Olympiad problems

2005 District Olympiad, 3

Tags: geometry , 3d geometry , tetrahedron , circumcircle , trigonometry , trig identities , law of sines

Prove that if the circumcircles of the faces of a tetrahedron $ABCD$ have equal radii, then $AB=CD$, $AC=BD$ and $AD=BC$.

1981 Polish MO Finals, 6

Tags: geometry , 3d geometry , tetrahedron , geometric inequalities

In a tetrahedron of volume $V$ the sum of the squares of the lengths of its edges equals $S$. Prove that $$V \le \frac{S\sqrt{S}}{72\sqrt{3}}$$

2007 Today's Calculation Of Integral, 170

Tags: calculus , integration , algebra , partial fractions , calculus computations , logarithm

Let $a,\ b$ be constant numbers such that $a^{2}\geq b.$ Find the following definite integrals. (1) $I=\int \frac{dx}{x^{2}+2ax+b}$ (2) $J=\int \frac{dx}{(x^{2}+2ax+b)^{2}}$

2020 Durer Math Competition Finals, 3

Tags: number theory , least common multiple , perfect square , consecutive

Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?

For a positive integer $n$, let $s(n)$ be the number of ways that $n$ can be written as the sum of strictly increasing perfect $2010^{\text{th}}$ powers. For instance, $s(2) = 0$ and $s(1^{2010} + 2^{2010}) = 1$. Show that for every real number $x$, there exists an integer $N$ such that for all $n > N$, \[\frac{\max_{1 \leq i \leq n} s(i)}{n} > x.\] [i]Alex Zhu.[/i]

1975 Canada National Olympiad, 6

Tags: rotation , geometry , geometric transformation

(i) 15 chairs are equally placed around a circular table on which are name cards for 15 guests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated. (ii) Give an example of an arrangement in which just one of the 15 guests is correctly seated and for which no rotation correctly places more than one person.

1966 AMC 12/AHSME, 34

Tags: geometry , geometric transformation , rotation

Let $r$ be the speed in miles per hour at which a wheel, $11$ feet in circumference, travels. If the time for a complete rotation of the wheel is shortened by $\tfrac{1}{4}$ of a second, the speed $r$ is increased by $5$ miles per hour. The $r$ is: $\text{(A)}\ 9\qquad \text{(B)}\ 10\qquad \text{(C)}\ 10\tfrac{1}{2}\qquad \text{(D)}\ 11\qquad \text{(E)}\ 12$

2003 Estonia National Olympiad, 4

Tags: number theory , sum , reciprocal , divisor

Call a positive integer [i]lonely [/i] if the sum of reciprocals of its divisors (including $1$ and the integer itself) is not equal to the sum of reciprocals of divisors of any other positive integer. Prove that a) all primes are lonely, b) there exist infinitely many non-lonely positive integers.

1964 All Russian Mathematical Olympiad, 054

Tags: number theory , perfect square

Find the smallest exact square with last digit not $0$, such that after deleting its last two digits we shall obtain another exact square.

2018 Germany Team Selection Test, 1

Tags: algebra , polynomial , ineqstd

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

2007 China Girls Math Olympiad, 5

Tags: circumcircle , geometric transformation , reflection , homothety , parallelogram , geometry

Point $D$ lies inside triangle $ABC$ such that $\angle DAC = \angle DCA = 30^{\circ}$ and $\angle DBA = 60^{\circ}$. Point $E$ is the midpoint of segment $BC$. Point $F$ lies on segment $AC$ with $AF = 2FC$. Prove that $DE \perp EF$.

2021 Azerbaijan EGMO TST, 4

Tags: algebra

Let $P(x), Q(x)$ be distinct polynomials of degree $2020$ with non-zero coefficients. Suppose that they have $r$ common real roots counting multiplicity and $s$ common coefficients. Determine the maximum possible value of $r + s$. [i]Demetres Christofides, Cyprus[/i]

2018 MMATHS, 2

Tags: number theory , geometry

Prove that if a triangle has integer side lengths and the area (in square units) equals the perimeter (in units), then the perimeter is not a prime number.

1998 Hong kong National Olympiad, 4

Tags: function , algebra

Define a function $f$ on positive real numbers to satisfy \[f(1)=1 , f(x+1)=xf(x) \textrm{ and } f(x)=10^{g(x)},\] where $g(x) $ is a function defined on real numbers and for all real numbers $y,z$ and $0\leq t \leq 1$, it satisfies \[g(ty+(1-t)z) \leq tg(y)+(1-t)g(z).\] (1) Prove: for any integer $n$ and $0 \leq t \leq 1$, we have \[t[g(n)-g(n-1)] \leq g(n+t)-g(n) \leq t[g(n+1)-g(n)].\] (2) Prove that \[\frac{4}{3} \leq f(\frac{1}{2}) \leq \frac{4}{3} \sqrt{2}.\]

2004 AMC 12/AHSME, 3

Tags: modular arithmetic , number theory

For how many ordered pairs of positive integers $ (x,y)$ is $ x \plus{} 2y \equal{} 100$? $ \textbf{(A)}\ 33 \qquad \textbf{(B)}\ 49 \qquad \textbf{(C)}\ 50 \qquad \textbf{(D)}\ 99 \qquad \textbf{(E)}\ 100$

2022 Stanford Mathematics Tournament, 1

Tags:

Compute \[\frac{5+\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{7+\sqrt{12}}{\sqrt{3}+\sqrt{4}}+\dots+\frac{63+\sqrt{992}}{\sqrt{31}+\sqrt{32}}.\]

2016 Dutch IMO TST, 3

Tags: number theory , sum of digits , digit

Let $k$ be a positive integer, and let $s(n)$ denote the sum of the digits of $n$. Show that among the positive integers with $k$ digits, there are as many numbers $n$ satisfying $s(n) < s(2n)$ as there are numbers $n$ satisfying $s(n) > s(2n)$.

2008 iTest Tournament of Champions, 2

Tags:

Find the value of $|xy|$ given that $x$ and $y$ are integers and \[6x^2y^2+5x^2-18y^2=17253.\]

2002 JBMO ShortLists, 13

Tags: inequalities , rectangle , geometry

Let $ A_1,A_2,...,A_{2002}$ be arbitrary points in the plane. Prove that for every circle of radius $ 1$ and for every rectangle inscribed in this circle, there exist $3$ vertices $ M,N,P$ of the rectangle such that $ MA_1 + MA_2 + \cdots + MA_{2002} + $ $NA_1 + NA_2 + \cdots + NA_{2002} + $ $PA_1 + PA_2 + \cdots + PA_{2002}\ge 6006$.

2004 Estonia National Olympiad, 3

Tags: combinatorics , combinatorial geometry

From $25$ points in a plane, both of whose coordinates are integers of the set $\{-2,-1, 0, 1, 2\}$, some $17$ points are marked. Prove that there are three points on one line, one of them is the midpoint of two others.

2005 VTRMC, Problem 3

Tags: recurrence relation , counting , combinatorics

We wish to tile a strip of $n$ $1$-inch by $1$-inch squares. We can use dominos which are made up of two tiles that cover two adjacent squares, or $1$-inch square tiles which cover one square. We may cover each square with one or two tiles and a tile can be above or below a domino on a square, but no part of a domino can be placed on any part of a different domino. We do not distinguish whether a domino is above or below a tile on a given square. Let $t(n)$ denote the number of ways the strip can be tiled according to the above rules. Thus for example, $t(1)=2$ and $t(2)=8$. Find a recurrence relation for $t(n)$, and use it to compute $t(6)$.

2016 Dutch BxMO TST, 3

Tags: geometry , incircle , collinear , right triangle

Let $\vartriangle ABC$ be a right-angled triangle with $\angle A = 90^o$ and circumcircle $\Gamma$. The inscribed circle is tangent to $BC$ in point $D$. Let $E$ be the midpoint of the arc $AB$ of $\Gamma$ not containing $C$ and let $F$ be the midpoint of the arc $AC$ of $\Gamma$ not containing $B$. (a) Prove that $\vartriangle ABC \sim \vartriangle DEF$. (b) Prove that $EF$ goes through the points of tangency of the incircle to $AB$ and $AC$.

2010 Contests, 3

Tags: set , set theory , interval , union , intersection

Let $I_1, I_2, I_3$ be three open intervals of $\mathbb{R}$ such that none is contained in another. If $I_1\cap I_2 \cap I_3$ is non-empty, then show that at least one of these intervals is contained in the union of the other two.

2025 Belarusian National Olympiad, 9.6

Tags: algebra , inequality

Numbers $a,b,c$ are lengths of sides of some triangle. Prove the inequality$$\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c} \geq \frac{a+b}{2c}+\frac{b+c}{2a}+\frac{c+a}{2b}$$ [i]M. Karpuk[/i]

2020 Peru EGMO TST, 5

Tags: angle bisector , geometry

Let $AD$ be the diameter of a circle $\omega$ and $BC$ is a chord of $\omega$ which is perpendicular to $AD$. Let $M,N,P$ be points on the segments $AB,AC,BC$ respectively, such that $MP\parallel AC$ and $PN\parallel AB$. The line $MN$ cuts the line $PD$ in the point $Q$ and the angle bisector of $\angle MPN$ in the point $R$. Prove that the points $B,R,Q,C$ are concyclic.