Olympiad problems

1953 AMC 12/AHSME, 3

The factors of the expression $ x^2\plus{}y^2$ are: $ \textbf{(A)}\ (x\plus{}y)(x\minus{}y) \qquad\textbf{(B)}\ (x\plus{}y)^2 \qquad\textbf{(C)}\ (x^{\frac{2}{3}}\plus{}y^{\frac{2}{3}})(x^{\frac{4}{3}}\plus{}y^{\frac{4}{3}}) \\ \textbf{(D)}\ (x\plus{}iy)(x\minus{}iy) \qquad\textbf{(E)}\ \text{none of these}$

1998 All-Russian Olympiad Regional Round, 9.5

Tags: algebra , trinomial

The roots of the two monic square trinomials are negative integers, and one of these roots is common. Can the values of these trinomials at some positive integer point equal 19 and 98?

2008 IMO Shortlist, 1

Tags: algebra , functional equation , IMO Shortlist , IMO , IMO 2008 , Hi

Find all functions $ f: (0, \infty) \mapsto (0, \infty)$ (so $ f$ is a function from the positive real numbers) such that \[ \frac {\left( f(w) \right)^2 \plus{} \left( f(x) \right)^2}{f(y^2) \plus{} f(z^2) } \equal{} \frac {w^2 \plus{} x^2}{y^2 \plus{} z^2} \] for all positive real numbers $ w,x,y,z,$ satisfying $ wx \equal{} yz.$ [i]Author: Hojoo Lee, South Korea[/i]

2009 Ukraine National Mathematical Olympiad, 4

Tags: function , algebra , functional equation , algebra unsolved

Find all functions $f : \mathbb R \to \mathbb R$ such that \[f\left(x+xy+f(y)\right)= \left( f(x)+\frac 12 \right) \left( f(y)+\frac 12 \right) \qquad \forall x,y \in \mathbb R.\]

1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 2

Tags: geometry , rectangle , quadratics

On the figure, the quadrilateral $ ABCD$ is a rectangle, $ P$ lies on $ AD$ and $ Q$ on $ AB.$ The triangles $ PAQ, QBC,$ and $ PCD$ all have the same areas, and $ BQ \equal{} 2.$ How long is $ AQ$? [img]http://i250.photobucket.com/albums/gg265/geometry101/NielsHenrikAbel1995Number2.jpg[/img] A. 7/2 B. $ \sqrt{7}$ C. $ 2 \sqrt{3}$ D. $ 1 \plus{} \sqrt{5}$ E. Not uniquely determined

2005 Finnish National High School Mathematics Competition, 3

Tags: absolute value , algebra unsolved , algebra

Solve the group of equations: \[\begin{cases} (x + y)^3 = z \\ (y + z)^3 = x \\ (z + x)^3 = y \end{cases}\]

2024 LMT Fall, A4

Tags: theme

In Brawl Stars, Rico can shoot his opponent directly or make his bullet bounce off the wall at the same angle. His opponent is $15$ feet in front of him and there are infinitely long walls $1$ feet to the left and right of Rico. If Rico's bullet travels $d$ feet before hitting the opponent, find the sum of all possible integer values of $d$.

2020 Macedonian Nationаl Olympiad, 4

Tags: combinatorics , set theory , national olympiad

Let $S$ be a nonempty finite set, and $\mathcal {F}$ be a collection of subsets of $S$ such that the following conditions are met: (i) $\mathcal {F}$ $\setminus$ {$S$} $\neq$ $\emptyset$ ; (ii) if $F_1, F_2 \in \mathcal {F}$, then $F_1 \cap F_2 \in \mathcal {F}$ and $F_1 \cup F_2 \in \mathcal {F}$. Prove that there exists $a \in S$ which belongs to at most half of the elements of $\mathcal {F}$.

2008 District Round (Round II), 2

Tags:

Two circles $U,V$ have distinct radii,tangent to each other externally at $T$.$A,B$ are points on $U,V$ respectively,both distinct from $T$,such that $\angle ATB=90$. (1)Prove that line $AB$ passes through a fixed point; (2)Find the locus of the midpoint of $AB$.

2011 Princeton University Math Competition, A7

Tags: number theory

Let $\{g_i\}_{i=0}^{\infty}$ be a sequence of positive integers such that $g_0=g_1=1$ and the following recursions hold for every positive integer $n$: \begin{align*} g_{2n+1} &= g_{2n-1}^2+g_{2n-2}^2 \\ g_{2n} &= 2g_{2n-1}g_{2n-2}-g_{2n-2}^2 \end{align*} Compute the remainder when $g_{2011}$ is divided by $216$.