Olympiad problems

2004 Iran MO (3rd Round), 2

$A$ is a compact convex set in plane. Prove that there exists a point $O \in A$, such that for every line $XX'$ passing through $O$, where $X$ and $X'$ are boundary points of $A$, then \[ \frac12 \leq \frac {OX}{OX'} \leq 2.\]

2015 Vietnam Team selection test, Problem 1

Tags: algebra

Let $\alpha$ be the positive root of the equation $x^2+x=5$. Let $n$ be a positive integer number, and let $c_0,c_1,\ldots,c_n\in \mathbb{N}$ be such that $ c_0+c_1\alpha+c_2\alpha^2+\cdots+c_n\alpha^n=2015. $ a. Prove that $c_0+c_1+c_2+\cdots+c_n\equiv 2 \pmod{3}$. b. Find the minimum value of the sum $c_0+c_1+c_2+\cdots+c_n$.

2015 Putnam, B3

Tags: putnam matrices

Let $S$ be the set of all $2\times 2$ real matrices \[M=\begin{pmatrix}a&b\\c&d\end{pmatrix}\] whose entries $a,b,c,d$ (in that order) form an arithmetic progression. Find all matrices $M$ in $S$ for which there is some integer $k>1$ such that $M^k$ is also in $S.$

2009 239 Open Mathematical Olympiad, 7

Tags: inequalities

In the triangle $ABC$, the cevians $AA_1$, $BB_1$ and $CC_1$ intersect at the point $O$. It turned out that $AA_1$ is the bisector, and the point $O$ is closer to the straight line $AB$ than to the straight lines $A_1C_1$ and $B_1A_1$. Prove that $\angle{BAC} > 120^{\circ}$.

2012 IFYM, Sozopol, 3

Tags: number theory , diophantine equation

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

LMT Guts Rounds, 25-27

Tags:

$25.$ Let $C$ be the answer to Problem $27.$ What is the $C$-th smallest positive integer with exactly four positive factors? $26.$ Let $A$ be the answer to Problem $25.$ Determine the absolute value of the difference between the two positive integer roots of the quadratic equation $x^2-Ax+48=0$ $27.$ Let $B$ be the answer to Problem $26.$ Compute the smallest integer greater than $\frac{B}{\pi}$

2012 Belarus Team Selection Test, 3

Tags: circumcircle , trigonometry , geometric transformation , geometry

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

2008 Romania Team Selection Test, 1

Tags: gauss , geometry

Let $ ABCD$ be a convex quadrilateral and let $ O \in AC \cap BD$, $ P \in AB \cap CD$, $ Q \in BC \cap DA$. If $ R$ is the orthogonal projection of $ O$ on the line $ PQ$ prove that the orthogonal projections of $ R$ on the sidelines of $ ABCD$ are concyclic.

2025 Belarusian National Olympiad, 10.6

Tags: combinatorics

For a sequence of zeros and ones Vasya makes move of the following form until the process doesn't halt 1. If the first digit of the sequence is zero, this digit is erased. 2. If the first digit is one and there are at least two digits, Vasya swaps first two digits, reverses the sequence and replaces ones with zeros and zeros with ones. 3. If the sequence is empty or consists of a single one, the process stops. Find the number of sequences of length 2025 starting from which Vasya can get to the empty sequence. [i]M. Zorka[/i]

MBMT Team Rounds, 2020.12

Tags:

Find the number of ways to partition $S = \{1, 2, 3, \dots, 2020\}$ into two disjoint sets $A$ and $B$ with $A \cup B = S$ so that if you choose an element $a$ from $A$ and an element $b$ from $B$, $a+b$ is never a multiple of $20$. $A$ or $B$ can be the empty set, and the order of $A$ and $B$ doesn't matter. In other words, the pair of sets $(A,B)$ is indistinguishable from the pair of sets $(B,A)$. [i]Proposed by Timothy Qian[/i]

2011 Kazakhstan National Olympiad, 2

Tags: circumcircle , symmetry , parallelogram , geometry

Let $w$-circumcircle of triangle $ABC$ with an obtuse angle $C$ and $C '$symmetric point of point $C$ with respect to $AB$. $M$ midpoint of $AB$. $C'M$ intersects $w$ at $N$ ($C '$ between $M$ and $N$). Let $BC'$ second crossing point $w$ in $F$, and $AC'$ again crosses the $w$ at point $E$. $K$-midpoint $EF$. Prove that the lines $AB, CN$ and$ KC'$are concurrent.

PEN J Problems, 16

Tags: divisor function

We say that an integer $m \ge 1$ is super-abundant if \[\frac{\sigma(m)}{m}>\frac{\sigma(k)}{k}\] for all $k \in \{1, 2,\cdots, m-1 \}$. Prove that there exists an infinite number of super-abundant numbers.

2016 Saudi Arabia IMO TST, 1

Tags: sequence , algebra , recurrence relation

Define the sequence $a_1, a_2,...$ as follows: $a_1 = 1$, and for every $n \ge 2$, $a_n = n - 2$ if $a_{n-1} = 0$ and $a_n = a_{n-1} - 1$, otherwise. Find the number of $1 \le k \le 2016$ such that there are non-negative integers $r, s$ and a positive integer $n$ satisfying $k = r + s$ and $a_{n+r} = a_n + s$.

2020 MBMT, 29

Tags:

The center of circle $\omega_1$ of radius $6$ lies on circle $\omega_2$ of radius $6$. The circles intersect at points $K$ and $W$. Let point $U$ lie on the major arc $\overarc{KW}$ of $\omega_2$, and point $I$ be the center of the largest circle that can be inscribed in $\triangle KWU$. If $KI+WI=11$, find $KI\cdot WI$. [i]Proposed by Bradley Guo[/i]

2013 ELMO Shortlist, 2

Tags: inequalities

Prove that for all positive reals $a,b,c$, \[\frac{1}{a+\frac{1}{b}+1}+\frac{1}{b+\frac{1}{c}+1}+\frac{1}{c+\frac{1}{a}+1}\ge \frac{3}{\sqrt[3]{abc}+\frac{1}{\sqrt[3]{abc}}+1}. \][i]Proposed by David Stoner[/i]

XMO (China) 2-15 - geometry, 6.2

Tags: complex number , algebra , inequalities , geometric inequality

Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let $$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$ (1) If $n = 6063$, find the maximum value of $S$. (2) If $n= 2021$, find the maximum value of $S$.

1976 IMO Longlists, 39

Tags: geometric transformation , homothety , geometry

In $ ABC$, the inscribed circle is tangent to side $BC$ at$ X$. Segment $ AX$ is drawn. Prove that the line joining the midpoint of $ AX$ to the midpoint of side $ BC$ passes through center $ I$ of the inscribed circle.

2017-IMOC, G5

Tags: concyclic , incenter , geometry

We have $\vartriangle ABC$ with $I$ as its incenter. Let $D$ be the intersection of $AI$ and $BC$ and define $E, F$ in a similar way. Furthermore, let $Y = CI \cap DE, Z = BI \cap DF$. Prove that if $\angle BAC = 120^o$, then $E, F, Y,Z$ are concyclic. [img]https://1.bp.blogspot.com/-5IFojUbPE3o/XnSKTlTISqI/AAAAAAAALd0/0OwKMl02KJgqPs-SDOlujdcWXM0cWJiegCK4BGAYYCw/s1600/imoc2017%2Bg5.png[/img]

1998 Gauss, 25

Tags: gauss

Two natural numbers, $p$ and $q$, do not end in zero. The product of any pair, p and q, is a power of 10 (that is, $10, 100, 1000, 10 000$ , ...). If $p >q$, the last digit of $p – q$ cannot be $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 9$

2022 APMO, 2

Tags: geometry , circumcircle

Let $ABC$ be a right triangle with $\angle B=90^{\circ}$. Point $D$ lies on the line $CB$ such that $B$ is between $D$ and $C$. Let $E$ be the midpoint of $AD$ and let $F$ be the seconf intersection point of the circumcircle of $\triangle ACD$ and the circumcircle of $\triangle BDE$. Prove that as $D$ varies, the line $EF$ passes through a fixed point.

2004 Iran MO (3rd Round), 2

2015 Vietnam Team selection test, Problem 1

2015 Putnam, B3

2009 239 Open Mathematical Olympiad, 7

2012 IFYM, Sozopol, 3

LMT Guts Rounds, 25-27

2012 Belarus Team Selection Test, 3

2008 Romania Team Selection Test, 1

2025 Belarusian National Olympiad, 10.6

MBMT Team Rounds, 2020.12

2011 Kazakhstan National Olympiad, 2

PEN J Problems, 16

2016 Saudi Arabia IMO TST, 1

2020 MBMT, 29

2013 ELMO Shortlist, 2

XMO (China) 2-15 - geometry, 6.2

1976 IMO Longlists, 39

2017-IMOC, G5

1998 Gauss, 25

2022 APMO, 2

1995 Moldova Team Selection Test, 4

2014 Harvard-MIT Mathematics Tournament, 7

1989 National High School Mathematics League, 6

2020 Jozsef Wildt International Math Competition, W3

2014 Singapore Senior Math Olympiad, 18