Olympiad problems

2006 Junior Balkan Team Selection Tests - Moldova, 3

Tags: geometry

The convex polygon $A_{1}A_{2}\ldots A_{2006}$ has opposite sides parallel $(A_{1}A_{2}||A_{1004}A_{1005}, \ldots)$. Prove that the diagonals $A_{1}A_{1004}, A_{2}A_{1005}, \ldots A_{1003}A_{2006}$ are concurrent if and only if opposite sides are equal.

2012 District Olympiad, 3

Tags: complex number , sequence , trigonometry , algebra

Let be a sequence of natural numbers $ \left( a_n \right)_{n\ge 1} $ such that $ a_n\le n $ for all natural numbers $ n, $ and $$ \sum_{j=1}^{k-1} \cos \frac{\pi a_j}{k} =0, $$ for all natural $ k\ge 2. $ [b]a)[/b] Find $ a_2. $ [b]b)[/b] Determine this sequence.

2005 USA Team Selection Test, 3

Tags: euler , percent , polynomial , probability , algebra , induction

We choose random a unitary polynomial of degree $n$ and coefficients in the set $1,2,...,n!$. Prove that the probability for this polynomial to be special is between $0.71$ and $0.75$, where a polynomial $g$ is called special if for every $k>1$ in the sequence $f(1), f(2), f(3),...$ there are infinitely many numbers relatively prime with $k$.

2019 Taiwan APMO Preliminary Test, P3

Tags: geometry inequality , triangle inequality , algebra

Let $\triangle ABC$ be an acute triangle, $H$ is its orthocenter. $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$ intersect $\triangle ABC$'s circumcircle at $A',B',C'$ respectively. Find the range (minimum value and the maximum upper bound) of $$\dfrac{AH}{AA'}+\dfrac{BH}{BB'}+\dfrac{CH}{CC'}$$

2012 Sharygin Geometry Olympiad, 16

Tags: geometry , circumcircle

Given right-angled triangle $ABC$ with hypothenuse $AB$. Let $M$ be the midpoint of $AB$ and $O$ be the center of circumcircle $\omega$ of triangle $CMB$. Line $AC$ meets $\omega$ for the second time in point $K$. Segment $KO$ meets the circumcircle of triangle $ABC$ in point $L$. Prove that segments $AL$ and $KM$ meet on the circumcircle of triangle $ACM$.

2013 Hanoi Open Mathematics Competitions, 11

Tags: minimum , maximum , polynomial

The positive numbers $a, b,c, d, p, q$ are such that $(x+a)(x+b)(x+c)(x+d) = x^4+4px^3+6x^2+4qx+1$ holds for all real numbers $x$. Find the smallest value of $p$ or the largest value of $q$.

2012 IMAC Arhimede, 6

Tags: algebra , inequalities , three variable inequality

Let $a,b,c$ be positive real numbers that satisfy the condition $a + b + c = 1$. Prove the inequality $$\frac{a^{-3}+b}{1-a}+\frac{b^{-3}+c}{1-b}+\frac{c^{-3}+a}{1-c}\ge 123$$

2007 Argentina National Olympiad, 5

Tags: sum of digits , number theory

We will say that a positive integer is [i]lucky [/i ]if the sum of its digits is divisible by $31$. What is the maximum possible difference between two consecutive [i]lucky [/i ] numbers?

2018 Dutch IMO TST, 3

Tags: right angle , geometry , equal segments

Let $ABC$ be an acute triangle, and let $D$ be the foot of the altitude through $A$. On $AD$, there are distinct points $E$ and $F$ such that $|AE| = |BE|$ and $|AF| =|CF|$. A point$ T \ne D$ satises $\angle BTE = \angle CTF = 90^o$. Show that $|TA|^2 =|TB| \cdot |TC|$.

2015 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: set , combinatorics

It is given set $A=\{1,2,3,...,2n-1\}$. From set $A$, at least $n-1$ numbers are expelled such that: $a)$ if number $a \in A$ is expelled, and if $2a \in A$ then $2a$ must be expelled $b)$ if $a,b \in A$ are expelled, and $a+b \in A$ then $a+b$ must be also expelled Which numbers must be expelled such that sum of numbers remaining in set stays minimal

Denmark (Mohr) - geometry, 1992.4

Tags: geometric inequality , median , geometry

Let $a, b$ and $c$ denote the side lengths and $m_a, m_b$ and $m_c$ of the median's lengths in an arbitrary triangle. Show that $$\frac34 < \frac{m_a + m_b + m_c}{a + b + c}<1$$ Also show that there is no narrower range that for each triangle that contains the fraction $$\frac{m_a + m_b + m_c}{a + b + c}$$

2008 India Regional Mathematical Olympiad, 4

Tags: divisibility , natural number , number theory

Determine all the natural numbers $n$ such that $21$ divides $2^{2^{n}}+2^n+1.$

1965 AMC 12/AHSME, 37

Tags: similar triangles , ratio , geometry

Point $ E$ is selected on side $ AB$ of triangle $ ABC$ in such a way that $ AE: EB \equal{} 1: 3$ and point $ D$ is selected on side $ BC$ such that $ CD: DB \equal{} 1: 2$. The point of intersection of $ AD$ and $ CE$ is $ F$. Then $ \frac {EF}{FC} \plus{} \frac {AF}{FD}$ is: $ \textbf{(A)}\ \frac {4}{5} \qquad \textbf{(B)}\ \frac {5}{4} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \frac {5}{2}$

2008 Spain Mathematical Olympiad, 2

Tags: trigonometry , inequalities

Let $a$ and $b$ be two real numbers, with $0<a,b<1$. Prove that \[\sqrt{ab^2+a^2b}+\sqrt{(1-a)(1-b)^2+(1-a)^2(1-b)}<\sqrt{2}\]

2016 Korea Winter Program Practice Test, 1

Tags: geometry , tangent circle

There is circle $\omega$ and $A, B$ on it. Circle $\gamma_1$ tangent to $\omega$ on $T$ and $AB$ on $D$. Circle $\gamma_2$ tangent to $\omega$ on $S$ and $AB$ on $E$. (like the figure below) Let $AB\cap TS=C$. Prove that $CA=CB$ iff $CD=CE$

1984 Iran MO (2nd round), 1

Tags: domain , limit , floor function , function , algebra

Let $f$ and $g$ be two functions such that \[f(x)=\frac{1}{\lfloor | x | \rfloor}, \quad g(x)=\frac{1}{|\lfloor x \rfloor |}.\] Find the domains of $f$ and $g$ and then prove that \[\lim_{x \to -1^+} f(x)= \lim_{x \to 1^- } g(x).\]

1965 AMC 12/AHSME, 28

Tags:

An escalator (moving staircase) of $ n$ uniform steps visible at all times descends at constant speed. Two boys, $ A$ and $ Z$, walk down the escalator steadily as it moves, $ A$ negotiating twice as many escalator steps per minute as $ Z$. $ A$ reaches the bottom after taking $ 27$ steps while $ Z$ reaches the bottom after taking $ 18$ steps. Then $ n$ is: $ \textbf{(A)}\ 63 \qquad \textbf{(B)}\ 54 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ 36 \qquad \textbf{(E)}\ 30$

2018 Moscow Mathematical Olympiad, 3

Tags: number theory

$a_1,a_2,...,a_k$ are positive integers and $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}>1$. Prove that equation $$[\frac{n}{a_1}]+[\frac{n}{a_2}]+...+[\frac{n}{a_k}]=n$$ has no more than $a_1*a_2*...*a_k$ postivie integer solutions in $n$.

2020 BMT Fall, 9

Tags: algebra , diophantine equation

There is a unique triple $(a,b,c)$ of two-digit positive integers $a,\,b,$ and $c$ that satisfy the equation $$a^3+3b^3+9c^3=9abc+1.$$ Compute $a+b+c$.

2006 All-Russian Olympiad Regional Round, 8.3

Tags: combinatorics

Four drivers took part in the round-robin racing. Their cars started simultaneously from one point and moved at constant speeds. It is known that after the start of the race, for any three cars there was a moment when they met. Prove that after the start of the race there will be a moment when all 4 cars meet. (We consider races to be infinitely long in time.)

1989 Federal Competition For Advanced Students, 3

Tags: inequalities

Let $ a$ be a real number. Prove that if the equation $ x^2\minus{}ax\plus{}a\equal{}0$ has two real roots $ x_1$ and $ x_2$, then: $ x_1^2\plus{}x_2^2 \ge 2(x_1\plus{}x_2).$

2013 India IMO Training Camp, 3

Tags: circumcircle , geometry , perpendicular bisector , angle bisector

In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.

1993 Polish MO Finals, 3

Tags: 3d geometry , geometry , circumcircle , tetrahedron

Find out whether it is possible to determine the volume of a tetrahedron knowing the areas of its faces and its circumradius.

1968 IMO Shortlist, 19

Tags: approximation , circles , combinatorics , algebra

We are given a fixed point on the circle of radius $1$, and going from this point along the circumference in the positive direction on curved distances $0, 1, 2, \ldots $ from it we obtain points with abscisas $n = 0, 1, 2, .\ldots$ respectively. How many points among them should we take to ensure that some two of them are less than the distance $\frac 15$ apart ?

2022 IMAR Test, 3

Tags: parallelogram , geometry

Given is a parallelogram $XYZT$, and the variable points $A, B, C, D$ lie on the sides $XY, XT, TZ, ZY$ respectively, so that $ABCD$ is cyclic with circumcenter $O$, $AC \parallel XT$, and $BD \parallel XY$. Let $P$ be the intersection point of the lines $AD$ and $BC$, and let $Q$ be the intersection of the lines $AB$ and $CD$. Prove that the circle $(POQ)$ passes through a fixed point.