Olympiad problems

1973 All Soviet Union Mathematical Olympiad, 188

Given $4$ points in three-dimensional space, not lying in one plane. What is the number of such a parallelepipeds (bricks), that each point is a vertex of each parallelepiped?

2019 Lusophon Mathematical Olympiad, 2

Tags: algebra , trinomial , Rational roots

Prove that for every $n$ nonzero integer , there are infinite triples of nonzero integers $a, b$ and $c$ that satisfy the conditions: 1. $a + b + c = n$ 2. $ax^2 + bx + c = 0$ has rational roots.

1997 French Mathematical Olympiad, Problem 3

Tags: geometry , 3D geometry

Let $C$ be a unit cube and let $p$ denote the orthogonal projection onto the plane. Find the maximum area of $p(C)$.

2013 Princeton University Math Competition, 12

Tags:

Let $D$ be a point on the side $BC$ of $\triangle ABC$. If $AB=8$, $AC=7$, $BD=2$, and $CD=1$, find $AD$.

CIME II 2018, 13

Tags: AIME II

Two lines, $l_1$ and $l_2$, are tangent to the parabola $x^2-4(x+y)+y^2=2xy+8$ such that they intersect at a point whose coordinates sum to $-32$. The minimum possible sum of the slopes of $l_1$ and $l_2$ can be written as $\frac{m}{n}$ for relatively prime integers $m$ and $n$. Find $m+n$. [I] Proposed by [b]AOPS12142015[/b][/I]

2008 Serbia National Math Olympiad, 6

Tags: geometry , trigonometry , inequalities , cyclic quadrilateral , geometry proposed

In a convex pentagon $ ABCDE$, let $ \angle EAB \equal{} \angle ABC \equal{} 120^{\circ}$, $ \angle ADB \equal{} 30^{\circ}$ and $ \angle CDE \equal{} 60^{\circ}$. Let $ AB \equal{} 1$. Prove that the area of the pentagon is less than $ \sqrt {3}$.

2011 Albania Team Selection Test, 4

Tags: modular arithmetic , number theory , number theory unsolved

Find all prime numbers p such that $2^p+p^2 $ is also a prime number.

2021 CMIMC, 2.2

Tags: algebra

Suppose $a,b$ are positive real numbers such that $a+a^2 = 1$ and $b^2+b^4=1$. Compute $a^2+b^2$. [i]Proposed by Thomas Lam[/i]

2005 Bundeswettbewerb Mathematik, 1

Tags: floor function , combinatorics proposed , combinatorics

Two players $A$ and $B$ have one stone each on a $100 \times 100$ chessboard. They move their stones one after the other, and a move means moving one's stone to a neighbouring field (horizontally or vertically, not diagonally). At the beginning of the game, the stone of $A$ lies in the lower left corner, and the one of $B$ in the lower right corner. Player $A$ starts. Prove: Player $A$ is, independently from that what $B$ does, able to reach, after finitely many steps, the field $B$'s stone is lying on at that moment.

2017 CCA Math Bonanza, L2.3

Tags:

Jack is jumping on the number line. He first jumps one unit and every jump after that he jumps one unit more than the previous jump. What is the least amount of jumps it takes to reach exactly $19999$ from his starting place? [i]2017 CCA Math Bonanza Lightning Round #2.3[/i]

2009 Greece Junior Math Olympiad, 1

Tags: number theory

If the number $K = \frac{9n^2+31}{n^2+7}$ is integer, find the possible values of $n \in Z$.

2007 China Team Selection Test, 1

Tags: algebra , polynomial , geometry proposed , geometry

When all vertex angles of a convex polygon are equal, call it equiangular. Prove that $ p > 2$ is a prime number, if and only if the lengths of all sides of equiangular $ p$ polygon are rational numbers, it is a regular $ p$ polygon.

2003 Baltic Way, 8

Tags: combinatorics unsolved , combinatorics , Game Theory

There are $2003$ pieces of candy on a table. Two players alternately make moves. A move consists of eating one candy or half of the candies on the table (the “lesser half” if there are an odd number of candies). At least one candy must be eaten at each move. The loser is the one who eats the last candy. Which player has a winning strategy?

1998 Portugal MO, 1

Tags: number theory

A chicken breeder went to check what price per chick he had charged the previous year. He found an invoice, half erased, which read: $72$ chickens sold for $*679*$ escudos” (the digits of the units and tens of thousands were illegible). What price did each chick sell for last year?

2022 All-Russian Olympiad, 4

Tags: combinatorics , algebra

There are $18$ children in the class. Parents decided to give children from this class a cake. To do this, they first learned from each child the area of the piece he wants to get. After that, they showed a square-shaped cake, the area of which is exactly equal to the sum of $18$ named numbers. However, when they saw the cake, the children wanted their pieces to be squares too. The parents cut the cake with lines parallel to the sides of the cake (cuts do not have to start or end on the side of the cake). For what maximum k the parents are guaranteed to cut out $k$ square pieces from the cake, which you can give to $k$ children so that each of them gets what they want?

2009 All-Russian Olympiad Regional Round, 9.7

Tags: geometry , Circumcenter , parallelogram

Given a parallelogram $ABCD$, in which the angle $\angle ABC$ is obtuse. Line $AD$ intersects the circle a second time $\omega$ circumscribed around triangle $ABC$, at the point $E$. Line $CD$ intersects second time circle $\omega$ at point $F$. Prove that the circumcenter of triangle $DEF$ lies on the circle $\omega$.

2013 Online Math Open Problems, 5

Tags: Online Math Open , probability , analytic geometry , number theory , relatively prime

A wishing well is located at the point $(11,11)$ in the $xy$-plane. Rachelle randomly selects an integer $y$ from the set $\left\{ 0, 1, \dots, 10 \right\}$. Then she randomly selects, with replacement, two integers $a,b$ from the set $\left\{ 1,2,\dots,10 \right\}$. The probability the line through $(0,y)$ and $(a,b)$ passes through the well can be expressed as $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [i]Proposed by Evan Chen[/i]

2019 South East Mathematical Olympiad, 3

Tags: combinatorics

$n$ symbols line up in a row, numbered as $1,2,...,n$ from left to right. Delete every symbol with squared numbers. Renumber the rest from left to right. Repeat the process until all $n$ symbols are deleted. Let $f(n)$ be the initial number of the last symbol deleted. Find $f(n)$ in terms of $n$ and find $f(2019)$.

2017 Moscow Mathematical Olympiad, 10

Tags: geometry

Point $D$ lies in $\triangle ABC$ and $BD=CD$,$\angle BDC=120$. Point $E$ lies outside $ABC$ and $AE=CE,\angle AEC=60$. Points $B$ and $E$ lies on different sides of $AC$. $F$ is midpoint $BE$. Prove, that $\angle AFD=90$

2011 Nordic, 2

Tags: ratio , geometry unsolved , geometry

In a triangle $ABC$ assume $AB = AC$, and let $D$ and $E$ be points on the extension of segment $BA$ beyond $A$ and on the segment $BC$, respectively, such that the lines $CD$ and $AE$ are parallel. Prove $CD \ge \frac{4h}{BC}CE$, where $h$ is the height from $A$ in triangle $ABC$. When does equality hold?

2022 Pan-African, 6

Tags: number theory

Does there exist positive integers $n_1, n_2, \dots, n_{2022}$ such that the number $$ \left( n_1^{2020} + n_2^{2019} \right)\left( n_2^{2020} + n_3^{2019} \right) \cdots \left( n_{2021}^{2020} + n_{2022}^{2019} \right)\left( n_{2022}^{2020} + n_1^{2019} \right) $$ is a power of $11$?

2014 Contests, 3

Tags: Diophantine equation , number theory proposed , number theory

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

VI Soros Olympiad 1999 - 2000 (Russia), 9.2

Tags: number theory , Diophantine equation , diophantine

Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds

2001 Moldova National Olympiad, Problem 3

Tags: geometry

For an arbitrary point $D$ on side $BC$ of an acute-angled triangle $ABC$, let $O_1$ and $O_2$ be the circumcenters of the triangles $ABD$ and $ACD$, and $O$ be the circumcenter of the triangle $AO_1O_2$. Find the locus of $O$ when $D$ moves across $BC$.

2023 Belarusian National Olympiad, 11.3

Tags: number theory

Prove that for any fixed integer $a$ equation $$(m!+a)^2=n!+a^2$$ has finitely many solutions in positive integers $m,n$