Found problems: 85335
Novosibirsk Oral Geo Oly VIII, 2022.7
The diagonals of the convex quadrilateral $ABCD$ intersect at the point $O$. The points $X$ and $Y$ are symmetrical to the point $O$ with respect to the midpoints of the sides $BC$ and $AD$, respectively. It is known that $AB = BC = CD$. Prove that the point of intersection of the perpendicular bisectors of the diagonals of the quadrilateral lies on the line $XY$.
2000 Austria Beginners' Competition, 4
Let $ABCDEFG$ be half of a regular dodecagon . Let $P$ be the intersection of the lines $AB$ and $GF$, and let $Q$ be the intersection of the lines $AC$ and $GE$. Prove that $Q$ is the circumcenter of the triangle $AGP$.
2021 Argentina National Olympiad, 5
Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.
1955 Miklós Schweitzer, 1
[b]1.[/b] Let $a_{1}, a_{2}, \dots , a_{n}$ and $b_{1}, b_{2}, \dots , b_{m}$ be $n+m$ unit vectors in the $r$-dimensional Euclidean space $E_{r} (n,m \leq r)$; let $a_{1}, a_{2}, \dots , a_{n}$ as well as $b_{1}, b_{2}, \dots , b_{m}$ be mutually orthogonal. For any vector $x \in E_{r}$, consider
$Tx= \sum_{i=1}^{n}\sum_{k=1}^{m}(x,a_{i})(a_{i},b_{k})b_{k}$
($(a,b)$ denotes the scalar product of $a$ and $b$). Show that the sequence $(T^{k}x)^{\infty}_{ k =0}$, where $T^{0} x= x$ and $T^{k} x = T(T^{k-1}x)$, is convergent and give a geometrical characterization of how the limit depends on $x$. [b](S. 14)[/b]
2000 AMC 10, 17
Boris has an incredible coin changing machine. When he puts in a quarter, it returns five nickels; when he puts in a nickel, it returns five pennies; and when he puts in a penny, it returns five quarters. Boris starts with just one penny. Which of the following amounts could Boris have after using the machine repeatedly?
$\text{(A)}\ \$3.63 \qquad\text{(B)}\ \$5.13\qquad\text{(C)}\ \$6.30 \qquad\text{(D)}\ \$7.45 \qquad\text{(E)}\ \$9.07$
2019 Bangladesh Mathematical Olympiad, 10
Given $2020*2020$ chessboard, what is the maximum number of warriors you can put on its cells such that no two warriors attack each other.
Warrior is a special chess piece which can move either $3$ steps forward and one step sideward and $2$ step forward and $2$ step sideward in any direction.
2003 India National Olympiad, 6
Each lottery ticket has a 9-digit numbers, which uses only the digits $1$, $2$, $3$. Each ticket is colored [color=red]red[/color],[color=blue] blue [/color]or [color=green]green[/color]. If two tickets have numbers which differ in all nine places, then the tickets have different colors. Ticket $122222222$ is red, and ticket $222222222$ is [color=green]green.[/color] What color is ticket $123123123$ ?
2003 Cono Sur Olympiad, 4
In an acute triangle $ABC$, the points $H$, $G$, and $M$ are located on $BC$ in such a way that $AH$, $AG$, and $AM$ are the height, angle bisector, and median of the triangle, respectively. It is known that $HG=GM$, $AB=10$, and $AC=14$. Find the area of triangle $ABC$.
I Soros Olympiad 1994-95 (Rus + Ukr), 10.3
Find any two consecutive natural numbers, each of which is divisible by the square of the sum of its digits.
1999 All-Russian Olympiad Regional Round, 10.3
There are $n$ points in general position in space (no three lie on the same straight line, no four lie in the same plane). A plane is drawn through every three of them. Prove that If you take any whatever $n-3$ points in space, there is a plane from those drawn that does not contain any of these $n - 3$ points.
2019 Taiwan TST Round 2, 2
Given a triangle $ \triangle{ABC} $. Denote its incircle and circumcircle by $ \omega, \Omega $, respectively. Assume that $ \omega $ tangents the sides $ AB, AC $ at $ F, E $, respectively. Then, let the intersections of line $ EF $ and $ \Omega $ to be $ P,Q $. Let $ M $ to be the mid-point of $ BC $. Take a point $ R $ on the circumcircle of $ \triangle{MPQ} $, say $ \Gamma $, such that $ MR \perp EF $. Prove that the line $ AR $, $ \omega $ and $ \Gamma $ intersect at one point.
2012 Today's Calculation Of Integral, 837
Let $f_n(x)=\sum_{k=1}^n (-1)^{k+1} \left(\frac{x^{2k-1}}{2k-1}+\frac{x^{2k}}{2k}\right).$
Find $\lim_{n\to\infty} f_n(1).$
1964 AMC 12/AHSME, 15
A line through the point $(-a,0)$ cuts from the second quadrant a triangular region with area $T$. The equation of the line is:
$ \textbf{(A)}\ 2Tx+a^2y+2aT=0 \qquad\textbf{(B)}\ 2Tx-a^2y+2aT=0 \qquad$
${{\textbf{(C)}\ 2Tx+a^2y-2aT=0 \qquad\textbf{(D)}\ 2Tx-a^2y-2aT=0 }\qquad\textbf{(E)}\ \text{none of these} } $
2004 Oral Moscow Geometry Olympiad, 4
Triangle $ABC$ is inscribed in a circle. Through points $A$ and $B$ tangents to this circle are drawn, which intersect at point $P$. Points $X$ and $Y$ are orthogonal projections of point $P$ onto lines $AC$ and $BC$. Prove that line $XY$ is perpendicular to the median of triangle $ABC$ from vertex $C$.
2006 Germany Team Selection Test, 1
Find all real solutions $x$ of the equation
$\cos\cos\cos\cos x=\sin\sin\sin\sin x$.
(Angles are measured in radians.)
2019 Durer Math Competition Finals, 11
What is the smallest $N$ for which $\sum_{k=1}^{N} k^{2018}$ is divisible by $2018$?
2014 National Olympiad First Round, 6
The numbers which contain only even digits in their decimal representations are written in ascending order such that \[2,4,6,8,20,22,24,26,28,40,42,\dots\] What is the $2014^{\text{th}}$ number in that sequence?
$
\textbf{(A)}\ 66480
\qquad\textbf{(B)}\ 64096
\qquad\textbf{(C)}\ 62048
\qquad\textbf{(D)}\ 60288
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2020 MBMT, 25
Let $\left \lfloor x \right \rfloor$ denote the greatest integer less than or equal to $x$. Find the sum of all positive integer solutions to $$\left \lfloor \frac{n^3}{27} \right \rfloor - \left \lfloor \frac{n}{3} \right \rfloor ^3=10.$$
[i]Proposed by Jason Hsu[/i]
1986 AMC 8, 17
Let $ o$ be an odd whole number and let $ n$ be any whole number. Which of the following statements about the whole number $ (o^2\plus{}no)$ is always true?
\[ \textbf{(A)}\ \text{it is always odd} \\
\textbf{(B)}\ \text{it is always even} \\
\textbf{(C)}\ \text{it is even only if n is even} \\
\textbf{(D)}\ \text{it is odd only if n is odd} \\
\textbf{(E)}\ \text{it is odd only if n is even}
\]
2008 Germany Team Selection Test, 1
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$.
[i]Author: Stephan Wagner, Austria[/i]
2023 Harvard-MIT Mathematics Tournament, 3
Let $ABCD$ be a convex quadrilateral such that $\angle{ABD}=\angle{BCD}=90^\circ,$ and let $M$ be the midpoint of segment $BD.$ Suppose that $CM=2$ and $AM=3.$ Compute $AD.$
2009 AIME Problems, 11
Consider the set of all triangles $ OPQ$ where $ O$ is the origin and $ P$ and $ Q$ are distinct points in the plane with nonnegative integer coordinates $ (x,y)$ such that $ 41x\plus{}y \equal{} 2009$. Find the number of such distinct triangles whose area is a positive integer.
1985 Iran MO (2nd round), 2
Let $x, y$ and $z$ be three positive real numbers for which
\[x^2+y^2+z^2=xy+yz+zx.\]
Find the value of $\frac{\sqrt x}{\sqrt x + \sqrt y+ \sqrt z}.$
1988 AMC 8, 15
The reciprocal of $ \left(\frac{1}{2}+\frac{1}{3}\right) $ is
$ \text{(A)}\ \frac{1}{6}\qquad\text{(B)}\ \frac{2}{5}\qquad\text{(C)}\ \frac{6}{5}\qquad\text{(D)}\ \frac{5}{2}\qquad\text{(E)}\ 5 $
2019 Turkey Team SeIection Test, 2
$(a_{n})_{n=1}^{\infty}$ is an integer sequence, $a_{1}=1$, $a_{2}=2$ and for $n\geq{1}$, $a_{n+2}=a_{n+1}^{2}+(n+2)a_{n+1}-a_{n}^{2}-na_{n}$.
$a)$ Prove that the set of primes that divides at least one term of the sequence can not be finite.
$b)$ Find 3 different prime numbers that do not divide any terms of this sequence.