Olympiad problems

1967 Polish MO Finals, 6

Given a sphere and a plane that has no common points with the sphere. Find the geometric locus of the centers of the circles of tangency with the sphere of those cones circumcribed on the sphere whose vertices lie on the given plane.

2022 Azerbaijan National Mathematical Olympiad, 4

Tags: system of equations , system , algebra

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

2015 ASDAN Math Tournament, 6

Tags: algebra test

Find all triples of integers $(x,y,z)$ which satisfy the equations \begin{align*} x^2-y-2z&=4\\ y^2-2z-3x&=-2\\ 2z^2-3x-5y&=-22.\\ \end{align*}

1986 AMC 8, 7

Tags:

How many whole numbers are between $ \sqrt{8}$ and $ \sqrt{80}$? \[ \textbf{(A)}\ 5 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 8 \qquad \textbf{(E)}\ 9 \]

2007 Iran Team Selection Test, 2

Tags: algebra , polynomial , search , number theory

Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication. [i]By Mohsen Jamali[/i]

2003 Junior Tuymaada Olympiad, 1

Tags: geometry , rectangle , combinatorics

A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares. What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices? [i]Proposed by A. Golovanov[/i]

2019 Sharygin Geometry Olympiad, 24

Tags: geometry

Two unit cubes have a common center. Is it always possible to number the vertices of each cube from $1$ to $8$ so that the distance between each pair of identically numbered vertices would be at most $4/5$? What about at most $13/16$?

2020 Dutch IMO TST, 4

Tags: combinatorics , tiles , tiling

Given are two positive integers $k$ and $n$ with $k \le n \le 2k - 1$. Julian has a large stack of rectangular $k \times 1$ tiles. Merlin calls a positive integer $m$ and receives $m$ tiles from Julian to place on an $n \times n$ board. Julian first writes on every tile whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number $m$ that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?

2017 239 Open Mathematical Olympiad, 5

Tags: geometry

Given a quadrilateral $ABCD$ in which$$\sqrt{2}(BC-BA)=AC.$$Let $X$ be the midpoint of $AC$. Prove that $2\angle BXD=\angle DAB - \angle DCB.$

1993 AMC 12/AHSME, 18

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Al and Barb start their new jobs on the same day. Al's schedule is $3$ work-days followed by $1$ rest-day. Barb's schedule is $7$ work-days followed by $3$ rest-days. On how many of their first $1000$ days do both have rest-days on the same day? $ \textbf{(A)}\ 48 \qquad\textbf{(B)}\ 50 \qquad\textbf{(C)}\ 72 \qquad\textbf{(D)}\ 75 \qquad\textbf{(E)}\ 100 $

2014 Contests, 2

Tags: geometry , circumcircle , perpendicular

The center of the circumcircle of the acute triangle $ABC$ is $M$, and the circumcircle of $ABM$ meets $BC$ and $AC$ at $P$ and $Q$ ($P\ne B$). Show that the extension of the line segment $CM$ is perpendicular to $PQ$.

1969 All Soviet Union Mathematical Olympiad, 120

Tags: number theory , relatively prime

Given natural $n$. Consider all the fractions $1/(pq)$, where $p$ and $q$ are relatively prime, $0<p<q\le n , p+q>n$. Prove that the sum of all such a fractions equals to $1/2$.

2023 239 Open Mathematical Olympiad, 2

Tags: number theory , divisibility

Let $1 < a_1 < a_2 < \cdots < a_N$ be natural numbers. It is known that for any $1\leqslant i\leqslant N$ the product of all these numbers except $a_i$ increased by one, is a multiple of $a_i$. Prove that $a_1\leqslant N$.

2010 Puerto Rico Team Selection Test, 6

Tags: inequalities , algebra

Find all values of $ r$ such that the inequality $$r (ab + bc + ca) + (3- r) \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right) \ge 9$$ is true for $a,b,c$ arbitrary positive reals

Kvant 2020, M818

Tags: combinatorics , coloring

Some $k{}$ vertices of a regular $n{}$-gon are colored red. We will call a coloring [i]uniform[/i] if for any $m$ the number of red vertices in any two sets of $m$ consecutive vertices of the $n{}$-gon coincide or differ by 1. Prove that a uniform coloring exists for any $k<n$ and is unique, up to rotations of the $n{}$-gon. [i]Proposed by M. Kontsevich[/i]

2015 BMT Spring, 6

Tags: number theory

An integer-valued function $f$ satisfies $f(2) = 4$ and $f(mn) = f(m)f(n)$ for all integers $m$ and $n$. If $f$ is an increasing function, determine $f(2015)$.

PEN A Problems, 75

Tags: modular arithmetic , divisibility theory

Find all triples $(a,b,c)$ of positive integers such that $2^{c}-1$ divides $2^{a}+2^{b}+1$.

2002 Polish MO Finals, 3

Tags: combinatorics

Three non-negative integers are written on a blackboard. A move is to replace two of the integers $k,m$ by $k+m$ and $|k-m|$. Determine whether we can always end with triplet which has at least two zeros

2023 IFYM, Sozopol, 5

Tags: algebra , polynomial

Let $a$ and $b$ be natural numbers. Prove that the number of polynomials $P(x)$ with integer coefficients such that $|P(n)| \leq a^n$ for every natural number $n \geq b$ is finite.

1988 Nordic, 3

Tags: geometry , 3d geometry , sphere

Two concentric spheres have radii $r$ and $R,r < R$. We try to select points $A, B$ and $C$ on the surface of the larger sphere such that all sides of the triangle $ABC$ would be tangent to the surface of the smaller sphere. Show that the points can be selected if and only if $R \le 2r$.

2022 CCA Math Bonanza, I6

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Let regular tetrahedron $ABCD$ have center $O$. Find $\tan^2(\angle AOB)$. [i]2022 CCA Math Bonanza Individual Round #6[/i]

1989 Flanders Math Olympiad, 2

Tags: ratio , trigonometry , geometry

When drawing all diagonals in a regular pentagon, one gets an smaller pentagon in the middle. What's the ratio of the areas of those pentagons?

2009 Iran Team Selection Test, 5

Tags: angle bisector , geometry

$ ABC$ is a triangle and $ AA'$ , $ BB'$ and $ CC'$ are three altitudes of this triangle . Let $ P$ be the feet of perpendicular from $ C'$ to $ A'B'$ , and $ Q$ is a point on $ A'B'$ such that $ QA \equal{} QB$ . Prove that : $ \angle PBQ \equal{} \angle PAQ \equal{} \angle PC'C$

2014 JBMO TST - Macedonia, 3

Tags: number theory , digit , divisibility

Find all positive integers $n$ which are divisible by 11 and satisfy the following condition: all the numbers which are generated by an arbitrary rearrangement of the digits of $n$, are also divisible by 11.

1999 Croatia National Olympiad, Problem 4

Tags: combinatorics , tournament

In a basketball competition, $n$ teams took part. Each pair of teams played exactly one match, and there were no draws. At the end of the competition the $i$-th team had $x_i$ wins and $y_i$ defeats $(i=1,\ldots,n)$. Prove that $x_1^2+x_2^2+\ldots+x_n^2=y_1^2+y_2^2+\ldots+y_n^2$.