Olympiad problems

Kyiv City MO 1984-93 - geometry, 1988.9.1

Each side of a convex quadrilateral is less than $20$ cm. Prove that you can specify the vertex of the quadrilateral, the distance from which to any point $Q$ inside the quadrilateral is less than $15$ cm.

2009 Turkey Team Selection Test, 3

Tags: search , symmetry , combinatorics unsolved , combinatorics

Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.

2012 Denmark MO - Mohr Contest, 4

Tags: number theory , Digits

Two two-digit numbers $a$ and b satisfy that the product $a \cdot b$ divides the four-digit number one gets by writing the two digits in $a$ followed by the two digits in $b$. Determine all possible values of $a$ and $b$.

2020 Peru Iberoamerican Team Selection Test, P7

Tags: combinatorics

The numbers $1, 2,\ldots ,50$ are written on a blackboard. Ana performs the following operations: she chooses any three numbers $a, b$ and $c$ from the board and replaces them with their sum $a + b + c$ and writes the number $(a + b) (b + c) (c + a)$ in the notebook. Ana performs these operations until there are only two numbers left on the board ($24$ operations in total). Then, she calculates the sum of the numbers written down in her notebook. Let $M$ and $m$ be the maximum and minimum possible of the sums obtained by Ana. Find the value of $\frac{M}{m}$.

2017 China Western Mathematical Olympiad, 4

Tags: combinatorics

Let $n$ and $k$ be given integers such that $n\ge k\ge 2$. Alice and Bob play a game on an $n$ by $n$ table with white cells. They take turns to pick a white cell and color it black. Alice moves first. The game ends as soon as there is at least one black cell in every $k$ by $k$ square after a player moves, who is declared the winner of the game. Who has the winning strategy?

1988 ITAMO, 7

Tags: number theory , greatest common divisor

Given $n \ge 3$ positive integers not exceeding $100$, let $d$ be their greatest common divisor. Show that there exist three of these numbers whose greatest common divisor is also equal to $d$.

2024 IFYM, Sozopol, 1

Tags: algebra

Let $ n \geq 2 $ be a positive integer. Find all $ n $-tuples $ (a_1, \ldots, a_n) $ of complex numbers such that the numbers $ a_1 - 2a_2 $, $ a_2 - 2a_3 $, $\ldots$ , $ a_{n-1} - 2a_n $, $ a_n - 2a_1 $ form a permutation of the numbers $ a_1, \ldots, a_n $.

2020 CMIMC Algebra & Number Theory, Estimation

Tags: algebra , number theory , 2020 , estimation

Vijay picks two random distinct primes $1\le p, q\le 10^4$. Let $r$ be the probability that $3^{2205403200}\equiv 1\bmod pq$. Estimate $r$ in the form $0.abcdef$, where $a, b, c, d, e, f$ are decimal digits.

2023/2024 Tournament of Towns, 3

Tags: number theory

3. Consider all 100-digit positive integers such that each decimal digit of these equals $2,3,4,5,6$, or 7 . How many of these integers are divisible by $2^{100}$ ? Pavel Kozhevnikov

1928 Eotvos Mathematical Competition, 2

Tags: combinatorics

Put the numbers $1,2,3,...,n$ on the circumference of a circle in such a way that the difference between neighboring numbers is at most $2$. Prove that there is just one solution (if regard is paid only to the order in which the numbers are arranged).

1997 Tournament Of Towns, (563) 4

Tags: geometry , combinatorial geometry , combinatorics , Regular

(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail? (b) The same question for regular pentagons. (A Kanel)

2004 Junior Balkan Team Selection Tests - Moldova, 2

Tags: algebra , inequalities

Let $n \in N^*$ . Let $a_1, a_2..., a_n$ be real such that $a_1 + a_2 +...+ a_n \ge 0$. Prove the inequality $\sqrt{a_1^2+1}+\sqrt{a_2^2+1}+...+\sqrt{a_1^2+1}\ge \sqrt{2n(a_1 + a_2 +...+ a_n )}$.

2011 Princeton University Math Competition, A7 / B8

Tags: combinatorics

At the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts $90$ people in the audience. Every minute afterwards, either one person enters the auditorium (due to waking up late) or leaves (in order to take a dreadful math contest). The speaker observes that in this time, exactly $100$ people enter the auditorium, $100$ leave, and $100$ was the largest audience size he saw. Find the largest integer $m$ such that $2^m$ divides the number of different possible sequences of entries and exits given the above information.

2022 Durer Math Competition Finals, 12

Tags: geometry

Csongi taught Benedek how to fold a duck in 8 steps from a $24$ cm $\times 24$ cm piece of paper. The paper is meant to be folded along the dashed line in the direction of the arrow. Once Benedek folded the duck, he undid all the steps, finding crease lines on the square sheet of paper. On one side of the paper, he drew in blue the folds which opened towards Benedek, and in red the folds which opened toward the table. How many cm is the difference between the total length of the blue lines and the red lines? [img]https://cdn.artofproblemsolving.com/attachments/0/1/358a3b2c3b959a85406b94e34c182fd1c2e28d.png[/img]

2006 China Northern MO, 1

Tags: projective geometry , geometry unsolved , geometry

$AB$ is the diameter of circle $O$, $CD$ is a non-diameter chord that is perpendicular to $AB$. Let $E$ be the midpoint of $OC$, connect $AE$ and extend it to meet the circle at point $P$. Let $DP$ and $BC$ meet at $F$. Prove that $F$ is the midpoint of $BC$.

2024 All-Russian Olympiad Regional Round, 11.7

Tags: quadratics , derivative , tangent , conics , parabola , algebra

Graph $G_1$ of a quadratic trinomial $y = px^2 + qx + r$ with real coefficients intersects the graph $G_2$ of a quadratic trinomial $y = x^2$ in points $A$, $B$. The intersection of tangents to $G_2$ in points $A$, $B$ is point $C$. If $C \in G_1$, find all possible values of $p$.

2015 Regional Olympiad of Mexico Southeast, 2

Tags: geometry , altitudes

In a acutangle triangle $ABC, \angle B>\angle C$. Let $D$ the foot of the altitude from $A$ to $BC$ and $E$ the foot of the perpendicular from $D$ to $AC$. Let $F$ a point in $DE$. Prove that $AF$ and $BF$ are perpendiculars if and only if $EF\cdot DC=BD\cdot DE$.

2017 Peru IMO TST, 13

Tags: IMO Shortlist , functional equation , algebra

Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$, \[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]

2016 Iran MO (3rd Round), 3

Tags: algebra , polynomial

Do there exists many infinitely points like $(x_1,y_1),(x_2,y_2),...$ such that for any sequences like {$b_1,b_2,...$} of real numbers there exists a polynomial $P(x,y)\in R[x,y]$ such that we have for all $i$ : $P(x_{i},y_{i})=b_{i}$

2018 Purple Comet Problems, 3

Tags: algebra

Find $x$ so that the arithmetic mean of $x, 3x, 1000$, and $3000$ is $2018$.

I Soros Olympiad 1994-95 (Rus + Ukr), 9.6

Tags: geometry , tangent circles

A circle can be drawn around the quadrilateral $ABCD$. $K$ is a point on the diagonal $BD$ . The straight line $CK$ intersects the side $AD$ at the point $M$. Prove that the circles circumscribed around the triangles $BCK$ and $ACM$ are tangent.

2014 South East Mathematical Olympiad, 7

Tags: number theory unsolved , number theory

Show that there are infinitely many triples of positive integers $(a_i,b_i,c_i)$, $i=1,2,3,\ldots$, satisfying the equation $a^2+b^2=c^4$, such that $c_n$ and $c_{n+1}$ are coprime for any positive integer $n$.

1989 IMO Shortlist, 18

Tags: geometry , algebra , convex polygon , area , IMO Shortlist

Given a convex polygon $ A_1A_2 \ldots A_n$ with area $ S$ and a point $ M$ in the same plane, determine the area of polygon $ M_1M_2 \ldots M_n,$ where $ M_i$ is the image of $ M$ under rotation $ R^{\alpha}_{A_i}$ around $ A_i$ by $ \alpha_i, i \equal{} 1, 2, \ldots, n.$

1998 Romania National Olympiad, 1

Tags: algebra , Integers , combinatorics

Let $n \ge 2$ be an integer and $M= \{1,2,\ldots,n\}.$ For each $k \in \{1,2,\ldots,n-1\}$ we define $$x_k= \frac{1}{n+1} \sum_{\substack{A \subset M \\ |A|=k}} (\min A + \max A).$$ Prove that the numbers $x_k$ are integers and not all of them are divisible by $4.$ [hide=Notations]$|A|$ is the cardinal of $A$ $\min A$ is the smallest element in $A$ $\max A$ is the largest element in $A$[/hide]

2017 ASDAN Math Tournament, 14

Tags: 2017 , Guts Round

What are the last two digits of $2017^{2017}$?