Found problems: 85335
2005 Greece National Olympiad, 1
Find the polynomial $P(x)$ with real coefficients such that $P(2)=12$ and $P(x^2)=x^2(x^2+1)P(x)$ for each $x\in\mathbb{R}$.
2016 Turkey EGMO TST, 2
In a simple graph, there are two disjoint set of vertices $A$ and $B$ where $A$ has $k$ and $B$ has $2016$ vertices. Four numbers are written to each vertex using the colors red, green, blue and black. There is no any edge at the beginning. For each vertex in $A$, we first choose a color and then draw all edges from this vertex to the vertices in $B$ having a larger number with the chosen color. It is known that for each vertex in $B$, the set of vertices in $A$ connected to this vertex are different. Find the minimal possible value of $k$.
1996 South africa National Olympiad, 6
The function $f$ is increasing and convex (i.e. every straight line between two points on the graph of $f$ lies above the graph) and satisfies $f(f(x))=3^x$ for all $x\in\mathbb{R}$. If $f(0)=0.5$ determine $f(0.75)$ with an error of at most $0.025$. The following are corrent to the number of digits given:
\[3^{0.25}=1.31607,\quad 3^{0.50}=1.73205,\quad 3^{0.75}=2.27951.\]
2005 Croatia National Olympiad, 4
Show that in any set of eleven integers there are six whose sum is divisible by $6$.
2002 USAMTS Problems, 4
The vertices of a cube have coordinates $(0,0,0),(0,0,4),(0,4,0),(0,4,4),(4,0,0)$,$(4,0,4),(4,4,0)$, and $(4,4,4)$. A plane cuts the edges of this cube at the points $(0,2,0),(1,0,0),(1,4,4)$, and two other points. Find the coordinates of the other two points.
2004 Iran MO (3rd Round), 5
assume that k,n are two positive integer $k\leq n$count the number of permutation $\{\ 1,\dots ,n\}\ $ st for any $1\leq i,j\leq k$and any positive integer m we have $f^m(i)\neq j$ ($f^m$ meas iterarte function,)
2024 Germany Team Selection Test, 3
Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$, $b_1,\dots,b_N$, and $c_1,\dots,c_N$ of $1,\dots,N$ such that \[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\] for every $k=1,2,\dots,N$.
2023 Belarus Team Selection Test, 3.2
Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define
$$x_{k+1} = \begin{cases}
x_k + d &\text{if } a \text{ does not divide } x_k \\
x_k/a & \text{if } a \text{ divides } x_k
\end{cases}$$
Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.
2017 India Regional Mathematical Olympiad, 1
Let \(AOB\) be a given angle less than \(180^{\circ}\) and let \(P\) be an interior point of the angular region determined by \(\angle AOB\). Show, with proof, how to construct, using only ruler and compass, a line segment \(CD\) passing through \(P\) such that \(C\) lies on the way \(OA\) and \(D\) lies on the ray \(OB\), and \(CP:PD=1:2\).
2001 China Second Round Olympiad, 1
Let $O,H$ be the circumcenter and orthocenter of $\triangle ABC,$ respectively. Line $AH$ and $BC$ intersect at $D,$ Line $BH$ and $AC$ intersect at $E,$ Line $CH$ and $AB$ intersect at $F,$ Line $AB$ and $ED$ intersect at $M,$ $AC$ and $FD$ intersect at $N.$ Prove that
$(1)OB\perp DF,OC\perp DE;$
$(2)OH\perp MN.$
2014 NIMO Problems, 6
Let $P(x)$ be a polynomial with real coefficients such that $P(12)=20$ and \[ (x-1) \cdot P(16x)= (8x-1) \cdot P(8x) \] holds for all real numbers $x$. Compute the remainder when $P(2014)$ is divided by $1000$.
[i]Proposed by Alex Gu[/i]
2013 China Western Mathematical Olympiad, 8
Find all positive integers $a$ such that for any positive integer $n\ge 5$ we have $2^n-n^2\mid a^n-n^a$.
2010 Harvard-MIT Mathematics Tournament, 10
Let $f(n)=\displaystyle\sum_{k=1}^n \dfrac{1}{k}$. Then there exists constants $\gamma$, $c$, and $d$ such that \[f(n)=\ln(x)+\gamma+\dfrac{c}{n}+\dfrac{d}{n^2}+O\left(\dfrac{1}{n^3}\right),\] where the $O\left(\dfrac{1}{n^3}\right)$ means terms of order $\dfrac{1}{n^3}$ or lower. Compute the ordered pair $(c,d)$.
2015 IMO Shortlist, N3
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
2014 Bosnia And Herzegovina - Regional Olympiad, 2
Solve the equation, where $x$ and $y$ are positive integers: $$ x^3-y^3=999$$
1991 IMTS, 4
Let $n$ points with integer coordinates be given in the $xy$-plane. What is the minimum value of $n$ which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area?
Novosibirsk Oral Geo Oly IX, 2017.3
Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.
2013 Brazil Team Selection Test, 4
Let $a$ and $b$ be positive integers, and let $A$ and $B$ be finite sets of integers satisfying
(i) $A$ and $B$ are disjoint;
(ii) if an integer $i$ belongs to either to $A$ or to $B$, then either $i+a$ belongs to $A$ or $i-b$ belongs to $B$.
Prove that $a\left\lvert A \right\rvert = b \left\lvert B \right\rvert$. (Here $\left\lvert X \right\rvert$ denotes the number of elements in the set $X$.)
2016 All-Russian Olympiad, 2
Diagonals $AC,BD$ of cyclic quadrilateral $ABCD$ intersect at $P$.Point $Q$ is on$BC$ (between$B$ and $C$) such that $PQ \perp AC$.Prove that the line passes through the circumcenters of triangles $APD$ and $BQD$ is parallel to $AD$.(A.Kuznetsov)
2019 Saudi Arabia BMO TST, 1
Let $19$ integer numbers are given. Let Hamza writes on the paper the greatest common divisor for each pair of numbers. It occurs that the difference between the biggest and smallest numbers written on the paper is less than $180$. Prove that not all numbers on the paper are different.
2014 BMT Spring, 2
A mathematician is walking through a library with twenty-six shelves, one for each letter of the alphabet. As he walks, the mathematician will take at most one book off each shelf. He likes symmetry, so if the letter of a shelf has at least one line of symmetry (e.g., M works, L does not), he will pick a book with probability $\frac12$. Otherwise he has a $\frac14$ probability of taking a book. What is the expected number of books that the mathematician will take?
1958 AMC 12/AHSME, 23
If, in the expression $ x^2 \minus{} 3$, $ x$ increases or decreases by a positive amount of $ a$, the expression changes by an amount:
$ \textbf{(A)}\ {\pm 2ax \plus{} a^2}\qquad
\textbf{(B)}\ {2ax \pm a^2}\qquad
\textbf{(C)}\ {\pm a^2 \minus{} 3} \qquad
\textbf{(D)}\ {(x \plus{} a)^2 \minus{} 3}\qquad\\
\textbf{(E)}\ {(x \minus{} a)^2 \minus{} 3}$
2015 Saint Petersburg Mathematical Olympiad, 2
$AB=CD,AD \parallel BC$ and $AD>BC$. $\Omega$ is circumcircle of $ABCD$. Point $E$ is on $\Omega$ such that $BE \perp AD$. Prove that $AE+BC>DE$
2020-2021 OMMC, 2
Sequences $a_n$ and $b_n$ are defined for all positive integers $n$ such that $a_1 = 5,$ $b_1 = 7,$
$$a_{n+1} = \frac{\sqrt{(a_n+b_n-1)^2+(a_n-b_n+1)^2}}{2},$$
and
$$b_{n+1} = \frac{\sqrt{(a_n+b_n+1)^2+(a_n-b_n-1)^2}}{2}.$$
$ $ \\
How many integers $n$ from 1 to 1000 satisfy the property that $a_n, b_n$ form the legs of a right triangle with a hypotenuse that has integer length?
1991 Arnold's Trivium, 60
Is there a solution of the Cauchy problem
\[x(x^2+y^2)\frac{\partial u}{\partial x}+y^3\frac{\partial u}{\partial y}=0,\;u|_{y=0}=1\]
in a neighbourhood of the point $(x_0,0)$ of the $x$-axis? Is it unique?