Found problems: 85335
2011 Princeton University Math Competition, A8
Calculate the sum of the coordinates of all pairs of positive integers $(n, k)$ such that $k\equiv 0, 3\pmod 4$, $n > k$, and $\displaystyle\sum^n_{i = k + 1} i^3 = (96^2\cdot3 - 1)\displaystyle\left(\sum^k_{i = 1} i\right)^2 + 48^2$
2002 China Team Selection Test, 3
Let $ p_i \geq 2$, $ i \equal{} 1,2, \cdots n$ be $ n$ integers such that any two of them are relatively prime. Let:
\[ P \equal{} \{ x \equal{} \sum_{i \equal{} 1}^{n} x_i \prod_{j \equal{} 1, j \neq i}^{n} p_j \mid x_i \text{is a non \minus{} negative integer}, i \equal{} 1,2, \cdots n \}
\]
Prove that the biggest integer $ M$ such that $ M \not\in P$ is greater than $ \displaystyle \frac {n \minus{} 2}{2} \cdot \prod_{i \equal{} 1}^{n} p_i$, and also find $ M$.
2022 IFYM, Sozopol, 4
Let $x_1,\dots ,x_n$ be real numbers. We look at all the $2^{n-1}$ possible sums between some of the numbers. If the number of different sums is at least $1.8^n$, prove that the number of sums equal to $2022$ is no more than $1.67^n$.
2023 Sharygin Geometry Olympiad, 9.8
Let $ABC$ be a triangle with $\angle A = 120^\circ$, $I$ be the incenter, and $M$ be the midpoint of $BC$. The line passing through $M$ and parallel to $AI$ meets the circle with diameter $BC$ at points $E$ and $F$ ($A$ and $E$ lie on the same semiplane with respect to $BC$). The line passing through $E$ and perpendicular to $FI$ meets $AB$ and $AC$ at points $P$ and $Q$ respectively. Find the value of $\angle PIQ$.
2018 BMT Spring, 8
Compute $\displaystyle \sum_{k=1}^{1009} (-1)^{k+1} \dbinom{2018-k}{k-1} 2^{2019 - 2k}$.
2017 Stars of Mathematics, 3
Let
$$ 2^{-n_1}+2^{-n_2}+2^{-n_3}+\cdots,\quad1\le n_1\le n_2\le n_3\le\cdots $$
be the binary representation of the golden ratio minus one.
Prove that $ n_k\le 2^{k-1}-2, $ for all integers $ k\ge 4. $
[i]American Mathematical Monthly[/i]
2012 South East Mathematical Olympiad, 1
Find a triple $(l, m, n)$ of positive integers $(1<l<m<n)$, such that $\sum_{k=1}^{l}k, \sum_{k=l+1}^{m}k, \sum_{k=m+1}^{n}k$ form a geometric sequence in order.
2014 BMT Spring, 9
Suppose $a_1, a_2, ...$ and $b_1, b_2,...$ are sequences satisfying $a_n + b_n = 7$, $a_n = 2b_{n-1} - a_{n-1}$, and $b_n = 2a_{n-1} - b_{n-1}$, for all $n$. If $a_1 = 2$, find $(a_{2014})^2 - (b_{2014})^2$.
.
2022 Stanford Mathematics Tournament, 2
Let $a$, $b$, $c$ be the solutions to $x^3+3x^2-1=0$. Define $S_n=a^n+b^n+c^n$. Given that there are integers $0\le i,j,k\le36$ such that $S_n\equiv i^n+j^n+k^n\pmod{37}$ for all positive integer $n$, determine the product $ijk$.
2015 British Mathematical Olympiad Round 1, 1
On Thursday 1st January 2015, Anna buys one book and one shelf. For the next two years she buys one book every day and one shelf on alternate Thursdays, so she next buys a shelf on 15th January. On how many days in the period Thursday 1st January 2015 until (and including) Saturday 31st December 2016 is it possible for Anna to put all her books on all her shelves, so that there is an equal number of books on each shelf?
2009 Tournament Of Towns, 7
At the entrance to a cave is a rotating round table. On top of the table are $n$ identical barrels, evenly spaced along its circumference. Inside each barrel is a herring either with its head up or its head down. In a move, Ali Baba chooses from $1$ to $n$ of the barrels and turns them upside down. Then the table spins around. When it stops, it is impossible to tell which barrels have been turned over. The cave will open if the heads of the herrings in all $n$ barrels are up or are all down. Determine all values of $n$ for which Ali Baba can open the cave in a fi nite number of moves.
[i](11 points)[/i]
2016 Indonesia MO, 6
For a quadrilateral $ABCD$, we call a square $amazing$ if all of its sides(extended if necessary) pass through distinct vertices of $ABCD$(no side passing through 2 vertices). Prove that for an arbitrary $ABCD$ such that its diagonals are not perpendicular, there exist at least 6 $amazing$ squares
2012-2013 SDML (Middle School), 8
A unit square is cut into four pieces that can be arranged to make an isosceles triangle as shown below. What is the perimeter of the triangle? Express your answer in simplest radical form.
[asy]
filldraw((0, 3)--(-1, 3)--(-2, 2)--(-1, 1)--cycle,lightgrey);
filldraw((0, 3)--(1, 3)--(2, 2)--(1, 1)--cycle,lightgrey);
filldraw((0, 4)--(-1, 3)--(1, 3)--cycle,grey);
draw((-1, 1)--(0,0)--(1, 1));
filldraw((4,1)--(3,2)--(2,0)--(3,0)--cycle,lightgrey);
filldraw((4,1)--(5,2)--(6,0)--(5,0)--cycle,lightgrey);
filldraw((4,1)--(3,0)--(5,0)--cycle,grey);
draw((3,2)--(4,4)--(5,2));
[/asy]
1980 IMO Longlists, 20
Let $S$ be a set of 1980 points in the plane such that the distance between every pair of them is at least 1. Prove that $S$ has a subset of 220 points such that the distance between every pair of them is at least $\sqrt{3}.$
2019 China Team Selection Test, 5
Let $M$ be the midpoint of $BC$ of triangle $ABC$. The circle with diameter $BC$, $\omega$, meets $AB,AC$ at $D,E$ respectively. $P$ lies inside $\triangle ABC$ such that $\angle PBA=\angle PAC, \angle PCA=\angle PAB$, and $2PM\cdot DE=BC^2$. Point $X$ lies outside $\omega$ such that $XM\parallel AP$, and $\frac{XB}{XC}=\frac{AB}{AC}$. Prove that $\angle BXC +\angle BAC=90^{\circ}$.
1997 Czech And Slovak Olympiad IIIA, 1
Let $ABC$ be a triangle with sides $a,b,c$ and corresponding angles $\alpha,\beta\gamma$ . Prove that if $\alpha = 3\beta$ then $(a^2 -b^2)(a-b) = bc^2$ . Is the converse true?
2013 USA TSTST, 9
Let $r$ be a rational number in the interval $[-1,1]$ and let $\theta = \cos^{-1} r$. Call a subset $S$ of the plane [i]good[/i] if $S$ is unchanged upon rotation by $\theta$ around any point of $S$ (in both clockwise and counterclockwise directions). Determine all values of $r$ satisfying the following property: The midpoint of any two points in a good set also lies in the set.
2023 HMNT, 6
Let $ABCD$ be a square of side length $5$. A circle passing through $A$ is tangent to segment $CD$ at $T$ and meets $AB$ and $AD$ again at $X\ne A$ and $Y\ne A$, respectively. Given that $XY = 6$, compute $AT$.
2012 Czech-Polish-Slovak Junior Match, 4
Prove that among any $51$ vertices of the $101$-regular polygon there are three that are the vertices of an isosceles triangle.
2018 Saint Petersburg Mathematical Olympiad, 3
Point $T$ lies on the bisector of $\angle B$ of acuteangled $\triangle ABC$. Circle $S$ with diameter $BT$ intersects $AB$ and $BC$ at points $P$ and $Q$. Circle, that goes through point $A$ and tangent to $S$ at $P$ intersects line $AC$ at $X$. Circle, that goes through point $C$ and tangent to $S$ at $Q$ intersects line $AC$ at $Y$. Prove, that $TX=TY$
2007 AMC 12/AHSME, 22
For each positive integer $ n,$ let $ S(n)$ denote the sum of the digits of $ n.$ For how many values of $ n$ is $ n \plus{} S(n) \plus{} S(S(n)) \equal{} 2007?$
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$
2014 Paraguay Mathematical Olympiad, 5
Let $ABC$ be a triangle with area $92$ square centimeters. Calculate the area of another triangle whose sides have the same lengths as the medians of triangle $ABC$.
2017 Costa Rica - Final Round, A1
Let $P (x)$ be a polynomial of degree $2n$, such that $P (k) =\frac{k}{k + 1}$ for $k = 0,...,2n$. Determine $P (2n + 1)$.
2001 India IMO Training Camp, 1
Let $x$ , $y$ , $z>0$. Prove that if $xyz\geq xy+yz+zx$, then $xyz \geq 3(x+
y+z)$.
2013 ELMO Shortlist, 5
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]