Found problems: 85335
2004 Singapore Team Selection Test, 1
Let $x_0, x_1, x_2, \ldots$ be the sequence defined by
$x_i= 2^i$ if $0 \leq i \leq 2003$
$x_i=\sum_{j=1}^{2004} x_{i-j}$ if $i \geq 2004$
Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by 2004.
1972 All Soviet Union Mathematical Olympiad, 165
Let $O$ be the intersection point of the diagonals of the convex quadrangle $ABCD$ . Prove that the line drawn through the points of intersection of the medians of triangles $AOB$ and $COD$ is orthogonal to the line drawn through the points of intersection of the heights of triangles $BOC$ and $AOD$ .
2012 Bosnia And Herzegovina - Regional Olympiad, 2
Let $a$, $b$, $c$ and $d$ be integers such that $ac$, $bd$ and $bc+ad$ are divisible with positive integer $m$. Show that numbers $bc$ and $ad$ are divisible with $m$
2003 Mexico National Olympiad, 1
Find all positive integers with two or more digits such that if we insert a $0$ between the units and tens digits we get a multiple of the original number.
2007 Tournament Of Towns, 2
Initially, the number $1$ and two positive numbers $x$ and $y$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write on the blackboard, in a finite number of moves, the number
[list][b]a)[/b] $x^2$;
[b]b)[/b] $xy$?[/list]
2017 VJIMC, 4
A positive integer $t$ is called a Jane's integer if $t = x^3+y^2$ for some positive integers $x$ and $y$. Prove
that for every integer $n \ge 2$ there exist infinitely many positive integers $m$ such that the set of $n^2$ consecutive
integers $\{m+1,m+2,\dotsc,m+n^2\}$ contains exactly $n + 1$ Jane's integers.
2018 Turkey Team Selection Test, 3
A Retired Linguist (R.L.) writes in the first move a word consisting of $n$ letters, which are all different. In each move, he determines the maximum $i$, such that the word obtained by reversing the first $i$ letters of the last word hasn't been written before, and writes this new word. Prove that R.L. can make $n!$ moves.
2017 Saudi Arabia Pre-TST + Training Tests, 9
Let $ABC$ be a triangle inscribed in circle $(O)$, with its altitudes $BH_b, CH_c$ intersect at orthocenter $H$ ($H_b \in AC$, $H_c \in AB$). $H_bH_c$ meets $BC$ at $P$. Let $N$ be the midpoint of $AH, L$ be the orthogonal projection of $O$ on the symmedian with respect to angle $A$ of triangle $ABC$. Prove that $\angle NLP = 90^o$.
2017 Junior Balkan Team Selection Tests - Moldova, Problem 6
Let $a,b$ and $c$ be real numbers such that $|a+b|+|b+c|+|c+a|=8.$
Find the maximum and minimum value of the expression $P=a^2+b^2+c^2.$
2025 6th Memorial "Aleksandar Blazhevski-Cane", P4
Let $ABCDE$ be a pentagon such that $\angle DCB < 90^{\circ} < \angle EDC$. The circle with diameter $BD$ intersects the line $BC$ again at $F$, and the circle with diameter $DE$ intersects the line $CE$ again at $G$. Prove that the second intersection ($\neq D$) of the circumcircle of $\triangle DFG$ and the circle with diameter $AD$ lies on $AC$.
Proposed by [i]Petar Filipovski[/i]
2022 Romania Team Selection Test, 3
Let $n\geq 2$ be an integer. Let $a_{ij}, \ i,j=1,2,\ldots,n$ be $n^2$ positive real numbers satisfying the following conditions:
[list=1]
[*]For all $i=1,\ldots,n$ we have $a_{ii}=1$ and,
[*]For all $j=2,\ldots,n$ the numbers $a_{ij}, \ i=1,\ldots, j-1$ form a permutation of $1/a_{ji}, \ i=1,\ldots, j-1.$
[/list]
Given that $S_i=a_{i1}+\cdots+a_{in}$, determine the maximum value of the sum $1/S_1+\cdots+1/S_n.$
2015 Czech-Polish-Slovak Junior Match, 4
Let $ABC$ ne a right triangle with $\angle ACB=90^o$. Let $E, F$ be respecitvely the midpoints of the $BC, AC$ and $CD$ be it's altitude. Next, let $P$ be the intersection of the internal angle bisector from $A$ and the line $EF$. Prove that $P$ is the center of the circle inscribed in the triangle $CDE$ .
1997 Romania Team Selection Test, 3
The vertices of a regular dodecagon are coloured either blue or red. Find the number of all possible colourings which do not contain monochromatic sub-polygons.
[i]Vasile Pop[/i]
2001 AMC 12/AHSME, 12
How many positive integers not exceeding 2001 are multiple of 3 or 4 but not 5?
$ \textbf{(A)} \ 768 \qquad \textbf{(B)} \ 801 \qquad \textbf{(C)} \ 934 \qquad \textbf{(D)} \ 1067 \qquad \textbf{(E)} \ 1167$
2009 Princeton University Math Competition, 2
It is known that a certain mechanical balance can measure any object of integer mass anywhere between 1 and 2009 (both included). This balance has $k$ weights of integral values. What is the minimum $k$ for which there exist weights that satisfy this condition?
1943 Eotvos Mathematical Competition, 3
Let $a < b < c < d$ be real numbers and $(x,y, z,t)$ be any permutation of $a$,$b$, $c$ and $d$. What are the maximum and minimum values of the expression $$(x - y)^2 + (y- z)^2 + (z - t)^2 + (t - x)^2?$$
2015 May Olympiad, 4
We say that a number is [i]superstitious [/i] when it is equal to $13$ times the sum of its digits . Find all superstitious numbers.
May Olympiad L2 - geometry, 2008.2
Let $ABCD$ be a rectangle and $P$ be a point on the side$ AD$ such that $\angle BPC = 90^o$. The perpendicular from $A$ on $BP$ cuts $BP$ at $M$ and the perpendicular from $D$ on $CP$ cuts $CP$ in $N$. Show that the center of the rectangle lies in the $MN$ segment.
1996 Polish MO Finals, 1
Find all pairs $(n,r)$ with $n$ a positive integer and $r$ a real such that $2x^2+2x+1$ divides $(x+1)^n - r$.
2014 Poland - Second Round, 5.
Circles $o_1$ and $o_2$ tangent to some line at points $A$ and $B$, respectively, intersect at points $X$ and $Y$ ($X$ is closer to the line $AB$). Line $AX$ intersects $o_2$ at point $P\neq X$. Tangent to $o_2$ at point $P$ intersects line $AB$ at point $Q$. Prove that $\sphericalangle XYB = \sphericalangle BYQ$.
1993 AMC 12/AHSME, 16
Consider the non-decreasing sequence of positive integers
\[ 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5,... \] in which the $n^{\text{th}}$ positive integer appears $n$ times. The remainder when the $1993^{\text{rd}}$ term is divided by $5$ is
$ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $
2010 LMT, 11
Carl, James, Saif, and Ted play several games of two-player For The Win on the Art of Problem Solving website. If, among these games, Carl wins $5$ and loses $0,$ James wins $4$ and loses $2,$ Saif wins $1$ and loses $6,$ and Ted wins $4,$ how many games does Ted lose?
2015 Caucasus Mathematical Olympiad, 3
The workers laid a floor of size $n\times n$ ($10 <n <20$) with two types of tiles: $2 \times 2$ and $5\times 1$. It turned out that they were able to completely lay the floor so that the same number of tiles of each type was used. For which $n$ could this happen? (You can’t cut tiles and also put them on top of each other.)
2016 Kosovo National Mathematical Olympiad, 3
Show that the sum $S=5+5^2+5^3+…+5^{2016}$ is divisible by $31$
2017 AMC 10, 20
The number $21!=51,090,942,171,709,440,000$ has over $60,000$ positive integer divisors. One of them is chosen at random. What is the probability that it is odd?
$\textbf{(A)} \frac{1}{21} \qquad \textbf{(B)} \frac{1}{19} \qquad \textbf{(C)} \frac{1}{18} \qquad \textbf{(D)} \frac{1}{2} \qquad \textbf{(E)} \frac{11}{21}$