Found problems: 3349
2013 USA TSTST, 9
Let $r$ be a rational number in the interval $[-1,1]$ and let $\theta = \cos^{-1} r$. Call a subset $S$ of the plane [i]good[/i] if $S$ is unchanged upon rotation by $\theta$ around any point of $S$ (in both clockwise and counterclockwise directions). Determine all values of $r$ satisfying the following property: The midpoint of any two points in a good set also lies in the set.
2009 Today's Calculation Of Integral, 430
For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$.
Answer the following questions.
(1) Find $ a_1$.
(2) Express $ a_{n\plus{}1}$ in terms of $ a_n$.
(3) Find $ \lim_{n\to\infty} a_n$.
(4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.
2010 Today's Calculation Of Integral, 535
Let $ C$ be the parameterized curve for a given positive number $ r$ and $ 0\leq t\leq \pi$,
$ C: \left\{\begin{array}{ll} x \equal{} 2r(t \minus{} \sin t\cos t) & \quad \\
y \equal{} 2r\sin ^ 2 t & \quad \end{array} \right.$
When the point $ P$ moves on the curve $ C$,
(1) Find the magnitude of acceleralation of the point $ P$ at time $ t$.
(2) Find the length of the locus by which the point $ P$ sweeps for $ 0\leq t\leq \pi$.
(3) Find the volume of the solid by rotation of the region bounded by the curve $ C$ and the $ x$-axis about the $ x$-axis.
Edited.
1989 AIME Problems, 12
Let $ABCD$ be a tetrahedron with $AB=41$, $AC=7$, $AD=18$, $BC=36$, $BD=27$, and $CD=13$, as shown in the figure. Let $d$ be the distance between the midpoints of edges $AB$ and $CD$. Find $d^{2}$.
[asy]
pair C=origin, D=(4,11), A=(8,-5), B=(16,0);
draw(A--B--C--D--B^^D--A--C);
draw(midpoint(A--B)--midpoint(C--D), dashed);
label("27", B--D, NE);
label("41", A--B, SE);
label("7", A--C, SW);
label("$d$", midpoint(A--B)--midpoint(C--D), NE);
label("18", (7,8), SW);
label("13", (3,9), SW);
pair point=(7,0);
label("$A$", A, dir(point--A));
label("$B$", B, dir(point--B));
label("$C$", C, dir(point--C));
label("$D$", D, dir(point--D));[/asy]
2006 Estonia National Olympiad, 4
Triangle $ ABC$ is isosceles with $ AC \equal{} BC$ and $ \angle{C} \equal{} 120^o$. Points $ D$ and $ E$ are chosen on segment $ AB$ so that $ |AD| \equal{} |DE| \equal{} |EB|$. Find the sizes of the angles of triangle $ CDE$.
1974 IMO Longlists, 49
Determine an equation of third degree with integral coefficients having roots $\sin \frac{\pi}{14}, \sin \frac{5 \pi}{14}$ and $\sin \frac{-3 \pi}{14}.$
1997 India National Olympiad, 1
Let $ABCD$ be a parallelogram. Suppose a line passing through $C$ and lying outside the parallelogram meets $AB$ and $AD$ produced at $E$ and $F$ respectively. Show that \[ AC^2 + CE \cdot CF = AB \cdot AE + AD \cdot AF . \]
2006 Turkey MO (2nd round), 3
Find all the triangles such that its side lenghts, area and its angles' measures (in degrees) are rational.
2005 National High School Mathematics League, 9
If $0<\alpha<\beta<\gamma<2\pi$, for all $x\in\mathbb{R}$, $\cos(x+\alpha)+\cos(x+\beta)+\cos(x+\gamma)=0$, then $\gamma-\alpha=$________.
1983 IMO Shortlist, 4
On the sides of the triangle $ABC$, three similar isosceles triangles $ABP \ (AP = PB)$, $AQC \ (AQ = QC)$, and $BRC \ (BR = RC)$ are constructed. The first two are constructed externally to the triangle $ABC$, but the third is placed in the same half-plane determined by the line $BC$ as the triangle $ABC$. Prove that $APRQ$ is a parallelogram.
2013 Romania Team Selection Test, 1
Let $n$ be a positive integer and let $x_1$, $\ldots$, $x_n$ be positive real numbers. Show that:
\[
\min\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right )\leq 2\cos \frac{\pi}{n+2}
\leq\max\left ( x_1,\frac{1}{x_1}+x_2, \cdots,\frac{1}{x_{n-1}}+x_n,\frac{1}{x_n} \right ). \]
2019 Jozsef Wildt International Math Competition, W. 10
If ${si}(x) =- \int \limits_{x}^{\infty}\left(\frac{\sin t}{t}\right)dt; x>0$ then $$\int \limits_{e}^{e^2} \left(\frac{1}{x}\left(si\left(e^4x\right)-si\left(e^3x\right)\right)\right)\,dx=\int \limits_{3}^{e^4} \left(\frac{1}{x}\left(\operatorname{si}\left(e^2x\right)-si\left(ex\right)\right)\right)dx$$
2017 Nordic, 2
Let $a, b, \alpha, \beta$ be real numbers such that $0 \leq a, b \leq 1$, and $0 \leq \alpha, \beta \leq \frac{\pi}{2}$. Show that if \[ ab\cos(\alpha - \beta) \leq \sqrt{(1-a^2)(1-b^2)}, \] then \[ a\cos\alpha + b\sin\beta \leq 1 + ab\sin(\beta - \alpha). \]
2010 Today's Calculation Of Integral, 537
Evaluate $ \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx$.
1966 IMO Shortlist, 61
Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) \[ \frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx} \]
2005 National Olympiad First Round, 9
Let $ABC$ be a triangle with circumradius $1$. If the center of the circle passing through $A$, $C$, and the orthocenter of $\triangle ABC$ lies on the circumcircle of $\triangle ABC$, what is $|AC|$?
$
\textbf{(A)}\ 2
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ \dfrac 32
\qquad\textbf{(D)}\ \sqrt 2
\qquad\textbf{(E)}\ \sqrt 3
$
2001 Iran MO (2nd round), 2
In triangle $ABC$, $AB>AC$. The bisectors of $\angle{B},\angle{C}$ intersect the sides $AC,AB$ at $P,Q$, respectively. Let $I$ be the incenter of $\Delta ABC$. Suppose that $IP=IQ$. How much isthe value of $\angle A$?
2019 India PRMO, 22
In parallelogram $ABCD$, $AC=10$ and $BD=28$. The points $K$ and $L$ in the plane of $ABCD$ move in such a way that $AK=BD$ and $BL=AC$. Let $M$ and $N$ be the midpoints of $CK$ and $DL$, respectively. What is the maximum walue of $\cot^2 (\tfrac{\angle BMD}{2})+\tan^2(\tfrac{\angle ANC}{2})$ ?
2004 India IMO Training Camp, 1
Prove that in any triangle $ABC$,
\[ 0 < \cot { \left( \frac{A}{4} \right)} - \tan{ \left( \frac{B}{4} \right) } - \tan{ \left( \frac{C}{4} \right) } - 1 < 2 \cot { \left( \frac{A}{2} \right) }. \]
2005 National Olympiad First Round, 29
Let $h_1$ and $h_2$ be the altitudes of a triangle drawn to the sides with length $5$ and $2\sqrt 6$, respectively. If $5+h_1 \leq 2\sqrt 6 + h_2$, what is the third side of the triangle?
$
\textbf{(A)}\ 5
\qquad\textbf{(B)}\ 7
\qquad\textbf{(C)}\ 2\sqrt 6
\qquad\textbf{(D)}\ 3\sqrt 6
\qquad\textbf{(E)}\ 5\sqrt 3
$
Russian TST 2022, P1
Let $a{}$ and $b{}$ be positive integers. Prove that for any real number $x{}$ \[\sum_{j=0}^a\binom{a}{j}\big(2\cos((2j-a)x)\big)^b=\sum_{j=0}^b\binom{b}{j}\big(2\cos((2j-b)x)\big)^a.\]
2010 Moldova National Olympiad, 11.4
Let $ a_n\equal{}1\plus{}\dfrac1{2^2}\plus{}\dfrac1{3^2}\plus{}\cdots\plus{}\dfrac1{n^2}$
Find $ \lim_{n\to\infty}a_n$
2005 Today's Calculation Of Integral, 63
For a positive number $x$, let $f(x)=\lim_{n\to\infty} \sum_{k=1}^n \left|\cos \left(\frac{2k+1}{2n}x\right)-\cos \left(\frac{2k-1}{2n}x\right)\right|$
Evaluate
\[\lim_{x\rightarrow\infty}\frac{f(x)}{x}\]
2012 Online Math Open Problems, 23
Let $ABC$ be an equilateral triangle with side length $1$. This triangle is rotated by some angle about its center to form triangle $DEF.$ The intersection of $ABC$ and $DEF$ is an equilateral hexagon with an area that is $\frac{4} {5}$ the area of $ABC.$ The side length of this hexagon can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
[i]Author: Ray Li[/i]
2010 Indonesia TST, 4
Let $ ABC$ be a non-obtuse triangle with $ CH$ and $ CM$ are the altitude and median, respectively. The angle bisector of $ \angle BAC$ intersects $ CH$ and $ CM$ at $ P$ and $ Q$, respectively. Assume that \[ \angle ABP\equal{}\angle PBQ\equal{}\angle QBC,\]
(a) prove that $ ABC$ is a right-angled triangle, and
(b) calculate $ \dfrac{BP}{CH}$.
[i]Soewono, Bandung[/i]