Olympiad problems

2014 PUMaC Geometry A, 8

$ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. $AB$ intersects $CD$ at $E$, $DA$ intersects $CB$ at $F$. $OE=13$, $OF=14$. Let $\cos\angle FOE=\dfrac pq$, with $p$, $q$ coprime. Find $p+q$.

1996 IMO, 4

Tags: modular arithmetic , number theory , quadratic reciprocity , perfect square

The positive integers $ a$ and $ b$ are such that the numbers $ 15a \plus{} 16b$ and $ 16a \minus{} 15b$ are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?

1981 National High School Mathematics League, 3

Tags: function

Let $\alpha$ be a real number and $\alpha\neq\frac{k\pi}{2} , k\in\mathbb{Z}$, $$T=\frac{\sin\alpha+\tan\alpha}{\cos\alpha+\cot\alpha}$$. $\text{(A)}$$T$ is negative. $\text{(B)}$$T$ is nonnegative. $\text{(C)}$$T$ is positive. $\text{(D)}$$T$ can be either positive or negative.

2011 Sharygin Geometry Olympiad, 24

Tags: geometry , minimal , sides of a triangle

Given is an acute-angled triangle $ABC$. On sides $BC, CA, AB$, find points $A', B', C'$ such that the longest side of triangle $A'B'C'$ is minimal.

2009 All-Russian Olympiad Regional Round, 10.1

Tags: polynomial , algebra

Square trinomial $f(x)$ is such that the polynomial $(f(x)) ^3- f(x)$ has exactly three real roots. Find the ordinate of the vertex of the graph of this trinomial.

2022 Indonesia MO, 1

Tags: functional equation , functional equation in r

Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that for any $x,y \in \mathbb{R}$ we have \[ f(f(f(x)) + f(y)) = f(y) - f(x) \]

2017 Balkan MO Shortlist, N4

Tags: number theory , divisibility

Find all pairs of positive integers $(x,y)$ , such that $x^2$ is divisible by $2xy^2 -y^3 +1$.

PEN P Problems, 31

Tags: quadratic , additive number theory

A finite sequence of integers $a_{0}, a_{1}, \cdots, a_{n}$ is called quadratic if for each $i \in \{1,2,\cdots,n \}$ we have the equality $\vert a_{i}-a_{i-1} \vert = i^2$. [list=a] [*] Prove that for any two integers $b$ and $c$, there exists a natural number $n$ and a quadratic sequence with $a_{0}=b$ and $a_{n}=c$. [*] Find the smallest natural number $n$ for which there exists a quadratic sequence with $a_{0}=0$ and $a_{n}=1996$. [/list]

1985 AMC 12/AHSME, 16

Tags: trigonometry

If $ A \equal{} 20^{\circ}$ and $ B \equal{} 25^{\circ}$, then the value of $ (1 \plus{} \tan A)(1 \plus{} \tan B)$ is $ \textbf{(A)} \sqrt3 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 1 \plus{} \sqrt2 \qquad \textbf{(D)}\ 2(\tan A \plus{} \tan B)$ $ \textbf{(E)}\ \text{ none of these}$

2003 Iran MO (3rd Round), 23

Tags: polynomial , algebra

Find all homogeneous linear recursive sequences such that there is a $ T$ such that $ a_n\equal{}a_{n\plus{}T}$ for each $ n$.

2009 USAMTS Problems, 2

Tags: quadratic , algebra , quadratic formula

Let $a, b, c, d$ be four real numbers such that \begin{align*}a + b + c + d &= 8, \\ ab + ac + ad + bc + bd + cd &= 12.\end{align*} Find the greatest possible value of $d$.

2021 Korea - Final Round, P4

Tags: easy , set , combinatorics

There are $n$($\ge 2$) clubs $A_1,A_2,...A_n$ in Korean Mathematical Society. Prove that there exist $n-1$ sets $B_1,B_2,...B_{n-1}$ that satisfy the condition below. (1) $A_1\cup A_2\cup \cdots A_n=B_1\cup B_2\cup \cdots B_{n-1}$ (2) for any $1\le i<j\le n-1$, $B_i\cap B_j=\emptyset, -1\le\left\vert B_i \right\vert -\left\vert B_j \right\vert\le 1$ (3) for any $1\le i \le n-1$, there exist $A_k,A_j $($1\le k\le j\le n$)such that $B_i\subseteq A_k\cup A_j$

1988 Austrian-Polish Competition, 7

Tags: coloring , octagon , combinatorics

Each side of a regular octagon is colored blue or yellow. In each step, the sides are simultaneously recolored as follows: if the two neighbors of a side have different colors, the side will be recolored blue, otherwise it will be recolored yellow. Show that after a finite number of moves all sides will be colored yellow. What is the least value of the number $N$ of moves that always lead to all sides being yellow?

1996 All-Russian Olympiad Regional Round, 9.1

Tags: algebra , trinomial

Find all pairs of square trinomials $x^2 + ax + b$, $ x^2 + cx + d$ such that $a$ and $b$ are the roots of the second trinomial, $c$ and $d$ are the roots of the first.

1999 Estonia National Olympiad, 5

Tags: tiling , combinatorics

There is a hole in the roof with dimensions $23 \times 19$ cm. Can August fill the the roof with tiles of dimensions $5 \times 24 \times 30$ cm?

2021 Yasinsky Geometry Olympiad, 3

Tags: equal angles , geometry , equal segments , angle

The segments $AC$ and $BD$ are perpendicular, and $AC$ is twice as large as $BD$ and intersects $BD$ in it in the midpoint. Find the value of the angle $BAD$, if we know that $\angle CAD = \angle CDB$. (Gregory Filippovsky)

1980 AMC 12/AHSME, 16

Tags: geometry , ratio , 3d geometry , tetrahedron , congruent triangles

Four of the eight vertices of a cube are the vertices of a regular tetrahedron. Find the ratio of the surface area of the cube to the surface area of the tetrahedron. $\text{(A)} \ \sqrt 2 \qquad \text{(B)} \ \sqrt 3 \qquad \text{(C)} \ \sqrt{\frac{3}{2}} \qquad \text{(D)} \ \frac{2}{\sqrt{3}} \qquad \text{(E)} \ 2$

2006 Pre-Preparation Course Examination, 1

Tags: function , combinatorics

a) Find the value of $\sum_{n=1}^{\infty}\frac{\phi(n)}{2^n-1}$; b) Show that $\sum_k {m\choose k}{{n+k}\choose m}=\sum_k {m\choose k} {n\choose k} 2^k$ for $m,n\geq 0$; c) Using the identity $(1-x)^{-\frac 12}(1-x)^{-\frac 12}=(1-x)^{-1}$ derive a combinatorial identity! d) Express the value of $\sum (2^{a_1}-1)\ldots (2^{a_k}-1)$ where the sum is over all $2^{n-1}$ ways of choosing $(a_1,a_2,\ldots,a_k)$ such that $a_1+a_2+\ldots +a_k=n$, as a function of some Fibonacci term.

2022 VIASM Summer Challenge, Problem 2

Tags: algebra , inequalities

Let $S$ be the set of real numbers $k$ with the following property: for all set of real numbers $(a,b,c)$ satisfying $ab+bc+ca=1$, we always have the inequality:$$\frac{a}{{\sqrt {{a^2} + ab + {b^2} + k} }} + \frac{b}{{\sqrt {{b^2} + bc + {c^2} + k} }} + \frac{c}{{\sqrt {{c^2} + ca + {a^2} + k} }} \ge \sqrt {\frac{3}{{k + 1}}} .$$ a) Assume that $k\in S$. Prove that: $k\ge 2$. b) Prove that: $2\in S$.

2004 Czech and Slovak Olympiad III A, 5

Tags: circumcircle , geometry

Let $L$ be an arbitrary point on the minor arc $CD$ of the circumcircle of square $ABCD$. Let $K,M,N$ be the intersection points of $AL,CD$; $CL,AD$; and $MK,BC$ respectively. Prove that $B,M,L,N$ are concyclic.

1956 AMC 12/AHSME, 49

Tags:

Triangle $ PAB$ is formed by three tangents to circle $ O$ and $ < APB \equal{} 40^{\circ}$; then angle $ AOB$ equals: $ \textbf{(A)}\ 45^{\circ} \qquad\textbf{(B)}\ 50^{\circ} \qquad\textbf{(C)}\ 55^{\circ} \qquad\textbf{(D)}\ 60^{\circ} \qquad\textbf{(E)}\ 70^{\circ}$

2014 ASDAN Math Tournament, 14

Tags: team test

Consider a round table on which $2014$ people are seated. Suppose that the person at the head of the table receives a giant plate containing all the food for supper. He then serves himself and passes the plate either right or left with equal probability. Each person, upon receiving the plate, will serve himself if necessary and similarly pass the plate either left or right with equal probability. Compute the probability that you are served last if you are seated $2$ seats away from the person at the head of the table.

2019 Purple Comet Problems, 15

Tags: equation , number theory

Let $a, b, c$, and $d$ be prime numbers with $a \le b \le c \le d > 0$. Suppose $a^2 + 2b^2 + c^2 + 2d^2 = 2(ab + bc - cd + da)$. Find $4a + 3b + 2c + d$.

2008 Czech and Slovak Olympiad III A, 1

Tags: algebra

Find all pairs of real numbers $(x,y)$ satisfying: \[x+y^2=y^3,\]\[y+x^2=x^3.\]

1969 AMC 12/AHSME, 22

Tags: geometry , trapezoid

Let $K$ be the measure of the area bounded by the $x$-axis, the line $x=8$, and the curve defined by \[f=\{(x,y)\,|\, y=x\text{ when }0\leq x\leq 5,\,y=2x-5\text{ when }5\leq x\leq 8\}.\] Then $K$ is: $\textbf{(A) }21.5\qquad \textbf{(B) }36.4\qquad \textbf{(C) }36.5\qquad \textbf{(D) }44\qquad$ $\textbf{ (E) }\text{less than 44 but arbitrarily close to it.}$