Found problems: 85335
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.
[hide]https://artofproblemsolving.com/community/c6h605458p3596629
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 $\angle EFA$?
2005 China Team Selection Test, 3
Let $a_1,a_2 \dots a_n$ and $x_1, x_2 \dots x_n$ be integers and $r\geq 2$ be an integer. It is known that \[\sum_{j=0}^{n} a_j x_j^k =0 \qquad \text{for} \quad k=1,2, \dots r.\]
Prove that
\[\sum_{j=0}^{n} a_j x_j^m \equiv 0 \pmod m, \qquad \text{for all}\quad m \in \{ r+1, r+2, \cdots, 2r+1 \}.\]
2023 MIG, 11
A [i]semi-palindrome[/i] is a four-digit number whose first two digits and last two digits are identical. For instance, $2323$ and $5757$ are semi-palindromes, but $1001$ and $2324$ are not. What is the difference between the largest semi-palindrome and smallest semi-palindrome?
$\textbf{(A) } 7979\qquad\textbf{(B) } 8080\qquad\textbf{(C) } 8181\qquad\textbf{(D) } 8484\qquad\textbf{(E) } 8989$