Found problems: 85335
1975 Chisinau City MO, 90
Construct a right-angled triangle along its two medians, starting from the acute angles.
2021 Tuymaada Olympiad, 5
Sines of three acute angles form an arithmetic progression, while the cosines of these angles form a geometric progression. Prove that all three angles are equal.
1997 Italy TST, 2
Let $ABC$ be a triangle with $AB = AC$. Suppose that the bisector of $\angle ABC$ meets the side $AC$ at point $D$ such that $BC = BD+AD$. Find the measure of $\angle BAC$.
2016 India IMO Training Camp, 1
Let $n$ be a natural number. We define sequences $\langle a_i\rangle$ and $\langle b_i\rangle$ of integers as follows. We let $a_0=1$ and $b_0=n$. For $i>0$, we let $$\left( a_i,b_i\right)=\begin{cases} \left(2a_{i-1}+1,b_{i-1}-a_{i-1}-1\right) & \text{if } a_{i-1}<b_{i-1},\\
\left( a_{i-1}-b_{i-1}-1,2b_{i-1}+1\right) & \text{if } a_{i-1}>b_{i-1},\\
\left(a_{i-1},b_{i-1}\right) & \text{if } a_{i-1}=b_{i-1}.\end{cases}$$
Given that $a_k=b_k$ for some natural number $k$, prove that $n+3$ is a power of two.
2011 AMC 10, 4
LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip it turned out that LeRoy had paid $A$ dollars and Bernardo had paid $B$ dollars, where $A < B$. How many dollars must LeRoy give to Bernardo so that they share the costs equally?
$ \textbf{(A)}\ \frac{A+B}{2} \qquad
\textbf{(B)}\ \frac{A-B}{2} \qquad
\textbf{(C)}\ \frac{B-A}{2} \qquad
\textbf{(D)}\ B-A \qquad
\textbf{(E)}\ A+B $
2015 AMC 12/AHSME, 4
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller?
$\textbf{(A) }\dfrac54\qquad\textbf{(B) }\dfrac32\qquad\textbf{(C) }\dfrac95\qquad\textbf{(D) }2\qquad\textbf{(E) }\dfrac52$
2000 JBMO ShortLists, 8
Find all positive integers $a,b$ for which $a^4+4b^4$ is a prime number.
2013 Princeton University Math Competition, 6
Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$. Find the remainder when $d$ is divided by $2013$.
1997 Tournament Of Towns, (557) 2
Let $a$ and $b$ be two sides of a triangle. How should the third side $c$ be chosen so that the points of contact of the incircle and the excircle with side $c$ divide that side into three equal segments? (The excircle corresponding to the side $c$ is the circle which is tangent to the side $c$ and to the extensions of the sides $a$ and $b$.)
(Folklore)
2023 Chile TST Ibero., 1
Given a non-negative integer \( n \), determine the values of \( c \) for which the sequence of numbers
\[
a_n = 4^n c + \frac{4^n - (-1)^n}{5}
\]
contains at least one perfect square.
2021 Azerbaijan EGMO TST, 4
Let $ABC$ be an acute, non isosceles with $I$ is its incenter. Denote $D, E$ as tangent points of $(I)$ on $AB,AC$, respectively. The median segments respect to vertex $A$ of triangles $ABE$ and $ACD$ meet$ (I)$ at$ P,Q,$ respectively. Take points $M, N$ on the line $DE$ such that $AM \parallel BE$ and $AN \parallel C D$ respectively.
a) Prove that $A$ lies on the radical axis of $(MIP)$ and $(NIQ)$.
b) Suppose that the orthocenter $H$ of triangle $ABC$ lies on $(I)$. Prove that there exists a line which is tangent to three circles of center $A, B, C$ and all pass through $H$.
2016 Latvia National Olympiad, 3
Is it possible to insert numbers $1, \ldots, 16$ into a table $4 \times 4$ (each cell should have a different number) so that every two adjacent cells (i.e. cells sharing a common side) have numbers $a$ and $b$ satisfying\\
(a) $|a-b| \geq 6$\\
(b) $|a-b| \geq 7$
2019 Baltic Way, 1
For all non-negative real numbers $x,y,z$ with $x \geq y$, prove the inequality
$$\frac{x^3-y^3+z^3+1}{6}\geq (x-y)\sqrt{xyz}.$$
2021 German National Olympiad, 2
Let $P$ on $AB$, $Q$ on $BC$, $R$ on $CD$ and $S$ on $AD$ be points on the sides of a convex quadrilateral $ABCD$. Show that the following are equivalent:
(1) There is a choice of $P,Q,R,S$, for which all of them are interior points of their side, such that $PQRS$ has minimal perimeter.
(2) $ABCD$ is a cyclic quadrilateral with circumcenter in its interior.
1991 USAMO, 3
Show that, for any fixed integer $\,n \geq 1,\,$ the sequence \[ 2, \; 2^2, \; 2^{2^2}, \; 2^{2^{2^2}}, \ldots (\mbox{mod} \; n) \] is eventually constant.
[The tower of exponents is defined by $a_1 = 2, \; a_{i+1} = 2^{a_i}$. Also $a_i \; (\mbox{mod} \; n)$ means the remainder which results from dividing $a_i$ by $n$.]
2008 Harvard-MIT Mathematics Tournament, 7
Given that $ x \plus{} \sin y \equal{} 2008$ and $ x \plus{} 2008 \cos y \equal{} 2007$, where $ 0 \leq y \leq \pi/2$, find the value of $ x \plus{} y$.
2017 Dutch BxMO TST, 2
Let define a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ such that :
$i)$$f(p)=1$ for all prime numbers $p$.
$ii)$$f(xy)=xf(y)+yf(x)$ for all positive integers $x,y$
find the smallest $n \geq 2016$ such that $f(n)=n$
2012 Tournament of Towns, 1
A treasure is buried under a square of an $8\times 8$ board. Under each other square is a message which indicates the minimum number of steps needed to reach the square with the treasure. Each step takes one from a square to another square sharing a common side. What is the minmum number of squares we must dig up in order to bring up the treasure for sure?
1949 Kurschak Competition, 2
$P$ is a point on the base of an isosceles triangle. Lines parallel to the sides through $P$ meet the sides at $Q$ and $R$. Show that the reflection of $P$ in the line $QR$ lies on the circumcircle of the triangle.
1999 Vietnam Team Selection Test, 2
Two polynomials $f(x)$ and $g(x)$ with real coefficients are called similar if there exist nonzero real number a such that $f(x) = q \cdot g(x)$ for all $x \in R$.
[b]I.[/b] Show that there exists a polynomial $P(x)$ of degree 1999 with real coefficients which satisfies the condition: $(P(x))^2 - 4$ and $(P'(x))^2 \cdot (x^2-4)$ are similar.
[b]II.[/b] How many polynomials of degree 1999 are there which have above mentioned property.
2020 Bangladesh Mathematical Olympiad National, Problem 9
Bristy wants to build a special set $A$. She starts with $A=\{0, 42\}$. At any step, she can add an integer $x$ to the set $A$ if it is a root of a polynomial which uses the already existing integers in $A$ as coefficients. She keeps doing this, adding more and more numbers to $A$. After she eventually runs out of numbers to add to $A$, how many numbers will be in $A$?
2016 NIMO Problems, 4
Let $f(x,y)$ be a function defined for all pairs of nonnegative integers $(x, y),$ such that $f(0,k)=f(k,0)=2^k$ and \[f(a,b)+f(a+1,b+1)=f(a+1,b)+f(a,b+1)\] for all nonnegative integers $a, b.$ Determine the number of positive integers $n\leq2016$ for which there exist two nonnegative integers $a, b$ such that $f(a,b)=n$.
[i]Proposed by Michael Ren[/i]
2009 AIME Problems, 9
Let $ m$ be the number of solutions in positive integers to the equation $ 4x\plus{}3y\plus{}2z\equal{}2009$, and let $ n$ be the number of solutions in positive integers to the equation $ 4x\plus{}3y\plus{}2z\equal{}2000$. Find the remainder when $ m\minus{}n$ is divided by $ 1000$.
2005 Switzerland - Final Round, 7
Let $n\ge 1$ be a natural number. Determine all positive integer solutions of the equation
$$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$
2024 Macedonian Mathematical Olympiad, Problem 3
Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfy the equation
$$f(f(x+y))=f(x+y)+f(x)f(y)-xy,$$
for any two real numbers $x$ and $y$.