Found problems: 85335
1996 Greece National Olympiad, 4
Find the number of functions $f : \{1, 2, . . . , n\} \to \{1995, 1996\}$ such that $f(1) + f(2) + ... + f(1996)$ is odd.
2024 Princeton University Math Competition, B2
Ben and Connor are playing a game of wallball. The first player to lead by $2$ points wins the game. Suppose Ben wins each point with probability $\tfrac{3}{4}$ and is gracious enough to allow Connor to start with a $1$ point lead. The probability that Ben wins the game is $\tfrac{m}{n}$ for coprime positive integers $m$ and $n.$ What is $m + n$?
2013 Iran Team Selection Test, 14
we are given $n$ rectangles in the plane. Prove that between $4n$ right angles formed by these rectangles there are at least $[4\sqrt n]$ distinct right angles.
2020 BMT Fall, 2
Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
2005 Portugal MO, 3
On a board with $a$ rows and $b$ columns, each square has a switch and an unlit light bulb. By pressing the switch of a house, the lamp in that house changes state, along with the lamps in the same row and those in the same column (those that are on go out and the that are off light up). For what values of $a$ and $b$ is it possible to have just one lamp on, by pressing a series of switches?
2004 Tournament Of Towns, 7
Let A and B be two rectangles such that it is possible to get rectangle similar to A by putting together rectangles equal to B. Show that it is possible to get rectangle similar to B by putting together rectangles equal to A.
2009 Spain Mathematical Olympiad, 6
Inside a circle of center $ O$ and radius $ r$, take two points $ A$ and $ B$ symmetrical about $ O$. We consider a variable point $ P$ on the circle and draw the chord $ \overline{PP'}\perp \overline{AP}$. Let $ C$ is the symmetric of $ B$ about $ \overline{PP'}$ ($ \overline{PP}'$ is the axis of symmetry) . Find the locus of point $ Q \equal{} \overline{PP'}\cap\overline{AC}$ when we change $ P$ in the circle.
2007 Irish Math Olympiad, 5
Let $ r$ and $ n$ be nonnegative integers such that $ r \le n$.
$ (a)$ Prove that: $ \frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}$ is an integer.
$ (b)$ Prove that: $ \displaystyle\sum_{r\equal{}0}^{[n/2]}\frac{n\plus{}1\minus{}2r}{n\plus{}1\minus{}r} \binom{n}{r}<2^{n\minus{}2}$ for all $ n \ge 9$.
2011 IberoAmerican, 1
Let $ABC$ be an acute-angled triangle, with $AC \neq BC$ and let $O$ be its circumcenter. Let $P$ and $Q$ be points such that $BOAP$ and $COPQ$ are parallelograms. Show that $Q$ is the orthocenter of $ABC$.
2004 India IMO Training Camp, 3
An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.
[i]Proposed by Hojoo Lee, Korea[/i]
2017 Romania National Olympiad, 2
Show that for every integer $n \ge 3$ there exists positive integers $x_1, x_2, . . . , x_n$, pairwise different, so that $\{2, n\} \subset \{x_1, x_2, . . . , x_n\}$ and
$$\frac{1}{x_1}+\frac{1}{x_2}+.. +\frac{1}{x_n}= 1.$$
2024 Singapore Junior Maths Olympiad, Q3
Seven triangles of area $7$ lie in a square of area $27$. Prove that among the $7$ triangles there are $2$ that intersect in a region of area not less than $1$.
2006 Indonesia MO, 8
Find the largest $ 85$-digit integer which has property: the sum of its digits equals to the product of its digits.
PEN H Problems, 24
Prove that if $n$ is a positive integer such that the equation \[x^{3}-3xy^{2}+y^{3}=n.\] has a solution in integers $(x,y),$ then it has at least three such solutions. Show that the equation has no solutions in integers when $n=2891$.
2021 Lusophon Mathematical Olympiad, 3
Let triangle $ABC$ be an acute triangle with $AB\neq AC$. The bisector of $BC$ intersects the lines $AB$ and $AC$ at points $F$ and $E$, respectively. The circumcircle of triangle $AEF$ has center $P$ and intersects the circumcircle of triangle $ABC$ at point $D$ with $D$ different to $A$.
Prove that the line $PD$ is tangent to the circumcircle of triangle $ABC$.
2009 Postal Coaching, 3
Let $N_0$ denote the set of nonnegative integers and $Z$ the set of all integers. Let a function $f : N_0 \times Z \to Z$ satisfy the conditions
(i) $f(0, 0) = 1$, $f(0, 1) = 1$
(ii) for all $k, k \ne 0, k \ne 1$, $f(0, k) = 0$ and
(iii) for all $n \ge 1$ and $k, f(n, k) = f(n -1, k) + f(n- 1, k - 2n)$. Find the value of
$$\sum_{k=0}^{2009 \choose 2} f(2008, k)$$
2000 Manhattan Mathematical Olympiad, 3
A pizza is divided into six slices. Each slice contains one olive. One plays the following game. At each move it is allowed to move an olive on a neighboring slice. Is it possible to bring all the olives on one slice by exactly $20$ moves?
2006 Austria Beginners' Competition, 1
Do integers $a, b$ exist such that $a^{2006} + b^{2006} + 1$ is divisible by $2006^2$?
2019 Jozsef Wildt International Math Competition, W. 36
For any $a$, $b$, $c > 0$ and for any $n \in \mathbb{N}^*$, prove the inequality$$(a - b)\left(\frac{a}{b}\right)^n+(b - c)\left(\frac{b}{c}\right)^n+(c - a)\left(\frac{c}{a}\right)^n\geq (a - b)\frac{a}{b}+(b - c)\frac{b}{c}+(c - a)\frac{c}{a}$$
2001 Mediterranean Mathematics Olympiad, 1
Let $P$ and $Q$ be points on a circle $k$. A chord $AC$ of $k$ passes through the midpoint $M$ of $PQ$. Consider a trapezoid $ABCD$ inscribed in $k$ with $AB \parallel PQ \parallel CD$. Prove that the intersection point $X$ of $AD$ and $BC$ depends only on $k$ and $P,Q.$
Kvant 2022, M2706
16 NHL teams in the first playoff round divided in pairs and to play series until 4 wins (thus the series could finish with score 4-0, 4-1, 4-2, or 4-3). After that 8 winners of the series play the second playoff round divided into 4 pairs to play series until 4 wins, and so on. After all the final round is over, it happens that $k$ teams have non-negative balance of wins (for example, the team that won in the first round with a score of 4-2 and lost in the second with a score of 4-3 fits the condition: it has $4+3=7$ wins and $2+4=6$ losses). Find the least possible $k$.
1991 Arnold's Trivium, 42
Do the medians of a triangle meet in a single point in the Lobachevskii plane? What about the altitudes?
1997 IMO Shortlist, 2
Let $ R_1,R_2, \ldots$ be the family of finite sequences of positive integers defined by the following rules: $ R_1 \equal{} (1),$ and if $ R_{n - 1} \equal{} (x_1, \ldots, x_s),$ then
\[ R_n \equal{} (1, 2, \ldots, x_1, 1, 2, \ldots, x_2, \ldots, 1, 2, \ldots, x_s, n).\]
For example, $ R_2 \equal{} (1, 2),$ $ R_3 \equal{} (1, 1, 2, 3),$ $ R_4 \equal{} (1, 1, 1, 2, 1, 2, 3, 4).$ Prove that if $ n > 1,$ then the $ k$th term from the left in $ R_n$ is equal to 1 if and only if the $ k$th term from the right in $ R_n$ is different from 1.
2010 QEDMO 7th, 3
An alphabet has $n$ letters. A word is called [i]differentiated [/i] if it has the following property fulfilled: No letter occurs more than once between two identical letters. For example with the alphabet $\{a, b, c, d\}$ the word [i]abbdacbdd [/i] is not, the word [i]bbacbadcdd [/i] is differentiated.
(a) Each differentiated word has a maximum of $3n$ letters.
(b) How many differentiated words with exactly $3n$ letters are ther
2003 Purple Comet Problems, 6
Evaluate:
\[\frac{1}{\log_2 (\frac{1}{6})} - \frac{1}{\log_3 (\frac{1}{6})} - \frac{1}{\log_4 (\frac{1}{6})}\]