Olympiad problems

2025 Malaysian IMO Training Camp, 1

Tags: geometry

Let $ABC$ be a triangle with $AB<AC$ and with its incircle touching the sides $AB$ and $BC$ at $M$ and $J$ respectively. A point $D$ lies on the extension of $AB$ beyond $B$ such that $AD=AC$. Let $O$ be the midpoint of $CD$. Prove that the points $J$, $O$, $M$ are collinear. [i](Proposed by Tan Rui Xuen)[/i]

2001 Vietnam National Olympiad, 3

Tags: trigonometry , integration , calculus , calculus computations

For real $a, b$ define the sequence $x_{0}, x_{1}, x_{2}, ...$ by $x_{0}= a, x_{n+1}= x_{n}+b \sin x_{n}$. If $b = 1$, show that the sequence converges to a finite limit for all $a$. If $b > 2$, show that the sequence diverges for some $a$.

2013 JBMO Shortlist, 1

Tags: number theory , JBMO Shortlist

$\boxed{N1}$ find all positive integers $n$ for which $1^3+2^3+\cdots+{16}^3+{17}^n$ is a perfect square.

Russian TST 2014, P2

Tags: number theory , prime numbers

Let $p,q$ and $s{}$ be prime numbers such that $2^sq =p^y-1$ where $y > 1.$ Find all possible values of $p.$

CNCM Online Round 2, 2

Tags: CNCM , Problem Discussion

There is a rectangle $ABCD$ such that $AB=12$ and $BC=7$. $E$ and $F$ lie on sides $AB$ and $CD$ respectively such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Call $X$ the intersection of $AF$ and $DE$. What is the area of pentagon $BCFXE$? Proposed by Minseok Eli Park (wolfpack)

2001 South africa National Olympiad, 2

Tags: trigonometry , algebra unsolved , algebra

Find all triples $(x,y,z)$ of real numbers that satisfy \[ \begin{aligned} & x\left(1 - y^2\right)\left(1 - z^2\right) + y\left(1 - z^2\right)\left(1 - x^2\right) + z\left(1 - x^2\right)\left(1 - y^2\right) \\ & = 4xyz \\ & = 4(x + y + z). \end{aligned} \]

2014 Thailand TSTST, 1

Tags: Diophantine equation , number theory

Find all triples of positive integers $(a, b, c)$ such that $$(2^a-1)(3^b-1)=c!.$$

1994 Poland - First Round, 11

Tags: probability , expected value

Given are natural numbers $n>m>1$. We draw $m$ numbers from the set $\{1,2,...,n\}$ one by one without putting the drawn numbers back. Find the expected value of the difference between the largest and the smallest of the drawn numbers.

1966 AMC 12/AHSME, 13

Tags: analytic geometry

The number of points with positive rational coordinates selected from the set of points in the xy-plane such that $x+y\leq 5$, is: $\text{(A)} \ 9 \qquad \text{(B)} \ 10 \qquad \text{(C)} \ 14 \qquad \text{(D)} \ 15 \qquad \text{(E)} \ \text{infinite}$

1999 Harvard-MIT Mathematics Tournament, 9

Tags: algebra

Evaluate $$\sum^{17}_{n=2} \frac{n^2+n+1}{n^4+2n^3-n^2-2n}.$$

2010 Today's Calculation Of Integral, 538

Tags: calculus , integration , logarithms , calculus computations

Evaluate $ \int_1^{\sqrt{2}} \frac{x^2\plus{}1}{x\sqrt{x^4\plus{}1}}\ dx$.

2021 Spain Mathematical Olympiad, 3

Tags: combinatorics , Spain , Parity

We have $2021$ colors and $2021$ chips of each color. We place the $2021^2$ chips in a row. We say that a chip $F$ is [i]bad[/i] if there is an odd number of chips that have a different color to $F$ both to the left and to the right of $F$. (a) Determine the minimum possible number of bad chips. (b) If we impose the additional condition that each chip must have at least one adjacent chip of the same color, determine the minimum possible number of bad chips.

2008 Vietnam Team Selection Test, 2

Tags: algebra , polynomial , algebra proposed

Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$, we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$.

2024 German National Olympiad, 2

Tags: geometry , geometry proposed , 3D geometry

Six quadratic mirrors are put together to form a cube $ABCDEFGH$ with a mirrored interior. At each of the eight vertices, there is a tiny hole through which a laser beam can enter and leave the cube. A laser beam enters the cube at vertex $A$ in a direction not parallel to any of the cube's sides. If the beam hits a side, it is reflected; if it hits an edge, the light is absorbed, and if it hits a vertex, it leaves the cube. For each positive integer $n$, determine the set of vertices where the laser beam can leave the cube after exactly $n$ reflections.

2015 Oral Moscow Geometry Olympiad, 4

Tags: geometry , circumcircle , equal angles , Tangents , midpoints

In triangle $ABC$, point $M$ is the midpoint of $BC, P$ is the intersection point of the tangents at points $B$ and $C$ of the circumscribed circle, $N$ is the midpoint of the segment $MP$. The segment $AN$ intersects the circumscribed circle at point $Q$. Prove that $\angle PMQ = \angle MAQ$.

1996 All-Russian Olympiad, 1

Tags: geometry , geometric transformation , reflection , extremal principle , number theory proposed , number theory

Can the number obtained by writing the numbers from 1 to $n$ in order ($n > 1$) be the same when read left-to-right and right-to-left? [i]N. Agakhanov[/i]

2010 Contests, 1

Tags: modular arithmetic , logarithms , number theory proposed , number theory

Prove that $ 7^{2^{20}} + 7^{2^{19}} + 1 $ has at least $ 21 $ distinct prime divisors.

2012 Tournament of Towns, 7

Tags: combinatorics

Konstantin has a pile of $100$ pebbles. In each move, he chooses a pile and splits it into two smaller ones until he gets $100 $ piles each with a single pebble. (a) Prove that at some point, there are $30$ piles containing a total of exactly $60$ pebbles. (b) Prove that at some point, there are $20$ piles containing a total of exactly $60$ pebbles. (c) Prove that Konstantin may proceed in such a way that at no point, there are $19$ piles containing a total of exactly $60$ pebbles.

1962 AMC 12/AHSME, 15

Tags: conics , parabola , ellipse , analytic geometry , ratio

Given triangle $ ABC$ with base $ AB$ fixed in length and position. As the vertex $ C$ moves on a straight line, the intersection point of the three medians moves on: $ \textbf{(A)}\ \text{a circle} \qquad \textbf{(B)}\ \text{a parabola} \qquad \textbf{(C)}\ \text{an ellipse} \qquad \textbf{(D)}\ \text{a straight line} \qquad \textbf{(E)}\ \text{a curve here not listed}$

2022 BMT, 11

Tags: number theory

Kylie is trying to count to $202250$. However, this would take way too long, so she decides to only write down positive integers from $1$ to $202250$, inclusive, that are divisible by $125$. How many times does she write down the digit $2$?

2021 Saudi Arabia Training Tests, 11

Tags: IGO , Iran , geometry

Three circles $\omega_1,\omega_2,\omega_3$ are tangent to line $l$ at points $A,B,C$ ($B$ lies between $A,C$) and $\omega_2$ is externally tangent to the other two. Let $X,Y$ be the intersection points of $\omega_2$ with the other common external tangent of $\omega_1,\omega_3$. The perpendicular line through $B$ to $l$ meets $\omega_2$ again at $Z$. Prove that the circle with diameter $AC$ touches $ZX,ZY$. [i]Proposed by Iman Maghsoudi - Siamak Ahmadpour[/i]

2014 VJIMC, Problem 1

Tags: complex numbers , algebra

Find all complex numbers $z$ such that $|z^3+2-2i|+z\overline z|z|=2\sqrt2.$

1976 Poland - Second Round, 4

Tags: geometry , Concyclic

Inside the circle $ S $ there is a circle $ T $ and circles $ K_1, K_2, \ldots, K_n $ tangent externally to $ T $ and internally to $ S $, and the circle $ K_1 $ is tangent to $ K_2 $, $ K_2 $ tangent to $ K_3 $ etc. Prove that the points of tangency of the circles $ K_1 $ with $ K_2 $, $ K_2 $ with $ K_3 $ etc. lie on the circle.

2015 ASDAN Math Tournament, 9

Tags: 2015 , Guts Test

Compute the sum of the digits of $101^6$.

2007 Purple Comet Problems, 10

Tags:

Tom can run to Beth's house in $63$ minutes. Beth can run to Tom's house in $84$ minutes. At noon Tom starts running from his house toward Beth's house while at the same time Beth starts running from her house toward Tom's house. When they meet, they both run at Beth's speed back to Beth's house. At how many minutes after noon will they arrive at Beth's house?