Found problems: 85335
1987 All Soviet Union Mathematical Olympiad, 444
Prove that $1^{1987} + 2^{1987} + ... + n^{1987}$ is divisible by $n+2$.
1994 China Team Selection Test, 3
Find the smallest $n \in \mathbb{N}$ such that if any 5 vertices of a regular $n$-gon are colored red, there exists a line of symmetry $l$ of the $n$-gon such that every red point is reflected across $l$ to a non-red point.
2011 NZMOC Camp Selection Problems, 3
There are $16$ competitors in a tournament, all of whom have different playing strengths and in any match between two players the stronger player always wins. Show that it is possible to find the strongest and second strongest players in $18$ matches.
2004 Harvard-MIT Mathematics Tournament, 5
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$.
2018 Iranian Geometry Olympiad, 5
$ABCD$ is a cyclic quadrilateral. A circle passing through $A,B$ is tangent to segment $CD$ at point $E$. Another circle passing through $C,D$ is tangent to $AB$ at point $F$. Point $G$ is the intersection point of $AE,DF$, and point $H$ is the intersection point of $BE$, $CF$. Prove that the incenters of triangles $AGF$, $BHF$, $CHE$, $DGE$ lie on a circle.
Proposed by Le Viet An (Vietnam)
1963 Kurschak Competition, 1
$mn$ students all have different heights. They are arranged in $m > 1$ rows of $n > 1$. In each row select the shortest student and let $A$ be the height of the tallest such. In each column select the tallest student and let $B$ be the height of the shortest such. Which of the following are possible: $A < B$, $A = B$, $A > B$? If a relation is possible, can it always be realized by a suitable arrangement of the students?
2014 HMNT, 9
For any positive integers $a$ and $b$, define $a \oplus b$ to be the result when adding $a$ to $b$ in binary (base $2$), neglecting any carry-overs. For example, $20 \oplus 14 = 10100_2 \oplus 1110_2 = 11010_2 = 26$.
(The operation $\oplus$ is called the [i]exclusive or.[/i])
Compute the sum $$\sum^{2^{2014} -1}_{k=0} \left( k \oplus \left\lfloor \frac{k}{2} \right \rfloor \right).$$ Here $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.
2014 Contests, 2
Roy's cat eats $\frac{1}{3}$ of a can of cat food every morning and $\frac{1}{4}$ of a can of cat food every evening. Before feeding his cat on Monday morning, Roy opened a box containing $6$ cans of cat food. On what day of the week did the cat finish eating all the cat food in the box?
${ \textbf{(A)}\ \text{Tuesday}\qquad\textbf{(B)}\ \text{Wednesday}\qquad\textbf{(C)}\ \text{Thursday}\qquad\textbf{(D)}}\ \text{Friday}\qquad\textbf{(E)}\ \text{Saturday}$
2018 Math Prize for Girls Problems, 7
For every positive integer $n$, let $T_n = \frac{n(n+1)}{2}$ be the $n^{\text{th}}$ triangular number. What is the $2018^{\text{th}}$ smallest positive integer $n$ such that $T_n$ is a multiple of 1000?
2002 Italy TST, 1
Given that in a triangle $ABC$, $AB=3$, $BC=4$ and the midpoints of the altitudes of the triangle are collinear, find all possible values of the length of $AC$.
2009 AMC 12/AHSME, 9
Triangle $ ABC$ has vertices $ A\equal{}(3,0)$, $ B\equal{}(0,3)$, and $ C$, where $ C$ is on the line $ x\plus{}y\equal{}7$. What is the area of $ \triangle ABC$?
$ \textbf{(A)}\ 6\qquad
\textbf{(B)}\ 8\qquad
\textbf{(C)}\ 10\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 14$
Estonia Open Junior - geometry, 2013.2.3
In an isosceles right triangle $ABC$ the right angle is at vertex $C$. On the side $AC$ points $K, L$ and on the side $BC$ points $M, N$ are chosen so that they divide the corresponding side into three equal segments. Prove that there is exactly one point $P$ inside the triangle $ABC$ such that $\angle KPL = \angle MPN = 45^o$.
2014 ISI Entrance Examination, 7
Let $f: [0,\infty)\to \mathbb{R}$ a non-decreasing function. Then show this inequality holds for all $x,y,z$ such that $0\le x<y<z$.
\begin{align*} & (z-x)\int_{y}^{z}f(u)\,\mathrm{du}\ge (z-y)\int_{x}^{z}f(u)\,\mathrm{du} \end{align*}
1966 IMO Shortlist, 17
Let $ABCD$ and $A^{\prime }B^{\prime}C^{\prime }D^{\prime }$ be two arbitrary parallelograms in the space, and let $M,$ $N,$ $P,$ $Q$ be points dividing the segments $AA^{\prime },$ $BB^{\prime },$ $CC^{\prime },$ $DD^{\prime }$ in equal ratios.
[b]a.)[/b] Prove that the quadrilateral $MNPQ$ is a parallelogram.
[b]b.)[/b] What is the locus of the center of the parallelogram $MNPQ,$ when the point $M$ moves on the segment $AA^{\prime }$ ?
(Consecutive vertices of the parallelograms are labelled in alphabetical order.
1949 Moscow Mathematical Olympiad, 157
a) Prove that if a planar polygon has several axes of symmetry, then all of them intersect at one point.
b) A finite solid body is symmetric about two distinct axes. Describe the position of the symmetry planes of the body.
1958 November Putnam, B7
Let $a_1 ,a_2 ,\ldots, a_n$ be a permutation of the integers $1,2,\ldots, n.$ Call $a_i$ a [i]big[/i] integer if $a_i >a_j$ for all $i<j.$ Find the mean number of big integers over all permutations on the first $n$ postive integers.
2014 Contests, 3
Say that a positive integer is [i]sweet[/i] if it uses only the digits 0, 1, 2, 4, and 8. For instance, 2014 is sweet. There are sweet integers whose squares are sweet: some examples (not necessarily the smallest) are 1, 2, 11, 12, 20, 100, 202, and 210. There are sweet integers whose cubes are sweet: some examples (not necessarily the smallest) are 1, 2, 10, 20, 200, 202, 281, and 2424. Prove that there exists a sweet positive integer $n$ whose square and cube are both sweet, such that the sum of all the digits of $n$ is 2014.
2023 Kazakhstan National Olympiad, 2
$a,b,c$ are positive real numbers such that $a+b+c\ge 3$ and $a^2+b^2+c^2=2abc+1$. Prove that $$a+b+c\le 2\sqrt{abc}+1$$
1963 AMC 12/AHSME, 7
Given the four equations:
$\textbf{(1)}\ 3y-2x=12 \qquad
\textbf{(2)}\ -2x-3y=10 \qquad
\textbf{(3)}\ 3y+2x=12 \qquad
\textbf{(4)}\ 2y+3x=10$
The pair representing the perpendicular lines is:
$\textbf{(A)}\ \text{(1) and (4)} \qquad
\textbf{(B)}\ \text{(1) and (3)} \qquad
\textbf{(C)}\ \text{(1) and (2)} \qquad
\textbf{(D)}\ \text{(2) and (4)} \qquad
\textbf{(E)}\ \text{(2) and (3)}$
2018 PUMaC Algebra B, 7
For $k \in \left \{ 0, 1, \ldots, 9 \right \},$ let $\epsilon_k \in \left \{-1, 1 \right \}$. If the minimum possible value of $\sum_{i = 1}^9 \sum_{j = 0}^{i -1} \epsilon_i \epsilon_j 2^{i + j}$ is $m$, find $|m|$.
1982 National High School Mathematics League, 12
Given a circle $C:x^2+y^2=r^2$ ($r$ is an odd number). $P(u,v)\in C$, satisfying: $u=p^m, v=q^n$($p,q$ are prime numbers, $m,n$ are integers, $u>v$).
Define $A,B,C,D,M,N:A(r,0),B(-r,0),C(0,-r),D(0,r),M(u,0),N(0,v)$.
Prove that $|AM|=1,|BM|=9,|CN|=8,|DN|=2$.
1977 Czech and Slovak Olympiad III A, 6
A cube $ABCDA'B'C'D',AA'\parallel BB'\parallel CC'\parallel DD'$ is given. Denote $S$ the center of square $ABCD.$ Determine all points $X$ lying on some edge such that the volumes of tetrahedrons $ABDX$ and $CB'SX$ are the same.
2018 Baltic Way, 7
On a $16 \times 16$ torus as shown all $512$ edges are colored red or blue. A coloring is [i]good [/i]if every vertex is an endpoint of an even number of red edges. A move consists of switching the color of each of the $4$ edges of an arbitrary cell. What is the largest number of good colorings so that none of them can be converted to another by a sequence of moves?
2000 Pan African, 1
Solve for $x \in R$:
\[ \sin^3{x}(1+\cot{x})+\cos^3{x}(1+\tan{x})=\cos{2x} \]
2010 Rioplatense Mathematical Olympiad, Level 3, 1
Let $r_2, r_3,\ldots, r_{1000}$ denote the remainders when a positive odd integer is divided by $2,3,\ldots,1000$, respectively. It is known that the remainders are pairwise distinct and one of them is $0$. Find all values of $k$ for which it is possible that $r_k = 0$.