Found problems: 85335
1957 Miklós Schweitzer, 6
[b]6.[/b] Let $f(x)$ be an arbitrary function, differentiable infinitely many times. Then the $n$th derivative of $f(e^{x})$ has the form
$\frac{d^{n}}{dx^{n}}f(e^{x})= \sum_{k=0}^{n} a_{kn}e^{kx}f^{(k)}(e^{x})$ ($n=0,1,2,\dots$).
From the coefficients $a_{kn}$ compose the sequence of polynomials
$P_{n}(x)= \sum_{k=0}^{n} a_{kn}x^{k}$ ($n=0,1,2,\dots$)
and find a closed form for the function
$F(t,x) = \sum_{n=0}^{\infty} \frac{P_{n}(x)}{n!}t^{n}.$
[b](S. 22)[/b]
2007 IberoAmerican, 3
Two teams, $ A$ and $ B$, fight for a territory limited by a circumference.
$ A$ has $ n$ blue flags and $ B$ has $ n$ white flags ($ n\geq 2$, fixed). They play alternatively and $ A$ begins the game. Each team, in its turn, places one of his flags in a point of the circumference that has not been used in a previous play. Each flag, once placed, cannot be moved.
Once all $ 2n$ flags have been placed, territory is divided between the two teams. A point of the territory belongs to $ A$ if the closest flag to it is blue, and it belongs to $ B$ if the closest flag to it is white. If the closest blue flag to a point is at the same distance than the closest white flag to that point, the point is neutral (not from $ A$ nor from $ B$). A team wins the game is their points cover a greater area that that covered by the points of the other team. There is a draw if both cover equal areas.
Prove that, for every $ n$, team $ B$ has a winning strategy.
2011 Pre-Preparation Course Examination, 6
We call a subset $S$ of vertices of graph $G$, $2$-dominating, if and only if for every vertex $v\notin S,v\in G$, $v$ has at least two neighbors in $S$. prove that every $r$-regular $(r\ge3)$ graph has a $2$-dominating set with size at most $\frac{n(1+\ln(r))}{r}$.(15 points)
time of this exam was 3 hours
2020 Iranian Geometry Olympiad, 5
Consider an acute-angled triangle $\triangle ABC$ ($AC>AB$) with its orthocenter $H$ and circumcircle $\Gamma$.Points $M$,$P$ are midpoints of $BC$ and $AH$ respectively.The line $\overline{AM}$ meets $\Gamma$ again at $X$ and point $N$ lies on the line $\overline{BC}$ so that $\overline{NX}$ is tangent to $\Gamma$.
Points $J$ and $K$ lie on the circle with diameter $MP$ such that $\angle AJP=\angle HNM$ ($B$ and $J$ lie one the same side of $\overline{AH}$) and circle $\omega_1$, passing through $K,H$, and $J$, and circle $\omega_2$ passing through $K,M$, and $N$, are externally tangent to each other. Prove that the common external tangents of $\omega_1$ and $\omega_2$ meet on the line $\overline{NH}$.
[i]Proposed by Alireza Dadgarnia[/i]
2012 NIMO Problems, 4
Parallel lines $\ell_1$ and $\ell_2$ are drawn in a plane. Points $A_1, A_2, \dots, A_n$ are chosen on $\ell_1$, and points $B_1, B_2, \dots, B_{n+1}$ are chosen on $\ell_2$. All segments $A_iB_j$ are drawn, such that $1 \le i \le n$ and $1 \le j \le n+1$. Let the number of total intersections between these segments (not including endpoints) be denoted by $Q$. Given that no three segments are concurrent, besides at endpoints, prove that $Q$ is divisible by 3.
[i]Proposed by Lewis Chen[/i]
2010 Finnish National High School Mathematics Competition, 3
Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.
2018 AMC 8, 10
The [i]harmonic mean[/i] of a set of non-zero numbers is the reciprocal of the average of the reciprocals of the numbers. What is the harmonic mean of 1, 2, and 4?
$\textbf{(A) }\frac{3}{7}\qquad\textbf{(B) }\frac{7}{12}\qquad\textbf{(C) }\frac{12}{7}\qquad\textbf{(D) }\frac{7}{4}\qquad \textbf{(E) }\frac{7}{3}$
2025 Greece National Olympiad, 2
Let $ABC$ be an acute triangle and $D$ be a point of side $ BC$. Consider points $E,Z$ on line $AD$ such that $EB \perp AB$ and $ZC \perp AC$, and points $H, T $ on line $BC$ such that $EH \parallel AC$ and $ZT \parallel AB$. Circumcircle of triangle $BHE$ intersects for second time line $AB$ at point $M$ ($M \ne B$) and circumcircle of triangle $CTZ$ intersects for second time line $AC$ at point $N$ ($N \ne C$). Prove that lines $MH$, $NT$ and $AD$ concur.
2014 BMO TST, 4
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(x)f(y)=f(x+y)+xy$ for all $x,y\in \mathbb{R}$.
1955 Miklós Schweitzer, 2
[b]2.[/b] Let $f_{1}(x), \dots , f_{n}(x)$ be Lebesgue integrable functions on $[0,1]$, with $\int_{0}^{1}f_{1}(x) dx= 0$ $ (i=1,\dots ,n)$. Show that, for every $\alpha \in (0,1)$, there existis a subset $E$ of $[0,1]$ with measure $\alpha$, such that $\int_{E}f_{i}(x)dx=0$. [b](R. 17)[/b]
2019 Final Mathematical Cup, 2
Let $m=\frac{-1+\sqrt{17}}{2}$. Let the polynomial $P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ is given, where $n$ is a positive integer, the coefficients $a_0,a_1,a_2,...,a_n$ are positive integers and $P(m) =2018$ . Prove that the sum $a_0+a_1+a_2+...+a_n$ is divisible by $2$ .
2014 Iran Team Selection Test, 1
Consider a tree with $n$ vertices, labeled with $1,\ldots,n$ in a way that no label is used twice. We change the labeling in the following way - each time we pick an edge that hasn't been picked before and swap the labels of its endpoints. After performing this action $n-1$ times, we get another tree with its labeling a permutation of the first graph's labeling.
Prove that this permutation contains exactly one cycle.
2004 Harvard-MIT Mathematics Tournament, 7
We have a polyhedron such that an ant can walk from one vertex to another, traveling only along edges, and traversing every edge exactly once. What is the smallest possible total number of vertices, edges, and faces of this polyhedron?
2005 ITAMO, 1
Let $ABC$ be a right angled triangle with hypotenuse $AC$, and let $H$ be the foot of the altitude from $B$ to $AC$. Knowing that there is a right-angled triangle with side-lengths $AB, BC, BH$, determine all the possible values of $\frac{AH}{CH}$
2015 Flanders Math Olympiad, 2
Consider two points $Y$ and $X$ in a plane and a variable point $P$ which is not on $XY$. Let the parallel line to $YP$ through $X$ intersect the internal angle bisector of $\angle XYP$ in $A$, and let the parallel line to $XP$ through $Y$ intersect the internal angle bisector of $\angle YXP$ in $B$. Let $AB$ intersect $XP$ and $YP$ in $S$ and $T$ respectively. Show that the product $|XS|*|YT|$ does not depend on the position of $P$.
PEN H Problems, 40
Determine all pairs of rational numbers $(x, y)$ such that \[x^{3}+y^{3}= x^{2}+y^{2}.\]
2021 Peru Cono Sur TST., P6
Prove that there are no positive integers $a_1, a_2, \ldots , a_{2021}$ (not necessarily distinct) such that for $k = 1, 2, 3, \ldots , 2021$ the number of elements in the set
$$A_k = \{ j \in \mathbb{N} : 1 \le j \le 2021 \text{ and } a_j|k \}$$
be exactly $a_k$.
2012 Purple Comet Problems, 4
The following diagram shows an equilateral triangle and two squares that share common edges. The area of each square is $75$. Find the distance from point $A$ to point $B$.
[asy]
size(175);
defaultpen(linewidth(0.8));
pair A=(-3,0),B=(3,0),C=rotate(60,A)*B,D=rotate(270,B)*C,E=rotate(90,C)*B,F=rotate(270,C)*A,G=rotate(90,A)*C;
draw(A--G--F--C--A--B--C--E--D--B);
label("$A$",F,N);
label("$B$",E,N);[/asy]
1954 Miklós Schweitzer, 5
[b]5.[/b] Let $\xi _{1},\xi _{2},\dots ,\xi _{n},... $ be independent random variables of uniform distribution in $(0,1)$. Show that the distribution of the random variable
$\eta _{n}= \sqrt[]{n}\prod_{k=1}^{n}(1-\frac{\xi _{k}}{k}) (n= 1,2,...)$
tends to a limit distribution for $n \to \infty $. [b](P. 6)[/b]
1988 All Soviet Union Mathematical Olympiad, 481
A polygonal line connects two opposite vertices of a cube with side $2$. Each segment of the line has length $3$ and each vertex lies on the faces (or edges) of the cube. What is the smallest number of segments the line can have?
2021 Austrian Junior Regional Competition, 4
Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$.
Prove that $m> p$.
(Karl Czakler)
2006 China National Olympiad, 6
Suppose $X$ is a set with $|X| = 56$. Find the minimum value of $n$, so that for any 15 subsets of $X$, if the cardinality of the union of any 7 of them is greater or equal to $n$, then there exists 3 of them whose intersection is nonempty.
2010 Contests, 3
Determine all possible values of positive integer $n$, such that there are $n$ different 3-element subsets $A_1,A_2,...,A_n$ of the set $\{1,2,...,n\}$, with $|A_i \cap A_j| \not= 1$ for all $i \not= j$.
1982 Miklós Schweitzer, 10
Let $ p_0,p_1,\ldots$ be a probability distribution on the set of nonnegative integers. Select a number according to this distribution and repeat the selection independently until either a zero or an already selected number is obtained. Write the selected numbers in a row in order of selection without the last one. Below this line, write the numbers again in increasing order. Let $ A_i$ denote the event that the number $ i$ has been selected and that it is in the same place in both lines. Prove that the events $ A_i \;(i\equal{}1,2,\ldots)$ are mutually independent, and $ P(A_i)\equal{}p_i$.
[i]T. F. Mori[/i]
2008 Postal Coaching, 3
Find all real polynomials $P(x, y)$ such that $P(x+y, x-y) = 2P(x, y)$, for all $x, y$ in $R$.