Found problems: 85335
1963 Putnam, A1
i) Show that a regular hexagon, six squares, and six equilateral triangles can be assembled without overlapping to form a regular dodecagon.
ii) Let $P_1 , P_2 ,\ldots, P_{12}$ be the vertices of a regular dodecagon. Prove that the three diagonals $P_{1}P_{9}, P_{2}P_{11}$ and $P_{4}P_{12}$ intersect.
2013 Kyiv Mathematical Festival, 2
For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of several distinct positive integers not exceeding $2n$?
1967 IMO Longlists, 11
Let $n$ be a positive integer. Find the maximal number of non-congruent triangles whose sides lengths are integers $\leq n.$
LMT Accuracy Rounds, 2022 S2
Let $a \spadesuit b = \frac{a^2-b^2}{2b-2a}$ . Given that $3 \spadesuit x = -10$, compute $x$.
2022 JHMT HS, 10
Let $R$ be the rectangle in the coordinate plane with corners $(0, 0)$, $(20, 0)$, $(20, 22)$, and $(0, 22)$, and partition $R$ into a $20\times 22$ grid of unit squares. For a given line in the coordinate plane, let its [i]pixelation[/i] be the set of grid squares in $R$ that contain part of the line in their interior. If $P$ is a point chosen uniformly at random in $R$, then compute the expected number of sets of grid squares that are pixelations of some line through $P$.
2000 Austrian-Polish Competition, 3
For each integer $n \ge 3$ solve in real numbers the system of equations:
$$\begin{cases} x_1^3 = x_2 + x_3 + 1 \\...\\x_{n-1}^3 = x_n+ x_1 + 1\\x_{n}^3 = x_1+ x_2 + 1 \end{cases}$$
1992 AMC 12/AHSME, 11
The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, then the radius of the larger circle is
[asy]
size(200);
defaultpen(linewidth(0.7)+fontsize(10));
pair O=origin, A=3*dir(180), B=3*dir(140), C=3*dir(0);
dot(O);
draw(Arc(origin,1,0,360));
draw(Arc(origin,3,0,360));
draw(A--B--C--A);
label("$A$", A, dir(O--A));
label("$B$", B, dir(O--B));
label("$C$", C, dir(O--C));
[/asy]
$ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 26 $
2018 CMIMC Combinatorics, 9
Compute the number of rearrangements $a_1, a_2, \dots, a_{2018}$ of the sequence $1, 2, \dots, 2018$ such that $a_k > k$ for $\textit{exactly}$ one value of $k$.
2021 Serbia Team Selection Test, P3
Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.
2023 BMT, 1
Compute the three-digit number that satisfies the following properties:
$\bullet$ The hundreds digit and ones digit are the same, but the tens digit is different.
$\bullet$ The number is divisible by $9$.
$\bullet$ When the number is divided by $5$, the remainder is $1$.
2012 Turkey MO (2nd round), 1
Find all polynomials with integer coefficients such that for all positive integers $n$ satisfies $P(n!)=|P(n)|!$
2012 AMC 12/AHSME, 20
Consider the polynomial \[P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024).\]
The coefficient of $x^{2012}$ is equal to $2^a$. What is $a$?
$ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24 $
1969 IMO Longlists, 3
$(BEL 3)$ Construct the circle that is tangent to three given circles.
2017 Ukrainian Geometry Olympiad, 4
Let $AD$ be the inner angle bisector of the triangle $ABC$. The perpendicular on the side $BC$ at the point $D$ intersects the outer bisector of $\angle CAB$ at point $I$. The circle with center $I$ and radius $ID$ intersects the sides $AB$ and $AC$ at points $F$ and $E$ respectively. $A$-symmedian of $\Delta AFE$ intersects the circumcircle of $\Delta AFE$ again at point $X$. Prove that the circumcircles of $\Delta AFE$ and $\Delta BXC$ are tangent.
2011 Princeton University Math Competition, A1
Find, with proof, all triples of positive integers $(x,y,z)$ satisfying the equation $3^x - 5^y = 4z^2$.
2017 Kazakhstan National Olympiad, 5
Consider all possible sets of natural numbers $(x_1, x_2, ..., x_{100})$ such that $1\leq x_i \leq 2017$ for every $i = 1,2, ..., 100$. We say that the set $(y_1, y_2, ..., y_{100})$ is greater than the set $(z_1, z_2, ..., z_{100})$ if $y_i> z_i$ for every $i = 1,2, ..., 100$. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?
2010 All-Russian Olympiad Regional Round, 10.8
Let's call it a [i] staircase of height [/i]$n$, a figure consisting from all square cells $n\times n$ lying no higher diagonals (the figure shows a [i]staircase of height [/i] $4$ ). In how many different ways can a [i]staircase of height[/i] $n$ can be divided into several rectangles whose sides go along the grid lines, but the areas are different in pairs?
[img]https://cdn.artofproblemsolving.com/attachments/f/0/f66d7e9ada0978e8403fbbd8989dc1b201f2cd.png[/img]
2015 Bosnia And Herzegovina - Regional Olympiad, 3
Let $ABC$ be a triangle with incenter $I$. Line $AI$ intersects circumcircle of $ABC$ in points $A$ and $D$, $(A \neq D)$. Incircle of $ABC$ touches side $BC$ in point $E$ . Line $DE$ intersects circumcircle of $ABC$ in points $D$ and $F$, $(D \neq F)$. Prove that $\angle AFI = 90^{\circ}$
2013 National Olympiad First Round, 5
Let $D$ be a point on side $[BC]$ of triangle $ABC$ where $|BC|=11$ and $|BD|=8$. The circle passing through the points $C$ and $D$ touches $AB$ at $E$. Let $P$ be a point on the line which is passing through $B$ and is perpendicular to $DE$. If $|PE|=7$, then what is $|DP|$?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ \text{None of above}
$
1903 Eotvos Mathematical Competition, 3
Let $A,B,C,D$ be the vertices of a rhombus, let $k_1$ be the circle through $B,C$ and $D$, let $k_2$ be the circle through $A,C$ and $D$, let $k_3$ be the circle through $A,B$ and $D$, let $k_4$ be the circle through $A,B$ and $C$. Prove that the tangents to $k_1$ and $k_3$ at $B$ form the same angle as the tangents to $k_2$ and $k_4$ at $A$.
2000 Belarus Team Selection Test, 7.3
A game is played by $n$ girls ($n \geq 2$), everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice [b]nice[/b] if at the end nobody has her own ball and it is called [b]tiresome[/b] if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.
2012 South East Mathematical Olympiad, 2
Find the least natural number $n$, such that the following inequality holds:$\sqrt{\dfrac{n-2011}{2012}}-\sqrt{\dfrac{n-2012}{2011}}<\sqrt[3]{\dfrac{n-2013}{2011}}-\sqrt[3]{\dfrac{n-2011}{2013}}$.
1996 Tournament Of Towns, (518) 1
Can one paint four vertices of a cube red and the other four points black so that any plane passing through three points of the same colour contains a vertex of the other colour?
(Mebius, Sharygin)
2019 Saint Petersburg Mathematical Olympiad, 2
Every two of the $n$ cities of Ruritania are connected by a direct flight of one from two airlines. Promonopoly Committee wants at least $k$ flights performed by one company. To do this, he can at least every day to choose any three cities and change the ownership of the three flights connecting these cities each other (that is, to take each of these flights from a company that performs it, and pass the other). What is the largest $k$ committee knowingly will be able to achieve its goal in no time, no matter how the flights are distributed hour?
2005 IMAR Test, 3
A flea moves in the positive direction on the real Ox axis, starting from the origin. He can only jump over distances equal with $\sqrt 2$ or $\sqrt{2005}$. Prove that there exists $n_0$ such that the flea can reach any interval $[n,n+1]$ with $n\geq n_0$.