Olympiad problems

2011 All-Russian Olympiad Regional Round, 9.4

$x$, $y$ and $z$ are positive real numbers. Prove the inequality \[\frac{x+1}{y+1}+\frac{y+1}{z+1}+\frac{z+1}{x+1}\leq\frac{x}{y}+\frac{y}{z}+\frac{z}{x}.\] (Authors: A. Khrabrov, B. Trushin)

2011 Middle European Mathematical Olympiad, 5

Tags: symmetry , geometry

Let $ABCDE$ be a convex pentagon with all five sides equal in length. The diagonals $AD$ and $EC$ meet in $S$ with $\angle ASE = 60^\circ$. Prove that $ABCDE$ has a pair of parallel sides.

I Soros Olympiad 1994-95 (Rus + Ukr), 9.8

Tags: algebra , inequalities

Let $f(x) =x^2-2x$. Find all $x$ for which $f(f(x))<3$.

1999 Tournament Of Towns, 2

Tags: ratio , square , geometry , right triangle

$ABC$ is a right-angled triangle. A square $ABDE$ is constructed on the opposite side of the hypothenuse $AB$ from $C$. The bisector of $\angle C$ cuts $DE$ at $F$. If $AC = 1$ and $BC = 3$, compute $\frac{DF}{EF}$. (A Blinkov)

2017 QEDMO 15th, 11

Tags: sum , algebra

Calculate $$\frac{(2^1+3^1)(2^2+3^2)(2^4+3^4)(2^8+3^8)...(2^{2048}+3^{2048})+2^{4096}}{3^{4096}}$$

2016 Hanoi Open Mathematics Competitions, 1

Tags: exponential equation , algebra

If $2016 = 2^5 + 2^6 + ...+ 2^m$ then $m$ is equal to (A): $8$ (B): $9$ (C): $10$ (D): $11$ (E): None of the above.

2003 National Olympiad First Round, 13

Tags: geometry , angle bisector

Let $ABC$ be a triangle such that $|AB|=8$ and $|AC|=2|BC|$. What is the largest value of altitude from side $[AB]$? $ \textbf{(A)}\ 3\sqrt 2 \qquad\textbf{(B)}\ 3\sqrt 3 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ \dfrac {16}3 \qquad\textbf{(E)}\ 6 $

2009 Postal Coaching, 1

Tags: sequence , power of 2 , recurrence relation , algebra

Let $a_1, a_2, a_3, . . . , a_n, . . . $ be an infinite sequence of natural numbers in which $a_1$ is not divisible by $5$. Suppose $a_{n+1} = a_n + b_n$ where bn is the last digit of $a_n$, for every $n$. Prove that the sequence $\{a_n\}$ contains infinitely many powers of 2.

2013 Saudi Arabia BMO TST, 1

Tags: lattice , combinatorial geometry , combinatorics

The set $G$ is defined by the points $(x,y)$ with integer coordinates, $1 \le x \le 5$ and $1 \le y \le 5$. Determine the number of five-point sequences $(P_1, P_2, P_3, P_4, P_5)$ such that for $1 \le i \le 5$, $P_i = (x_i,i)$ is in $G$ and $|x_1 - x_2| = |x_2 - x_3| = |x_3 - x_4|=|x_4 - x_5| = 1$.

2004 Germany Team Selection Test, 1

Tags: vector , geometric transformation , similar triangles , geometry

Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$. Show that the points $B$, $X$, $H$, $Y$ lie on one circle.

2022/2023 Tournament of Towns, P5

Tags: number theory , sum of digits

In an infinite arithmetic progression of positive integers there are two integers with the same sum of digits. Will there necessarily be one more integer in the progression with the same sum of digits? [i]Proposed by A. Shapovalov[/i]

2023 Malaysian Squad Selection Test, 7

Tags: number theory

Find all polynomials with integer coefficients $P$ such that for all positive integers $n$, the sequence $$0, P(0), P(P(0)), \cdots$$ is eventually constant modulo $n$. [i]Proposed by Ivan Chan Kai Chin[/i]

1984 IMO Longlists, 23

Tags: combinatorics

A $2\times 2\times 12$ box fixed in space is to be filled with twenty-four $1 \times 1 \times 2$ bricks. In how many ways can this be done?

2023 Korea Summer Program Practice Test, P3

Tags: geometry , incenter , circumcircle

$\triangle ABC$ is a triangle such that $\angle A = 60^{\circ}$. The incenter of $\triangle ABC$ is $I$. $AI$ intersects with $BC$ at $D$, $BI$ intersects with $CA$ at $E$, and $CI$ intersects with $AB$ at $F$, respectively. Also, the circumcircle of $\triangle DEF$ is $\omega$. The tangential line of $\omega$ at $E$ and $F$ intersects at $T$. Show that $\angle BTC \ge 60^{\circ}$

2000 Harvard-MIT Mathematics Tournament, 4

Tags: probability

Five positive integers from $1$ to $15$ are chosen without replacement. What is the probability that their sum is divisible by $3$?

PEN E Problems, 6

Tags: polynomial , algebra

Find a factor of $2^{33}-2^{19}-2^{17}-1$ that lies between $1000$ and $5000$.

2015 Swedish Mathematical Competition, 4

Tags: algebra , system of equations , system , logarithm

Solve the system of equations $$ \left\{\begin{array}{l} x \log x+y \log y+z \log x=0\\ \\ \dfrac{\log x}{x}+\dfrac{\log y}{y}+\dfrac{\log z}{z}=0 \end{array} \right. $$

2012 Grand Duchy of Lithuania, 4

Tags: algebra , sequence

Let $m$ be a positive integer. Find all bounded sequences of integers $a_1, a_2, a_3,... $for which $a_n + a_{n+1} + a_{n+m }= 0$ for all $n \in N$.

2014 Contests, 1

Tags:

Compute $1+2\cdot3^4$. [i]Proposed by Evan Chen[/i]

Ukraine Correspondence MO - geometry, 2020.8

Tags: geometry , perpendicular

Let $ABC$ be an acute triangle, $D$ be the midpoint of $BC$. Bisectors of angles $ADB$ and $ADC$ intersect the circles circumscribed around the triangles $ADB$ and $ADC$ at points $E$ and $F$, respectively. Prove that $EF\perp AD$.

2022 JHMT HS, 1

Tags: calculus , derivative , integration

Compute the value of \[ \frac{d}{dx}\int_{1}^{10} x^3\,dx. \]

2017 BMT Spring, 6

Tags: geometry

Given a cube with side length $ 1$, we perform six cuts as follows: one cut parallel to the $xy$-plane, two cuts parallel to the $yz$-plane, and three cuts parallel to the $xz$-plane, where the cuts are made uniformly independent of each other. What is the expected value of the volume of the largest piece?

2008 Tournament Of Towns, 3

Tags: geometry , circumcircle , equal segments , right triangle

In triangle $ABC, \angle A = 90^o$. $M$ is the midpoint of $BC$ and $H$ is the foot of the altitude from $A$ to $BC$. The line passing through $M$ and perpendicular to $AC$ meets the circumcircle of triangle $AMC$ again at $P$. If $BP$ intersects $AH$ at $K$, prove that $AK = KH$.

Kyiv City MO Juniors Round2 2010+ geometry, 2018.8.3

Tags: equal segments , angle , geometry

In the triangle $ABC$ it is known that $\angle ACB> 90 {} ^ \circ$, $\angle CBA> 45 {} ^ \circ$. On the sides $AC$ and $AB$, respectively, there are points $P$ and $T$ such that $ABC$ and $PT = BC$. The points ${{P} _ {1}}$ and ${{T} _ {1}}$ on the sides $AC$ and $AB$ are such that $AP = C {{P} _ {1}}$ and $AT = B {{T} _ {1}}$. Prove that $\angle CBA- \angle {{P} _ {1}} {{T} _ {1}} A = 45 {} ^ \circ$. (Anton Trygub)

2003 Belarusian National Olympiad, 8

Tags: area , convex , pentagon , geometry theorems

Given a convex pentagon $ABCDE$ with $AB=BC, CD=DE, \angle ABC=150^o, \angle CDE=30^o, BD=2$. Find the area of $ABCDE$. (I.Voronovich)