Found problems: 85335
2013 Czech-Polish-Slovak Junior Match, 4
Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle ABC =\angle BCD > 90^o$. The circle circumscribed around the triangle $ABC$ intersects the sides $AD$ and $CD$ at points $K$ and $L$, respectively, different from any vertex of the quadrilateral $ABCD$ . Segments $AL$ and $CK$ intersect at point $P$. Prove that $\angle ADB =\angle PDC$.
2001 Brazil Team Selection Test, Problem 3
For which positive integers $n$ is there a permutation $(x_1,x_2,\ldots,x_n)$ of $1,2,\ldots,n$ such that all the differences $|x_k-k|$, $k = 1,2,\ldots,n$, are distinct?
2015 BMT Spring, 7
Evaluate $\sum_{k=0}^{37}(-1)^k\binom{75}{2k}$.
2001 China Western Mathematical Olympiad, 4
Let $ x, y, z$ be real numbers such that $ x \plus{} y \plus{} z \geq xyz$. Find the smallest possible value of $ \frac {x^2 \plus{} y^2 \plus{} z^2}{xyz}$.
1976 Chisinau City MO, 125
From twenty different books on mathematics and physics, sets are made containing $5$ books on mathematics and $5$ books on physics each. How many math books should there be for the largest number of possible sets?
2010 AIME Problems, 9
Let $ (a,b,c)$ be the real solution of the system of equations $ x^3 \minus{} xyz \equal{} 2$, $ y^3 \minus{} xyz \equal{} 6$, $ z^3 \minus{} xyz \equal{} 20$. The greatest possible value of $ a^3 \plus{} b^3 \plus{} c^3$ can be written in the form $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m \plus{} n$.
2001 National Olympiad First Round, 29
Let $ABCD$ be a isosceles trapezoid such that $AB || CD$ and all of its sides are tangent to a circle. $[AD]$ touches this circle at $N$. $NC$ and $NB$ meet the circle again at $K$ and $L$, respectively. What is $\dfrac {|BN|}{|BL|} + \dfrac {|CN|}{|CK|}$?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 6
\qquad\textbf{(C)}\ 8
\qquad\textbf{(D)}\ 9
\qquad\textbf{(E)}\ 10
$
2002 ITAMO, 5
Prove that if $m=5^n+3^n+1$ is a prime, then $12$ divides $n$.
2013 Thailand Mathematical Olympiad, 1
Find the largest integer that divides $p^4 - 1$ for all primes $p > 4$
2016 Flanders Math Olympiad, 4
Prove that there exists a unique polynomial function f with positive integer coefficients such that $f(1) = 6$ and $f(2) = 2016$.
2017 Math Prize for Girls Problems, 1
A bag contains 4 red marbles, 5 yellow marbles, and 6 blue marbles. Three marbles are to be picked out randomly (without replacement). What is the probability that exactly two of them have the same color?
May Olympiad L1 - geometry, 2001.2
Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm.
We do three folds:
1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$.
A right trapezoid $BCDQ$ is then formed.
2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed.
3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$.
After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$.
Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.
1994 Spain Mathematical Olympiad, 5
Let $21$ pieces, some white and some black, be placed on the squares of a $3\times 7$ rectangle. Prove that there always exist four pieces of the same color standing at the vertices of a rectangle.
2021 MOAA, 21
King William is located at $(1, 1)$ on the coordinate plane. Every day, he chooses one of the eight lattice points closest to him and moves to one of them with equal probability. When he exits the region bounded by the $x, y$ axes and $x+y = 4$, he stops moving and remains there forever. Given that after an arbitrarily large amount of time he must exit the region, the probability he ends up on $x+y = 4$ can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
[i]Proposed by Andrew Wen[/i]
2023 ISL, C4
Let $n\geqslant 2$ be a positive integer. Paul has a $1\times n^2$ rectangular strip consisting of $n^2$ unit squares, where the $i^{\text{th}}$ square is labelled with $i$ for all $1\leqslant i\leqslant n^2$. He wishes to cut the strip into several pieces, where each piece consists of a number of consecutive unit squares, and then [i]translate[/i] (without rotating or flipping) the pieces to obtain an $n\times n$ square satisfying the following property: if the unit square in the $i^{\text{th}}$ row and $j^{\text{th}}$ column is labelled with $a_{ij}$, then $a_{ij}-(i+j-1)$ is divisible by $n$.
Determine the smallest number of pieces Paul needs to make in order to accomplish this.
2010 ITAMO, 1
In a mathematics test number of participants is $N < 40$. The passmark is fixed at $65$. The test results are
the following:
The average of all participants is $66$, that of the promoted $71$ and that of the repeaters $56$.
However, due to an error in the wording of a question, all scores are increased by $5$. At this point
the average of the promoted participants becomes $75$ and that of the non-promoted $59$.
(a) Find all possible values of $N$.
(b) Find all possible values of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$.
Russian TST 2022, P3
Determine the largest integer $N$ for which there exists a table $T$ of integers with $N$ rows and $100$ columns that has the following properties:
$\text{(i)}$ Every row contains the numbers $1$, $2$, $\ldots$, $100$ in some order.
$\text{(ii)}$ For any two distinct rows $r$ and $s$, there is a column $c$ such that $|T(r,c) - T(s, c)|\geq 2$. (Here $T(r,c)$ is the entry in row $r$ and column $c$.)
1996 Irish Math Olympiad, 3
Suppose that $ p$ is a prime number and $ a$ and $ n$ positive integers such that: $ 2^p\plus{}3^p\equal{}a^n$. Prove that $ n\equal{}1$.
2017 South Africa National Olympiad, 2
Let $ABCD$ be a rectangle with side lengths $AB = CD = 5$ and $BC = AD = 10$. $W, X, Y, Z$ are points on $AB, BC, CD$ and $DA$ respectively chosen in such a way that $WXYZ$ is a kite, where $\angle ZWX$ is a right angle. Given that $WX = WZ = \sqrt{13}$ and $XY = ZY$, determine the length of $XY$.
LMT Team Rounds 2021+, 7
Jerry writes down all binary strings of length $10$ without any two consecutive $1$s. How many $1$s does Jerry write?
1998 Argentina National Olympiad, 4
Determine all possible values of the expression$$x-\left [\frac{x}{2}\right ]-\left [\frac{x}{3}\right ]-\left [\frac{x} {6}\right ]$$by varying $x$ in the real numbers.
Clarification: The brackets indicate the integer part of the number they enclose.
2019 JBMO Shortlist, C3
A $5 \times 100$ table is divided into $500$ unit square cells, where $n$ of them are coloured black and the rest are coloured white. Two unit square cells are called [i]adjacent[/i] if they share a common side. Each of the unit square cells has at most two adjacent black unit square cells. Find the largest possible value of $n$.
2020 Bosnia and Herzegovina Junior BMO TST, 4
Determine the largest positive integer $n$ such that the following statement holds:
If $a_1,a_2,a_3,a_4,a_5,a_6$ are six distinct positive integers less than or equal to $n$, then there exist $3$ distinct positive integers ,from these six, say $a,b,c$ s.t. $ab>c,bc>a,ca>b$.
2018 Malaysia National Olympiad, B3
Let $n$ be an integer greater than $1$, such that $3n + 1$ is a perfect square. Prove that $n + 1$ can be expressed as a sum of three perfect squares.
2001 South africa National Olympiad, 6
The unknown real numbers $x_1,x_2,\dots,x_n$ satisfy $x_1 < x_2 < \cdots < x_n,$ where $n \geq 3$. The numbers $s$, $t$ and $d_1,d_2,\dots,d_{n - 2}$ are given, such that \[ \begin{aligned} s & = \sum\limits_{i = 1}^nx_i, \\ t & = \sum\limits_{i = 1}^nx_i^2,\\ d_i & = x_{i + 2} - x_i,\ \ i = 1,2,\dots,n - 2. \end{aligned} \] For which $n$ is this information always sufficient to determine $x_1,x_2,\dots,x_n$ uniquely?