Found problems: 85335
2022 Dutch BxMO TST, 5
In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the $k$ people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of $k$?
2018 China Team Selection Test, 1
Let $\omega_1,\omega_2$ be two non-intersecting circles, with circumcenters $O_1,O_2$ respectively, and radii $r_1,r_2$ respectively where $r_1 < r_2$. Let $AB,XY$ be the two internal common tangents of $\omega_1,\omega_2$, where $A,X$ lie on $\omega_1$, $B,Y$ lie on $\omega_2$. The circle with diameter $AB$ meets $\omega_1,\omega_2$ at $P$ and $Q$ respectively. If $$\angle AO_1P+\angle BO_2Q=180^{\circ},$$ find the value of $\frac{PX}{QY}$ (in terms of $r_1,r_2$).
1985 IMO Longlists, 64
Let $p$ be a prime. For which $k$ can the set $\{1, 2, \dots , k\}$ be partitioned into $p$ subsets with equal sums of elements ?
1966 IMO Longlists, 15
Given four points $A,$ $B,$ $C,$ $D$ on a circle such that $AB$ is a diameter and $CD$ is not a diameter. Show that the line joining the point of intersection of the tangents to the circle at the points $C$ and $D$ with the point of intersection of the lines $AC$ and $BD$ is perpendicular to the line $AB.$
2013 May Olympiad, 4
Pablo wrote $5$ numbers on one sheet and then wrote the numbers $6,7,8,8,9,9,10,10,11$ and $ 12$ on another sheet that he gave Sofia, indicating that those numbers are the possible sums of two of the numbers that he had hidden. Decide if with this information Sofia can determine the five numbers Pablo wrote .
1991 Austrian-Polish Competition, 6
Suppose that there is a point $P$ inside a convex quadrilateral $ABCD$ such that the triangles $PAB$, $PBC$, $PCD$, $PDA$ have equal areas. Prove that one of the diagonals bisects the area of $ABCD$.
2011 Greece Team Selection Test, 3
Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold:
$$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$
$$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$
for all $x,y \in \mathbb{Q}$.
2010 Contests, 4
Find all integer solutions $(a,b)$ of the equation \[ (a+b+3)^2 + 2ab = 3ab(a+2)(b+2)\]
2018 Regional Olympiad of Mexico Center Zone, 3
Consider $n$ lines in the plane in general position, that is, there are not three of the $n$ lines that pass through the same point. Determine if it is possible to label the $k$ points where these lines are inserted with the numbers $1$ through $k$ (using each number exactly once), so that on each line, the labels of the $n-1$ points of that line are arranged in increasing order (in one of the two directions in which they can be traversed).
2017 Sharygin Geometry Olympiad, P22
Let $P$ be an arbitrary point on the diagonal $AC$ of cyclic quadrilateral $ABCD$, and $PK, PL, PM, PN, PO$ be the perpendiculars from $P$ to $AB, BC, CD, DA, BD$ respectively. Prove that the distance from $P$ to $KN$ is equal to the distance from $O$ to $ML$.
1990 ITAMO, 1
A cube of edge length $3$ consists of $27$ unit cubes. Find the number of lines passing through exactly three centers of these $27$ cubes, as well as the number of those passing through exactly two such centers.
1995 Denmark MO - Mohr Contest, 1
A trapezoid has side lengths as indicated in the figure (the sides with length $11$ and $36$ are parallel). Calculate the area of the trapezoid.[img]https://1.bp.blogspot.com/-5PKrqDG37X4/XzcJtCyUv8I/AAAAAAAAMY0/tB0FObJUJdcTlAJc4n6YNEaVIDfQ91-eQCLcBGAsYHQ/s0/1995%2BMohr%2Bp1.png[/img]
2016 Oral Moscow Geometry Olympiad, 1
Angles are equal in a hexagon, three main diagonals are equal and the other six diagonals are also equal. Is it true that it has equal sides?
1999 IMC, 5
Suppose that $2n$ points of an $n\times n$ grid are marked. Show that for some $k > 1$ one can select $2k$ distinct marked points, say $a_1,...,a_{2k}$, such that $a_{2i-1}$ and $a_{2i}$ are in the same row, $a_{2i}$ and $a_{2i+1}$ are in the same column, $\forall i$, indices taken mod 2n.
2014 Iran Team Selection Test, 6
Consider $n$ segments in the plane which no two intersect and between their $2n$ endpoints no three are collinear. Is the following statement true?
Statement: There exists a simple $2n$-gon such that it's vertices are the $2n$ endpoints of the segments and each segment is either completely inside the polygon or an edge of the polygon.
2009 Kosovo National Mathematical Olympiad, 3
Prove that $\sqrt 2$ is irrational.
2013 BMT Spring, 10
If five squares of a $3 \times 3$ board initially colored white are chosen at random and blackened, what is the expected number of edges between two squares of the same color?
1999 Czech And Slovak Olympiad IIIA, 4
In a certain language there are only two letters, $A$ and $B$. We know that
(i) There are no words of length $1$, and the only words of length $2$ are $AB$ and $BB$.
(ii) A segment of length $n > 2$ is a word if and only if it can be obtained from a word of length less than $n$ by replacing each letter $B$ by some (not necessarily the same) word.
Prove that the number of words of length $n$ is equal to $\frac{2^n +2\cdot (-1)^n}{3}$
2011 Romanian Master of Mathematics, 6
The cells of a square $2011 \times 2011$ array are labelled with the integers $1,2,\ldots, 2011^2$, in such a way that every label is used exactly once. We then identify the left-hand and right-hand edges, and then the top and bottom, in the normal way to form a torus (the surface of a doughnut).
Determine the largest positive integer $M$ such that, no matter which labelling we choose, there exist two neighbouring cells with the difference of their labels at least $M$.
(Cells with coordinates $(x,y)$ and $(x',y')$ are considered to be neighbours if $x=x'$ and $y-y'\equiv\pm1\pmod{2011}$, or if $y=y'$ and $x-x'\equiv\pm1\pmod{2011}$.)
[i](Romania) Dan Schwarz[/i]
1948 Putnam, A6
Answer either (i) or (ii):
(i) A force acts on the element $ds$ of a closed plane curve. The magnitude of this force is $r^{-1} ds$ where $r$ is the radius of curvature at the point considered, and the direction of the force is perpendicular to the curve, it points to the convex side. Show that the system of such forces acting on all elements of the curve keep it in equilibrium.
(ii) Show that
$$x+ \frac{2}{3}x^{3}+ \frac{2\cdot 4}{3\cdot 5} x^5 +\frac{2\cdot 4\cdot 6}{3\cdot 5\cdot 7} x^7 + \ldots= \frac{ \arcsin x}{\sqrt{1-x^{2}}}.$$
2022 Israel National Olympiad, P3
Let $w$ be a circle of diameter $5$. Four lines were drawn dividing $w$ into $5$ "strips", each of width $1$. The strips were colored orange and purple alternatingly, as depicted. Which area is larger: the orange or the purple?
2011 Hanoi Open Mathematics Competitions, 6
Find all positive integers $(m,n)$ such that $m^2 + n^2 + 3 = 4(m + n)$
2009 National Olympiad First Round, 12
How many 8-digit numbers are there such that exactly 7 digits are all different?
$\textbf{(A)}\ {{9}\choose{3}}^2 \cdot 6! \cdot 3 \qquad\textbf{(B)}\ {{8}\choose{3}}^2 \cdot 7! \qquad\textbf{(C)}\ {{8}\choose{3}}^2 \cdot 7! \cdot 3 \\ \qquad\textbf{(D)}\ {{7}\choose{3}}^2 \cdot 7! \qquad\textbf{(E)}\ {{9}\choose{4}}^2 \cdot 6! \cdot 8$
2021 USAMTS Problems, 2
Find, with proof, the minimum positive integer n with the following property: for
any coloring of the integers $\{1, 2, . . . , n\}$ using the colors red and blue (that is, assigning the
color “red” or “blue” to each integer in the set), there exist distinct integers a, b, c between
1 and n, inclusive, all of the same color, such that $2a + b = c.$
2020 USA IMO Team Selection Test, 4
For a finite simple graph $G$, we define $G'$ to be the graph on the same vertex set as $G$, where for any two vertices $u \neq v$, the pair $\{u,v\}$ is an edge of $G'$ if and only if $u$ and $v$ have a common neighbor in $G$.
Prove that if $G$ is a finite simple graph which is isomorphic to $(G')'$, then $G$ is also isomorphic to $G'$.
[i]Mehtaab Sawhney and Zack Chroman[/i]