Olympiad problems

2018 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics , chessboard

What is the maximum number of rooks one can place on a chessboard such that any rook attacks exactly two other rooks? (We say that two rooks attack each other if they are on the same line or on the same column and between them there are no other rooks.) Alexandru Mihalcu

2015 India IMO Training Camp, 1

Tags: number theory

Find all positive integers $a,b$ such that $\frac{a^2+b}{b^2-a}$ and $\frac{b^2+a}{a^2-b}$ are also integers.

2019 India PRMO, 16

Tags: algebra

Let $N$ denote the number of all natural numbers $n$ such that $n$ is divisible by a prime $p> \sqrt{n}$ and $p<20$. What is the value of $N$ ?

1983 Canada National Olympiad, 2

Tags: function , functional equation , algebra , logarithm

For each $r\in\mathbb{R}$ let $T_r$ be the transformation of the plane that takes the point $(x, y)$ into the point $(2^r x; r2^r x+2^r y)$. Let $F$ be the family of all such transformations (i.e. $F = \{T_r : r\in\mathbb{R}\}$). Find all curves $y = f(x)$ whose graphs remain unchanged by every transformation in $F$.

2021 Math Prize for Girls Problems, 12

Tags: parabola , conic

Let $P_1$, $P_2$, $P_3$, $P_4$, $P_5$, and $P_6$ be six parabolas in the plane, each congruent to the parabola $y = x^2/16$. The vertices of the six parabolas are evenly spaced around a circle. The parabolas open outward with their axes being extensions of six of the circle's radii. Parabola $P_1$ is tangent to $P_2$, which is tangent to $P_3$, which is tangent to $P_4$, which is tangent to $P_5$, which is tangent to $P_6$, which is tangent to $P_1$. What is the diameter of the circle?

2022 MMATHS, 2

Tags: combinatorics

How many ways are there to fill in a three by three grid of cells with $0$’s and $2$’s, one number in each cell, such that each two by two contiguous subgrid contains exactly three $2$’s and one $0$?

2009 Hanoi Open Mathematics Competitions, 2

Tags: number theory

Show that there is a natural number $n$ such that the number $a = n!$ ends exactly in $2009$ zeros.

1999 India Regional Mathematical Olympiad, 2

Tags:

Find the number of positive integers which divide $10^{999}$ but not $10^{998}$.

2017 ELMO Shortlist, 3

Tags: geometry , combinatorial geometry

Call the ordered pair of distinct circles $(\omega, \gamma)$ scribable if there exists a triangle with circumcircle $\omega$ and incircle $\gamma$. Prove that among $n$ distinct circles there are at most $(n/2)^2$ scribable pairs. [i]Proposed by Daniel Liu

2019 Benelux, 4

Tags: number theory

An integer $m>1$ is [i]rich[/i] if for any positive integer $n$, there exist positive integers $x,y,z$ such that $n=mx^2-y^2-z^2$. An integer $m>1$ is [i]poor[/i] if it is not rich. [list=a] [*]Find a poor integer.[/*] [*]Find a rich integer.[/*] [/list]

2019 IMO Shortlist, G2

Tags: geometry

Let $ABC$ be an acute-angled triangle and let $D, E$, and $F$ be the feet of altitudes from $A, B$, and $C$ to sides $BC, CA$, and $AB$, respectively. Denote by $\omega_B$ and $\omega_C$ the incircles of triangles $BDF$ and $CDE$, and let these circles be tangent to segments $DF$ and $DE$ at $M$ and $N$, respectively. Let line $MN$ meet circles $\omega_B$ and $\omega_C$ again at $P \ne M$ and $Q \ne N$, respectively. Prove that $MP = NQ$. (Vietnam)

2019 Malaysia National Olympiad, 6

Tags: prime factorization , algebra

It is known that $2018(2019^{39}+2019^{37}+...+2019)+1$ is prime. How many positive factors does $2019^{41}+1$ have?

2021 AMC 10 Fall, 1

Tags:

What is the value of $\frac{(2112-2021)^2}{169}$? $\textbf{(A) }7\qquad\textbf{(B) }21\qquad\textbf{(C) }49\qquad\textbf{(D) }64\qquad\textbf{(E) }91$

2021 Macedonian Balkan MO TST, Problem 2

Tags: number theory , sequence

Define a sequence: $x_0=1$ and for all $n\ge 0$, $x_{2n+1}=x_{n}$ and $x_{2n+2}=x_{n}+x_{n+1}$. Prove that for any relatively prime positive integers $a$ and $b$, there is a non-negative integer $n$ such that $a=x_n$ and $b=x_{n+1}$.

2007 ISI B.Math Entrance Exam, 6

Tags: isi entrance , combinatorics

In $ISI$ club each member is on two committees and any two committees have exactly one member in common . There are 5 committees . How many members does $ISI$ club have????

1980 IMO Shortlist, 1

Tags: trigonometry , geometry , triangle , perpendicular bisector

Let $\alpha, \beta$ and $\gamma$ denote the angles of the triangle $ABC$. The perpendicular bisector of $AB$ intersects $BC$ at the point $X$, the perpendicular bisector of $AC$ intersects it at $Y$. Prove that $\tan(\beta) \cdot \tan(\gamma) = 3$ implies $BC= XY$ (or in other words: Prove that a sufficient condition for $BC = XY$ is $\tan(\beta) \cdot \tan(\gamma) = 3$). Show that this condition is not necessary, and give a necessary and sufficient condition for $BC = XY$.

IV Soros Olympiad 1997 - 98 (Russia), 11.2

Tags: number theory , algebra , inequalities

Find all values of the parameter $a$ for which there are exactly $1998$ integers $x$ satisfying the inequality $$x^2 -\pi x +a < 0.$$

1969 IMO Longlists, 46

Tags: geometry , perimeter , polygon , area , optimization

$(NET 1)$ The vertices of an $(n + 1)-$gon are placed on the edges of a regular $n-$gon so that the perimeter of the $n-$gon is divided into equal parts. How does one choose these $n + 1$ points in order to obtain the $(n + 1)-$gon with $(a)$ maximal area; $(b)$ minimal area?

2013 Iran Team Selection Test, 18

Tags: geometry , parallelogram , rotation , combinatorics

A special kind of parallelogram tile is made up by attaching the legs of two right isosceles triangles of side length $1$. We want to put a number of these tiles on the floor of an $n\times n$ room such that the distance from each vertex of each tile to the sides of the room is an integer and also no two tiles overlap. Prove that at least an area $n$ of the room will not be covered by the tiles. [i]Proposed by Ali Khezeli[/i]

2017 China Team Selection Test, 2

Tags: graph theory , combinatorics

$2017$ engineers attend a conference. Any two engineers if they converse, converse with each other in either Chinese or English. No two engineers converse with each other more than once. It is known that within any four engineers, there was an even number of conversations and furthermore within this even number of conversations: i) At least one conversation is in Chinese. ii) Either no conversations are in English or the number of English conversations is at least that of Chinese conversations. Show that there exists $673$ engineers such that any two of them conversed with each other in Chinese.

1982 National High School Mathematics League, 5

Tags: function

For any$\varphi\in(0,\frac{\pi}{2})$, we have $\text{(A)}\sin\sin\varphi<\cos\varphi<\cos\cos\varphi\qquad\text{(B)}\sin\sin\varphi>\cos\varphi>\cos\cos\varphi$ $\text{(C)}\sin\cos\varphi>\cos\varphi>\cos\sin\varphi\qquad\text{(D)}\sin\cos\varphi<\cos\varphi<\cos\sin\varphi$

2011 Postal Coaching, 1

Tags: ratio , parallelogram , trigonometry , geometry

Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$. Let \[AO = 5, BO =6, CO = 7, DO = 8.\] If $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, determine $\frac{OM}{ON}$ .

2021 Iranian Geometry Olympiad, 3

Tags: geometry , igo p3 , inversion , angle chasing

Consider a triangle $ABC$ with altitudes $AD, BE$, and $CF$, and orthocenter $H$. Let the perpendicular line from $H$ to $EF$ intersects $EF, AB$ and $AC$ at $P, T$ and $L$, respectively. Point $K$ lies on the side $BC$ such that $BD=KC$. Let $\omega$ be a circle that passes through $H$ and $P$, that is tangent to $AH$. Prove that circumcircle of triangle $ATL$ and $\omega$ are tangent, and $KH$ passes through the tangency point.

2009 Balkan MO Shortlist, N1

Tags: number theory

Solve the given equation in integers \begin{align*} y^3=8x^6+2x^3y-y^2 \end{align*}

2015 Mathematical Talent Reward Programme, MCQ: P 12

Tags: trigonometry , algebra , trigonometric equation

Maximum value of $\sin^4\theta +\cos^6\theta $ will be ? [list=1] [*] $\frac{1}{2\sqrt{2}}$ [*] $\frac{1}{2}$ [*] $\frac{1}{\sqrt{2}}$ [*] 1 [/list]