Found problems: 85335
2003 Tuymaada Olympiad, 4
Given are polynomial $f(x)$ with non-negative integral coefficients and positive integer $a.$ The sequence $\{a_{n}\}$ is defined by $a_{1}=a,$ $a_{n+1}=f(a_{n}).$ It is known that the set of primes dividing at least one of the terms of this sequence is finite.
Prove that $f(x)=cx^{k}$ for some non-negative integral $c$ and $k.$
[i]Proposed by F. Petrov[/i]
[hide="For those of you who liked this problem."]
Check [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=62259]this thread[/url] out.[/hide]
1989 China National Olympiad, 5
Given $1989$ points in the space, any three of them are not collinear. We divide these points into $30$ groups such that the numbers of points in these groups are different from each other. Consider those triangles whose vertices are points belong to three different groups among the $30$. Determine the numbers of points of each group such that the number of such triangles attains a maximum.
2018 239 Open Mathematical Olympiad, 8-9.3
Is it possible to divide all non-empty subsets of a set of 10 elements into triples so that in each triple, two of the subsets do not intersect and in their union give the third?
[i]Proposed by Vladislav Frank[/i]
2021 Kyiv City MO Round 1, 9.2
Roma wrote on the board each of the numbers $2018, 2019, 2020$, $100$ times each. Let us denote by $S(n)$ the sum of digits of positive integer $n$. In one action, Roma can choose any positive integer $k$ and instead of any three numbers $a, b, c$ written on the board write the numbers $2S(a + b) + k, 2S(b + c) + k$ and $2S(c + a) + k$. Can Roma after several such actions make $299$ numbers on the board equal, and the last one differing from them by $1$?
[i]Proposed by Oleksii Masalitin[/i]
2019 LIMIT Category A, Problem 7
The digit in unit place of $1!+2!+\ldots+99!$ is
$\textbf{(A)}~3$
$\textbf{(B)}~0$
$\textbf{(C)}~1$
$\textbf{(D)}~7$
1988 Irish Math Olympiad, 12
Prove that if $n$ is a positive integer ,then \[cos^4\frac{\pi}{2n+1}+cos^4\frac{2\pi}{2n+1}+\cdots+cos^4\frac{n\pi}{2n+1}=\frac{6n-5}{16}.\]
2006 Romania Team Selection Test, 3
For which pairs of positive integers $(m,n)$ there exists a set $A$ such that for all positive integers $x,y$, if $|x-y|=m$, then at least one of the numbers $x,y$ belongs to the set $A$, and if $|x-y|=n$, then at least one of the numbers $x,y$ does not belong to the set $A$?
[i]Adapted by Dan Schwarz from A.M.M.[/i]
1984 IMO Longlists, 27
The function $f(n)$ is defined on the nonnegative integers $n$ by: $f(0) = 0, f(1) = 1$, and
\[f(n) = f\left(n -\frac{1}{2}m(m - 1)\right)-f\left(\frac{1}{2}m(m+ 1)-n\right)\]
for $\frac{1}{2}m(m - 1) < n \le \frac{1}{2}m(m+ 1), m \ge 2$. Find the smallest integer $n$ for which $f(n) = 5$.
1950 Miklós Schweitzer, 8
A coastal battery sights an enemy cruiser lying one kilometer off the coast and opens fire on it at the rate of one round per minute. After the first shot, the cruiser begins to move away at a speed of $ 60$ kilometers an hour. Let the probability of a hit be $ 0.75x^{ \minus{} 2}$, where $ x$ denotes the distance (in kilometers) between the cruiser and the coast ($ x\geq 1$), and suppose that the battery goes on firing till the cruiser either sinks or disappears. Further, let the probability of the cruiser sinking after $ n$ hits be $ 1 \minus{} \frac {1}{4^n}$ ($ n \equal{} 0,1,...$). Show that the probability of the cruiser escaping is $ \frac {2\sqrt {2}}{3\pi}$
2010 Today's Calculation Of Integral, 608
For $a>0$, find the minimum value of $\int_0^1 \frac{ax^2+(a^2+2a)x+2a^2-2a+4}{(x+a)(x+2)}dx.$
2010 Gakusyuin University entrance exam/Science
2014 PUMaC Algebra A, 7
$x$, $y$, and $z$ are positive real numbers that satisfy $x^3+2y^3+6z^3=1$. Let $k$ be the maximum possible value of $2x+y+3z$. Let $n$ be the smallest positive integer such that $k^n$ is an integer. Find the value of $k^n+n$.
2009 Puerto Rico Team Selection Test, 5
The [i]weird [/i] mean of two numbers $ a$ and $ b$ is defined as $ \sqrt {\frac {2a^2 + 3b^2}{5}}$. $ 2009$ positive integers are placed around a circle such that each number is equal to the the weird mean of the two numbers beside it. Show that these $ 2009$ numbers must be equal.
2007 Harvard-MIT Mathematics Tournament, 23
In triangle $ABC$, $\angle ABC$ is obtuse. Point $D$ lies on side $AC$ such that $\angle ABD$ is right, and point $E$ lies on side $AC$ between $A$ and $D$ such that $BD$ bisects $\angle EBC$. Find $CE$ given that $AC=35$, $BC=7$, and $BE=5$.
1967 IMO Shortlist, 6
Three disks of diameter $d$ are touching a sphere in their centers. Besides, every disk touches the other two disks. How to choose the radius $R$ of the sphere in order that axis of the whole figure has an angle of $60^\circ$ with the line connecting the center of the sphere with the point of the disks which is at the largest distance from the axis ? (The axis of the figure is the line having the property that rotation of the figure of $120^\circ$ around that line brings the figure in the initial position. Disks are all on one side of the plane, passing through the center of the sphere and orthogonal to the axis).
2007 Today's Calculation Of Integral, 168
Prove that $\sum_{n=1}^{\infty}\int_{\frac{1}{n+1}}^{\frac{1}{n}}{\left|\frac{1}{x}\sin \frac{\pi}{x}\right| dx}$ diverge for $x>0.$
1984 Austrian-Polish Competition, 4
A regular heptagon $A_1A_2... A_7$ is inscribed in circle $C$. Point $P$ is taken on the shorter arc $A_7A_1$.
Prove that $PA_1+PA_3+PA_5+PA_7 = PA_2+PA_4+PA_6$.
2002 Junior Balkan MO, 1
The triangle $ABC$ has $CA = CB$. $P$ is a point on the circumcircle between $A$ and $B$ (and on the opposite side of the line $AB$ to $C$). $D$ is the foot of the perpendicular from $C$ to $PB$. Show that $PA + PB = 2 \cdot PD$.
2018 Germany Team Selection Test, 2
Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.
1983 Putnam, A1
How many positive integers $n$ are there such that $n$ is an exact divisors of at least one of the numbers $10^{40}$ and $20^{30}$?
2019 Hong Kong TST, 2
Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.
2014 Tajikistan Team Selection Test, 1
Given the polynomial $p(x) = x^2 + x - 70$, do there exist integers $0<m<n$, so that $p(m)$ is divisible by $n$ and $p(m+1)$ is divisible by $n+1$?
[i]Proposed by Nairy Sedrakyan[/i]
1978 Vietnam National Olympiad, 2
Find all values of the parameter $m$ such that the equations $x^2 = 2^{|x|} + |x| - y - m = 1 - y^2$ have only one root.
1991 China National Olympiad, 1
We are given a convex quadrilateral $ABCD$ in the plane.
([i]i[/i]) If there exists a point $P$ in the plane such that the areas of $\triangle ABP, \triangle BCP, \triangle CDP, \triangle DAP$ are equal, what condition must be satisfied by the quadrilateral $ABCD$?
([i]ii[/i]) Find (with proof) the maximum possible number of such point $P$ which satisfies the condition in ([i]i[/i]).
2018 Korea Junior Math Olympiad, 5
Let there be an acute scalene triangle $ABC$ with circumcenter $O$. Denote $D,E$ be the reflection of $O$ with respect to $AB,AC$, respectively. The circumcircle of $ADE$ meets $AB$, $AC$, the circumcircle of $ABC$ at points $K,L,M$, respectively, and they are all distinct from $A$. Prove that the lines $BC,KL,AM$ are concurrent.
2010 F = Ma, 22
A balloon filled with helium gas is tied by a light string to the floor of a car; the car is sealed so that the motion of the car does not cause air from outside to affect the balloon. If the car is traveling with constant speed along a circular path, in what direction will the balloon on the string lean towards?
[asy]
size(300);
draw(circle((0,0),7));
path A=(1,2)--(1,-2)--(-1,-2)--(-1,2)--cycle;
filldraw(shift(7*left)*A,lightgray);
draw((-7,0)--(-7,5),EndArrow(size=21));
label(scale(1.5)*"A",(-8,2),2*N);
label(scale(1.5)*"B",(-8,0),2*W);
label(scale(1.5)*"C",(-7,-2),3*S);
label(scale(1.5)*"D",(-6,0),2*E);
[/asy]
(A) A
(B) B
(C) C
(D) D
(E) Remains vertical