Found problems: 85335
1999 AMC 12/AHSME, 2
Which of the following statements is false?
$ \textbf{(A)}\ \text{All equilateral triangles are congruent to each other.}$
$ \textbf{(B)}\ \text{All equilateral triangles are convex.}$
$ \textbf{(C)}\ \text{All equilateral triangles are equilangular.}$
$ \textbf{(D)}\ \text{All equilateral triangles are regular polygons.}$
$ \textbf{(E)}\ \text{All equilateral triangles are similar to each other.}$
2016 CMIMC, 5
Let $\mathcal{S}$ be a regular 18-gon, and for two vertices in $\mathcal{S}$ define the $\textit{distance}$ between them to be the length of the shortest path along the edges of $\mathcal{S}$ between them (e.g. adjacent vertices have distance 1). Find the number of ways to choose three distinct vertices from $\mathcal{S}$ such that no two of them have distance 1, 8, or 9.
2012 China National Olympiad, 3
Prove for any $M>2$, there exists an increasing sequence of positive integers $a_1<a_2<\ldots $ satisfying:
1) $a_i>M^i$ for any $i$;
2) There exists a positive integer $m$ and $b_1,b_2,\ldots ,b_m\in\left\{ -1,1\right\}$, satisfying $n=a_1b_1+a_2b_2+\ldots +a_mb_m$ if and only if $n\in\mathbb{Z}/ \{0\}$.
2021 AMC 10 Fall, 17
An architect is building a structure that will place vertical pillars at the vertices of regular hexagon $ABCDEF$, which is lying horizontally on the ground. The six pillars will hold up a flat solar panel that will not be parallel to the ground. The heights of the pillars at $A$, $B$, and $C$ are $12,9,$ and $10$ meters, respectively. What is the height, in meters, of the pillar at $E$?
$\textbf{(A) }9\qquad\textbf{(B) }6\sqrt3\qquad\textbf{(C) }8\sqrt3\qquad\textbf{(D) }17\qquad\textbf{(E) }12\sqrt3$
2015 Geolympiad Summer, 1.
Show in an acute triangle $ABC$ that $\cot A + \cot B + \cot C \ge \dfrac{12[ABC]}{a^2+b^2+c^2}$.
2016 IOM, 3
Let $A_1A_2 . . . A_n$ be a cyclic convex polygon whose circumcenter is strictly in its interior. Let $B_1, B_2, ..., B_n$ be arbitrary points on the sides $A_1A_2, A_2A_3, ..., A_nA_1$, respectively, other than the vertices. Prove that
$\frac{B_1B_2}{A_1A_3}+ \frac{B_2B_3}{A_2A_4}+...+\frac{B_nB_1}{A_nA_2}>1$.
2016 Online Math Open Problems, 11
Let $f$ be a random permutation on $\{1, 2, \dots, 100\}$ satisfying $f(1) > f(4)$ and $f(9)>f(16)$. The probability that $f(1)>f(16)>f(25)$ can be written as $\frac mn$ where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$.
Note: In other words, $f$ is a function such that $\{f(1), f(2), \ldots, f(100)\}$ is a permutation of $\{1,2, \ldots, 100\}$.
[i]Proposed by Evan Chen[/i]
2024 Bangladesh Mathematical Olympiad, P5
Let $I$ be the incenter of $\triangle ABC$ and $P$ be a point such that $PI$ is perpendicular to $BC$ and $PA$ is parallel to $BC$. Let the line parallel to $BC$, which is tangent to the incircle of $\triangle ABC$, intersect $AB$ and $AC$ at points $Q$ and $R$ respectively. Prove that $\angle BPQ = \angle CPR$.
2003 May Olympiad, 5
We have a $4 \times 4$ squared board. We define the [i]separation [/i] between two squares as the least number of moves that a chess knight must take to go from one square to the other (using moves of the knight). Three boxes $A, B, C$ form a good trio if the three separations between $A$ and $B$, between $A$ and $C$ and between $B$ and $C$ are equal. Determines the number of good trios that are formed on the board.
Clarification: In each move the knight moves $2$ squares in the horizontal direction plus one square in the vertical direction or moves $2$ squares in the vertical direction plus one square in the horizontal direction.
1967 IMO Longlists, 42
Decompose the expression into real factors:
\[E = 1 - \sin^5(x) - \cos^5(x).\]
2023 Czech-Polish-Slovak Match, 2
Let $a_1, a_2, \ldots, a_n$ be reals such that for all $k=1,2, \ldots, n$, $na_k \geq a_1^2+a_2^2+ \ldots+a_k^2$. Prove that there exist at least $\frac{n} {10}$ indices $k$, such that $a_k \leq 1000$.
2015 Hanoi Open Mathematics Competitions, 3
Suppose that $a > b > c > 1$. One of solutions of the equation
$\frac{(x - a)(x - b)}{(c - a)(c - b)}+\frac{(x - b)(x - c)}{(a - b)(a - c)}+\frac{(x - c)(x - a)}{(b - c)(b - a)}= x$ is
(A): $-1$, (B): $-2$, (C): $0$, (D): $1$, (E): None of the above.
2016 International Zhautykov Olympiad, 2
A convex hexagon $ABCDEF$ is given such that $AB||DE$, $BC||EF$, $CD||FA$. The point $M, N, K$ are common points of the lines $BD$ and $AE$, $AC$ and $DF$, $CE$ and $BF$ respectively. Prove that perpendiculars drawn from $M, N, K$ to lines $AB, CD, EF$ respectively concurrent.
1979 IMO Shortlist, 23
Find all natural numbers $n$ for which $2^8 +2^{11} +2^n$ is a perfect square.
2020 AMC 12/AHSME, 9
A three-quarter sector of a circle of radius $4$ inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping together along the two radii shown. What is the volume of the cone in cubic inches?
[asy]
draw(Arc((0,0), 4, 0, 270));
draw((0,-4)--(0,0)--(4,0));
label("$4$", (2,0), S);
[/asy]
$\textbf{(A)}\ 3\pi \sqrt5 \qquad\textbf{(B)}\ 4\pi \sqrt3 \qquad\textbf{(C)}\ 3 \pi \sqrt7 \qquad\textbf{(D)}\ 6\pi \sqrt3 \qquad\textbf{(E)}\ 6\pi \sqrt7$
2022 Bulgarian Autumn Math Competition, Problem 11.4
The number $2022$ is written on the white board. Ivan and Peter play a game, Ivan starts and they alternate. On a move, Ivan erases the number $b$, written on the board, throws a dice which shows some number $a$, and writes the residue of $(a+b) ^2$ modulo $5$. Similarly, Peter throws a dice which shows some number $a$, and changes the previously written number $b$ to the residue of $a+b$ modulo $3$. The first player to write a $0$ wins. What is the probability of Ivan winning the game?
2010 Contests, 1b
The edges of the square in the figure have length $1$.
Find the area of the marked region in terms of $a$, where $0 \le a \le 1$.
[img]https://cdn.artofproblemsolving.com/attachments/2/2/f2b6ca973f66c50e39124913b3acb56feff8bb.png[/img]
2003 AMC 12-AHSME, 20
Part of the graph of $ f(x) \equal{} x^3 \plus{} bx^2 \plus{} cx \plus{} d$ is shown. What is $ b$?
[asy]import graph;
unitsize(1.5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=3;
real y(real x)
{
return (x-1)*(x+1)*(x-2);
}
path bounds=(-1.5,-1)--(1.5,-1)--(1.5,2.5)--(-1.5,2.5)--cycle;
pair[] points={(-1,0),(0,2),(1,0)};
draw(bounds,white);
draw(graph(y,-1.5,1.5));
drawline((0,0),(1,0));
drawline((0,0),(0,1));
dot(points);
label("$(-1,0)$",(-1,0),SE);
label("$(1,0)$",(1,0),SW);
label("$(0,2)$",(0,2),NE);
clip(bounds);[/asy]$ \textbf{(A)}\minus{}\!4 \qquad \textbf{(B)}\minus{}\!2 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 4$
2013 Canadian Mathematical Olympiad Qualification Repechage, 3
A positive integer $n$ has the property that there are three positive integers $x, y, z$ such that $\text{lcm}(x, y) = 180$, $\text{lcm}(x, z) = 900$, and $\text{lcm}(y, z) = n$, where $\text{lcm}$ denotes the lowest common multiple. Determine the number of positive integers $n$ with this property.
2017 Pan African, Problem 6
Let $ABC$ be a triangle with $H$ its orthocenter. The circle with diameter $[AC]$ cuts the circumcircle of triangle $ABH$ at $K$. Prove that the point of intersection of the lines $CK$ and $BH$ is the midpoint of the segment $[BH]$
2021 Lotfi Zadeh Olympiad, 1
In the inscribed quadrilateral $ABCD$, $P$ is the intersection point of diagonals and $M$ is the midpoint of arc $AB$. Prove that line $MP$ passes through the midpoint of segment $CD$, if and only if lines $AB, CD$ are parallel.
2021 Princeton University Math Competition, B1
Parallelogram $ABCD$ is given such that $\angle ABC$ equals $30^o$ . Let $X$ be the foot of the perpendicular from $A$ onto $BC$, and $Y$ the foot of the perpendicular from $C$ to $AB$. If $AX = 20$ and $CY = 22$, find the area of the parallelogram.
2009 Today's Calculation Of Integral, 413
Find the maximum and minimum value of $ F(x) \equal{} \frac {1}{2}x \plus{} \int_0^x (t \minus{} x)\sin t\ dt$ for $ 0\leq x\leq \pi$.
2015 Olympic Revenge, 5
Given a triangle $A_1 A_2 A_3$, let $a_i$ denote the side opposite to $A_i$, where indices are taken modulo 3. Let $D_1 \in a_1$. For $D_i \in A_i$, let $\omega_i$ be the incircle of the triangle formed by lines $a_i, a_{i+1}, A_iD_i$, and $D_{i+1} \in a_{i+1}$ with $A_{i+1} D_{i+1}$ tangent to $\omega_i$. Show that the set $\{D_i: i \in \mathbb{N}\}$ is finite.
1948 Moscow Mathematical Olympiad, 149
Let $R$ and $r$ be the radii of the circles circumscribed and inscribed, respectively, in a triangle. Prove that $R \ge 2r$, and that $R = 2r$ only for an equilateral triangle.