Olympiad problems

1977 AMC 12/AHSME, 16

Tags: trigonometry , function

If $i^2 = -1$, then the sum \[ \cos{45^\circ} + i\cos{135^\circ} + \cdots + i^n\cos{(45 + 90n)^\circ} \] \[ + \cdots + i^{40}\cos{3645^\circ} \] equals \[ \text{(A)}\ \frac{\sqrt{2}}{2} \qquad \text{(B)}\ -10i\sqrt{2} \qquad \text{(C)}\ \frac{21\sqrt{2}}{2} \] \[ \text{(D)}\ \frac{\sqrt{2}}{2}(21 - 20i) \qquad \text{(E)}\ \frac{\sqrt{2}}{2}(21 + 20i) \]

2011 Saudi Arabia Pre-TST, 3.1

Tags: number theory , divide , divisible

Let $n$ be a positive integer such that $2011^{2011}$ divides $n!$. Prove that $2011^{2012} $divides $n!$ .

2014 South East Mathematical Olympiad, 4

Tags: inequalities

Let $x_1,x_2,\cdots,x_n$ be non-negative real numbers such that $x_ix_j\le 4^{-|i-j|}$ $(1\le i,j\le n)$. Prove that\[x_1+x_2+\cdots+x_n\le \frac{5}{3}.\]

2009 Ukraine National Mathematical Olympiad, 3

Tags: circumcircle , parallelogram , cyclic quadrilateral , geometric transformation , geometry

In triangle $ABC$ points $M, N$ are midpoints of $BC, CA$ respectively. Point $P$ is inside $ABC$ such that $\angle BAP = \angle PCA = \angle MAC .$ Prove that $\angle PNA = \angle AMB .$

2012 Romanian Masters In Mathematics, 3

Tags: function , algebra , additive combinatorics , combinatorics

Each positive integer is coloured red or blue. A function $f$ from the set of positive integers to itself has the following two properties: (a) if $x\le y$, then $f(x)\le f(y)$; and (b) if $x,y$ and $z$ are (not necessarily distinct) positive integers of the same colour and $x+y=z$, then $f(x)+f(y)=f(z)$. Prove that there exists a positive number $a$ such that $f(x)\le ax$ for all positive integers $x$. [i](United Kingdom) Ben Elliott[/i]

1996 Korea National Olympiad, 8

Tags: geometry , angle bisector , circumcircle

Let $\triangle ABC$ be the acute triangle such that $AB\ne AC.$ Let $V$ be the intersection of $BC$ and angle bisector of $\angle A.$ Let $D$ be the foot of altitude from $A$ to $BC.$ Let $E,F$ be the intersection of circumcircle of $\triangle AVD$ and $CA,AB$ respectively. Prove that the lines $AD, BE,CF$ is concurrent.

LMT Team Rounds 2021+, 1

Tags: number theory , algebra

Given the following system of equations: $$\begin{cases} R I +G +SP = 50 \\ R I +T + M = 63 \\ G +T +SP = 25 \\ SP + M = 13 \\ M +R I = 48 \\ N = 1 \end{cases}$$ Find the value of L that makes $LMT +SPR I NG = 2023$ true.

1988 IberoAmerican, 6

Tags: algorithm , combinatorics

Consider all sets of $n$ distinct positive integers, no three of which form an arithmetic progression. Prove that among all such sets there is one which has the largest sum of the reciprocals of its elements.

2008 Turkey MO (2nd round), 2

Tags: quadratic , number theory

$ a \minus{} )$ Find all prime $ p$ such that $ \dfrac{7^{p \minus{} 1} \minus{} 1}{p}$ is a perfect square $ b \minus{} )$ Find all prime $ p$ such that $ \dfrac{11^{p \minus{} 1} \minus{} 1}{p}$ is a perfect square

2024 Junior Balkan Team Selection Tests - Romania, P4

Tags: combinatorics

Let $n\geqslant 2$ be an integer. A [i]Welsh darts board[/i] is a disc divided into $2n$ equal sectors, half of them being red and the other half being white. Two Welsh darts boards are [i]matched[/i] if they have the same radius and they are superimposed so that each sector of the first board comes exactly over a sector of the second board. Suppose that two given Welsh darts boards can be matched so that more than half of the paurs of superimposed sectors have different colours. Prove that these Welsh darts boards can be matched so that at least $2\lfloor n/2\rfloor +2$ pairs of superimposed sectors have the same colour.

2013 Kyiv Mathematical Festival, 1

Tags: inequality , 4-variable inequality , inequalities , algebra

For every positive $a, b, c, d$ such that $a + c\le ac$ and $b + d \le bd$ prove that $ab + cd \ge 8$.

2015 Vietnam National Olympiad, 2

Tags: induction , calculus , derivative , inequalities

If $a,b,c$ are nonnegative real numbers, then \[{ 3(a^2+b^2+c^2) \geq (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2 \geq (a+b+c)^2.}\]

2014 Math Prize for Girls Olympiad, 2

Tags: function , trigonometry

Let $f$ be the function defined by $f(x) = 4x(1 - x)$. Let $n$ be a positive integer. Prove that there exist distinct real numbers $x_1$, $x_2$, $\ldots\,$, $x_n$ such that $x_{i + 1} = f(x_i)$ for each integer $i$ with $1 \le i \le n - 1$, and such that $x_1 = f(x_n)$.

2017 IFYM, Sozopol, 2

Tags: geometry

Point $F$ lies on the circumscribed circle around $\Delta ABC$, $P$ and $Q$ are projections of point $F$ on $AB$ and $AC$ respectively. Prove that, if $M$ and $N$ are the middle points of $BC$ and $PQ$ respectively, then $MN$ is perpendicular to $FN$.

2024 ELMO Shortlist, A6

Tags: algebra

Let $\mathbb R^+$ denote the set of positive real numbers. Find all functions $f:\mathbb R^+\to\mathbb R$ and $g:\mathbb R^+\to\mathbb R$ such that for all $x,y\in\mathbb R^+$, $g(x)-g(y)=(x-y)f(xy)$. [i]Linus Tang[/i]

May Olympiad L2 - geometry, 2020.4

Tags: geometry

Let $ABC$ be a right triangle, right at $B$, and let $M$ be the midpoint of the side $BC$. Let $P$ be the point in bisector of the angle $ \angle BAC$ such that $PM$ is perpendicular to $BC (P$ is outside the triangle $ABC$). Determine the triangle area $ABC$ if $PM = 1$ and $MC = 5$.

The Golden Digits 2024, P3

Tags: geometry

Let $ABC$ be an acute scalene triangle with orthocentre $H{}$ and circumcentre $O.{}$ Let $P{}$ be an arbitrary point on the segment $OH$ and $O_a$ be the circumcentre of $PBC.{}$ The line $PO_a$ intersects the line $HA$ at $X_a.{}$ Define $X_b$ and $X_c$ similarly. Let $Q{}$ be the isogonal conjugate of $P{}$ and $X{}$ be the circumcentre of $X_aX_bX_c.{}$ Prove that $PQ$ and $HX$ are parallel. [i]Proposed by David Anghel[/i]

2016 Korea National Olympiad, 2

Tags: geometry , incenter

A non-isosceles triangle $\triangle ABC$ has its incircle tangent to $BC, CA, AB$ at points $D, E, F$. Let the incenter be $I$. Say $AD$ hits the incircle again at $G$, at let the tangent to the incircle at $G$ hit $AC$ at $H$. Let $IH \cap AD = K$, and let the foot of the perpendicular from $I$ to $AD$ be $L$. Prove that $IE \cdot IK= IC \cdot IL$.

2007 Iran MO (3rd Round), 2

Tags: floor function , number theory

We call the mapping $ \Delta:\mathbb Z\backslash\{0\}\longrightarrow\mathbb N$, a degree mapping if and only if for each $ a,b\in\mathbb Z$ such that $ b\neq0$ and $ b\not|a$ there exist integers $ r,s$ such that $ a \equal{} br\plus{}s$, and $ \Delta(s) <\Delta(b)$. a) Prove that the following mapping is a degree mapping: \[ \delta(n)\equal{}\mbox{Number of digits in the binary representation of }n\] b) Prove that there exist a degree mapping $ \Delta_{0}$ such that for each degree mapping $ \Delta$ and for each $ n\neq0$, $ \Delta_{0}(n)\leq\Delta(n)$. c) Prove that $ \delta \equal{}\Delta_{0}$ [img]http://i16.tinypic.com/4qntmd0.png[/img]

2014 India Regional Mathematical Olympiad, 1

Tags: geometry , circumcircle , asymptote , perpendicular bisector

Let $ABC$ be a triangle with $\angle ABC $ as the largest angle. Let $R$ be its circumcenter. Let the circumcircle of triangle $ARB$ cut $AC$ again at $X$. Prove that $RX$ is perpendicular to $BC$.

2023 Durer Math Competition Finals, 4

Tags: algebra

Benedek wrote down the following numbers: $1$ piece of one, $2$ pieces of twos, $3$ pieces of threes, $... $, $50$ piecies of fifties. How many digits did Benedek write down?

2018 Saudi Arabia BMO TST, 2

Tags: combinatorics , sum , product

Suppose that $2018$ numbers $1$ and $-1$ are written around a circle. For every two adjacent numbers, their product is taken. Suppose that the sum of all $2018$ products is negative. Find all possible values of sum of $2018$ given numbers.

2017 Brazil Undergrad MO, 1

Tags: polynomial , algebra

A polynomial is called positivist if it can be written as a product of two non-constant polynomials with non-negative real coefficients. Let $f(x)$ be a polynomial of degree greater than one such that $f(x^n)$ is positivist for some positive integer $n$. Show that $f(x)$ is positivist.

2019 Yasinsky Geometry Olympiad, p2

Tags: geometry , perimeter , 3d geometry , pyramid , rectangle

The base of the quadrilateral pyramid $SABCD$ lies the $ABCD$ rectangle with the sides $AB = 1$ and $AD = 10$. The edge $SA$ of the pyramid is perpendicular to the base, $SA = 4$. On the edge of $AD$, find a point $M$ such that the perimeter of the triangle of $SMC$ was minimal.

2010 Princeton University Math Competition, 5

Tags:

Given that $x$, $y$, and $z$ are positive integers such that $\displaystyle{\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 2}$. Find the number of all possible $x$ values.