Olympiad problems

2024 Belarusian National Olympiad, 9.3

Tags: geometry

On the side $AC$ of triangle $ABC$ point $D$ is chosen. The perpendicular bisector of segment $BD$ intersects the circumcircle $\Omega$ of triangle $ABC$ at $P$, $Q$. Point $E$ lies on the arc $AC$ of circle $\Omega$, that doesn't contain point $B$, such that $\angle ABD=\angle CBE$. Prove that the orthocenter of the triangle $PQE$ lies on the line $AC$ [i]M. Zorka[/i]

1959 AMC 12/AHSME, 42

Tags: greatest common divisor , relatively prime , least common multiple , number theory

Given three positive integers $a,b,$ and $c$. Their greatest common divisor is $D$; their least common multiple is $m$. Then, which two of the following statements are true? $ \text{(1)}\ \text{the product MD cannot be less than abc} \qquad$ $\text{(2)}\ \text{the product MD cannot be greater than abc}\qquad$ $\text{(3)}\ \text{MD equals abc if and only if a,b,c are each prime}\qquad$ $\text{(4)}\ \text{MD equals abc if and only if a,b,c are each relatively prime in pairs}$ $\text{ (This means: no two have a common factor greater than 1.)}$ $ \textbf{(A)}\ 1,2 \qquad\textbf{(B)}\ 1,3\qquad\textbf{(C)}\ 1,4\qquad\textbf{(D)}\ 2,3\qquad\textbf{(E)}\ 2,4 $

2023 Estonia Team Selection Test, 4

Tags: geometry

A convex quadrilateral $ABCD$ has $\angle BAC = \angle ADC$. Let $M{}$ be the midpoint of the diagonal $AC$. The side $AD$ contains a point $E$ such that $ABME$ is a parallelogram. Let $N{}$ be the midpoint of the line segment $AE{}$. Prove that the line $AC$ touches the circumcircle of the triangle $DMN$ at point $M{}$.

2011 Princeton University Math Competition, B1

Tags: combinatorial geometry

If the plane is partitioned into a grid of congruent equilateral triangles, prove that there does not exist a square with vertices at the vertices of this grid.

2012 Centers of Excellency of Suceava, 3

Tags: real analysis , factorial , series , definite integral , integral sequence

Consider the sequence $ \left( I_n \right)_{n\ge 1} , $ where $ I_n=\int_0^{\pi/4} e^{\sin x\cos x} (\cos x-\sin x)^{2n} (\cos x+\sin x )dx, $ for any natural number $ n. $ [b]a)[/b] Find a relation between any two consecutive terms of $ I_n. $ [b]b)[/b] Calculate $ \lim_{n\to\infty } nI_n. $ [i]c)[/i] Show that $ \sum_{i=1}^{\infty }\frac{1}{(2i-1)!!} =\int_0^{\pi/4} e^{\sin x\cos x} (\cos x+\sin x )dx. $ [i]Cătălin Țigăeru[/i]

1996 Argentina National Olympiad, 1

Tags: algebra

$100$ numbers were written around a circle. The sum of the $100$ numbers is equal to $100$ and the sum of six consecutive numbers is always less than or equal to $6$. The first number is $6$. Find all the numbers.

2007 Germany Team Selection Test, 2

Tags: graph theory , combinatorics , extremal combinatorics , tournament graphs

An $ (n, k) \minus{}$ tournament is a contest with $ n$ players held in $ k$ rounds such that: $ (i)$ Each player plays in each round, and every two players meet at most once. $ (ii)$ If player $ A$ meets player $ B$ in round $ i$, player $ C$ meets player $ D$ in round $ i$, and player $ A$ meets player $ C$ in round $ j$, then player $ B$ meets player $ D$ in round $ j$. Determine all pairs $ (n, k)$ for which there exists an $ (n, k) \minus{}$ tournament. [i]Proposed by Carlos di Fiore, Argentina[/i]

1982 Spain Mathematical Olympiad, 5

Tags: square , construction , geometry

Construct a square knowing the sum of the diagonal and the side.

2024 Indonesia TST, 1

Tags: geometry

Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.

2009 All-Russian Olympiad Regional Round, 10.4

Tags: collinear , geometry

Circles $\omega_1$ and $\omega_2$ touch externally at the point $O$. Points $A$ and $B$ on the circle $\omega_1$ and points $C$ and $D$ on the circle $\omega_2$ are such that $AC$ and $BD$ are common external tangents to circles. Line $AO$ intersects segment $CD$ at point $M$ and straight line $CO$ intersexts $\omega_1$ again at point $N$. Prove that the points $B$, $M$ and $N$ lie on the same straight line.

1994 Irish Math Olympiad, 2

Tags: algebra , polynomial

Let $ p,q,r$ be distinct real numbers that satisfy: $ q\equal{}p(4\minus{}p), \, r\equal{}q(4\minus{}q), \, p\equal{}r(4\minus{}r).$ Find all possible values of $ p\plus{}q\plus{}r$.

1997 Abels Math Contest (Norwegian MO), 3a

Tags: sum , combinatorics , algebra

Each subset of $97$ out of $1997$ given real numbers has positive sum. Show that the sum of all the $1997$ numbers is positive.

2014 India National Olympiad, 4

Tags: calculus , quadratic , analytic geometry , algebra , probability , polynomial , integration

Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

2008 Bosnia Herzegovina Team Selection Test, 1

Tags: derivative , geometry , trigonometry , inequalities , calculus

Prove that in an isosceles triangle $ \triangle ABC$ with $ AC\equal{}BC\equal{}b$ following inequality holds $ b> \pi r$, where $ r$ is inradius.

2008 Canada National Olympiad, 2

Tags: algebra , function , functional equation

Determine all functions $ f$ defined on the set of rational numbers that take rational values for which \[ f(2f(x) \plus{} f(y)) \equal{} 2x \plus{} y, \] for each $ x$ and $ y$.

2017 CMIMC Number Theory, 10

Tags: factorial , greatest common divisor , number theory

For each positive integer $n$, define \[g(n) = \gcd\left\{0! n!, 1! (n-1)!, 2 (n-2)!, \ldots, k!(n-k)!, \ldots, n! 0!\right\}.\] Find the sum of all $n \leq 25$ for which $g(n) = g(n+1)$.

2011 239 Open Mathematical Olympiad, 4

Tags: geometry

In convex quadrilateral $ABCD$, where $AB=AD$, on $BD$ point $K$ is chosen. On $KC$ point $L$ is such that $\bigtriangleup BAD \sim \bigtriangleup BKL$. Line parallel to $DL$ and passes through $K$, intersect $CD$ at $P$. Prove that $\angle APK = \angle LBC$. @below edited

2024 BMT, 7

Tags: geometry

In parallelogram $ABCD,$ $E$ is a point on $\overline{AD}$ such that $\overline{CE} \perp \overline{AD},$ $F$ is a point on $\overline{CD}$ such that $\overline{AF} \perp \overline{CD},$ and $G$ is a point on $\overline{BC}$ such that $\overline{AG} \perp \overline{BC}.$ Let $H$ be a point on $\overline{GF}$ such that $\overline{AH} \perp \overline{GF},$ and let $J$ be the intersection of lines $EF$ and $BC.$ Given that $AH=8, AE=6,$ and $EF=4,$ compute $CJ.$

1996 Tuymaada Olympiad, 2

Tags: algebra , set , real , set theory

Given a finite set of real numbers $A$, not containing $0$ and $1$ and possessing the property: if the number a belongs to $A$, then numbers $\frac{1}{a}$ and $1-a$ also belong to $A$. How many numbers are in the set $A$?

1989 Greece National Olympiad, 1

Tags: algebra , system of equations

Let $a,b,c,d x,y,z, w$ be real numbers such that $$\begin{matrix} ax -by-c z-dw =0\\ b x +a y -d z +cw=0\\ c x+ d y +a z -b w=0\\ dx-c y+bz+aw=0 \end{matrix}$$ prove that $$a=b=c=d=0, \ \ or \ \ x=y=z=w=0$$

2021 Nordic, 1

Tags: prime , number theory

On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.

III Soros Olympiad 1996 - 97 (Russia), 11.3

Tags: algebra , system of equations

Find the greatest $a$ for which there is $b$ such that the system $$\begin{cases} y=x^4+a \\ x=\dfrac{1}{y^4}+b \end{cases}$$ has exactly two solutions.

MOAA Team Rounds, 2018.2

Tags: team , algebra

If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.

2016 Nigerian Senior MO Round 2, Problem 9

Tags: geometry , parallelogram , ratio

$ABCD$ is a parallelogram, line $DF$ is drawn bisecting $BC$ at $E$ and meeting $AB$ (extended) at $F$ from vertex $C$. Line $CH$ is drawn bisecting side $AD$ at $G$ and meeting $AB$ (extended) at $H$. Lines $DF$ and $CH$ intersect at $I$. If the area of parallelogram $ABCD$ is $x$, find the area of triangle $HFI$ in terms of $x$.

2014 Contests, 4

Tags: binomial coefficient , combinatorics

Square $ABCD$ is divided into $n^2$ equal small squares by lines parallel to its sides.A spider starts from $A$ and moving only rightward or upwards,tries to reach $C$.Every "movement" of the spider consists of $k$ steps rightward and $m$ steps upwards or $m$ steps rightward and $k$ steps upwards(it can follow any possible order for the steps of each "movement").The spider completes $l$ "movements" and afterwards it moves without limitation (it still moves rightwards and upwards only).If $n=m\cdot l$,find the number of the possible paths the spider can follow to reach $C$.Note that $n,m,k,l\in \mathbb{N^{*}}$ with $k<m$.