Found problems: 85335
2010 Today's Calculation Of Integral, 664
For a positive integer $n$, let $I_n=\int_{-\pi}^{\pi} \left(\frac{\pi}{2}-|x|\right)\cos nx\ dx$.
Find $I_1+I_2+I_3+I_4$.
[i]1992 University of Fukui entrance exam/Medicine[/i]
2010 SEEMOUS, Problem 2
Inside a square consider circles such that the sum of their circumferences is twice the perimeter of the square.
a) Find the minimum number of circles having this property.
b) Prove that there exist infinitely many lines which intersect at least 3 of these circles.
2018 German National Olympiad, 2
We are given a tetrahedron with two edges of length $a$ and the remaining four edges of length $b$ where $a$ and $b$ are positive real numbers. What is the range of possible values for the ratio $v=a/b$?
1981 Swedish Mathematical Competition, 3
Find all polynomials $p(x)$ of degree $5$ such that $p(x) + 1$ is divisible by $(x-1)^3$ and $p(x) - 1$ is divisible by $(x+1)^3$.
1982 All Soviet Union Mathematical Olympiad, 335
Three numbers $a,b,c$ belong to $[0,\pi /2]$ interval with $$\cos a = a, \sin(\cos b) = b, \cos(\sin c ) = c$$ Sort those numbers in increasing order.
2021 Stars of Mathematics, 1
For every integer $n\geq 3$, let $s_n$ be the sum of all primes (strictly) less than $n$. Are there infinitely many integers $n\geq 3$ such that $s_n$ is coprime to $n$?
[i]Russian Competition[/i]
2009 International Zhautykov Olympiad, 1
On the plane, a Cartesian coordinate system is chosen. Given points $ A_1,A_2,A_3,A_4$ on the parabola $ y \equal{} x^2$, and points $ B_1,B_2,B_3,B_4$ on the parabola $ y \equal{} 2009x^2$. Points $ A_1,A_2,A_3,A_4$ are concyclic, and points $ A_i$ and $ B_i$ have equal abscissas for each $ i \equal{} 1,2,3,4$.
Prove that points $ B_1,B_2,B_3,B_4$ are also concyclic.
2004 Estonia National Olympiad, 2
Draw a line passing through a point $M$ on the angle bisector of the angle $\angle AOB$, that intersects $OA$ and $OB$ at points $K$ and $L$ respectively. Prove that the valus of the sum $\frac{1}{|OK|}+\frac{1}{|OL|}$ does not depend on the choice of the straight line passing through $M$, i.e. is defined by the size of the angle AOB and the selection of the point $M$ only.
2023 CMIMC Combo/CS, 7
Max has a light bulb and a defective switch. The light bulb is initially off, and on the $n$th time the switch is flipped, the light bulb has a $\tfrac 1{2(n+1)^2}$ chance of changing its state (i.e. on $\to$ off or off $\to$ on). If Max flips the switch 100 times, find the probability the light is on at the end.
[i]Proposed by Connor Gordon[/i]
2002 USAMTS Problems, 2
Find four distinct positive integers, $a$, $b$, $c$, and $d$, such that each of the four sums $a+b+c$, $a+b+d$,$a+c+d$, and $b+c+d$ is the square of an integer. Show that infinitely many quadruples $(a,b,c,d)$ with this property can be created.
2023 Canadian Mathematical Olympiad Qualification, 3
Let circles $\Gamma_1$ and $\Gamma_2$ have radii $r_1$ and $r_2$, respectively. Assume that $r_1 < r_2$. Let $T$ be an intersection point of $\Gamma_1$ and $\Gamma_2$, and let $S$ be the intersection of the common external tangents of $\Gamma_1$ and $\Gamma_2$. If it is given that the tangents to $\Gamma_1$ and $ \Gamma_2$ at $T$ are perpendicular, determine the length of $ST$ in terms of $r_1$ and $r_2$.
2013 239 Open Mathematical Olympiad, 1
Consider all permutations of natural numbers from $1$ to $100$. A permutation is called $\emph{double}$ when it has the following property: If you write this permutation twice in a row, then delete $100$ numbers from them you get the remaining numbers $1, 2, 3, \ldots , 100$ in order. How many $\emph{double}$ permutations are there?
1989 Tournament Of Towns, (237) 1
Is it possible to choose a sphere, a triangular pyramid and a plane so that every plane, parallel to the chosen one, intersects the sphere and the pyramid in sections of equal area?
(Problem from Latvia)
1984 IMO Longlists, 45
Let $X$ be an arbitrary nonempty set contained in the plane and let sets $A_1, A_2,\cdots,A_m$ and $B_1, B_2,\cdots, B_n$ be its images under parallel translations. Let us suppose that $A_1\cup A_2 \cup \cdots\cup A_m \subset B_1 \cup B_2 \cup\cdots\cup B_n$ and that the sets $A_1, A_2,\cdots,A_m$ are disjoint. Prove that $m \le n$.
2010 Indonesia TST, 2
Let $T$ be a tree with$ n$ vertices. Choose a positive integer $k$ where $1 \le k \le n$ such that $S_k$ is a subset with $k$ elements from the vertices in $T$. For all $S \in S_k$, define $c(S)$ to be the number of component of graph from $S$ if we erase all vertices and edges in $T$, except all vertices and edges in $S$. Determine $\sum_{S\in S_k} c(S)$, expressed in terms of $n$ and $k$.
2014 Singapore MO Open, 4
Fill up each square of a $50$ by $50$ grid with an integer. Let $G$ be the configuration of $8$ squares obtained by taking a $3$ by $3$ grid and removing the central square. Given that for any such $G$ in the $50$ by $50$ grid, the sum of integers in its squares is positive, show there exist a $2$ by $2$ square such that the sum of its entries is also positive.
2005 Thailand Mathematical Olympiad, 18
Compute the sum $$\sum_{k=0}^{1273}\frac{1}{1 + tan^{2548}\left(\frac{k\pi}{2548}\right)}$$
2013 Harvard-MIT Mathematics Tournament, 13
Find the smallest positive integer $n$ such that $\dfrac{5^{n+1}+2^{n+1}}{5^n+2^n}>4.99$.
2013 USAJMO, 3
In triangle $ABC$, points $P$, $Q$, $R$ lie on sides $BC$, $CA$, $AB$ respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $AQR$, $BRP$, $CPQ$, respectively. Given the fact that segment $AP$ intersects $\omega_A$, $\omega_B$, $\omega_C$ again at $X$, $Y$, $Z$, respectively, prove that $YX/XZ=BP/PC$.
2009 India Regional Mathematical Olympiad, 1
Let $ ABC$ be a triangle in which $ AB \equal{} AC$ and let $ I$ be its in-centre. Suppose $ BC \equal{} AB \plus{} AI$. Find $ \angle{BAC}$
2005 India National Olympiad, 4
All possible $6$-digit numbers, in each of which the digits occur in nonincreasing order (from left to right, e.g. $877550$) are written as a sequence in increasing order. Find the $2005$-th number in this sequence.
2019 Bosnia and Herzegovina Junior BMO TST, 2
$2.$ Let $ABC$ be a triangle and $AD$ the angle bisector ($D\in BC$). The perpendicular from $B$ to $AD$ cuts the circumcircle of triangle $ABD$ at $E$. If $O$ is the center of the circle around $ABC$ , prove $A,O,E$ are collinear.
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https://artofproblemsolving.com/community/c6h1294020p6857833[/hide]
2019 Peru MO (ONEM), 2
Find all the real numbers $k$ that have the following property: For any non-zero real numbers $a$ and $b$, it is true that at least one of the following numbers: $$a, b,\frac{5}{a^2}+\frac{6}{b^3}$$is less than or equal to $k$.
1988 IMO Longlists, 19
Let $Z_{m,n}$ be the set of all ordered pairs $(i,j)$ with $i \in {1, \ldots, m}$ and $j \in {1, \ldots, n}.$ Also let $a_{m,n}$ be the number of all those subsets of $Z_{m,n}$ that contain no 2 ordered pairs $(i_1,j_1)$ and $(i_2,j_2)$ with $|i_1 - i_2| + |j_1 - j_2| = 1.$ Then show, for all positive integers $m$ and $k,$ that \[ a^2_{m, 2 \cdot k} \leq a_{m, 2 \cdot k - 1} \cdot a_{m, 2 \cdot k + 1}. \]
2006 Princeton University Math Competition, 1
$A,B,C,D,E$, and $F$ are points of a convex hexagon, and there is a circle such that $A,B,C,D,E$, and $F$ are all on the circle. If $\angle ABC = 72^o$, $\angle BCD = 96^o$, $\angle CDE = 118^o$, and $\angle DEF = 104^o$, what is $\angle EFA$?