Found problems: 85335
2002 China Team Selection Test, 1
Find all natural numbers $n (n \geq 2)$ such that there exists reals $a_1, a_2, \dots, a_n$ which satisfy \[ \{ |a_i - a_j| \mid 1\leq i<j \leq n\} = \left\{1,2,\dots,\frac{n(n-1)}{2}\right\}. \]
Let $A=\{1,2,3,4,5,6\}, B=\{7,8,9,\dots,n\}$. $A_i(i=1,2,\dots,20)$ contains eight numbers, three of which are chosen from $A$ and the other five numbers from $B$. $|A_i \cap A_j|\leq 2, 1\leq i<j\leq 20$. Find the minimum possible value of $n$.
1999 Harvard-MIT Mathematics Tournament, 2
For what single digit $n$ does $91$ divide the $9$-digit number $12345n789$?
1999 Mongolian Mathematical Olympiad, Problem 6
Let $f$ be a map of the plane into itself with the property that if $d(A,B)=1$, then $d(f(A),f(B))=1$, where $d(X,Y)$ denotes the distance between points $X$ and $Y$. Prove that for any positive integer $n$, $d(A,B)=n$ implies $d(f(A),f(B))=n$.
2016 Regional Olympiad of Mexico West, 3
A circle $\omega$ with center $O$ and radius $r$ is constructed. A point $P$ is chosen on the circumference $\omega$ and a point A is taken inside it, such that is outside the line that passes through $P$ and $O$. Point $B$ is constructed, the reflection of $A$ wrt $O$. and $P'$ is another point on the circumference such that the chord $PP'$ is perpendicular to $PA$. Let $Q$ be the point on the line $PP'$ that minimizes the sum of distances from $A$ to $Q$ and from $Q$ to $B$. Show that the value of the sum of the lengths $AQ+QB$ does not depend on the choice of points $P$ or $A$
2022 Thailand TST, 2
Let $ABCD$ be a quadrilateral inscribed in a circle $\Omega.$ Let the tangent to $\Omega$ at $D$ meet rays $BA$ and $BC$ at $E$ and $F,$ respectively. A point $T$ is chosen inside $\triangle ABC$ so that $\overline{TE}\parallel\overline{CD}$ and $\overline{TF}\parallel\overline{AD}.$ Let $K\ne D$ be a point on segment $DF$ satisfying $TD=TK.$ Prove that lines $AC,DT,$ and $BK$ are concurrent.
Kyiv City MO 1984-93 - geometry, 1989.9.1
The perimeter of the triangle $ABC$ is equal to $2p$, the length of the side$ AC$ is equal to $b$, the angle $ABC$ is equal to $\beta$. A circle with center at point $O$, inscribed in this triangle, touches the side $BC$ at point $K$. Calculate the area of the triangle $BOK$.
2001 Estonia National Olympiad, 2
Find the minimum value of $n$ such that, among any $n$ integers, there are three whose sum is divisible by $3$.
2009 Princeton University Math Competition, 6
Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is $\frac{1734}{274}$, then you would submit 1734274).
2019 NMTC Junior, 8
A circular disc is divided into $12$ equal sectors and one of $6$ different colours is used to colour each sector. No two adjacent sectors can have the same colour. Find the number of such distinct colorings possible.
2001 JBMO ShortLists, 8
Prove that no three points with integer coordinates can be the vertices of an equilateral triangle.
2016 Macedonia JBMO TST, 1
Solve the following equation in the set of integers
$x_{1}^4 + x_{2}^4 +...+ x_{14}^4=2016^3 - 1$.
2013 Stanford Mathematics Tournament, 10
A unit circle is centered at the origin and a tangent line to the circle is constructed in the first quadrant such that it makes an angle $5\pi/6$ with the $y$-axis. A series of circles centered on the $x$-axis are constructed such that each circle is both tangent to the previous circle and the original tangent line. Find the total area of the series of circles.
2021 Brazil National Olympiad, 3
Find all positive integers \(k\) for which there is an irrational \(\alpha>1\) and a positive integer \(N\) such that \(\left\lfloor\alpha^{n}\right\rfloor\) is a perfect square minus \(k\) for every integer \(n\) with \(n>N\).
2009 AMC 12/AHSME, 23
A region $ S$ in the complex plane is defined by \[ S \equal{} \{x \plus{} iy: \minus{} 1\le x\le1, \minus{} 1\le y\le1\}.\] A complex number $ z \equal{} x \plus{} iy$ is chosen uniformly at random from $ S$. What is the probability that $ \left(\frac34 \plus{} \frac34i\right)z$ is also in $ S$?
$ \textbf{(A)}\ \frac12\qquad \textbf{(B)}\ \frac23\qquad \textbf{(C)}\ \frac34\qquad \textbf{(D)}\ \frac79\qquad \textbf{(E)}\ \frac78$
2012 Indonesia TST, 2
An $m \times n$ chessboard where $m \le n$ has several black squares such that no two rows have the same pattern. Determine the largest integer $k$ such that we can always color $k$ columns red while still no two rows have the same pattern.
2011 Puerto Rico Team Selection Test, 1
The product of 22 integers is 1. Show that their sum can not be 0.
2017 CCA Math Bonanza, L5.2
Compute $e^{\pi}+\pi^e$. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\frac{4}{\pi}\arctan\left(\frac{1}{\left|C-A\right|}\right)$ (note that the closer you are to the right answer, the higher your score is).
[i]2017 CCA Math Bonanza Lightning Round #5.2[/i]
KoMaL A Problems 2020/2021, A. 780
We colored the $n^2$ unit squares of an $n\times n$ square lattice such that in each $2\times 2$ square, at least two of the four unit squares have the same color. What is the largest number of colors we could have used?
[i]Based on a problem of the Dürer Competition[/i]
1985 Polish MO Finals, 6
There is a convex polyhedron with $k$ faces.
Show that if more than $k/2$ of the faces are such that no two have a common edge,
then the polyhedron cannot have an inscribed sphere.
2007 Purple Comet Problems, 8
Penelope plays a game where she adds $25$ points to her score each time she wins a game and deducts $13$ points from her score each time she loses a game. Starting with a score of zero, Penelope plays $m$ games and has a total score of $2007$ points. What is the smallest possible value for $m$?
2013 Princeton University Math Competition, 5
Define a "digitized number" as a ten-digit number $a_0a_1\ldots a_9$ such that for $k=0,1,\ldots, 9$, $a_k$ is equal to the number of times the digit $k$ occurs in the number. Find the sum of all digitized numbers.
2003 Junior Balkan MO, 1
Let $n$ be a positive integer. A number $A$ consists of $2n$ digits, each of which is 4; and a number $B$ consists of $n$ digits, each of which is 8. Prove that $A+2B+4$ is a perfect square.
2016 Irish Math Olympiad, 10
Let $AE$ be a diameter of the circumcircle of triangle $ABC$. Join $E$ to the orthocentre, $H$, of $\triangle ABC$ and extend $EH$ to meet the circle again at $D$. Prove that the nine point circle of $\triangle ABC$ passes through the midpoint of $HD$.
Note. The nine point circle of a triangle is a circle that passes through the midpoints of the sides, the feet of the altitudes and the midpoints of the line segments that join the orthocentre to the vertices.
2010 China National Olympiad, 2
Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have
\[a_n =
\begin{cases}
a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\
2n & \text{if } (a_{n-1},n) > 1
\end{cases}
\]
Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.
2007 Peru IMO TST, 2
Let $a,b,c$ be positive real numbers, such that: $a+b+c \geq \frac{1}{a}+\frac{1}{b}+\frac{1}{c}.$
Prove that:
\[a+b+c \geq \frac{3}{a+b+c}+\frac{2}{abc}. \]