Olympiad problems

2011 Balkan MO, 1

Let $ABCD$ be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at $E$. The midpoints of $AB$ and $CD$ are $F$ and $G$ respectively, and $\ell$ is the line through $G$ parallel to $AB$. The feet of the perpendiculars from E onto the lines $\ell$ and $CD$ are $H$ and $K$, respectively. Prove that the lines $EF$ and $HK$ are perpendicular.

1992 AMC 12/AHSME, 27

Tags: trigonometry , trig identities , law of cosines

A circle of radius $r$ has chords $\overline{AB}$ of length $10$ and $\overline{CD}$ of length $7$. When $\overline{AB}$ and $\overline{CD}$ are extended through $B$ and $C$, respectively, they intersect at $P$, which is outside the circle. If $\angle APD = 60^{\circ}$ and $BP = 8$, then $r^{2} =$ $ \textbf{(A)}\ 70\qquad\textbf{(B)}\ 71\qquad\textbf{(C)}\ 72\qquad\textbf{(D)}\ 73\qquad\textbf{(E)}\ 74 $

2009 Iran MO (3rd Round), 6

Tags: induction , algorithm , function , trigonometry , number theory

Let $z$ be a complex non-zero number such that $Re(z),Im(z)\in \mathbb{Z}$. Prove that $z$ is uniquely representable as $a_0+a_1(1+i)+a_2(1+i)^2+\dots+a_n(1+i)^n$ where $n\geq 0$ and $a_j \in \{0,1\}$ and $a_n=1$. Time allowed for this problem was 1 hour.

2005 National Olympiad First Round, 17

Tags: geometry , trigonometry

Construct outer squares $ABMN$, $BCKL$, $ACPQ$ on sides $[AB]$, $[BC]$, $[CA]$ of triangle $ABC$, respectively. Construct squares $NQZT$ and $KPYX$ on segments $[NQ]$ and $[KP]$. If $Area(ABMN) - Area(BCKL)=1$, what is $Area(NQZT)-Area(KPYX)$? $ \textbf{(A)}\ \dfrac 34 \qquad\textbf{(B)}\ \dfrac 53 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

2013 Princeton University Math Competition, 6

Tags: function , trigonometry , induction , modular arithmetic , floor function , trig identities

Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$. Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.

2000 Harvard-MIT Mathematics Tournament, 18

Tags: trigonometry , limit

What is the value of $ \sum_{n=1}^\infty (\tan^{-1}\sqrt{n}-\tan^{-1}\sqrt{n+1})$?

2014 Contests, 2

Tags: trigonometry , geometry , geometric transformation , reflection , projective geometry , trig identities , law of sines

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

2005 Tuymaada Olympiad, 4

Tags: inequalities , trigonometry , 3d geometry , tetrahedron , triangle inequality , geometry

In a triangle $ABC$, let $A_{1}$, $B_{1}$, $C_{1}$ be the points where the excircles touch the sides $BC$, $CA$ and $AB$ respectively. Prove that $A A_{1}$, $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle. [i]Proposed by L. Emelyanov[/i]

1966 IMO Longlists, 50

Tags: inequalities , trigonometry , geometry

For any quadrilateral with the side lengths $a,$ $b,$ $c,$ $d$ and the area $S,$ prove the inequality$S\leq \frac{a+c}{2}\cdot \frac{b+d}{2}.$

2021 BMT, 15

Tags: trigonometry

Compute $$\frac{\cos \left(\frac{\pi}{12}\right)\cos \left(\frac{\pi}{24}\right)\cos \left(\frac{\pi}{48}\right)\cos \left(\frac{\pi}{96}\right)...}{\cos \left(\frac{\pi}{4}\right)\cos \left(\frac{\pi}{8}\right)\cos \left(\frac{\pi}{16}\right)\cos \left(\frac{\pi}{32}\right)...}$$

1967 IMO Longlists, 15

Tags: trigonometry , number theory , diophantine equation , trigonometric equation

Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.

1977 Vietnam National Olympiad, 2

Tags: trigonometric equation , trigonometry , algebra

Show that there are $1977$ non-similar triangles such that the angles $A, B, C$ satisfy $\frac{\sin A + \sin B + \sin C}{\cos A +\cos B + \cos C} = \frac{12}{7}$ and $\sin A \sin B \sin C = \frac{12}{25}$.

1998 Vietnam National Olympiad, 2

Tags: analytic geometry , trigonometry , inequalities

Find minimum value of $F(x,y)=\sqrt{(x+1)^{2}+(y-1)^{2}}+\sqrt{(x-1)^{2}+(y+1)^{2}}+\sqrt{(x+2)^{2}+(y+2)^{2}}$, where $x,y\in\mathbb{R}$.

2011 Kazakhstan National Olympiad, 1

Tags: ratio , trigonometry , geometry

Inscribed in a triangle $ABC$ with the center of the circle $I$ touch the sides $AB$ and $AC$ at points $C_{1}$ and $B_{1}$, respectively. The point $M$ divides the segment $C_{1}B_{1}$ in a 3:1 ratio, measured from $C_{1}$. $N$ - the midpoint of $AC$. Prove that the points $I, M, B_{1}, N$ lie on a circle, if you know that $AC = 3 (BC-AB)$.

2012 Today's Calculation Of Integral, 797

Tags: calculus , integration , geometry , 3d geometry , tetrahedron , trigonometry , calculus computations

In the $xyz$-space take four points $P(0,\ 0,\ 2),\ A(0,\ 2,\ 0),\ B(\sqrt{3},-1,\ 0),\ C(-\sqrt{3},-1,\ 0)$. Find the volume of the part satifying $x^2+y^2\geq 1$ in the tetrahedron $PABC$. 50 points

2010 Today's Calculation Of Integral, 643

Tags: calculus , integration , trigonometry , calculus computations

Evaluate \[\int_0^{\pi} \frac{x}{\sqrt{1+\sin ^ 3 x}}\{(3\pi \cos x+4\sin x)\sin ^ 2 x+4\}dx.\] Own

2005 Today's Calculation Of Integral, 19

Tags: calculus , integration , trigonometry , calculus computations , logarithm

Calculate the following indefinite integrals. [1] $\int \tan ^ 3 x dx$ [2] $\int a^{mx+n}dx\ (a>0,a\neq 1, mn\neq 0)$ [3] $\int \cos ^ 5 x dx$ [4] $\int \sin ^ 2 x\cos ^ 3 x dx$ [5]$ \int \frac{dx}{\sin x}$

2009 Today's Calculation Of Integral, 469

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_0^1 \frac{t}{(1\plus{}t^2)(1\plus{}2t\minus{}t^2)}\ dt$.

1994 Turkey MO (2nd round), 6

Tags: incenter , perimeter , trigonometry , trapezoid , angle bisector , geometry

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AC$ at $E$. Let $K$ be the point on $CB$ with $CK=BD$, and $L$ be the point on $CA$ with $AE=CL$. Lines $AK$ and $BL$ meet at $P$. If $Q$ is the midpoint of $BC$, $I$ the incenter, and $G$ the centroid of $\triangle ABC$, show that: $(a)$ $IQ$ and $AK$ are parallel, $(b)$ the triangles $AIG$ and $QPG$ have equal area.

2005 Today's Calculation Of Integral, 86

Tags: calculus , integration , trigonometry , calculus computations

Prove \[\left[\int_\pi^\infty \frac{\cos x}{x}\ dx\right]^2< \frac{1}{{\pi}^2}\]

2002 Moldova National Olympiad, 4

Tags: trigonometry

Let $ x\in \mathbb R$. Find the minimum and maximum values of the expresion: $ E\equal{}\dfrac{(1\plus{}x)^8\plus{}16x^4}{(1\plus{}x^2)^4}$

2005 Today's Calculation Of Integral, 70

Tags: calculus , integration , trigonometry , function , calculus computations

Find the number of root for $\int_0^{\frac{\pi}{2}} e^x\cos (x+a)\ dx=0$ at $0\leq a <2\pi$

2014 CentroAmerican, 2

Tags: geometry , incenter , geometric transformation , reflection , trigonometry , circumcircle , angle bisector

Points $A$, $B$, $C$ and $D$ are chosen on a line in that order, with $AB$ and $CD$ greater than $BC$. Equilateral triangles $APB$, $BCQ$ and $CDR$ are constructed so that $P$, $Q$ and $R$ are on the same side with respect to $AD$. If $\angle PQR=120^\circ$, show that \[\frac{1}{AB}+\frac{1}{CD}=\frac{1}{BC}.\]

2017 Kosovo National Mathematical Olympiad, 4

Tags: trigonometry , algebra

Prove that : $\cos36-\sin18=\frac{1}{2}$

2011 AMC 12/AHSME, 25

Tags: geometry , incenter , circumcircle , trigonometry , function , inequalities , calculus

Triangle $ABC$ has $\angle BAC=60^\circ$, $\angle CBA \le 90^\circ$, $BC=1$, and $AC \ge AB$. Let $H$, $I$, and $O$ be the orthocenter, incenter, and circumcenter of $\triangle ABC$, respectively. Assume that the area of the pentagon $BCOIH$ is the maximum possible. What is $\angle CBA$? $\textbf{(A)}\ 60 ^\circ \qquad \textbf{(B)}\ 72 ^\circ\qquad \textbf{(C)}\ 75 ^\circ \qquad \textbf{(D)}\ 80 ^\circ\qquad \textbf{(E)}\ 90 ^\circ$