Found problems: 3349
2008 ITest, 62
Find the number of values of $x$ such that the number of square units in the area of the isosceles triangle with sides $x$, $65$, and $65$ is a positive integer.
2006 Baltic Way, 12
Let $ABC$ be a triangle, let $B_{1}$ be the midpoint of the side $AB$ and $C_{1}$ the midpoint of the side $AC$. Let $P$ be the point of intersection, other than $A$, of the circumscribed circles around the triangles $ABC_{1}$ and $AB_{1}C$. Let $P_{1}$ be the point of intersection, other than $A$, of the line $AP$ with the circumscribed circle around the triangle $AB_{1}C_{1}$. Prove that $2AP=3AP_{1}$.
2013 ELMO Shortlist, 7
Let $ABC$ be a triangle inscribed in circle $\omega$, and let the medians from $B$ and $C$ intersect $\omega$ at $D$ and $E$ respectively. Let $O_1$ be the center of the circle through $D$ tangent to $AC$ at $C$, and let $O_2$ be the center of the circle through $E$ tangent to $AB$ at $B$. Prove that $O_1$, $O_2$, and the nine-point center of $ABC$ are collinear.
[i]Proposed by Michael Kural[/i]
2012 Kazakhstan National Olympiad, 2
Given an inscribed quadrilateral $ABCD$, which marked the midpoints of the points $M, N, P, Q$ in this order. Let diagonals $AC$ and $BD$ intersect at point $O$. Prove that the triangle $OMN, ONP, OPQ, OQM$ have the same radius of the circles
2007 ISI B.Stat Entrance Exam, 5
Show that
\[-2 \le \cos \theta\left(\sin \theta + \sqrt{\sin ^2 \theta +3}\right) \le 2\]
for all values of $\theta$.
2014 AIME Problems, 15
In $ \triangle ABC $, $ AB = 3 $, $ BC = 4 $, and $ CA = 5 $. Circle $\omega$ intersects $\overline{AB}$ at $E$ and $B$, $\overline{BC}$ at $B$ and $D$, and $\overline{AC}$ at $F$ and $G$. Given that $EF=DF$ and $\tfrac{DG}{EG} = \tfrac{3}{4}$, length $DE=\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is a positive integer not divisible by the square of any prime. Find $a+b+c$.
Estonia Open Senior - geometry, 2007.2.5
Consider triangles whose each side length squared is a rational number. Is it true
that
(a) the square of the circumradius of every such triangle is rational;
(b) the square of the inradius of every such triangle is rational?
2010 Today's Calculation Of Integral, 601
Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$.
created by kunny
2000 Pan African, 1
Solve for $x \in R$:
\[ \sin^3{x}(1+\cot{x})+\cos^3{x}(1+\tan{x})=\cos{2x} \]
1985 Vietnam Team Selection Test, 2
Find all real values of a for which the equation $ (a \minus{} 3x^2 \plus{} \cos \frac {9\pi x}{2})\sqrt {3 \minus{} ax} \equal{} 0$ has an odd number of solutions in the interval $ [ \minus{} 1,5]$
1990 IMO Longlists, 86
Given function $f(x) = \sin x + \sin \pi x$ and positive number $d$. Prove that there exists real number $p$ such that $|f(x + p) - f(x)| < d$ holds for all real numbers $x$, and the value of $p$ can be arbitrarily large.
2009 Today's Calculation Of Integral, 443
Evaluate $ \int_1^{e^2} \frac{(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)\plus{}(e^{\sqrt{x}}\plus{}e^{\minus{}\sqrt{x}})\cos \left(e^{\sqrt{x}}\minus{}e^{\minus{}\sqrt{x}}\plus{}\frac{\pi}{4}\right)}{\sqrt{x}}\ dx.$
2014 AMC 12/AHSME, 25
What is the sum of all positive real solutions $x$ to the equation \[2\cos(2x)\left(\cos(2x)-\cos\left(\frac{2014\pi^2}{x}\right)\right)=\cos(4x)-1?\]
$\textbf{(A) }\pi\qquad
\textbf{(B) }810\pi\qquad
\textbf{(C) }1008\pi\qquad
\textbf{(D) }1080\pi\qquad
\textbf{(E) }1800\pi\qquad$
2006 Purple Comet Problems, 10
An equilateral triangle with side length $6$ has a square of side length $6$ attached to each of its edges as shown. The distance between the two farthest vertices of this figure (marked $A$ and $B$ in the figure) can be written as $m + \sqrt{n}$ where $m$ and $n$ are positive integers. Find $m + n$.
[asy]
draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle);
draw((1,0)--(1+sqrt(3)/2,1/2)--(1/2+sqrt(3)/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
draw((0,0)--(-sqrt(3)/2,1/2)--(-sqrt(3)/2+1/2,1/2+sqrt(3)/2)--(1/2,sqrt(3)/2));
dot((-sqrt(3)/2+1/2,1/2+sqrt(3)/2));
label("A", (-sqrt(3)/2+1/2,1/2+sqrt(3)/2), N);
draw((1,0)--(1,-1)--(0,-1)--(0,0));
dot((1,-1));
label("B", (1,-1), SE);
[/asy]
2014 AMC 12/AHSME, 20
In $\triangle BAC$, $\angle BAC=40^\circ$, $AB=10$, and $AC=6$. Points $D$ and $E$ lie on $\overline{AB}$ and $\overline{AC}$ respectively. What is the minimum possible value of $BE+DE+CD$?
$\textbf{(A) }6\sqrt 3+3\qquad
\textbf{(B) }\dfrac{27}2\qquad
\textbf{(C) }8\sqrt 3\qquad
\textbf{(D) }14\qquad
\textbf{(E) }3\sqrt 3+9\qquad$
2019 USMCA, 3
Let $ABC$ be a scalene triangle. The incircle of $ABC$ touches $\overline{BC}$ at $D$. Let $P$ be a point on $\overline{BC}$ satisfying $\angle BAP = \angle CAP$, and $M$ be the midpoint of $\overline{BC}$. Define $Q$ to be on $\overline{AM}$ such that $\overline{PQ} \perp \overline{AM}$. Prove that the circumcircle of $\triangle AQD$ is tangent to $\overline{BC}$.
1988 IMO Longlists, 44
Let $-1 < x < 1.$ Show that \[ \sum^{6}_{k=0} \frac{1 - x^2}{1 - 2 \cdot x \cdot \cos \left( \frac{2 \cdot \pi \cdot k }{7} \right) + x^2} = \frac{7 \cdot \left( 1 + x^7 \right)}{\left( 1 - x^7 \right)}. \] Deduce that \[ \csc^2\left( x + \frac{\pi}{7} \right) + \csc^2\left(2 \cdot x + \frac{\pi}{7} \right) + \csc^2\left(3 \cdot x + \frac{\pi}{7} \right) = 8. \]
2010 Today's Calculation Of Integral, 600
Evaluate $\int_{-a}^a \left(x+\frac{1}{\sin x+\frac{1}{e^x-e^{-x}}}\right)dx\ (a>0)$.
created by kunny
2000 Tuymaada Olympiad, 3
Polynomial $ P(t)$ is such that for all real $ x$,
\[ P(\sin x) \plus{} P(\cos x) \equal{} 1.
\]
What can be the degree of this polynomial?
2018 CMIMC Geometry, 3
Let $ABC$ be a triangle with side lengths $5$, $4\sqrt 2$, and $7$. What is the area of the triangle with side lengths $\sin A$, $\sin B$, and $\sin C$?
I Soros Olympiad 1994-95 (Rus + Ukr), 10.8
Find all $x$ for which the inequality holds
$$\sqrt{7+8x-16x^2} \ge 2^{\cos^2 \pi x}+2^{\sin ^2 \pi x}$$
2009 Harvard-MIT Mathematics Tournament, 6
Let $x$ and $y$ be positive real numbers and $\theta$ an angle such that $\theta \neq \frac{\pi}{2}n$ for any integer $n$. Suppose
\[\frac{\sin\theta}{x}=\frac{\cos\theta}{y}\]
and
\[
\frac{\cos^4 \theta}{x^4}+\frac{\sin^4\theta}{y^4}=\frac{97\sin2\theta}{x^3y+y^3x}.
\]
Compute $\frac xy+\frac yx.$
2013 Today's Calculation Of Integral, 863
For $0<t\leq 1$, let $F(t)=\frac{1}{t}\int_0^{\frac{\pi}{2}t} |\cos 2x|\ dx.$
(1) Find $\lim_{t\rightarrow 0} F(t).$
(2) Find the range of $t$ such that $F(t)\geq 1.$
1967 IMO Shortlist, 5
Show that a triangle whose angles $A$, $B$, $C$ satisfy the equality
\[ \frac{\sin^2 A + \sin^2 B + \sin^2 C}{\cos^2 A + \cos^2 B + \cos^2 C} = 2 \]
is a rectangular triangle.
2013 ISI Entrance Examination, 2
For $x\ge 0$, define
\[f(x)=\frac1{x+2\cos x}\]
Find the set $\{ y \in \mathbb{R}: y=f(x), x\ge 0\}$