Olympiad problems

2010 Macedonia National Olympiad, 1

Solve the equation \[ x^3+2y^3-4x-5y+z^2=2012, \] in the set of integers.

2019 China National Olympiad, 2

Tags: number theory

Call a set of 3 positive integers $\{a,b,c\}$ a [i]Pythagorean[/i] set if $a,b,c$ are the lengths of the 3 sides of a right-angled triangle. Prove that for any 2 Pythagorean sets $P,Q$, there exists a positive integer $m\ge 2$ and Pythagorean sets $P_1,P_2,\ldots ,P_m$ such that $P=P_1, Q=P_m$, and $\forall 1\le i\le m-1$, $P_i\cap P_{i+1}\neq \emptyset$.

MOAA Team Rounds, 2018.4

Tags: combinatorics , team

Michael and Andrew are playing the game Bust, which is played as follows: Michael chooses a positive integer less than or equal to $99$, and writes it on the board. Andrew then makes a move, which consists of him choosing a positive integer less than or equal to $ 8$ and increasing the integer on the board by the integer he chose. Play then alternates in this manner, with each person making exactly one move, until the integer on the board becomes greater than or equal to $100$. The person who made the last move loses. Let S be the sum of all numbers for which Michael could choose initially and win with both people playing optimally. Find S.

2004 IMC, 3

Tags: limit , function , inequalities

Let $A_n$ be the set of all the sums $\displaystyle \sum_{k=1}^n \arcsin x_k $, where $n\geq 2$, $x_k \in [0,1]$, and $\displaystyle \sum^n_{k=1} x_k = 1$. a) Prove that $A_n$ is an interval. b) Let $a_n$ be the length of the interval $A_n$. Compute $\displaystyle \lim_{n\to \infty} a_n$.

2016 Japan MO Preliminary, 2

Tags: number theory

For $1\leq n\leq 2016$, how many integers $n$ satisfying the condition: the reminder divided by $20$ is smaller than the one divided by $16$.

1999 Romania Team Selection Test, 15

Tags: combinatorics

The participants to an international conference are native and foreign scientist. Each native scientist sends a message to a foreign scientist and each foreign scientist sends a message to a native scientist. There are native scientists who did not receive a message. Prove that there exists a set $S$ of native scientists such that the outer $S$ scientists are exactly those who received messages from those foreign scientists who did not receive messages from scientists belonging to $S$. [i]Radu Niculescu[/i]

1976 AMC 12/AHSME, 15

Tags: modular arithmetic

If $r$ is the remainder when each of the numbers $1059,~1417,$ and $2312$ is divided by $d$, where $d$ is an integer greater than $1$, then $d-r$ equals $\textbf{(A) }1\qquad\textbf{(B) }15\qquad\textbf{(C) }179\qquad\textbf{(D) }d-15\qquad \textbf{(E) }d-1$

2002 IMO Shortlist, 7

Tags: combinatorics , graph theory , extremal graph theory , extremal combinatorics

Among a group of 120 people, some pairs are friends. A [i]weak quartet[/i] is a set of four people containing exactly one pair of friends. What is the maximum possible number of weak quartets ?

2009 JBMO Shortlist, 3

Tags: geometry

Parallelogram ${ABCD}$with obtuse angle $\angle ABC$ is given. After rotation of the triangle ${ACD}$ around the vertex ${C}$, we get a triangle ${CD'A'}$, such that points $B,C$ and ${D'}$are collinear. Extensions of median of triangle ${CD'A'}$ that passes through ${D'}$intersects the straight line ${BD}$ at point ${P}$. Prove that ${PC}$is the bisector of the angle $\angle BP{D}'$.

2021 Durer Math Competition Finals, 1

Tags: algebra

In Sixcountry there are $ 12$ months, but each month consists of $6$ weeks. The month are named the same way we do, from January to December, but in each month the weeks have different lengths. In the $k$-th month the weeks consist of $6^{k-1}$ days. What is the number of days of the spring (March, April, May together)?

2004 Romania Team Selection Test, 12

Tags: inequalities , induction , n-variable inequality

Let $n\geq 2$ be an integer and let $a_1,a_2,\ldots,a_n$ be real numbers. Prove that for any non-empty subset $S\subset \{1,2,3,\ldots, n\}$ we have \[ \left( \sum_{i \in S} a_i \right)^2 \leq \sum_{1\leq i \leq j \leq n } (a_i + \cdots + a_j ) ^2 . \] [i]Gabriel Dospinescu[/i]

2005 May Olympiad, 5

Tags: combinatorics , combinatorial geometry

The enemy ship has landed on a $9\times 9$ board that covers exactly $5$ squares of the board, like this: [img]https://cdn.artofproblemsolving.com/attachments/2/4/ae5aa95f5bb5e113fd5e25931a2bf8eb872dbe.png[/img] The ship is invisible. Each defensive missile covers exactly one square, and destroys the ship if it hits one of the $5$ squares that it occupies. Determine the minimum number of missiles needed to destroy the enemy ship with certainty .

2022 Saudi Arabia JBMO TST, 3

Tags: geometry , incenter

Let $BB_1$ and $CC_1$ be the altitudes of acute-angled triangle $ABC$, and $A_0$ is the midpoint of $BC$. Lines $A_0B_1$ and $A_0C_1$ meet the line passing through $A$ and parallel to $BC$ at points $P$ and $Q$. Prove that the incenter of triangle $PA_0Q$ lies on the altitude of triangle $ABC$.

2013 India PRMO, 11

Tags: algebra , system of equations

Three real numbers $x, y, z$ are such that $x^2 + 6y = -17, y^2 + 4z = 1$ and $z^2 + 2x = 2$. What is the value of $x^2 + y^2 + z^2$?

2022 Iberoamerican, 2

Tags: number theory

Let $S=\{13, 133, \cdots\}$ be the set of the positive integers of the form $133 \cdots 3$. Consider a horizontal row of $2022$ cells. Ana and Borja play a game: they alternatively write a digit on the leftmost empty cell, starting with Ana. When the row is filled, the digits are read from left to right to obtain a $2022$-digit number $N$. Borja wins if $N$ is divisible by a number in $S$, otherwise Ana wins. Find which player has a winning strategy and describe it.

1986 China Team Selection Test, 1

Tags: incenter , rectangle , cyclic quadrilateral , angle chasing , geometry

If $ABCD$ is a cyclic quadrilateral, then prove that the incenters of the triangles $ABC$, $BCD$, $CDA$, $DAB$ are the vertices of a rectangle.

2022 Thailand TST, 2

Tags: algebra , real analysis , inequalities

Let $n\geq 2$ be an integer and let $a_1, a_2, \ldots, a_n$ be positive real numbers with sum $1$. Prove that $$\sum_{k=1}^n \frac{a_k}{1-a_k}(a_1+a_2+\cdots+a_{k-1})^2 < \frac{1}{3}.$$

2019 Greece Junior Math Olympiad, 3

Tags: number theory

Determine all positive integers equal to 13 times the sum of their digits.

1984 Iran MO (2nd round), 3

Tags: function , calculus , derivative , geometric transformation , algebra , geometry

Let $f : \mathbb R \to \mathbb R$ be a function such that \[f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R\] Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$

2008 Sharygin Geometry Olympiad, 20

Tags: perimeter , geometry

(A.Zaslavsky, 10--11) a) Some polygon has the following property: if a line passes through two points which bisect its perimeter then this line bisects the area of the polygon. Is it true that the polygon is central symmetric? b) Is it true that any figure with the property from part a) is central symmetric?

2020 Germany Team Selection Test, 2

Tags: triangle , geometry

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

2020 APMO, 3

Tags: combinatorics , number theory

Determine all positive integers $k$ for which there exist a positive integer $m$ and a set $S$ of positive integers such that any integer $n > m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.

2023 CCA Math Bonanza, TB1

Tags:

$\text{Find }\left(\sum_{k=1}^{2023}{(k^{42432})}\right)\text{ mod 2023}$ [i]Tiebreaker #1[/i]

2019 BMT Spring, 1

Tags: algebra

Find the maximum integral value of $k$ such that $0 \le k \le 2019$ and $|e^{2\pi i \frac{k}{2019}} - 1|$ is maximal.

1996 May Olympiad, 1

Tags: rectangle , area , geometry

Let $ABCD$ be a rectangle. A line $r$ moves parallel to $AB$ and intersects diagonal $AC$ , forming two triangles opposite the vertex, inside the rectangle. Prove that the sum of the areas of these triangles is minimal when $r$ passes through the midpoint of segment $AD$ .