Found problems: 3349
1997 Balkan MO, 1
Suppose that $O$ is a point inside a convex quadrilateral $ABCD$ such that \[ OA^2 + OB^2 + OC^2 + OD^2 = 2\mathcal A[ABCD] , \] where by $\mathcal A[ABCD]$ we have denoted the area of $ABCD$. Prove that $ABCD$ is a square and $O$ is its center.
[i]Yugoslavia[/i]
2009 Postal Coaching, 3
Find all real polynomial functions $f : R \to R$ such that $f(\sin x) = f(\cos x)$.
2005 Today's Calculation Of Integral, 54
evaluate
\[\int_{-1}^0 \sqrt{\frac{1+x}{1-x}}dx\]
2003 National Olympiad First Round, 5
Let $ABC$ be a triangle and $D$ be the foot of the altitude from $C$ to $AB$. If $|CH|=|HD|$ where $H$ is the orthocenter, what is $\tan \widehat {A} \cdot \tan \widehat{B}$?
$
\textbf{(A)}\ 1
\qquad\textbf{(B)}\ \sqrt 2
\qquad\textbf{(C)}\ 3/2
\qquad\textbf{(D)}\ \sqrt 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$
2005 Thailand Mathematical Olympiad, 21
Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.
2013 Uzbekistan National Olympiad, 5
Let $SABC$ is pyramid, such that $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and $AC\le 8$.
Find max value capacity(volume) of the pyramid $SABC$.
2002 District Olympiad, 2
In the $xOy$ system, consider the points $A_n(n,n^3)$ with $n\in \mathbb{N}^*$ and the point $B(0,1)$. Prove that
a) for any positive integers $k>j>i\ge 1$, the points $A_i,A_j,A_k$ cannot be collinear.
b) for any positive integers $i_k>i_{k-1}>\ldots>i_1\ge 1$, we have
\[\mu(\widehat{A_{i_1}OB})+\mu(\widehat{A_{i_2}OB})+\cdots+\mu(\widehat{A_{i_k}OB})<\frac{\pi}{2}\]
[i]***[/i]
1979 IMO Longlists, 68
We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which \[ \frac{QP+PR}{QR} \] is maximum.
2012 AMC 12/AHSME, 23
Let $S$ be the square one of whose diagonals has endpoints $(0.1,0.7)$ and $(-0.1,-0.7)$. A point $v=(x,y)$ is chosen uniformly at random over all pairs of real numbers $x$ and $y$ such that $0\le x \le 2012$ and $0 \le y \le 2012$. Let $T(v)$ be a translated copy of $S$ centered at $v$. What is the probability that the square region determined by $T(v)$ contains exactly two points with integer coordinates in its interior?
$ \textbf{(A)}\ 0.125\qquad\textbf{(B)}\ 0.14\qquad\textbf{(C)}\ 0.16\qquad\textbf{(D)}\ 0.25\qquad\textbf{(E)}\ 0.32 $
2006 Peru MO (ONEM), 1
Find all integer values can take $n$ such that $$\cos(2x)=\cos^nx - \sin^nx$$ for every real number $x$.
2010 Today's Calculation Of Integral, 539
Evaluate $ \int_0^{\frac{\pi}{4}} \frac{\sin ^ 2 x}{\cos ^ 3 x}\ dx$.
2012 South East Mathematical Olympiad, 4
Let $a, b, c, d$ be real numbers satisfying inequality $a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1$ holds for arbitrary real number $x$. Find the maximal value of $a+b-c+d$ and determine the values of $a,b,c,d$ when that maximum is attained.
2001 JBMO ShortLists, 11
Consider a triangle $ABC$ with $AB=AC$, and $D$ the foot of the altitude from the vertex $A$. The point $E$ lies on the side $AB$ such that $\angle ACE= \angle ECB=18^{\circ}$.
If $AD=3$, find the length of the segment $CE$.
1996 Romania Team Selection Test, 3
Let $ x,y\in \mathbb{R} $. Show that if the set $ A_{x,y}=\{ \cos {(n\pi x)}+\cos {(n\pi y)} \mid n\in \mathbb{N}\} $ is finite then $ x,y \in \mathbb{Q} $.
[i]Vasile Pop[/i]
1984 IMO Longlists, 52
Construct a scalene triangle such that
\[a(\tan B - \tan C) = b(\tan A - \tan C)\]
1996 South africa National Olympiad, 2
Find all real numbers for which $3^x+4^x=5^x$.
2011 Switzerland - Final Round, 2
Let $\triangle{ABC}$ be an acute-angled triangle and let $D$, $E$, $F$ be points on $BC$, $CA$, $AB$, respectively, such that \[\angle{AFE}=\angle{BFD}\mbox{,}\quad\angle{BDF}=\angle{CDE}\quad\mbox{and}\quad\angle{CED}=\angle{AEF}\mbox{.}\] Prove that $D$, $E$ and $F$ are the feet of the perpendiculars through $A$, $B$ and $C$ on $BC$, $CA$ and $AB$, respectively.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 2)[/i]
1972 IMO Longlists, 40
Prove the inequalities
\[\frac{u}{v}\le \frac{\sin u}{\sin v}\le \frac{\pi}{2}\times\frac{u}{v},\text{ for }0 \le u < v \le \frac{\pi}{2}\]
1999 Romania Team Selection Test, 9
Let $O,A,B,C$ be variable points in the plane such that $OA=4$, $OB=2\sqrt3$ and $OC=\sqrt {22}$. Find the maximum value of the area $ABC$.
[i]Mihai Baluna[/i]
1998 Putnam, 3
Let $H$ be the unit hemisphere $\{(x,y,z):x^2+y^2+z^2=1,z\geq 0\}$, $C$ the unit circle $\{(x,y,0):x^2+y^2=1\}$, and $P$ the regular pentagon inscribed in $C$. Determine the surface area of that portion of $H$ lying over the planar region inside $P$, and write your answer in the form $A \sin\alpha + B \cos\beta$, where $A,B,\alpha,\beta$ are real numbers.
2009 Today's Calculation Of Integral, 412
Let the definite integral $ I_n\equal{}\int_0^{\frac{\pi}{4}} \frac{dx}{(\cos x)^n}\ (n\equal{}0,\ \pm 1,\ \pm 2,\ \cdots )$.
(1) Find $ I_0,\ I_{\minus{}1},\ I_2$.
(2) Find $ I_1$.
(3) Express $ I_{n\plus{}2}$ in terms of $ I_n$.
(4) Find $ I_{\minus{}3},\ I_{\minus{}2},\ I_3$.
(5) Evaluate the definite integrals $ \int_0^1 \sqrt{x^2\plus{}1}\ dx,\ \int_0^1 \frac{dx}{(x^2\plus{}1)^2}\ dx$ in using the avobe results.
You are not allowed to use the formula of integral for $ \sqrt{x^2\plus{}1}$ directively here.
2012 Iran Team Selection Test, 3
Suppose $ABCD$ is a parallelogram. Consider circles $w_1$ and $w_2$ such that $w_1$ is tangent to segments $AB$ and $AD$ and $w_2$ is tangent to segments $BC$ and $CD$. Suppose that there exists a circle which is tangent to lines $AD$ and $DC$ and externally tangent to $w_1$ and $w_2$. Prove that there exists a circle which is tangent to lines $AB$ and $BC$ and also externally tangent to circles $w_1$ and $w_2$.
[i]Proposed by Ali Khezeli[/i]
2005 Today's Calculation Of Integral, 4
Calculate the following indefinite integrals.
[1] $\int \frac{x}{\sqrt{5-x}}dx$
[2] $\int \frac{\sin x \cos ^2 x}{1+\cos x}dx$
[3] $\int (\sin x+\cos x)^2dx$
[4] $\int \frac{x-\cos ^2 x}{x\cos^ 2 x}dx$
[5]$\int (\sin x+\sin 2x)^2 dx$
2013 District Olympiad, 4
Let $n\in {{\mathbb{N}}^{*}}$. Prove that $2\sqrt{{{2}^{n}}}\cos \left( n\arccos \frac{\sqrt{2}}{4} \right)$ is an odd integer.
1985 AIME Problems, 4
A small square is constructed inside a square of area 1 by dividing each side of the unit square into $n$ equal parts, and then connecting the vertices to the division points closest to the opposite vertices. Find the value of $n$ if the the area of the small square is exactly 1/1985.
[asy]
size(200);
pair A=(0,1), B=(1,1), C=(1,0), D=origin;
draw(A--B--C--D--A--(1,1/6));
draw(C--(0,5/6)^^B--(1/6,0)^^D--(5/6,1));
pair point=( 0.5 , 0.5 );
//label("$A$", A, dir(point--A));
//label("$B$", B, dir(point--B));
//label("$C$", C, dir(point--C));
//label("$D$", D, dir(point--D));
label("$1/n$", (11/12,1), N, fontsize(9));[/asy]