Found problems: 85335
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 $y$.
2010 Saudi Arabia BMO TST, 3
Find all functions $f : R \to R$ such that $$xf(x+xy)= xf(x)+ f(x^2)f(y)$$ for all $x,y \in R$.
Kvant 2023, M2757
Let $p{}$ be a prime number. There are $p{}$ integers $a_0,\ldots,a_{p-1}$ around a circle. In one move, it is allowed to select some integer $k{}$ and replace the existing numbers via the operation $a_i\mapsto a_i-a_{i+k}$ where indices are taken modulo $p{}.$ Find all pairs of natural numbers $(m, n)$ with $n>1$ such that for any initial set of $p{}$ numbers, after performing any $m{}$ moves, the resulting $p{}$ numbers will all be divisible by $n{}.$
[i]Proposed by P. Kozhevnikov[/i]
2023 Assam Mathematics Olympiad, 5
What is the least possible value of $x^2 + y^2 - x - y - xy$ where $x, y$ are real numbers ?
2014 Postal Coaching, 2
Let $O$ be the centre of the square $ABCD$. Let $P,Q,R$ be respectively on the segments $OA,OB,OC$ such that $OP=3,OQ=5,OR=4$. Suppose $S$ is on $OD$ such that $X=AB\cap PQ,Y=BC\cap QR$ and $Z=CD\cap RS$ are collinear. Find $OS$.
2014 Contests, 2
Consider the following sequence
$$(a_n)_{n=1}^{\infty}=(1,1,2,1,2,3,1,2,3,4,1,2,3,4,5,1,\dots)$$
Find all pairs $(\alpha, \beta)$ of positive real numbers such that $\lim_{n\to \infty}\frac{\displaystyle\sum_{k=1}^n a_k}{n^{\alpha}}=\beta$.
(Proposed by Tomas Barta, Charles University, Prague)