Olympiad problems

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]

2010 Junior Balkan Team Selection Tests - Romania, 2

Tags: midpoint , equal angles , angle , geometry

Let $ABC$ be a triangle and $D, E, F$ the midpoints of the sides $BC, CA, AB$ respectively. Show that $\angle DAC = \angle ABE$ if and only if $\angle AFC = \angle BDA$

2006 China Team Selection Test, 3

Tags: number theory

Given positive integers $m$ and $n$ so there is a chessboard with $mn$ $1 \times 1$ grids. Colour the grids into red and blue (Grids that have a common side are not the same colour and the grid in the left corner at the bottom is red). Now the diagnol that goes from the left corner at the bottom to the top right corner is coloured into red and blue segments (Every segment has the same colour with the grid that contains it). Find the sum of the length of all the red segments.

2000 Brazil National Olympiad, 1

Tags: geometric transformation , reflection , geometry

A rectangular piece of paper has top edge $AD$. A line $L$ from $A$ to the bottom edge makes an angle $x$ with the line $AD$. We want to trisect $x$. We take $B$ and $C$ on the vertical ege through $A$ such that $AB = BC$. We then fold the paper so that $C$ goes to a point $C'$ on the line $L$ and $A$ goes to a point $A'$ on the horizontal line through $B$. The fold takes $B$ to $B'$. Show that $AA'$ and $AB'$ are the required trisectors.

1990 IMO Longlists, 17

Tags: combinatorics , graph theory , extremal graph theory , clique number

1990 mathematicians attend a meeting, every mathematician has at least 1327 friends (the relation of friend is reciprocal). Prove that there exist four mathematicians among them such that any two of them are friends.

2010 ELMO Shortlist, 2

Tags: parallelogram , trapezoid , geometry

Given a triangle $ABC$, a point $P$ is chosen on side $BC$. Points $M$ and $N$ lie on sides $AB$ and $AC$, respectively, such that $MP \parallel AC$ and $NP \parallel AB$. Point $P$ is reflected across $MN$ to point $Q$. Show that triangle $QMB$ is similar to triangle $CNQ$. [i]Brian Hamrick.[/i]

2023 Purple Comet Problems, 1

Tags: number theory , algebra

Find the sum of the four least positive integers each of whose digits add to $12$.