Found problems: 85335
2019 Regional Olympiad of Mexico Southeast, 6
Let $p\geq 3$ a prime number, $a$ and $b$ integers such that $\gcd(a, b)=1$. Let $n$ a natural number such that $p$ divides $a^{2^n}+b^{2^n}$, prove that $2^{n+1}$ divides $p-1$.
2008 Bosnia And Herzegovina - Regional Olympiad, 4
Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that:
$i)$ $a(0)=0$
$ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$.
If exists prove:
$a)$ $a(k)\geq a(k-1)$
$b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.
2014 ITAMO, 5
Prove that there exists a positive integer that can be written, in at least two ways, as a sum of $2014$-th powers of $2015$ distinct positive integers $x_1 <x_2 <\cdots <x_{2015}$.
2014 China Northern MO, 6
Let $x,y,z,w $ be real numbers such that $x+2y+3z+4w=1$. Find the minimum of $x^2+y^2+z^2+w^2+(x+y+z+w)^2$.
2007 Mathematics for Its Sake, 1
Prove that the parity of each term of the sequence $ \left( \left\lfloor \left( \lfloor \sqrt q \rfloor +\sqrt{q} \right)^n \right\rfloor \right)_{n\ge 1} $ is opposite to the parity of its index, where $ q $ is a squarefree natural number.
2005 China Team Selection Test, 1
Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.
2018 India PRMO, 30
Let $P(x)$ = $a_0+a_1x+a_2x^2+\cdots +a_nx^n$ be a polynomial in which $a_i$ is non-negative integer for each $i \in$ {$0,1,2,3,....,n$} . If $P(1) = 4$ and $P(5) = 136$, what is the value of $P(3)$?
1999 National High School Mathematics League, 10
$P$ is a point on hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$, if the distance from $P$ to right directrix is the arithmetic mean of the distance from $P$ to two focal points, then the $x$-axis of $P$ is________.
2007 Peru Iberoamerican Team Selection Test, P1
Solve in the set of real numbers, the system:
$$x(3y^2+1)=y(y^2+3)$$
$$y(3z^2+1)=z(z^2+3)$$
$$z(3x^2+1)=x(x^2+3)$$
1974 IMO, 1
Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).
2017 F = ma, 22
22) A particle of mass m moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$.
The fractional energy transfer is the absolute value of the final kinetic energy of $M$ divided by the initial kinetic energy of $m$.
If the collision is perfectly elastic, under what condition will the fractional energy transfer between the two objects be a maximum?
A) $\frac{m}{M} \ll 1$
B) $0.5 < \frac{m}{M} < 1$
C) $m = M$
D) $1 < \frac{m}{M} < 2$
E) $\frac{m}{M} \gg 1$
2016 Junior Balkan Team Selection Tests - Moldova, 6
Determine all pairs $(x, y)$ of natural numbers satisfying the equation $5^x=y^4+4y+1$.
2001 Slovenia National Olympiad, Problem 4
Cross-shaped tiles are to be placed on a $8\times8$ square grid without overlapping. Find the largest possible number of tiles that can be placed.
[img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMy8zL2EyY2Q4MDcyMWZjM2FmZGFhODkxYTk5ZmFiMmMwNzk0MzZmYmVjLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0wNyBhdCA2LjIzLjU4IEFNLnBuZw[/img]
2007 Iran MO (3rd Round), 1
Consider two polygons $ P$ and $ Q$. We want to cut $ P$ into some smaller polygons and put them together in such a way to obtain $ Q$. We can translate the pieces but we can not rotate them or reflect them. We call $ P,Q$ equivalent if and only if we can obtain $ Q$ from $ P$(which is obviously an equivalence relation).
[img]http://i3.tinypic.com/4lrb43k.png[/img]
a) Let $ P,Q$ be two rectangles with the same area(their sides are not necessarily parallel). Prove that $ P$ and $ Q$ are equivalent.
b) Prove that if two triangles are not translation of each other, they are not equivalent.
c) Find a necessary and sufficient condition for polygons $ P,Q$ to be equivalent.
1970 Regional Competition For Advanced Students, 1
Let $x,y,z$ be positive real numbers such that $x+y+z=1$ Prove that always $\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64$
When does equality hold?
2000 AMC 8, 7
What is the minimum possible product of three different numbers of the set $\{-8,-6,-4,0,3,5,7\}$?
$\text{(A)}\ -336 \qquad \text{(B)}\ -280 \qquad \text{(C)}\ -210 \qquad \text{(D)}\ -192 \qquad \text{(E)}\ 0$
2021 China Team Selection Test, 4
Proof that
$$ \sum_{m=1}^n5^{\omega (m)} \le \sum_{k=1}^n\lfloor \frac{n}{k} \rfloor \tau (k)^2 \le \sum_{m=1}^n5^{\Omega (m)} .$$
2020 Jozsef Wildt International Math Competition, W17
Let $(K,+,\cdot)$ be a field with the property $-x=x^{-1},\forall x\in K,x\ne0$. Prove that:
$$(K,+,\cdot)\simeq(\mathbb Z_2,+,\cdot)$$
[i]Proposed by Ovidiu Pop[/i]
2003 JHMMC 8, 9
Compute the product of the integers from $-5$ to $5$, inclusive.
1974 Bundeswettbewerb Mathematik, 2
Seven polygons of area $1$ lie in the interior of a square with side length $2$. Show that there are two of these polygons whose intersection has an area of at least $1\slash 7.$
2020 LMT Fall, A19
Euhan and Minjune are playing a game. They choose a number $N$ so that they can only say integers up to $N$. Euhan starts by saying the $1$, and each player takes turns saying either $n+1$ or $4n$ (if possible), where $n$ is the last number said. The player who says $N$ wins. What is the smallest number larger than $2019$ for which Minjune has a winning strategy?
[i]Proposed by Janabel Xia[/i]
2000 Stanford Mathematics Tournament, 3
A twelve foot tree casts a five foot shadow. How long is Henry's shadow (at the same time of day) if he is five and a half feet tall?
2022 Brazil National Olympiad, 5
Let $n$ be a positive integer number. Define $S(n)$ to be the least positive integer such that $S(n) \equiv n \pmod{2}$, $S(n) \geq n$, and such that there are [b]not[/b] positive integers numbers $k,x_1,x_2,...,x_k$ such that $n=x_1+x_2+...+x_k$ and $S(n)=x_1^2+x_2^2+...+x_k^2$. Prove that there exists a real constant $c>0$ and a positive integer $n_0$ such that, for all $n \geq n_0$, $S(n) \geq cn^{\frac{3}{2}}$.
LMT Theme Rounds, 5
Pixar Prison, for Pixar villains, is shaped like a 600 foot by 1000 foot rectangle with a 300 foot by 500 foot rectangle removed from it, as shown below. The warden separates the prison into three congruent polygonal sections for villains from The Incredibles, Finding Nemo, and Cars. What is the perimeter of each of these sections?
[asy]
draw((0,0)--(0,6)--(10,6)--(10,0)--(8,0)--(8,3)--(3,3)--(3,0)--(0,0));
label("600", (1,3.5));
label("1000", (5.5,6.5));
label("300", (4,1.5));
label("500", (5.5,3.5));
label("300", (1.5,-0.5));
[/asy]
[i]Proposed by Peter Rowley
1991 Vietnam National Olympiad, 3
Prove that:
$ \frac {x^{2}y}{z} \plus{} \frac {y^{2}z}{x} \plus{} \frac {z^{2}x}{y}\geq x^{2} \plus{} y^{2} \plus{} z^{2}$
where $ x;y;z$ are real numbers saisfying $ x \geq y \geq z \geq 0$