Found problems: 85335
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.
Today's calculation of integrals, 888
In the coordinate plane, given a circle $K: x^2+y^2=1,\ C: y=x^2-2$. Let $l$ be the tangent line of $K$ at $P(\cos \theta,\ \sin \theta)\ (\pi<\theta <2\pi).$ Find the minimum area of the part enclosed by $l$ and $C$.
2003 Dutch Mathematical Olympiad, 1
A Pythagorean triangle is a right triangle whose three sides are integers.
The best known example is the triangle with rectangular sides $3$ and $4$ and hypotenuse $5$.
Determine all Pythagorean triangles whose area is twice the perimeter.
2020 Peru Iberoamerican Team Selection Test, P1
In a classroom there are $m$ students. During the month of July each of them visited the library at least once but none of them visited the library twice in the same day. It turned out that during the month of July each student visited the library a different number of times, furthermore for any two students $A$ and $B$ there was a day in which $A$ visited the library and $B$ did not and there was also a day when $B$ visited the library and $A$ did not do so.
Determine the largest possible value of $m$.