Found problems: 85335
2010 Turkey Team Selection Test, 3
Let $\Lambda$ be the set of points in the plane whose coordinates are integers and let $F$ be the collection of all functions from $\Lambda$ to $\{1,-1\}.$ We call a function $f$ in $F$ [i]perfect[/i] if every function $g$ in $F$ that differs from $f$ at finitely many points satisfies the condition
\[ \sum_{0<d(P,Q)<2010} \frac{f(P)f(Q)-g(P)g(Q)}{d(P,Q)} \geq 0 \]
where $d(P,Q)$ denotes the distance between $P$ and $Q.$ Show that there exist infinitely many [i]perfect[/i] functions that are not translates of each other.
2025 AMC 8, 21
The Konigsberg School has assigned grades $1$ through $7$ to pods $A$ through $G$, one grade per pod. The school noticed that each pair of connected pods has been assigned grades differing by $2$ or more grade levels. (For example, grades $1$ and $2$ will not be in pods directly connected by a walkway.) What is the sum of the grade levels assigned to pods $C, E,$ and $F$?
$\textbf{(A)}\ 12\qquad \textbf{(B)}\ 13\qquad \textbf{(C)}\ 14\qquad \textbf{(D)}\ 15\qquad \textbf{(E)}\ 16$\\
1971 Bundeswettbewerb Mathematik, 1
The numbers $1,2,...,1970$ are written on a board. One is allowed to remove $2$ numbers and to write down their difference instead. When repeated often enough, only one number remains. Show that this number is odd.
1991 All Soviet Union Mathematical Olympiad, 538
A lottery ticket has $50$ cells into which one must put a permutation of $1, 2, 3, ... , 50$. Any ticket with at least one cell matching the winning permutation wins a prize. How many tickets are needed to be sure of winning a prize?
2016 APMO, 1
We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$.
[i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]
2016 China Team Selection Test, 5
Does there exist two infinite positive integer sets $S,T$, such that any positive integer $n$ can be uniquely expressed in the form
$$n=s_1t_1+s_2t_2+\ldots+s_kt_k$$
,where $k$ is a positive integer dependent on $n$, $s_1<\ldots<s_k$ are elements of $S$, $t_1,\ldots, t_k$ are elements of $T$?
1989 AIME Problems, 2
Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices?
2018 Brazil Team Selection Test, 3
Let $q$ be a real number. Gugu has a napkin with ten distinct real numbers written on it, and he writes the following three lines of real numbers on the blackboard:
[list]
[*]In the first line, Gugu writes down every number of the form $a-b$, where $a$ and $b$ are two (not necessarily distinct) numbers on his napkin.
[*]In the second line, Gugu writes down every number of the form $qab$, where $a$ and $b$ are
two (not necessarily distinct) numbers from the first line.
[*]In the third line, Gugu writes down every number of the form $a^2+b^2-c^2-d^2$, where $a, b, c, d$ are four (not necessarily distinct) numbers from the first line.
[/list]
Determine all values of $q$ such that, regardless of the numbers on Gugu's napkin, every number in the second line is also a number in the third line.
1999 Czech and Slovak Match, 2
The altitudes through the vertices $A,B,C$ of an acute-angled triangle $ABC$ meet the opposite sides at $D,E,F,$ respectively. The line through $D$ parallel to $EF$ meets the lines $AC$ and $AB$ at $Q$ and $R$, respectively. The line $EF$ meets $BC$ at $P$. Prove that the circumcircle of the triangle $PQR$ passes through the midpoint of $BC$.
2014 ASDAN Math Tournament, 15
Antoine, Benoît, Claude, Didier, Étienne, and Françoise go to the cinéma together to see a movie. The six of them want to sit in a single row of six seats. But Antoine, Benoît, and Claude are mortal enemies and refuse to sit next to either of the other two. How many different arrangements are possible?
2022 3rd Memorial "Aleksandar Blazhevski-Cane", P4
Let $ABC$ be an acute triangle with incircle $\omega$, incenter $I$, and $A$-excircle $\omega_{a}$. Let $\omega$ and $\omega_{a}$ meet $BC$ at $X$ and $Y$, respectively. Let $Z$ be the intersection point of $AY$ and $\omega$ which is closer to $A$. The point $H$ is the foot of the altitude from $A$. Show that $HZ$, $IY$ and $AX$ are concurrent.
[i]Proposed by Nikola Velov[/i]
2016 Purple Comet Problems, 5
A 2 meter long bookshelf is filled end-to-end with 46 books. Some of the books are 3 centimeters thick while all the others are 5 centimeters thick. Find the number of books on the shelf that are 3 centimeters thick.
2009 Mathcenter Contest, 1
For any natural $n$ , define $n!!=(n!)!$ e.g. $3!!=(3!)!=6!=720$.
Let $a_1,a_2,...,a_n$ be a positive integer Prove that $$\frac{(a_1+a_2+\cdots+a_n)!!}{a_1!!a_2!!\cdots a_n!!}$$ is an integer.
[i](nooonuii)[/i]
2015 India PRMO, 2
$2.$ The equations $x^2-4x+k=0$ and $x^2+kx-4=0,$ where $k$ is a real number, have exactly one common root. What is the value of $k ?$
1947 Moscow Mathematical Olympiad, 139
In the numerical triangle
$................1..............$
$...........1 ...1 ...1.........$
$......1... 2... 3 ... 2 ... 1....$
$.1...3...6...7...6...3...1$
$...............................$
each number is equal to the sum of the three nearest to it numbers from the row above it; if the number is at the beginning or at the end of a row then it is equal to the sum of its two nearest numbers or just to the nearest number above it (the lacking numbers above the given one are assumed to be zeros). Prove that each row, starting with the third one, contains an even number.
1984 IMO Longlists, 18
Let $c$ be the inscribed circle of the triangle $ABC$, $d$ a line tangent to $c$ which does not pass through the vertices of triangle $ABC$. Prove the existence of points $A_1,B_1, C_1$, respectively, on the lines $BC,CA,AB$ satisfying the following two properties:
$(i)$ Lines $AA_1,BB_1$, and $CC_1$ are parallel.
$(ii)$ Lines $AA_1,BB_1$, and $CC_1$ meet $d$ respectively at points $A' ,B'$, and $C'$ such that
\[\frac{A'A_1}{A' A}=\frac{B'B_1}{B 'B}=\frac{C'C_1}{C'C}\]
2011 Hanoi Open Mathematics Competitions, 3
What is the largest integer less than to $\sqrt[3]{(2011)^3 + 3 \times (2011)^2 + 4 \times 2011+ 5}$ ?
(A) $2010$, (B) $2011$, (C) $2012$, (D) $2013$, (E) None of the above.
2017 Romanian Master of Mathematics, 1
[b](a)[/b] Prove that every positive integer $n$ can be written uniquely in the form \[n=\sum_{j=1}^{2k+1}(-1)^{j-1}2^{m_j},\] where $k\geq 0$ and $0\le m_1<m_2\cdots <m_{2k+1}$ are integers.
This number $k$ is called [i]weight[/i] of $n$.
[b](b)[/b] Find (in closed form) the difference between the number of positive integers at most $2^{2017}$ with even weight and the number of positive integers at most $2^{2017}$ with odd weight.
2016 Israel Team Selection Test, 4
A regular 60-gon is given. What is the maximum size of a subset of its vertices containing no isosceles triangles?
1992 USAMO, 3
For a nonempty set $\, S \,$ of integers, let $\, \sigma(S) \,$ be the sum of the elements of $\, S$. Suppose that $\, A = \{a_1, a_2, \ldots, a_{11} \} \,$ is a set of positive integers with $\, a_1 < a_2 < \cdots < a_{11} \,$ and that, for each positive integer $\, n\leq 1500, \,$ there is a subset $\, S \,$ of $\, A \,$ for which $\, \sigma(S) = n$. What is the smallest possible value of $\, a_{10}$?
2014 Sharygin Geometry Olympiad, 4
A square is inscribed into a triangle (one side of the triangle contains two vertices and each of two remaining sides contains one vertex. Prove that the incenter of the triangle lies inside the square.
2009 Mediterranean Mathematics Olympiad, 2
Let $ABC$ be a triangle with $90^\circ \ne \angle A \ne 135^\circ$. Let $D$ and $E$ be external points to the triangle $ABC$ such that $DAB$ and $EAC$ are isoscele triangles with right angles at $D$ and $E$. Let $F = BE \cap CD$, and let $M$ and $N$ be the midpoints of $BC$ and $DE$, respectively.
Prove that, if three of the points $A$, $F$, $M$, $N$ are collinear, then all four are collinear.
2015 USAJMO, 4
Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ such that\[f(x)+f(t)=f(y)+f(z)\]for all rational numbers $x<y<z<t$ that form an arithmetic progression. ($\mathbb{Q}$ is the set of all rational numbers.)
2012 China Team Selection Test, 2
Given a scalene triangle $ABC$. Its incircle touches $BC,AC,AB$ at $D,E,F$ respectvely. Let $L,M,N$ be the symmetric points of $D$ with $EF$,of $E$ with $FD$,of $F$ with $DE$,respectively. Line $AL$ intersects $BC$ at $P$,line $BM$ intersects $CA$ at $Q$,line $CN$ intersects $AB$ at $R$. Prove that $P,Q,R$ are collinear.
2019 Brazil Team Selection Test, 2
We say that a distribution of students lined upen in collumns is $\textit{bacana}$ when there are no two friends in the same column. We know that all contestants in a math olympiad can be arranged in a $\textit{bacana}$ configuration with $n$ columns, and that this is impossible with $n-1$ columns. Show that we can choose competitors $M_1, M_2, \cdots, M_n$ in such a way that $M_i$ is on the $i$-th column, for each $i = 1, 2, 3, \ldots, n$ and $M_i$ is a friend of $M_{i+1}$ for each $i = 1, 2, \ldots, n - 1$.