Olympiad problems

2004 AIME Problems, 13

The polynomial \[P(x)=(1+x+x^2+\cdots+x^{17})^2-x^{17}\] has 34 complex roots of the form $z_k=r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34$, with $0<a_1\le a_2\le a_3\le\cdots\le a_{34}<1$ and $r_k>0$. Given that $a_1+a_2+a_3+a_4+a_5=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

1997 Slovenia National Olympiad, Problem 4

Tags: logic

In an enterprise, no two employees have jobs of the same difficulty and no two of them take the same salary. Every employee gave the following two claims: (i) Less than $12$ employees have a more difficult work; (ii) At least $30$ employees take a higher salary. Assuming that an employee either always lies or always tells the truth, find how many employees are there in the enterprise.

2008 China Northern MO, 7

Tags: geometry , combinatorics , combinatorial geometry

Given an equilateral triangle lattice composed of $\frac{n(n+1)}{2}$ points (as shown in the figure), record the number of equilateral triangles with three points in the lattice as vertices as $f(n)$. Find an expression for $f(n)$. [img]https://cdn.artofproblemsolving.com/attachments/7/f/1de27231e8ef9c1c6a3dfd590a7c71adc508d6.png[/img]

2016 Puerto Rico Team Selection Test, 2

Tags: number theory , digit

Determine all $6$-digit numbers $(abcdef)$ such that $(abcdef) = (def)^2$ where $(x_1x_2...x_n)$ is not a multiplication but a number of $n$ digits.

2018 IFYM, Sozopol, 1

Tags: geometry

In a quadrilateral $ABCD$ diagonal $AC$ is a bisector of $\angle BAD$ and $\angle ADC = \angle ACB$. The points $X$ and $Y$ are the feet of the perpendiculars from $A$ to $BC$ and $CD$ respectively. Prove that the orthocenter of $\triangle AXY$ lies on the line $BD$.

2024 Harvard-MIT Mathematics Tournament, 4

Tags: geometry

Let $ABCD$ be a square, and let $l$ be a line passing through the midpoint of segment $AB$ that intersects segment $BC$. Given that the distances from $A$ and $C$ to $l$ are $4$ and $7$, respectively, compute the area of $ABCD$.

2019 Saint Petersburg Mathematical Olympiad, 7

Tags: combinatorics , game , minimum , game strategy

In a circle there are $2019$ plates, on each lies one cake. Petya and Vasya are playing a game. In one move, Petya points at a cake and calls number from $1$ to $16$, and Vasya moves the specified cake to the specified number of check clockwise or counterclockwise (Vasya chooses the direction each time). Petya wants at least some $k$ pastries to accumulate on one of the plates and Vasya wants to stop him. What is the largest $k$ Petya can succeed?

2006 AMC 12/AHSME, 4

Tags:

A digital watch displays hours and minutes with $ \text c{AM}$ and $ \text c{PM}$. What is the largest possible sum of the digits in the display? $ \textbf{(A) } 17\qquad \textbf{(B) } 19\qquad \textbf{(C) } 21\qquad \textbf{(D) } 22\qquad \textbf{(E) } 23$

2006 Sharygin Geometry Olympiad, 10.6

Tags: cyclic quadrilateral , tangential quadrilateral , construction , incenter , circumcenter , geometry

A quadrangle was drawn on the board, that you can inscribe and circumscribe a circle. Marked are the centers of these circles and the intersection point of the lines connecting the midpoints of the opposite sides, after which the quadrangle itself was erased. Restore it with a compass and ruler.

2015 BMT Spring, 1

Tags: combinatorics

The boba shop sells four different types of milk tea, and William likes to get tea each weekday. If William refuses to have the same type of tea on successive days, how many different combinations could he get, Monday through Friday?

1987 AMC 12/AHSME, 24

Tags: function , polynomial , algebra

How many polynomial functions $f$ of degree $\ge 1$ satisfy \[ f(x^2)=[f(x)]^2=f(f(x)) \ ? \] $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ \text{finitely many but more than 2} \\ \qquad\textbf{(E)}\ \text{infinitely many} $

2018 Greece Team Selection Test, 3

Tags: function , number theory

Find all functions $f:\mathbb{Z}_{>0}\mapsto\mathbb{Z}_{>0}$ such that $$xf(x)+(f(y))^2+2xf(y)$$ is perfect square for all positive integers $x,y$. **This problem was proposed by me for the BMO 2017 and it was shortlisted. We then used it in our TST.

1966 AMC 12/AHSME, 12

Tags:

The number of real values of $x$ that satisfy the equation\[ (2^{6x+3})(4^{3x+6})=8^{4x+5} \]is: $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ \text{greater than 3}$

2021 Dutch IMO TST, 2

Tags: algebra , system of equations , system

Find all quadruplets $(x_1, x_2, x_3, x_4)$ of real numbers such that the next six equalities apply: $$\begin{cases} x_1 + x_2 = x^2_3 + x^2_4 + 6x_3x_4\\ x_1 + x_3 = x^2_2 + x^2_4 + 6x_2x_4\\ x_1 + x_4 = x^2_2 + x^2_3 + 6x_2x_3\\ x_2 + x_3 = x^2_1 + x^2_4 + 6x_1x_4\\ x_2 + x_4 = x^2_1 + x^2_3 + 6x_1x_3 \\ x_3 + x_4 = x^2_1 + x^2_2 + 6x_1x_2 \end{cases}$$

2013 German National Olympiad, 3

Tags: geometry , circles , tangent

Given two circles $k_1$ and $k_2$ which intersect at $Q$ and $Q'.$ Let $P$ be a point on $k_2$ and inside of $k_1 $ such that the line $PQ$ intersects $k_1$ in a point $X\ne Q$ and such that the tangent to $k_1$ at $X$ intersects $k_2$ in points $A$ and $B.$ Let $k$ be the circle through $A,B$ which is tangent to the line through $P$ parallel to $AB.$ Prove that the circles $k_1$ and $k$ are tangent.

2025 Iran MO (2nd Round), 4

Tags: geometry , circumcircle , reflection , power of a point , radical axis

Given is an acute and scalene triangle $ABC$ with circumcenter $O$. $BO$ and $CO$ intersect the altitude from $A$ to $BC$ at points $P$ and $Q$ respectively. $X$ is the circumcenter of triangle $OPQ$ and $O'$ is the reflection of $O$ over $BC$. $Y$ is the second intersection of circumcircles of triangles $BXP$ and $CXQ$. Show that $X,Y,O'$ are collinear.

2023 Federal Competition For Advanced Students, P2, 4

Tags: algebra

The number $2023$ is written $2023$ times on a blackboard. On one move, you can choose two numbers $x, y$ on the blackboard, delete them and write $\frac{x+y} {4}$ instead. Prove that when one number remains, it is greater than $1$.

2012 Harvard-MIT Mathematics Tournament, 8

Tags: geometry , hmmt , circumcircle , incircle , hexagon

Hexagon $ABCDEF$ has a circumscribed circle and an inscribed circle. If $AB = 9$, $BC = 6$, $CD = 2$, and $EF = 4$. Find $\{DE, FA\}$.

2011 Math Prize For Girls Problems, 6

Tags: geometry

Two circles each have radius 1. No point is inside both circles. The circles are contained in a square. What is the area of the smallest such square?

1970 IMO Longlists, 6

Tags: function , algebra

There is an equation $\sum_{i=1}^{n}{\frac{b_i}{x-a_i}}=c$ in $x$, where all $b_i >0$ and $\{a_i\}$ is a strictly increasing sequence. Prove that it has $n-1$ roots such that $x_{n-1}\le a_n$, and $a_i \le x_i$ for each $i\in\mathbb{N}, 1\le i\le n-1$.

2005 China Team Selection Test, 2

Tags: number theory

Given prime number $p$. $a_1,a_2 \cdots a_k$ ($k \geq 3$) are integers not divible by $p$ and have different residuals when divided by $p$. Let \[ S_n= \{ n \mid 1 \leq n \leq p-1, (na_1)_p < \cdots < (na_k)_p \} \] Here $(b)_p$ denotes the residual when integer $b$ is divided by $p$. Prove that $|S|< \frac{2p}{k+1}$.

2021 BMT, 16

Tags: algebra , combinatorics

Jason and Valerie agree to meet for game night, which runs from $4:00$ PM to $5:00$ PM. Jason and Valerie each choose a random time from $4:00$ PM to $5:00$ PM to show up. If Jason arrives first, he will wait $20$ minutes for Valerie before leaving. If Valerie arrives first, she will wait $10$ minutes for Jason before leaving. What is the probability that Jason and Valerie successfully meet each other for game night?

2016 Baltic Way, 3

Tags: number theory , diophantine equation

For which integers $n = 1, \ldots , 6$ does the equation $$a^n + b^n = c^n + n$$ have a solution in integers?

2024 Macedonian Balkan MO TST, Problem 3

Tags: number theory , prime number

Let $p \neq 5$ be a prime number. Prove that $p^5-1$ has a prime divisor of the form $5x+1$.

2016 Sharygin Geometry Olympiad, P16

Tags: geometry , circumcircle , euler line

Let $BB_1$ and $CC_1$ be altitudes of triangle $ABC$. The tangents to the circumcircle of $AB_1C_1$ at $B_1$ and $C_1$ meet AB and $AC$ at points $M$ and $N$ respectively. Prove that the common point of circles $AMN$ and $AB_1C_1$ distinct from $A$ lies on the Euler line of $ABC$.