Olympiad problems

2000 Saint Petersburg Mathematical Olympiad, 11.4

Let $P(x)=x^{2000}-x^{1000}+1$. Prove that there don't exist 8002 distinct positive integers $a_1,\dots,a_{8002}$ such that $a_ia_ja_k|P(a_i)P(a_j)P(a_k)$ for all $i\neq j\neq k$. [I]Proposed by A. Baranov[/i]

2015 Brazil National Olympiad, 1

Tags: geometry , circumcircle

Let $\triangle ABC$ be an acute-scalene triangle, and let $N$ be the center of the circle wich pass trough the feet of altitudes. Let $D$ be the intersection of tangents to the circumcircle of $\triangle ABC$ at $B$ and $C$. Prove that $A$, $D$ and $N$ are collinear iff $\measuredangle BAC = 45º$.

2010 Contests, 1

Tags: vector , analytic geometry , function , combinatorics

There are ten coins a line, which are indistinguishable. It is known that two of them are false and have consecutive positions on the line. For each set of positions, you may ask how many false coins it contains. Is it possible to identify the false coins by making only two of those questions, without knowing the answer to the first question before making the second?

2022 Nigerian MO round 3, Problem 4

Tags: geometry , nigeria mo

Let $PT$ and $PB$ be two tangents to a circle, $T$ and $B$ on the circle. $AB$ is the diameter of the circle through $B$ and $TH$ is the perpendicular from $T$ to $AB$, $H$ on $AB$. Prove that $AP$ bisects $TH$.

2005 Today's Calculation Of Integral, 64

Tags: calculus , integration , algebra , polynomial , trigonometry , calculus computations

Let $f(t)$ be the cubic polynomial for $t$ such that $\cos 3x=f(\cos x)$ holds for all real number $x$. Evaluate \[\int_0^1 \{f(t)\}^2 \sqrt{1-t^2}dt\]

1999 Greece National Olympiad, 3

Tags: circumcircle , geometry

In an acute-angled triangle $ABC$, $AD,BE$ and $CF$ are the altitudes and $H$ the orthocentre. Lines $EF$ and $BC$ meet at $N$. The line passing through $D$ and parallel to $FE$ meets lines $AB$ and $AC$ at $K$ and $L$, respectively. Prove that the circumcircle of the triangle $NKL$ bisects the side $BC$.

2014 India IMO Training Camp, 1

Tags: algebra , binomial theorem , pigeonhole principle

Prove that in any set of $2000$ distinct real numbers there exist two pairs $a>b$ and $c>d$ with $a \neq c$ or $b \neq d $, such that \[ \left| \frac{a-b}{c-d} - 1 \right|< \frac{1}{100000}. \]

2014 Thailand TSTST, 2

Tags: number theory , diophantine equation

Prove that the equation $x^8 = n! + 1$ has finitely many solutions in positive integers.

2023 Polish Junior MO Second Round, 1.

Tags: geometry

On the sides $AB$ and $BC$ of triangle $ABC$, there are points $D$ and $E$, respectively, such that \[\angle ADC=\angle BDE\quad\text{and}\quad \angle BCD=\angle AED.\] Prove that $AE=BE$.

1958 AMC 12/AHSME, 2

Tags:

If $ \frac {1}{x} \minus{} \frac {1}{y} \equal{} \frac {1}{z}$, then $ z$ equals: $ \textbf{(A)}\ y \minus{} x\qquad \textbf{(B)}\ x \minus{} y\qquad \textbf{(C)}\ \frac {y \minus{} x}{xy}\qquad \textbf{(D)}\ \frac {xy}{y \minus{} x}\qquad \textbf{(E)}\ \frac {xy}{x \minus{} y}$