Found problems: 85335
1998 Turkey Team Selection Test, 1
Suppose $n$ houses are to be assigned to $n$ people. Each person ranks the houses in the order of preference, with no ties. After the assignment is made, it is observed that every other assignment would assign to at least one person a less preferred house. Prove that there is at least one person who received the house he/she preferred most under this assignment.
2018 Korea National Olympiad, 1
Let there be an acute triangle $\triangle ABC$ with incenter $I$. $E$ is the foot of the perpendicular from $I$ to $AC$. The line which passes through $A$ and is perpendicular to $BI$ hits line $CI$ at $K$. The line which passes through $A$ and is perpendicular to $CI$ hits the line which passes through $C$ and is perpendicular to $BI$ at $L$. Prove that $E, K, L$ are colinear.
2019 PUMaC Combinatorics B, 2
Suppose Alan, Michael, Kevin, Igor, and Big Rahul are in a running race. It is given that exactly one pair of people tie (for example, two people both get second place), so that no other pair of people end in the same position. Each competitor has equal skill; this means that each outcome of the race, given that exactly two people tie, is equally likely. The probability that Big Rahul gets first place (either by himself or he ties for first) can be expressed in the form $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
2018-IMOC, N1
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$x+f^{f(x)}(y)\mid2(x+y)$$for all $x,y\in\mathbb N$.
2015 Saint Petersburg Mathematical Olympiad, 3
All cells of $2015 \times 2015$ table colored in one of $4$ colors. We count number of ways to place one tetris T-block in table. Prove that T-block has cell of all $4$ colors in less than $51\%$ of total number of ways.
2018 OMMock - Mexico National Olympiad Mock Exam, 5
Let $ABC$ be a triangle with circumcirle $\Gamma$, and let $M$ and $N$ be the respective midpoints of the minor arcs $AB$ and $AC$ of $\Gamma$. Let $P$ and $Q$ be points such that $AB=BP$, $AC=CQ$, and $P$, $B$, $C$, $Q$ lie on $BC$ in that order. Prove that $PM$ and $QN$ meet at a point on $\Gamma$.
[i]Proposed by Victor DomÃnguez[/i]
2013 QEDMO 13th or 12th, 10
Let $p$ be a prime number gretater then $3$. What is the number of pairs $(m, n)$ of integers with $0 <m <n <p$, for which the polynomial $x^p + px^n + px^m +1$ is not a product of two non-constant polynomials with integer coefficients can be written?
1975 Putnam, A5
Let $I\subset \mathbb{R}$ be an interval and $f(x)$ a continuous real-valued function on $I$. Let $y_1$ and $y_2$ be linearly independent solutions of $y''=f(x)y$ taking positive values on $I$. Show that for some positive number $k$ the function $k\cdot\sqrt{y_1 y_2}$ is a solution of $y''+\frac{1}{y^{3}}=f(x)y$.
2021 Novosibirsk Oral Olympiad in Geometry, 4
Angle bisectors $AD$ and $BE$ are drawn in triangle $ABC$. It turned out that $DE$ is the bisector of triangle $ADC$. Find the angle $BAC$.
2009 Hungary-Israel Binational, 2
Let $ x$, $ y$ and $ z$ be non negative numbers. Prove that \[ \frac{x^2\plus{}y^2\plus{}z^2\plus{}xy\plus{}yz\plus{}zx}{6}\le \frac{x\plus{}y\plus{}z}{3}\cdot\sqrt{\frac{x^2\plus{}y^2\plus{}z^2}{3}}\]
2009 Moldova Team Selection Test, 1
[color=darkred]For any $ m \in \mathbb{N}^*$ solve the ecuation
\[ \left\{\left( x \plus{} \frac {1}{m}\right) ^3\right\} \equal{} x^3
\]
[/color]
2024 Rioplatense Mathematical Olympiad, 5
Let $n$ be a positive integer. Ana and Beto play a game on a $2 \times n$ board (with 2 rows and $n$ columns). First, Ana writes a digit from 1 to 9 in each cell of the board such that in each column the two written digits are different. Then, Beto erases a digit from each column. Reading from left to right, a number with $n$ digits is formed. Beto wins if this number is a multiple of $n$; otherwise, Ana wins. Determine which of the two players has a winning strategy in the following cases:
$\bullet$ (a) $n = 1001$.
$\bullet$ (b) $n = 1003$.
1993 Chile National Olympiad, 2
Given a rectangle, circumscribe a rectangle of maximum area.
Swiss NMO - geometry, 2008.5
Let $ABCD$ be a square with side length $1$.
Find the locus of all points $P$ with the property $AP\cdot CP + BP\cdot DP = 1$.
2014 Contests, 2
find all polynomials with integer coefficients that $P(\mathbb{Z})= ${$p(a):a\in \mathbb{Z}$} has a Geometric progression.
1961 Polish MO Finals, 2
Prove that if $ a + b = 1 $, then $$
a^5 + b^5 \geq \frac{1}{16}$$
2018 lberoAmerican, 5
Let $n$ be a positive integer. For a permutation $a_1, a_2, \dots, a_n$ of the numbers $1, 2, \dots, n$ we define
$$b_k = \min_{1 \leq i \leq k} a_i + \max_{1 \leq j \leq k} a_j$$
We say that the permutation $a_1, a_2, \dots, a_n$ is [i]guadiana[/i] if the sequence $b_1, b_2, \dots, b_n$ does not contain two consecutive equal terms. How many guadiana permutations exist?
2013 Paraguay Mathematical Olympiad, 5
Let $ABC$ be an obtuse triangle, with $AB$ being the largest side.
Draw the angle bisector of $\measuredangle BAC$. Then, draw the perpendiculars to this angle bisector from vertices $B$ and $C$, and call their feet $P$ and $Q$, respectively.
$D$ is the point in the line $BC$ such that $AD \perp AP$.
Prove that the lines $AD$, $BQ$ and $PC$ are concurrent.
2023 LMT Fall, 10
A square has vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(10,10)$ on the $x-y$ coordinate plane. A second quadrilateral is constructed with vertices $(0,10)$, $(0, 0)$, $(10, 0)$, and $(15,15)$. Find the positive difference between the areas of the original square and the second quadrilateral.
[i]Proposed byWilliam Hua[/i]
1988 IMO Longlists, 78
It is proposed to partition a set of positive integers into two disjoint subsets $ A$ and $ B$ subject to the conditions
[b]i.)[/b] 1 is in $ A$
[b]ii.)[/b] no two distinct members of $ A$ have a sum of the form $ 2^k \plus{} 2, k \equal{} 0,1,2, \ldots;$ and
[b]iii.)[/b] no two distinct members of B have a sum of that form.
Show that this partitioning can be carried out in unique manner and determine the subsets to which 1987, 1988 and 1989 belong.
2019 Hong Kong TST, 3
Find an integral solution of the equation
\[ \left \lfloor \frac{x}{1!} \right \rfloor + \left \lfloor \frac{x}{2!} \right \rfloor + \left \lfloor \frac{x}{3!} \right \rfloor + \dots + \left \lfloor \frac{x}{10!} \right \rfloor = 2019. \]
(Note $\lfloor u \rfloor$ stands for the greatest integer less than or equal to $u$.)
2016 HMNT, 31-33
31. Define a number to be an anti-palindrome if, when written in base $3$ as $a_na_{n-1}\ldots a_0$, then $a_i+a_{n-i} = 2$ for any $0 \le i \le n$. Find the number of anti-palindromes less than $3^{12}$ such that no two consecutive digits in base 3 are equal.
32. Let $C_{k,n}$ denote the number of paths on the Cartesian plane along which you can travel from $(0, 0)$ to $(k, n)$, given the following rules: 1) You can only travel directly upward or directly rightward 2) You can only change direction at lattice points 3) Each horizontal segment in the path must be at most $99$ units long.
Find $$\sum_{j=0}^\infty C_{100j+19,17}$$
33. Camille the snail lives on the surface of a regular dodecahedron. Right now he is on vertex $P_1$ of the face with vertices $P_1, P_2, P_3, P_4, P_5$. This face has a perimeter of $5$. Camille wants to get to the point on the dodecahedron farthest away from $P_1$. To do so, he must travel along the surface a distance at least $L$. What is $L^2$?
2004 Purple Comet Problems, 20
A circle with area $40$ is tangent to a circle with area $10$. Let R be the smallest rectangle containing both circles. The area of $R$ is $\frac{n}{\pi}$. Find $n$.
[asy]
defaultpen(linewidth(0.7)); size(120);
real R = sqrt(40/pi), r = sqrt(10/pi);
draw(circle((0,0), R)); draw(circle((R+r,0), r));
draw((-R,-R)--(-R,R)--(R+2*r,R)--(R+2*r,-R)--cycle);[/asy]
2024 CMIMC Algebra and Number Theory, 1
Connor is thinking of a two-digit number $n$, which satisfies the following properties:
[list]
[*] If $n>70$, then $n$ is a perfect square.
[*] If $n>40$, then $n$ is prime.
[*] If $n<80$, then the sum of the digits of $n$ is $14$.
[/list]
What is Connor's number?
[i]Proposed by Connor Gordon[/i]
2025 Kyiv City MO Round 2, Problem 2
For some positive integer \( n \), Katya wrote the numbers from \( 1 \) to \( 2^n \) in a row in increasing order. Oleksii rearranged Katya's numbers and wrote the new sequence directly below the first row. Then, they calculated the sum of the two numbers in each column.
Katya calculated \( N \), the number of powers of two among the results, while Oleksii calculated \( K \), the number of distinct powers of two among the results. What is the maximum possible value of \( N + K \)?
[i]Proposed by Oleksii Masalitin[/i]