Found problems: 85335
2016 PUMaC Geometry A, 3
Let $C$ be a right circular cone with apex $A$. Let $P_1, P_2, P_3, P_4$ and $P_5$ be points placed evenly along the circular base in that order, so that $P_1P_2P_3P_4P_5$ is a regular pentagon. Suppose that the shortest path from $P_1$ to $P_3$ along the curved surface of the cone passes through the midpoint of $AP_2$. Let $h$ be the height of $C$, and $r$ be the radius of the circular base of $C$. If $\left(\frac{h}{r}\right)^2$ can be written in simplest form as $\frac{a}{b}$ , fi nd $a + b$.
2015 NIMO Problems, 6
Let $\triangle ABC$ be a triangle with $BC = 4, CA= 5, AB= 6$, and let $O$ be the circumcenter of $\triangle ABC$. Let $O_b$ and $O_c$ be the reflections of $O$ about lines $CA$ and $AB$ respectively. Suppose $BO_b$ and $CO_c$ intersect at $T$, and let $M$ be the midpoint of $BC$. Given that $MT^2 = \frac{p}{q}$ for some coprime positive integers $p$ and $q$, find $p+q$.
[i]Proposed by Sreejato Bhattacharya[/i]
2023 Sharygin Geometry Olympiad, 3
A circle touches the lateral sides of a trapezoid $ABCD$ at points $B$ and $C$, and its center lies on $AD$. Prove that the diameter of the circle is less than the medial line of the trapezoid.
2011 Korea National Olympiad, 2
Let $x, y$ be positive integers such that $\gcd(x,y)=1$ and $x+3y^2$ is a perfect square. Prove that $x^2+9y^4$ can't be a perfect square.
2013 Online Math Open Problems, 12
There are 25 ants on a number line; five at each of the coordinates $1$, $2$, $3$, $4$, and $5$. Each minute, one ant moves from its current position to a position one unit away. What is the minimum number of minutes that must pass before it is possible for no two ants to be on the same coordinate?
[i]Ray Li[/i]
2024 Polish Junior MO Finals, 4
Let $ABC$ be an isosceles triangle with $AC=BC$. Let $P,Q,R$ be points on the sides $AB, BC, CA$ of the triangle such that $CQPR$ is a parallelogram. Show that the reflection of $P$ over $QR$ lies on the circumcircle of $ABC$.
1957 Polish MO Finals, 5
Given a line $ m $ and a segment $ AB $ parallel to it. Divide the segment $ AB $ into three equal parts using only a ruler, i.e. drawing only the lines.
2020 Peru Iberoamerican Team Selection Test, P4
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.
2013 NIMO Problems, 5
Let $d$ and $n$ be positive integers such that $d$ divides $n$, $n > 1000$, and $n$ is not a perfect square. The minimum possible value of $\left\lvert d - \sqrt{n} \right\rvert$ can be written in the form $a\sqrt{b} + c$, where $b$ is a positive integer not divisible by the square of any prime, and $a$ and $c$ are nonzero integers (not necessarily positive). Compute $a+b+c$.
[i]Proposed by Matthew Lerner-Brecher[/i]
2013 Swedish Mathematical Competition, 6
Let $a, b, c$, be real numbers such that $$a^2b^2 + 18 abc > 4b^3+4a^3c+27c^2 .$$
Prove that $a^2>3b$.
2008 Purple Comet Problems, 4
While driving his car, Ken pulled off the road to get gasoline when he was of $\frac{7}{12}$ the way through his trip. After driving another eleven miles, he noticed that he was $\frac{13}{20}$ of the way through his trip. How many miles long was his entire trip?
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 $C$.