Olympiad problems

1998 National High School Mathematics League, 11

If ellipse $x^2+4(y-a)^2=4$ and parabola $x^2=2y$ have intersections, then the range value of $a$ is________.

2008 Thailand Mathematical Olympiad, 7

Tags: diophantine equation , lcm , gcd , diophantine

Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.

2001 Vietnam Team Selection Test, 3

Tags: algebra , number theory

Let a sequence $\{a_n\}$, $n \in \mathbb{N}^{*}$ given, satisfying the condition \[0 < a_{n+1} - a_n \leq 2001\] for all $n \in \mathbb{N}^{*}$ Show that there are infinitely many pairs of positive integers $(p, q)$ such that $p < q$ and $a_p$ is divisor of $a_q$.

2009 China Team Selection Test, 2

Tags: algebra , polynomial

Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.

2014-2015 SDML (High School), 7

Tags: polynomial , algebra

Let $a$, $b$, and $c$ be the roots of the polynomial $$x^3+4x^2-7x-1.$$ Which of the following has roots $ab$, $bc$, and $ac$? $\text{(A) }x^3-4x^2+7x-1\qquad\text{(B) }x^3-7x^2+4x-1\qquad\text{(C) }x^3+7x^2-4x-1\qquad\text{(D) }x^3-4x^2+7x+1\qquad\text{(E) }x^3+7x^2-4x+1$

1994 Denmark MO - Mohr Contest, 1

Tags: 3d geometry , cylinder , sphere , geometry

A wine glass with a cross section as shown has the property of an orange in shape as a sphere with a radius of $3$ cm just can be placed in the glass without protruding above glass. Determine the height $h$ of the glass. [img]https://1.bp.blogspot.com/-IuLm_IPTvTs/XzcH4FAjq5I/AAAAAAAAMYY/qMi4ng91us8XsFUtnwS-hb6PqLwAON_jwCLcBGAsYHQ/s0/1994%2BMohr%2Bp1.png[/img]

2014 Balkan MO Shortlist, N3

Tags: perimeter , geometry

$\boxed{N3}$Prove that there exist infinitely many non isosceles triangles with rational side lengths$,$rational lentghs of altitudes and$,$ perimeter equal to $3.$

2017 Vietnamese Southern Summer School contest, Problem 4

Tags: locus , locus problems , geometry

Let $ABC$ be a triangle. A point $P$ varies inside $BC$. Let $Q, R$ be the points on $AC, AB$ in that order, such that $PQ\parallel AB, PR\parallel AC$. 1. Prove that, when $P$ varies, the circumcircle of triangle $AQR$ always passes through a fixed point $X$ other than $A$. 2. Extend $AX$ so that it cuts the circumcircle of $ABC$ a second time at point $K$. Prove that $AX=XK$.

2019 Hong Kong TST, 5

Tags: hong kong , combinatorics , combinatorial geometry

Is it is possible to choose 24 distinct points in the space such that no three of them lie on the same line and choose 2019 distinct planes in a way that each plane passes through at least 3 of the chosen points and each triple belongs to one of the chosen planes?

2009 JBMO TST - Macedonia, 2

Tags: number theory

Let $ a $ and $ b $ be integer numbers. Let $ a = a^{2}+b^{2}-8b-2ab+16$. Prove that $ a $ is a perfect square.

2006 All-Russian Olympiad, 1

Tags: combinatorics

Given a $15\times 15$ chessboard. We draw a closed broken line without self-intersections such that every edge of the broken line is a segment joining the centers of two adjacent cells of the chessboard. If this broken line is symmetric with respect to a diagonal of the chessboard, then show that the length of the broken line is $\leq 200$.

1997 Kurschak Competition, 1

Tags: geometry , linear algebra , number theory , matrix

Let $p>2$ be a prime number and let $L=\{0,1,\dots,p-1\}^2$. Prove that we can find $p$ points in $L$ with no three of them collinear.

1968 IMO Shortlist, 2

Tags: algebra , triangle , triangle inequality , trigonometry

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

2024 LMT Fall, 26

Tags: guts

Let $P$ be a point in the interior of square $ABCD$ such that $\angle APB+\angle CPD=180^\circ$ and $\angle APB$ $ <$ $\angle CPD$. If $PC=7$ and $PD=5$, find $\tfrac{PA}{PB}$.

2019 Junior Balkan Team Selection Tests - Romania, 3

Tags: circumcenter , equal angles , excenter , incenter , geometry

In the acute triangle $ABC$ point $I$ is the incenter, $O$ is the circumcenter, while $I_a$ is the excenter opposite the vertex $A$. Point $A'$ is the reflection of $A$ across the line $BC$. Prove that angles $\angle IOI_a$ and $\angle IA'I_a$ are equal.

2015 Macedonia National Olympiad, Problem 2

Tags: inequalities

Let $a,b,c \in \mathbb{R}^{+}$ such that $abc=1$. Prove that: $$a^2b + b^2c + c^2a \ge \sqrt{(a+b+c)(ab + bc +ca)}$$

1992 AMC 12/AHSME, 14

Tags: conic

Which of the following equations have the same graph? $I.\ y = x - 2$ $II.\ y = \frac{x^{2} - 4}{x + 2}$ $III.\ (x + 2)y = x^{2} - 4$ $ \textbf{(A)}\ \text{I and II only} $ $ \textbf{(B)}\ \text{I and III only} $ $ \textbf{(C)}\ \text{II and III only} $ $ \textbf{(D)}\ \text{I, II and III} $ $ \textbf{(E)}\ \text{None. All equations have different graphs} $

2001 National Olympiad First Round, 13

Tags: geometry , trigonometry

Let $ABC$ be a triangle such that $|BC|=7$ and $|AB|=9$. If $m(\widehat{ABC}) = 2m(\widehat{BCA})$, then what is the area of the triangle? $ \textbf{(A)}\ 14\sqrt 5 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 10\sqrt 6 \qquad\textbf{(D)}\ 20 \sqrt 2 \qquad\textbf{(E)}\ 12 \sqrt 3 $

1997 Czech And Slovak Olympiad IIIA, 6

Tags: cyclic , locus , geometry , diagonal , parallelogram

In a parallelogram $ABCD$, triangle $ABD$ is acute-angled and $\angle BAD = \pi /4$. Consider all possible choices of points $K,L,M,N$ on sides $AB,BC, CD,DA$ respectively, such that $KLMN$ is a cyclic quadrilateral whose circumradius equals those of triangles $ANK$ and $CLM$. Find the locus of the intersection of the diagonals of $KLMN$

1963 AMC 12/AHSME, 28

Tags: quadratic

Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is: $\textbf{(A)}\ \dfrac{16}{9} \qquad \textbf{(B)}\ \dfrac{16}{3}\qquad \textbf{(C)}\ \dfrac{4}{9} \qquad \textbf{(D)}\ \dfrac{4}{3} \qquad \textbf{(E)}\ -\dfrac{4}{3}$

1971 Swedish Mathematical Competition, 6

Tags: combinatorics

$99$ cards each have a label chosen from $1,2,\dots,99$, such that no (non-empty) subset of the cards has labels with total divisible by $100$. Show that the labels must all be equal.

2021 USMCA, 1

Tags:

Let $a_1, a_2, \ldots, a_{2021}$ be a sequence, where each $a_i$ is a positive factor of $2021$. How many possible values are there for the product $a_1 a_2 \cdots a_{2021}$?

2021 Thailand TST, 3

Tags: iteration , functional equation , algebra

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ satisfying \[f^{a^{2} + b^{2}}(a+b) = af(a) +bf(b)\] for all integers $a$ and $b$

2016 Harvard-MIT Mathematics Tournament, 1

Tags:

Dodecagon $QWARTZSPHINX$ has all side lengths equal to $2$, is not self-intersecting (in particular, the twelve vertices are all distinct), and moreover each interior angle is either $90^{\circ}$ or $270^{\circ}$. What are all possible values of the area of $\triangle SIX$?

2007 Today's Calculation Of Integral, 208

Tags: calculus computations , function , trigonometry , integration , calculus

Find the values of real numbers $a,\ b$ for which the function $f(x)=a|\cos x|+b|\sin x|$ has local minimum at $x=-\frac{\pi}{3}$ and satisfies $\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\{f(x)\}^{2}dx=2$.