty$,$ then $S$ contains an "elegant" set.

2012 Federal Competition For Advanced Students, Part 2, 1

Tags: induction , arithmetic sequence , algebra unsolved , algebra

Given a sequence $<a_1,a_2,a_3,\cdots >$ of real numbers, we define $m_n$ as the arithmetic mean of the numbers $a_1$ to $a_n$ for $n\in\mathbb{Z}^+$. If there is a real number $C$, such that \[ (i-j)m_k+(j-k)m_i+(k-i)m_j=C\] for every triple $(i,j,k)$ of distinct positive integers, prove that the sequence $<a_1,a_2,a_3,\cdots >$ is an arithmetic progression.

2006 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , diophantine , Diophantine equation

. Find all triplets $(x, y, z)$, $x > y > z$ of positive integers such that $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1$

1952 Moscow Mathematical Olympiad, 223

Tags: geometry , incircle , tangent circles

In a convex quadrilateral $ABCD$, let $AB + CD = BC + AD$. Prove that the circle inscribed in $ABC$ is tangent to the circle inscribed in $ACD$.

2014 JBMO Shortlist, 4

Tags: combinatorics

$A=1\cdot4\cdot7\cdots2014$.Find the last non-zero digit of $A$ if it is known that $A\equiv 1\mod3$.

2010 Tuymaada Olympiad, 2

Tags: geometry , circumcircle , geometric transformation , reflection , parallelogram , calculus , trigonometry

Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram. Show that $\angle BPC > \angle BAC$.

1993 Romania Team Selection Test, 2

Tags: geometry unsolved , geometry

Suppose that $ D,E,F$ are points on sides $ BC,CA,AB$ of a triangle $ ABC$ respectively such that $ BD\equal{}CE\equal{}AF$ and $ \angle BAD\equal{}\angle CBE\equal{}\angle ACF$.Prove that the triangle $ ABC$ is equilateral.

1980 Bundeswettbewerb Mathematik, 1

Tags: game , number theory , Divisibility , Digits

Six free cells are given in a row. Players $A$ and $B$ alternately write digits from $0$ to $9$ in empty cells, with $A$ starting. When all the cells are filled, one considers the obtained six-digit number $z$. Player $B$ wins if $z$ is divisible by a given natural number $n$, and loses otherwise. For which values of $n$ not exceeding $20$ can $B$ win independently of his opponent’s moves?

PEN E Problems, 17

Tags: blogs

Let $a$, $b$, and $n$ be positive integers with $\gcd (a, b)=1$. Without using Dirichlet's theorem, show that there are infinitely many $k \in \mathbb{N}$ such that $\gcd(ak+b, n)=1$.

1961 Putnam, A5

Tags: Putnam , Subsets , Union of set

Let $\Omega$ be a set of $n$ points, where $n>2$. Let $\Sigma$ be a nonempty subcollection of the $2^n$ subsets of $\Omega$ that is closed with respect to the unions, intersections and complements. If $k$ is the number of elements of $\Sigma,$ what are the possible values of $k?$

2020 USMCA, 20

Tags:

Let $\Omega$ be a circle centered at $O$. Let $ABCD$ be a quadrilateral inscribed in $\Omega$, such that $AB = 12$, $AD = 18$, and $AC$ is perpendicular to $BD$. The circumcircle of $AOC$ intersects ray $DB$ past $B$ at $P$. Given that $\angle PAD = 90^\circ$, find $BD^2$.

2007 Thailand Mathematical Olympiad, 4

Tags: number theory , Perfect Square

Find all primes $p$ such that $\frac{2^{p-1}-1}{p}$ is a perfect square.

2014 Hanoi Open Mathematics Competitions, 8

Tags: geometry , right triangle , isosceles , Isosceles Triangle

Let $ABC$ be a triangle. Let $D,E$ be the points outside of the triangle so that $AD=AB,AC=AE$ and $\angle DAB =\angle EAC =90^o$. Let $F$ be at the same side of the line $BC$ as $A$ such that $FB = FC$ and $\angle BFC=90^o$. Prove that the triangle $DEF$ is a right- isosceles triangle.

2002 Korea Junior Math Olympiad, 2

Tags: number theory , prime numbers

Find all prime number $p$ such that $p^{2002}+2003^{p-1}-1$ is a multiple of $2003p$.

2022 Paraguay Mathematical Olympiad, 4

Tags: algebra , number theory

Karina, Leticia and Milena paint glass bottles and sell them as decoration. they had $100$ bottles, and they decorated them in such a way that each bottle was painted by a single person. After the finished, they put all the bottles on a table. In an oversight one of them pushed the table, falling and breaking exactly $\frac18$ of the bottles that Karina painted, $\frac13$ of the bottles that Milena, painted and $\frac16$ of the bottles that Leticia painted. In total, $82$ painted bottles remained unbroken. Knowing that the number of broken bottles that Milena had painted is equal to the average of the amounts of broken bottles painted by Karina and Leticia, how many bottles did each of them paint?

2007 National Olympiad First Round, 35

Tags:

What is the third digit after the decimal point of the decimal representation of $\sqrt[3]{2+\sqrt 5} + \sqrt[3]{2-\sqrt 5}$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None of the above} $

2010 QEDMO 7th, 4

Tags: geometry , areas , Squares

Let $ABCD$ and $A'B'C'D'$ be two squares, both are oriented clockwise. In addition, it is assumed that all points are arranged as shown in the figure.Then it has to be shown that the sum of the areas of the quadrilaterals $ABB'A'$ and $CDD'C'$ equal to the sum of the areas of the quadrilaterals $BCC'B'$ and $DAA'D'$. [img]https://cdn.artofproblemsolving.com/attachments/0/2/6f7f793ded22fe05a7b0a912ef6c4e132f963e.png[/img]

2005 AMC 12/AHSME, 19

Tags:

Let $ x$ and $ y$ be two-digit integers such that $ y$ is obtained by reversing the digits of $ x$. The integers $ x$ and $ y$ satisfy $ x^2 \minus{} y^2 \equal{} m^2$ for some positive integer $ m$. What is $ x \plus{} y \plus{} m$? $ \textbf{(A)}\ 88\qquad \textbf{(B)}\ 112\qquad \textbf{(C)}\ 116\qquad \textbf{(D)}\ 144\qquad \textbf{(E)}\ 154$