Found problems: 85335
2014 Estonia Team Selection Test, 3
Three line segments, all of length $1$, form a connected figure in the plane. Any two different line segments can intersect only at their endpoints. Find the maximum area of the convex hull of the figure.
2024 Brazil National Olympiad, 2
Let \( ABC \) be a scalene triangle. Let \( E \) and \( F \) be the midpoints of sides \( AC \) and \( AB \), respectively, and let \( D \) be any point on segment \( BC \). The circumcircles of triangles \( BDF \) and \( CDE \) intersect line \( EF \) at points \( K \neq F \), and \( L \neq E \), respectively, and intersect at points \( X \neq D \). The point \( Y \) is on line \( DX \) such that \( AY \) is parallel to \( BC \). Prove that points \( K \), \( L \), \( X \), and \( Y \) lie on the same circle.
2001 Estonia National Olympiad, 4
It is known that the equation$ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$.
2017 Princeton University Math Competition, 16
Robert is a robot who can move freely on the unit circle and its interior, but is attached to the origin by a retractable cord such that at any moment the cord lies in a straight line on the ground connecting Robert to the origin. Whenever his movement is counterclockwise (relative to the origin), the cord leaves a coating of black paint on the ground, and whenever his movement is clockwise, the cord leaves a coating of orange paint on the ground. The paint is dispensed regardless of whether there is already paint on the ground. The paints covers $1$ gallon/unit $^2$, and Robert starts at $(1, 0)$. Each second, he moves in a straight line from the point $(\cos(\theta),\sin(\theta))$ to the point $(\cos(\theta+a),\sin(\theta+a))$, where a changes after each movement. a starts out as $253^o$ and decreases by $2^o$ each step. If he takes $89$ steps, then the difference, in gallons, between the amount of black paint used and orange paint used can be written as $\frac{\sqrt{a}- \sqrt{b}}{c} \cot 1^o$, where $a, b$ and $c$ are positive integers and no prime divisor of $c$ divides both $a$ and $b$ twice. Find $a + b + c$.
1956 Czech and Slovak Olympiad III A, 1
Find all $x,y\in\left(0,\frac{\pi}{2}\right)$ such that
\begin{align*}
\frac{\cos x}{\cos y}&=2\cos^2 y, \\
\frac{\sin x}{\sin y}&=2\sin^2 y.
\end{align*}
2021 Bundeswettbewerb Mathematik, 3
We are given a circle $k$ and a point $A$ outside of $k$. Next we draw three lines through $A$: one secant intersecting the circle $k$ at points $B$ and $C$, and two tangents touching the circle$k$ at points $D$ and $E$. Let $F$ be the midpoint of $DE$.
Show that the line $DE$ bisects the angle $\angle BFC$.
1990 IMO Shortlist, 27
Find all natural numbers $ n$ for which every natural number whose decimal representation has $ n \minus{} 1$ digits $ 1$ and one digit $ 7$ is prime.
2024 Abelkonkurransen Finale, 3a
Determine the smallest constant $N$ so that the following may hold true:
Geostan has deployed secret agents in Combostan. All pairs of agents can communicate, either directly or through other agents. The distance between two agents is the smallest number of agents in a communication chain between the two agents.
Andreas and Edvard are among these agents, and Combostan has given Noah the task of determining the distance between Andreas and Edvard. Noah has a list of numbers, one for each agent. The number of an agent describes the longest of the two distances from the agent to Andreas and Edvard. However, Noah does not know which number corresponds to which agent, or which agents have direct contact.
Given this information, he can write down $N$ numbers and prove that the distance between Andreas and Edvard is one of these $N$ numbers. The number $N$ is independent of the agents’ communication network.
2013 Albania Team Selection Test, 4
It is given a triangle $ABC$ whose circumcenter is $O$ and orthocenter $H$.
If $AO=AH$ find the angle $\hat{BAC}$ of that triangle.
2018 Thailand TST, 2
A sequence of real numbers $a_1,a_2,\ldots$ satisfies the relation
$$a_n=-\max_{i+j=n}(a_i+a_j)\qquad\text{for all}\quad n>2017.$$
Prove that the sequence is bounded, i.e., there is a constant $M$ such that $|a_n|\leq M$ for all positive integers $n$.
2016 Germany Team Selection Test, 1
Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.
2012 NIMO Problems, 12
The NEMO (National Electronic Math Olympiad) is similar to the NIMO Summer Contest, in that there are fifteen problems, each worth a set number of points. However, the NEMO is weighted using Fibonacci numbers; that is, the $n^{\text{th}}$ problem is worth $F_n$ points, where $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n \ge 3$. The two problem writers are fair people, so they make sure that each of them is responsible for problems worth an equal number of total points. Compute the number of ways problem writing assignments can be distributed between the two writers.
[i]Proposed by Lewis Chen[/i]
2025 International Zhautykov Olympiad, 1
Let $a, b$ be positive reals such that $a^3 + b^3 = ab + 1$. Prove that \[(a-b)^2 + a + b \geq 2\]
2023 LMT Fall, 21
Let $(a_1,a_2,a_3,a_4,a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of $$\sum^5_{i=1} |a_i -i | = |a_1 -1|+|a_2 -2|+|a_3 -3|+|a_4 -4|+|a_5 -5|.$$
[i]Proposed by Muztaba Syed[/i]
2011 AMC 12/AHSME, 17
Let $f\left(x\right)=10^{10x}, g\left(x\right)=\log_{10}\left(\frac{x}{10}\right), h_1\left(x\right)=g\left(f\left(x\right)\right),$ and $h_n\left(x\right)=h_1\left(h_{n-1}\left(x\right)\right)$ for integers $n \ge 2$. What is the sum of the digits of $h_{2011}\left(1\right)$?
$ \textbf{(A)}\ 16,081 \qquad
\textbf{(B)}\ 16,089 \qquad
\textbf{(C)}\ 18,089 \qquad
\textbf{(D)}\ 18,098 \qquad
\textbf{(E)}\ 18,099 $
2011 Kosovo National Mathematical Olympiad, 2
Is it possible that by using the following transformations:
\[ f(x) \mapsto x^2 \cdot f \left(\frac{1}{x}+1 \right) \ \ \ \text{or} \ \ \ f(x) \mapsto (x-1)^2 \cdot f\left(\frac{1}{x-1} \right)\]
the function $f(x)=x^2+5x+4$ is sent to the function $g(x)=x^2+10x+8$ ?
2019 HMNT, 7
Consider sequences $a$ of the form $a = (a_1, a_2, ... , a_{20})$ such that each term $a_i$ is either $0$ or $1$. For each such sequence $a$, we can produce a sequence $b = (b_1, b_2, ..., b_{20})$, where $$b_i\begin{cases} a_i + a_{i+1} & i = 1 \\ a_{i-1} + a_i + a_{i+1} & 1 < i < 20\\ a_{i-1} + a_i &i = 20 \end{cases}$$
2007 Thailand Mathematical Olympiad, 17
Compute the product of positive integers $n$ such that $n^2 + 59n + 881$ is a perfect square.
2000 Harvard-MIT Mathematics Tournament, 5
Show that it is impossible to find a triangle in the plane with all integer coordinates such that the lengths of the sides are all odd.
1986 Bulgaria National Olympiad, Problem 2
Let $f(x)$ be a quadratic polynomial with two real roots in the interval $[-1,1]$. Prove that if the maximum value of $|f(x)|$ in the interval $[-1,1]$ is equal to $1$, then the maximum value of $|f'(x)|$ in the interval $[-1,1]$ is not less than $1$.
PEN K Problems, 17
Find all functions $h: \mathbb{Z}\to \mathbb{Z}$ such that for all $x,y\in \mathbb{Z}$: \[h(x+y)+h(xy)=h(x)h(y)+1.\]
2018 CMIMC Algebra, 3
Let $P(x)=x^2+4x+1$. What is the product of all real solutions to the equation $P(P(x))=0$?
2005 AMC 12/AHSME, 1
Two is $ 10 \%$ of $ x$ and $ 20 \%$ of $ y$. What is $ x \minus{} y$?
$ \textbf{(A)}\ 1 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ 5 \qquad
\textbf{(D)}\ 10 \qquad
\textbf{(E)}\ 20$
2015 IMO Shortlist, C4
Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are:
(i) A player cannot choose a number that has been chosen by either player on any previous turn.
(ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn.
(iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game.
The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies.
[i]Proposed by Finland[/i]
2001 All-Russian Olympiad Regional Round, 9.3
In parallelogram $ABCD$, points $M$ and $N$ are selected on sides $AB$ and $BC$ respectively so that $AM = NC$, $Q$ is the intersection point of segments $AN$ and $CM$. Prove that $DQ$ is the bisector of angle $D$.