Olympiad problems

2021 CMIMC, 2.8 1.4

Tags: combinatorics

Suppose you have a $6$ sided dice with $3$ faces colored red, $2$ faces colored blue, and $1$ face colored green. You roll this dice $20$ times and record the color that shows up on top. What is the expected value of the product of the number of red faces, blue faces, and green faces? [i]Proposed by Daniel Li[/i]

2019 Saudi Arabia JBMO TST, 4

Tags: inequalities

Let $n$ be positive integer and let $a_1, a_2,...,a_n$ be real numbers. Prove that there exist positive integers $m, k$ $<=n$ , $|$ $(a_1+a_2+...+a_m)$ $-$ $(a_{m+1}+a_{m+2}+...+a_n)$ $|$ $<=$ $|$ $a_k$ $|$

2013 Sharygin Geometry Olympiad, 7

Tags: geometry , circles , rectangle

In the plane, four points are marked. It is known that these points are the centers of four circles, three of which are pairwise externally tangent, and all these three are internally tangent to the fourth one. It turns out, however, that it is impossible to determine which of the marked points is the center of the fourth (the largest) circle. Prove that these four points are the vertices of a rectangle.

2019 Pan-African Shortlist, N5

Tags: number theory

Let $n > 1$ be a positive integer. Prove that every term of the sequence $$ n - 1, n^n - 1, n^{n^2} - 1, n^{n^3} - 1, \dots $$ has a prime divisor that does not divide any of the previous terms.

2012 AIME Problems, 6

Tags: inequalities , geometry , geometric transformation , reflection , triangle inequality

Let $z = a + bi$ be the complex number with $|z| = 5$ and $b > 0$ such that the distance between $(1 + 2i)z^3$ and $z^5$ is maximized, and let $z^4 = c + di$. Find $c+d$.

1998 Hungary-Israel Binational, 2

Tags: parallelogram , geometry

On the sides of a convex hexagon $ ABCDEF$ , equilateral triangles are constructd in its exterior. Prove that the third vertices of these six triangles are vertices of a regular hexagon if and only if the initial hexagon is [i]affine regular[/i]. (A hexagon is called affine regular if it is centrally symmetric and any two opposite sides are parallel to the diagonal determine by the remaining two vertices.)

2020 LIMIT Category 2, 12

Tags: gcd , number theory , limit

Let $A$ be the set $\{k^{19}-k: 1<k<20, k\in N\}$. Let $G$ be the GCD of all elements of $A$. Then the value of $G$ is?

2021 CHMMC Winter (2021-22), 5

Tags: algebra

Find all functions $f : R \to R$ such that $$f(f(x) + f(y)^2) = f(x)^2 +y^2f(y)^3.$$ Here $R$ denotes the usual real numbers.

2001 Portugal MO, 1

Tags: combinatorics

To be able to play the Game of Glory, seven friends have to form four teams which, obviously, may not have the same number of members. In how many different ways can they form these teams? (The order of people within teams is not important, but each person can only be part of one team)

2010 Mathcenter Contest, 1

Tags: polynomial , algebra

Let $a,b,c\in\mathbb{N}$ prove that if there is a polynomial $P,Q,R\in\mathbb{C}[x]$, which have no common factors and satisfy $$P^a+Q^b=R^c$$ and $$\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}>1.$$ [i](tatari/nightmare)[/i]

1999 Harvard-MIT Mathematics Tournament, 5

Tags: combinatorics

If $a$ and $b$ are randomly selected real numbers between $0$ and $1$, find the probability that the nearest integer to $\frac{a-b}{a+b}$ is odd.

2013 China Team Selection Test, 1

Tags: symmetry , number theory

For a positive integer $N>1$ with unique factorization $N=p_1^{\alpha_1}p_2^{\alpha_2}\dotsb p_k^{\alpha_k}$, we define \[\Omega(N)=\alpha_1+\alpha_2+\dotsb+\alpha_k.\] Let $a_1,a_2,\dotsc, a_n$ be positive integers and $p(x)=(x+a_1)(x+a_2)\dotsb (x+a_n)$ such that for all positive integers $k$, $\Omega(P(k))$ is even. Show that $n$ is an even number.

Kyiv City MO 1984-93 - geometry, 1984.9.5

Tags: geometry , construction

Using a ruler with a length of $20$ cm and a compass with a maximum deviation of $10$ cm to connect the segment given two points lying at a distance of $1$ m.

2025 Israel TST, P1

Tags: number theory

For a positive integer $ n \geq 2 $, does there exist positive integer solutions to the following system of equations? \[ \begin{cases} a^n - 2b^n = 1, \\ b^n - 2c^n = 1. \end{cases} \]

2024 Azerbaijan JBMO TST, 4

Tags: algebra , inequality

Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that $$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$

2008 Puerto Rico Team Selection Test, 3

Tags:

A sack contains blue and red marbles*. Consider the following game: marbles are taken out of the sack, one-by-one, until there is an equal number of blue and red marbles; once the number of blue marbles equals the number of red marbles, the game is over. In an instance of this game, it is observed that, at the end, $ 10$ marbles were taken out of the bag, and no $ 3$ consecutive marbles were all of the same color. Prove that, in said instance of the game, the fifth and sixth marbles were of different color. *The original problem involved "stones."

2022 District Olympiad, P3

Tags: sequence , limit

Let $(x_n)_{n\geq 1}$ be the sequence defined recursively as such: \[x_1=1, \ x_{n+1}=\frac{x_1}{n+1}+\frac{x_2}{n+2}+\cdots+\frac{x_n}{2n} \ \forall n\geq 1.\]Consider the sequence $(y_n)_{n\geq 1}$ such that $y_n=(x_1^2+x_2^2+\cdots x_n^2)/n$ for all $n\geq 1.$ Prove that [list=a] [*]$x_{n+1}^2<y_n/2$ and $y_{n+1}<(2n+1)/(2n+2)\cdot y_n$ for all $n\geq 1;$ [*]$\lim_{n\to\infty}x_n=0.$ [/list]

2011 239 Open Mathematical Olympiad, 4

Tags: geometry

In convex quadrilateral $ABCD$, where $AB=AD$, on $BD$ point $K$ is chosen. On $KC$ point $L$ is such that $\bigtriangleup BAD \sim \bigtriangleup BKL$. Line parallel to $DL$ and passes through $K$, intersect $CD$ at $P$. Prove that $\angle APK = \angle LBC$. @below edited

2002 Brazil National Olympiad, 1

Tags: perfect square , number theory , perfect cube , perfect power , sum , subset

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

2016 Kazakhstan National Olympiad, 5

Tags: point , combinatorics , coloring , combinatorial geometry

$101$ blue and $101$ red points are selected on the plane, and no three lie on one straight line. The sum of the pairwise distances between the red points is $1$ (that is, the sum of the lengths of the segments with ends at red points), the sum of the pairwise distances between the blue ones is also $1$, and the sum of the lengths of the segments with the ends of different colors is $400$. Prove that you can draw a straight line separating everything red dots from all blue ones.

2022 New Zealand MO, 1

Tags: number theory , diophantine equation , diophantine

Find all integers $a, b$ such that $$a^2 + b = b^{2022}.$$

1999 Romania National Olympiad, 1

Tags: algebra , system

Solve the system $$\begin{cases} \displaystyle 4^{-x}+27^{-y}= \frac{5}{6} \\ \displaystyle 27^y-4^x \le 1 \\ \displaystyle \log_{27}y-\log_4 x \ge \frac{1}{6} \end{cases}.$$

2022 Latvia Baltic Way TST, P15

Tags: number theory , first digit

Let $d_i$ be the first decimal digit of $2^i$ for every non-negative integer $i$. Prove that for each positive integer $n$ there exists a decimal digit other than $0$ which can be found in the sequence $d_0, d_1, \dots, d_{n-1}$ strictly less than $\frac{n}{17}$ times.

2023 Purple Comet Problems, 8

Tags: combinatorics , number theory

Find the number of ways to write $24$ as the sum of at least three positive integer multiples of $3$. For example, count $3 + 18 + 3$, $18 + 3 + 3$, and $3 + 6 + 3 + 9 + 3$, but not $18 + 6$ or $24$.

1982 AMC 12/AHSME, 4

Tags: perimeter , geometry

The perimeter of a semicircular region, measured in centimeters, is numerically equal to its area, measured in square centimeters. The radius of the semicircle, measured in centimeters, is $\text{(A)} \pi \qquad \text{(B)} \frac{2}{\pi} \qquad \text{(C)} 1 \qquad \text{(D)} \frac{1}{2} \qquad \text{(E)} \frac{4}{\pi} + 2$