Olympiad problems

2014 IMO Shortlist, G3

Let $\Omega$ and $O$ be the circumcircle and the circumcentre of an acute-angled triangle $ABC$ with $AB > BC$. The angle bisector of $\angle ABC$ intersects $\Omega$ at $M \ne B$. Let $\Gamma$ be the circle with diameter $BM$. The angle bisectors of $\angle AOB$ and $\angle BOC$ intersect $\Gamma$ at points $P$ and $Q,$ respectively. The point $R$ is chosen on the line $P Q$ so that $BR = MR$. Prove that $BR\parallel AC$. (Here we always assume that an angle bisector is a ray.) [i]Proposed by Sergey Berlov, Russia[/i]

2014 India PRMO, 19

Tags: sum , algebra

Let $x_1,x_2,... ,x_{2014}$ be real numbers different from $1$, such that $x_1 + x_2 +...+x_{2014} = 1$ and $\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}+...+\frac{x_{2014}}{1-x_{2014}}=1$ What is the value of $\frac{x^2_1}{1-x_1}+\frac{x^2_2}{1-x_2}+...+\frac{x^2_{2014}}{1-x_{2014}}$ ?

2007 Nicolae Păun, 2

Tags: function , linear algebra , commutative algebra , matrix

For a given natural number, $ n\ge 2, $ consider two matrices $ A,B\in\mathcal{M}_n(\mathbb{C}) $ that commute and such that $ A $ is invertible and that the function $ M:\mathbb{C}\longrightarrow\mathbb{C} ,M(x)=\det (A+xB) $ is bounded above or below. Prove that $ B^n=0. $ [i]Sorin Rădulescu[/i] and [i]Ion Savu[/i]

OMMC POTM, 2024 7

Tags: concyclic , geometric inequality , geometry

Let $A$ and $B$ be two points on the same line $\ell$. If the points $P$ and $Q$ are two points $X$ on $\ell$ that mazimize and minimize the ratio $\frac{AX}{BX}$ respectively, prove that $A,B,P$ and $Q$ are concyclic.

2022 Purple Comet Problems, 15

Tags: trigonometry , algebra

Let $a$ be a real number such that $$5 \sin^4 \left( \frac{a}{2} \right)+ 12 \cos a = 5 cos^4 \left( \frac{a}{2} \right)+ 12 \sin a.$$ There are relatively prime positive integers $m$ and $n$ such that $\tan a = \frac{m}{n}$ . Find $10m + n$.

1996 Korea National Olympiad, 1

Tags: combinatorics , pigeonhole principle

If you draw $4$ points on the unit circle, prove that you can always find two points where their distance between is less than $\sqrt{2}.$

2024 CCA Math Bonanza, L3.1

Tags: probability

Byan rolls a $12$-sided die, a $14$-sided die, a $20$-sided die, and a $24$-sided die. The probability the sum of the numbers the die landed on is divisible by $7$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Lightning 3.1[/i]

2025 All-Russian Olympiad, 9.2

Tags: geometry , circumcircle

The diagonals of a convex quadrilateral $ABCD$ intersect at point $E$. The points of tangency of the circumcircles of triangles $ABE$ and $CDE$ with their common external tangents lie on a circle $\omega$. The points of tangency of the circumcircles of triangles $ADE$ and $BCE$ with their common external tangents lie on a circle $\gamma$. Prove that the centers of circles $\omega$ and $\gamma$ coincide.

2013 Junior Balkan Team Selection Tests - Romania, 1

Tags: inequalities , algebra

If $a, b, c > 0$ satisfy $a + b + c = 3$, then prove that $$\frac{a^2(b + 1)}{ ab + a + b} + \frac{b^2(c + 1)}{ bc + b + c} + \frac{c^2(a + 1)}{ ca + c + a} \ge 2$$ Mathematical Excalibur P322/Vol.14, no.2

2013 Stanford Mathematics Tournament, 1

Tags: probability

Andrew flips a fair coin $5$ times, and counts the number of heads that appear. Beth flips a fair coin $6$ times and also counts the number of heads that appear. Compute the probability Andrew counts at least as many heads as Beth.

2010 Poland - Second Round, 3

Tags: inequalities , prime number , number theory

Positive integer numbers $k$ and $n$ satisfy the inequality $k > n!$. Prove that there exist pairwisely different prime numbers $p_1, p_2, \ldots, p_n$ which are divisors of the numbers $k+1, k+2, \ldots, k+n$ respectively (i.e. $p_i|k+i$).

1992 IMO Shortlist, 2

Tags: quadratic , algebra , functional equation , recurrence relation

Let $ \mathbb{R}^\plus{}$ be the set of all non-negative real numbers. Given two positive real numbers $ a$ and $ b,$ suppose that a mapping $ f: \mathbb{R}^\plus{} \mapsto \mathbb{R}^\plus{}$ satisfies the functional equation: \[ f(f(x)) \plus{} af(x) \equal{} b(a \plus{} b)x.\] Prove that there exists a unique solution of this equation.

2018 Malaysia National Olympiad, A2

Tags: digit , number theory

An integer has $2018$ digits and is divisible by $7$. The first digit is $d$, while all the other digits are $2$. What is the value of $d$?

2018 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , perfect square

Prove that a positive integer $A$ is a perfect square if and only if, for all positive integers $n$, at least one of the numbers $(A + 1)^2 - A, (A + 2)^2 - A, (A + 3)^2 - A,.., (A + n)^2- A$ is a multiple of $n$.

2010 Denmark MO - Mohr Contest, 5

Tags: trisector , geometry , semicircle , equilateral

An equilateral triangle $ABC$ is given. With $BC$ as diameter, a semicircle is drawn outside the triangle. On the semicircle, points $D$ and $E$ are chosen such that the arc lengths $BD, DE$ and $EC$ are equal. Prove that the line segments $AD$ and $AE$ divide the side $BC$ into three equal parts. [img]https://1.bp.blogspot.com/-hQQV-Of96Ls/XzXCZjCledI/AAAAAAAAMV0/SwXa4mtEEm04onYbFGZiTc5NSpkoyvJLwCLcBGAsYHQ/s0/2010%2BMohr%2Bp5.png[/img]

2001 BAMO, 2

Tags: collinear , geometry , perpendicular

Let $JHIZ$ be a rectangle, and let $A$ and $C$ be points on sides $ZI$ and $ZJ,$ respectively. The perpendicular from $A$ to $CH$ intersects line $HI$ in $X$ and the perpendicular from $C$ to $AH$ intersects line $HJ$ in $Y.$ Prove that $X,$ $Y,$ and $Z$ are collinear (lie on the same line).

2002 Junior Balkan Team Selection Tests - Moldova, 10

Tags: geometry , equal angles , circles , angle

The circles $C_1$ and $C_2$ intersect at the distinct points $M$ and $N$. Points $A$ and $B$ belong respectively to the circles $C_1$ and $C_2$ so that the chords $[MA]$ and $[MB]$ are tangent at point $M$ to the circles $C_2$ and $C_1$, respectively. To prove it that the angles $\angle MNA$ and $\angle MNB$ are equal.

2015 USAJMO, 2

Tags: diophantine equation , algebra , number theory

Solve in integers the equation \[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]

2013 Indonesia MO, 6

Tags: quadratic , number theory

A positive integer $n$ is called "strong" if there exists a positive integer $x$ such that $x^{nx} + 1$ is divisible by $2^n$. a. Prove that $2013$ is strong. b. If $m$ is strong, determine the smallest $y$ (in terms of $m$) such that $y^{my} + 1$ is divisible by $2^m$.

2021-IMOC, G1

Tags: geometry , altitude , antipode , cyclic quadrilateral

Let $\overline{BE}$ and $\overline{CF}$ be altitudes of triangle $ABC$, and let $D$ be the antipodal point of $A$ on the circumcircle of $ABC$. The lines $\overleftrightarrow{DE}$ and $\overleftrightarrow{DF}$ intersect $\odot(ABC)$ again at $Y$ and $Z$, respectively. Show that $\overleftrightarrow{YZ}$, $\overleftrightarrow{EF}$ and $\overleftrightarrow{BC}$ intersect at a point.

1996 IMO Shortlist, 9

Tags: floor function , modular arithmetic , algebra , sequence , recurrence relation

Let the sequence $ a(n), n \equal{} 1,2,3, \ldots$ be generated as follows with $ a(1) \equal{} 0,$ and for $ n > 1:$ \[ a(n) \equal{} a\left( \left \lfloor \frac{n}{2} \right \rfloor \right) \plus{} (\minus{}1)^{\frac{n(n\plus{}1)}{2}}.\] 1.) Determine the maximum and minimum value of $ a(n)$ over $ n \leq 1996$ and find all $ n \leq 1996$ for which these extreme values are attained. 2.) How many terms $ a(n), n \leq 1996,$ are equal to 0?

2005 Today's Calculation Of Integral, 50

Tags: calculus , integration , limit , polynomial , derivative , logarithm , algebra

Let $a,b$ be real numbers such that $a<b$. Evaluate \[\lim_{b\rightarrow a} \frac{\displaystyle\int_a^b \ln |1+(x-a)(b-x)|dx}{(b-a)^3}\].

2005 Taiwan TST Round 1, 3

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

Find all positive integer triples $(x,y,z)$ such that $x<y<z$, $\gcd (x,y)=6$, $\gcd (y,z)=10$, $\gcd (x,z)=8$, and lcm$(x,y,z)=2400$. Note that the problems of the TST are not arranged in difficulty (Problem 1 of day 1 was probably the most difficult!)

2012 JHMT, 4

Tags: geometry

Circle $O$ has radius $18$. From diameter $AB$, there exists a point $C$ such that $BC$ is tangent to $O$ and $AC$ intersects $O$ at a point $D$, with $AD = 24$. What is the length of $BC$?

1982 Czech and Slovak Olympiad III A, 2

Tags: algebra , inequalities

Given real numbers $x_1$, $x_2$, $x_3$, $x_4$, $x_5$, $x_6$. Let $M$ denote the maximum of their absolute values. Prove that it is valid $$ | x_1x_4-x_1x_5 +x_2x_5 -x_2x_6+x_3x_6-x_3x_4| \le 4M^2$$