Olympiad problems

2009 Jozsef Wildt International Math Competition, W. 29

Prove that for all triangle $\triangle ABC$ holds the following inequality $$\sum \limits_{cyc} \left (1-\sqrt{\sqrt{3}\tan \frac{A}{2}}+\sqrt{3}\tan \frac{A}{2}\right )\left (1-\sqrt{\sqrt{3}\tan \frac{B}{2}}+\sqrt{3}\tan \frac{B}{2}\right )\geq 3$$

2010 Middle European Mathematical Olympiad, 9

Tags: incenter , geometry , trigonometry

The incircle of the triangle $ABC$ touches the sides $BC$, $CA$, and $AB$ in the points $D$, $E$ and $F$, respectively. Let $K$ be the point symmetric to $D$ with respect to the incenter. The lines $DE$ and $FK$ intersect at $S$. Prove that $AS$ is parallel to $BC$. [i](4th Middle European Mathematical Olympiad, Team Competition, Problem 5)[/i]

2016 India National Olympiad, P5

Tags: equation , trigonometry , inradius , geometry

Let $ABC$ be a right-angle triangle with $\angle B=90^{\circ}$. Let $D$ be a point on $AC$ such that the inradii of the triangles $ABD$ and $CBD$ are equal. If this common value is $r^{\prime}$ and if $r$ is the inradius of triangle $ABC$, prove that \[ \cfrac{1}{r'}=\cfrac{1}{r}+\cfrac{1}{BD}. \]

2002 Canada National Olympiad, 4

Tags: geometry , circumcircle , perpendicular bisector , angle bisector , trigonometry

Let $\Gamma$ be a circle with radius $r$. Let $A$ and $B$ be distinct points on $\Gamma$ such that $AB < \sqrt{3}r$. Let the circle with centre $B$ and radius $AB$ meet $\Gamma$ again at $C$. Let $P$ be the point inside $\Gamma$ such that triangle $ABP$ is equilateral. Finally, let the line $CP$ meet $\Gamma$ again at $Q$. Prove that $PQ = r$.

Ukrainian TYM Qualifying - geometry, V.8

Tags: equilateral , geometric inequality , trigonometry , geometry

Let $X$ be a point inside an equilateral triangle $ABC$ such that $BX+CX <3 AX$. Prove that $$3\sqrt3 \left( \cot \frac{\angle AXC}{2}+ \cot \frac{\angle AXB}{2}\right) +\cot \frac{\angle AXC}{2} \cot \frac{\angle AXB}{2} >5$$

1998 National Olympiad First Round, 1

Tags: quadratic , trigonometry

If $ \left|BC\right| \equal{} a$, $ \left|AC\right| \equal{} b$, $ \left|AB\right| \equal{} c$, $ 3\angle A \plus{} \angle B \equal{} 180{}^\circ$ and $ 3a \equal{} 2c$, then find $ b$ in terms of $ a$. $\textbf{(A)}\ \frac {3a}{2} \qquad\textbf{(B)}\ \frac {5a}{4} \qquad\textbf{(C)}\ a\sqrt {2} \qquad\textbf{(D)}\ a\sqrt {3} \qquad\textbf{(E)}\ \frac {2a\sqrt {3} }{3}$

1992 Taiwan National Olympiad, 1

Tags: trigonometry , quadratic , geometry

Let $A,B$ be two points on a give circle, and $M$ be the midpoint of one of the arcs $AB$ . Point $C$ is the orthogonal projection of $B$ onto the tangent $l$ to the circle at $A$. The tangent at $M$ to the circle meets $AC,BC$ at $A',B'$ respectively. Prove that if $\hat{BAC}<\frac{\pi}{8}$ then $S_{ABC}<2S_{A'B'C'}$.

2019 Jozsef Wildt International Math Competition, W. 47

Tags: inequalities , trigonometry , limit

[list=1] [*] If $a$, $b$, $c$, $d > 0$, show inequality:$$\left(\tan^{-1}\left(\frac{ad-bc}{ac+bd}\right)\right)^2\geq 2\left(1-\frac{ac+bd}{\sqrt{\left(a^2+b^2\right)\left(c^2+d^2\right)}}\right)$$ [*] Calculate $$\lim \limits_{n \to \infty}n^{\alpha}\left(n- \sum \limits_{k=1}^n\frac{n^+k^2-k}{\sqrt{\left(n^2+k^2\right)\left(n^2+(k-1)^2\right)}}\right)$$where $\alpha \in \mathbb{R}$ [/list]

1993 IberoAmerican, 1

Tags: geometry , trigonometry

Let $ABC$ be an equilateral triangle and $\Gamma$ its incircle. If $D$ and $E$ are points on the segments $AB$ and $AC$ such that $DE$ is tangent to $\Gamma$, show that $\frac{AD}{DB}+\frac{AE}{EC}=1$.

2003 IberoAmerican, 2

Tags: power of a point , geometry , radical axis , reflection , geometric transformation , trigonometry , circumcircle

In a square $ABCD$, let $P$ and $Q$ be points on the sides $BC$ and $CD$ respectively, different from its endpoints, such that $BP=CQ$. Consider points $X$ and $Y$ such that $X\neq Y$, in the segments $AP$ and $AQ$ respectively. Show that, for every $X$ and $Y$ chosen, there exists a triangle whose sides have lengths $BX$, $XY$ and $DY$.

2013 APMO, 1

Tags: circumcircle , ratio , equal area , trigonometry , analytic geometry , parallelogram , geometry

Let $ABC$ be an acute triangle with altitudes $AD$, $BE$, and $CF$, and let $O$ be the center of its circumcircle. Show that the segments $OA$, $OF$, $OB$, $OD$, $OC$, $OE$ dissect the triangle $ABC$ into three pairs of triangles that have equal areas.

2005 National Olympiad First Round, 9

Tags: trigonometry , geometry , circumcircle

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 $

2000 Canada National Olympiad, 4

Tags: reflection , geometric transformation , trigonometry , geometry , circumcircle , perpendicular bisector

Let $ABCD$ be a convex quadrilateral with $\angle CBD = 2 \angle ADB$, $\angle ABD = 2 \angle CDB$ and $AB = CB$. Prove that $AD = CD$.

1941 Moscow Mathematical Olympiad, 080

Tags: analysis , root , trigonometry , algebra

How many roots does equation $\sin x = \frac{x}{100}$ have?

1991 APMO, 1

Tags: geometry , trigonometry

Let $G$ be the centroid of a triangle $ABC$, and $M$ be the midpoint of $BC$. Let $X$ be on $AB$ and $Y$ on $AC$ such that the points $X$, $Y$, and $G$ are collinear and $XY$ and $BC$ are parallel. Suppose that $XC$ and $GB$ intersect at $Q$ and $YB$ and $GC$ intersect at $P$. Show that triangle $MPQ$ is similar to triangle $ABC$.

2003 Brazil National Olympiad, 1

Tags: inequalities , triangle inequality , geometry , trigonometry

Given a circle and a point $A$ inside the circle, but not at its center. Find points $B$, $C$, $D$ on the circle which maximise the area of the quadrilateral $ABCD$.

2012 AIME Problems, 9

Tags: ratio , function , quadratic , trigonometry , relatively prime , number theory

Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

2009 Today's Calculation Of Integral, 453

Tags: calculus computations , trigonometry , calculus , integration

Find the minimum value of $ \int_0^{\frac{\pi}{2}} |x\sin t\minus{}\cos t|\ dt\ (x>0).$

2019 IFYM, Sozopol, 4

Tags: geometry , trigonometry

For a quadrilateral $ABCD$ is given that $\angle CBD=2\angle ADB$, $\angle ABD=2\angle CDB$, and $AB=CB$. Prove that $AD=CD$.

1945 Moscow Mathematical Olympiad, 104

Tags: nested radical , radical , trigonometry , algebra

The numbers $a_1, a_2, ..., a_n$ are equal to $1$ or $-1$. Prove that $$2 \sin \left(a_1+\frac{a_1a_2}{2}+\frac{a_1a_2a_3}{4}+...+\frac{a_1a_2...a_n}{2^{n-1}}\right)\frac{\pi}{4}=a_1\sqrt{2+a_2\sqrt{2+a_3\sqrt{2+...+a_n\sqrt2}}}$$ In particular, for $a_1 = a_2 = ... = a_n = 1$ we have $$2 \sin \left(1+\frac{1}{2}+\frac{1}{4}+...+\frac{1}{2^{n-1}}\right)\frac{\pi}{4}=2\cos \frac{\pi}{2^{n+1}}= \sqrt{2+\sqrt{2+\sqrt{2+...+\sqrt2}}}$$

2002 Estonia Team Selection Test, 5

Tags: inequalities , trigonometry , algebra

Let $0 < a < \frac{\pi}{2}$ and $x_1,x_2,...,x_n$ be real numbers such that $\sin x_1 + \sin x_2 +... + \sin x_n \ge n \cdot sin a $. Prove that $\sin (x_1 - a) + \sin (x_2 - a) + ... + \sin (x_n - a) \ge 0$ .

2013 North Korea Team Selection Test, 1

Tags: circumcircle , north korea , trigonometry , geometry

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $ BC, CA, AB$ at $ A_1 , B_1 , C_1 $ respectively. The line $AI$ meets the circumcircle of $ABC$ at $A_2 $. The line $B_1 C_1 $ meets the line $BC$ at $A_3 $ and the line $A_2 A_3 $ meets the circumcircle of $ABC$ at $A_4 (\ne A_2 ) $. Define $B_4 , C_4 $ similarly. Prove that the lines $ AA_4 , BB_4 , CC_4 $ are concurrent.

1970 IMO Longlists, 15

Tags: perimeter , circumcircle , trigonometry , geometry , inequalities

Given $\triangle ABC$, let $R$ be its circumradius and $q$ be the perimeter of its excentral triangle. Prove that $q\le 6\sqrt{3} R$. Typesetter's Note: the excentral triangle has vertices which are the excenters of the original triangle.

2011 Today's Calculation Of Integral, 762

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

Define a function $f_n(x)\ (n=0,\ 1,\ 2,\ \cdots)$ by \[f_0(x)=\sin x,\ f_{n+1}(x)=\int_0^{\frac{\pi}{2}} f_n\prime (t)\sin (x+t)dt.\] (1) Let $f_n(x)=a_n\sin x+b_n\cos x.$ Express $a_{n+1},\ b_{n+1}$ in terms of $a_n,\ b_n.$ (2) Find $\sum_{n=0}^{\infty} f_n\left(\frac{\pi}{4}\right).$

the 11th XMO, 9

Tags: algebra , inequalities , trigonometry

$x,y\in\mathbb{R},(4x^3-3x)^2+(4y^3-3y)^2=1.\text { Find the maximum of } x+y.$