Found problems: 85335
2011 IMAR Test, 1
Let $A_0A_1A_2$ be a triangle and let $P$ be a point in the plane, not situated on the circle $A_0A_1A_2$. The line $PA_k$ meets again the circle $A_0A_1A_2$ at point $B_k, k = 0, 1, 2$. A line $\ell$ through the point $P$ meets the line $A_{k+1}A_{k+2}$ at point $C_k, k = 0, 1, 2$. Show that the lines $B_kC_k, k = 0, 1, 2$, are concurrent and determine the locus of their concurrency point as the line $\ell$ turns about the point $P$.
2013 South East Mathematical Olympiad, 6
$n>1$ is an integer. The first $n$ primes are $p_1=2,p_2=3,\dotsc, p_n$. Set $A=p_1^{p_1}p_2^{p_2}...p_n^{p_n}$.
Find all positive integers $x$, such that $\dfrac Ax$ is even, and $\dfrac Ax$ has exactly $x$ divisors
2012 National Olympiad First Round, 34
If $10$ divides the number $1\cdot2^1+2\cdot2^2+3\cdot2^3+\dots+n\cdot2^n$, what is the least integer $n\geq 2012$?
$ \textbf{(A)}\ 2012 \qquad \textbf{(B)}\ 2013 \qquad \textbf{(C)}\ 2014 \qquad \textbf{(D)}\ 2015 \qquad \textbf{(E)}\ 2016$
2016 Danube Mathematical Olympiad, 1
Let $S=x_1x_2+x_3x_4+\cdots+x_{2015}x_{2016},$ where $x_1,x_2,\ldots,x_{2016}\in\{\sqrt{3}-\sqrt{2},\sqrt{3}+\sqrt{2}\}.$ Can $S$ be equal to $2016?$
[i]Cristian Lazăr[/i]
2015 India IMO Training Camp, 1
In a triangle $ABC$, a point $D$ is on the segment $BC$, Let $X$ and $Y$ be the incentres of triangles $ACD$ and $ABD$ respectively. The lines $BY$ and $CX$ intersect the circumcircle of triangle $AXY$ at $P\ne Y$ and $Q\ne X$, respectively. Let $K$ be the point of intersection of lines $PX$ and $QY$. Suppose $K$ is also the reflection of $I$ in $BC$ where $I$ is the incentre of triangle $ABC$. Prove that $\angle BAC=\angle ADC=90^{\circ}$.
1953 Putnam, B6
Let $P$ and $Q$ be any points inside a circle $C$ with center $O$ such that $OP=OQ.$ Determine the location of a point $Z$ on $C$ such that $PZ+QZ$ is minimal.
2009 Germany Team Selection Test, 3
The 16 fields of a $4 \times 4$ checker board can be arranged in 18 lines as follows: the four lines, the four columns, the five diagonals from north west to south east and the five diagonals from north east to south west. These diagonals consists of 2,3 or 4 edge-adjacent fields of same colour; the corner fields of the chess board alone do not form a diagonal. Now, we put a token in 10 of the 16 fields. Each of the 18 lines contains an even number of tokens contains a point. What is the highest possible point number when can be achieved by optimal placing of the 10 tokens. Explain your answer.
1993 Greece National Olympiad, 8
Let $S$ be a set with six elements. In how many different ways can one select two not necessarily distinct subsets of $S$ so that the union of the two subsets is $S$? The order of selection does not matter; for example, the pair of subsets $\{a, c\}$, $\{b, c, d, e, f\}$ represents the same selection as the pair $\{b, c, d, e, f\}$, $\{a, c\}$.
1976 Vietnam National Olympiad, 4
Find all three digit integers $\overline{abc} = n$, such that $\frac{2n}{3} = a! b! c!$
2012 Belarus Team Selection Test, 1
Find all primes numbers $p$ such that $p^2-p-1$ is the cube of some integer.
2007 ITest, 4
Star flips a quarter four times. Find the probability that the quarter lands heads exactly twice.
$\textbf{(A) }\dfrac18\hspace{14em}\textbf{(B) }\dfrac3{16}\hspace{14em}\textbf{(C) }\dfrac38$
$\textbf{(D) }\dfrac12$
2009 Jozsef Wildt International Math Competition, W. 16
Prove that $$\sum \limits_{k=1}^n \frac{1}{d(k)}>\sqrt{n+1}-1$$ For every $n\geq 1$, $d(n)$ is the number of divisors of $n$
2024 239 Open Mathematical Olympiad, 3
There are $169$ non-zero digits written around a circle. Prove that they can be split into $14$ non-empty blocks of consecutive digits so that among the $14$ natural numbers formed by the digits in those blocks, at least $13$ of them are divisible by $13$ (the digits in each block are read in clockwise direction).
PEN H Problems, 58
Solve in positive integers the equation $10^{a}+2^{b}-3^{c}=1997$.
2000 Croatia National Olympiad, Problem 3
Let $j$ and $k$ be integers. Prove that the inequality
$$\lfloor(j+k)\alpha\rfloor+\lfloor(j+k)\beta\rfloor\ge\lfloor j\alpha\rfloor+\lfloor j\beta\rfloor+\lfloor k(\alpha+\beta)\rfloor$$holds for all real numbers $\alpha,\beta$ if and only if $j=k$.
1992 India Regional Mathematical Olympiad, 1
Determine the set of integers $n$ for which $n^2+19n+92$ is a square.
2024 Nepal TST, P3
Prove that there are infinitely many integers $k\geqslant 2024$ for which there exists a set $\{a_1,\ldots,a_k\}$ with the following properties:[list]
[*]$a_1{}$ is a positive integer and $a_{i+1}=a_i+1$ for all $1\leqslant i<k,$ and
[*]$2(a_1\cdots a_{k-2}-1)^2$ is divisible by $2(a_1+\cdots+a_k)+a_1-a_1^2.$
[/list][i](Proposed by Prajit Adhikari, Nepal)[/i]
2009 Kyiv Mathematical Festival, 3
Let $AB$ be a segment of a plane. Is it possible to paint the plane in $2009$ colors in such a way that both of the following conditions are satisfied?
1) Every two points of the same color can be connected by a polygonal line.
2) For any point $C$ of $AB$, every $n \in N$ and every $k\in \{1,2,3,...,2009\}$ , there exists a point $D$, painted in $k$-th color such that the length of $CD$ is less than $0,0...01$, where all the zeros after the decimal point are exactly $n$.
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P1
A $6 \times 6$ board is given such that each unit square is either red or green. It is known that there are no $4$ adjacent unit squares of the same color in a horizontal, vertical, or diagonal line. A $2 \times 2$ subsquare of the board is [i]chesslike[/i] if it has one red and one green diagonal. Find the maximal possible number of chesslike squares on the board.
[i]Proposed by Nikola Velov[/i]
1974 All Soviet Union Mathematical Olympiad, 198
Given points $D$ and $E$ on the legs $[CA]$ and $[CB]$, respectively, of the isosceles right triangle. $|CD| = |CE|$. The extensions of the perpendiculars from $D$ and $C$ to the line $AE$ cross the hypotenuse $AB$ in the points $K$ and $L$. Prove that $|KL| = |LB|$
1973 Canada National Olympiad, 5
For every positive integer $n$, let \[h(n) = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}.\] For example, $h(1) = 1$, $h(2) = 1+\frac{1}{2}$, $h(3) = 1+\frac{1}{2}+\frac{1}{3}$. Prove that for $n=2,3,4,\ldots$ \[n+h(1)+h(2)+h(3)+\cdots+h(n-1) = nh(n).\]
2012 IMO, 6
Find all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \ldots, a_n$ such that
\[
\frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} =
\frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1.
\]
[i]Proposed by Dusan Djukic, Serbia[/i]
2011 Belarus Team Selection Test, 3
Let $x_1, \ldots , x_{100}$ be nonnegative real numbers such that $x_i + x_{i+1} + x_{i+2} \leq 1$ for all $i = 1, \ldots , 100$ (we put $x_{101 } = x_1, x_{102} = x_2).$ Find the maximal possible value of the sum $S = \sum^{100}_{i=1} x_i x_{i+2}.$
[i]Proposed by Sergei Berlov, Ilya Bogdanov, Russia[/i]
2001 Irish Math Olympiad, 4
Find all nonnegative real numbers $ x$ for which $ \sqrt[3]{13\plus{}\sqrt{x}}\plus{}\sqrt[3]{13\minus{}\sqrt{x}}$ is an integer.
Russian TST 2014, P1
A regular 1001-gon is drawn on a board, the vertiecs of which are numbered with $1,2,\ldots,1001.$ Is it possible to label the vertices of a cardboard 1001-gon with the numbers $1,2,\ldots,1001$ such that for any overlap between the two 1001-gons, there are two vertices with the same number one over the other? Note that the cardboard polygon can be inverted.