Olympiad problems

2012 Iran Team Selection Test, 2

Tags: geometry , circumcircle , incenter , trigonometry , angle bisector , power of a point , config geo

Consider $\omega$ is circumcircle of an acute triangle $ABC$. $D$ is midpoint of arc $BAC$ and $I$ is incenter of triangle $ABC$. Let $DI$ intersect $BC$ in $E$ and $\omega$ for second time in $F$. Let $P$ be a point on line $AF$ such that $PE$ is parallel to $AI$. Prove that $PE$ is bisector of angle $BPC$. [i]Proposed by Mr.Etesami[/i]

2006 Germany Team Selection Test, 1

Tags: trigonometry , percent , function , algebra

Find all real solutions $x$ of the equation $\cos\cos\cos\cos x=\sin\sin\sin\sin x$. (Angles are measured in radians.)

1978 Austrian-Polish Competition, 3

Tags: trigonometry , inequalities

Prove that $$\sqrt[44]{\tan 1^\circ\cdot \tan 2^\circ\cdot \dots\cdot \tan 44^\circ}<\sqrt 2-1<\frac{\tan 1^\circ+ \tan 2^\circ+\dots+\tan 44^\circ}{44}.$$

2014 India Regional Mathematical Olympiad, 3

Tags: trigonometry , geometry , angle bisector

Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$.

2018 AIME Problems, 15

Tags: trigonometry , geometry , cyclic quadrilateral

David found four sticks of different lengths that can be used to form three non-congruent convex cyclic quadrilaterals, $A$, $B$, $C$, which can each be inscribed in a circle with radius $1$. Let $\varphi_A$ denote the measure of the acute angle made by the diagonals of quadrilateral $A$, and define $\varphi_B$ and $\varphi_C$ similarly. Suppose that $\sin\varphi_A=\frac{2}{3}$, $\sin\varphi_B=\frac{3}{5}$, and $\sin\varphi_C=\frac{6}{7}$. All three quadrilaterals have the same area $K$, which can be written in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2008 Sharygin Geometry Olympiad, 2

Tags: ratio , trigonometry , circumcircle , geometry

(F.Nilov) Given right triangle $ ABC$ with hypothenuse $ AC$ and $ \angle A \equal{} 50^{\circ}$. Points $ K$ and $ L$ on the cathetus $ BC$ are such that $ \angle KAC \equal{} \angle LAB \equal{} 10^{\circ}$. Determine the ratio $ CK/LB$.

2012 Today's Calculation Of Integral, 828

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

Find a function $f(x)$, which is differentiable and $f'(x) $ is continuous, such that $\int_0^x f(t)\cos (x-t)\ dt=xe^{2x}.$

2008 Vietnam National Olympiad, 2

Tags: trigonometry , geometry

Given a triangle with acute angle $ \angle BEC,$ let $ E$ be the midpoint of $ AB.$ Point $ M$ is chosen on the opposite ray of $ EC$ such that $ \angle BME \equal{} \angle ECA.$ Denote by $ \theta$ the measure of angle $ \angle BEC.$ Evaluate $ \frac{MC}{AB}$ in terms of $ \theta.$

2005 Sharygin Geometry Olympiad, 4

Tags: geometry , convex polygon , trigonometry

At what smallest $n$ is there a convex $n$-gon for which the sines of all angles are equal and the lengths of all sides are different?

III Soros Olympiad 1996 - 97 (Russia), 9.8

Tags: trigonometry , algebra , geometric inequality

The two sides of the triangle are equal to $1$ and $x$, and $ x \ge 1$. The values $a$ and $b$ are the largest and smallest angles of this triangle, respectively. Find the greatest value of $\cos a$ and the smallest value of $\cos b$.

2004 National Olympiad First Round, 13

Tags: geometry , trigonometry

If the tangents of all interior angles of a triangle are integers, what is the sum of these integers? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 9 \qquad\textbf{(E)}\ \text{None of above} $

2009 ISI B.Math Entrance Exam, 4

Tags: trigonometry , algebra

Find the values of $x,y$ for which $x^2+y^2$ takes the minimum value where $(x+5)^2+(y-12)^2=14$.

2011 Purple Comet Problems, 14

Tags: trigonometry , pythagorean theorem , geometry

The lengths of the three sides of a right triangle form a geometric sequence. The sine of the smallest of the angles in the triangle is $\tfrac{m+\sqrt{n}}{k}$ where $m$, $n$, and $k$ are integers, and $k$ is not divisible by the square of any prime. Find $m + n + k$.

2010 Contests, 525

Tags: calculus , integration , geometry , vector , inequalities , quadratic , trigonometry

Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

1977 Polish MO Finals, 2

Tags: trigonometry , ratio , geometric sequence , geometry

Let $s \geq 3$ be a given integer. A sequence $K_n$ of circles and a sequence $W_n$ of convex $s$-gons satisfy: \[ K_n \supset W_n \supset K_{n+1} \] for all $n = 1, 2, ...$ Prove that the sequence of the radii of the circles $K_n$ converges to zero.

2009 China Team Selection Test, 1

Tags: geometry , circumcircle , trigonometry , geometric transformation , reflection , trapezoid , symmetry

Given that points $ D,E$ lie on the sidelines $ AB,BC$ of triangle $ ABC$, respectively, point $ P$ is in interior of triangle $ ABC$ such that $ PE \equal{} PC$ and $ \bigtriangleup DEP\sim \bigtriangleup PCA.$ Prove that $ BP$ is tangent of the circumcircle of triangle $ PAD.$

2010 Stanford Mathematics Tournament, 9

Tags: geometry , circumcircle , incenter , trigonometry , system of equations , algebra

For an acute triangle $ABC$ and a point $X$ satisfying $\angle{ABX}+\angle{ACX}=\angle{CBX}+\angle{BCX}$. Find the minimum length of $AX$ if $AB=13$, $BC=14$, and $CA=15$.

2000 Moldova National Olympiad, Problem 1

Tags: trigonometry , trig , algebra

Suppose that real numbers $x,y,z$ satisfy $$\frac{\cos x+\cos y+\cos z}{\cos(x+y+z)}=\frac{\sin x+\sin y+\sin z}{\sin(x+y+z)}=p.$$Prove that $\cos(x+y)+\cos(y+z)+\cos(x+z)=p$.

2024 Nigerian MO Round 3, Problem 3

Tags: geometry , circumcircle , trigonometry

Let $ABC$ be a triangle, and let $O$ be its circumcenter. Let $\overline{CO}\cap AB\equiv D$. Let $\angle BAC=\alpha$, and $\angle CBA=\beta$. Prove that $$\dfrac{OD}{OC}=\Bigg|\dfrac{\cos(\alpha+\beta)}{\cos(\alpha-\beta)}\Bigg|$$\\ For clarification, $\overline{CO}$ represents the line $CO$, and $AC$ represents the segment $AC$. Cases in which $D$ doesn't exist should be ignored.

PEN I Problems, 19

Tags: floor function , function , trigonometry

Let $a, b, c$, and $d$ be real numbers. Suppose that $\lfloor na\rfloor +\lfloor nb\rfloor =\lfloor nc\rfloor +\lfloor nd\rfloor $ for all positive integers $n$. Show that at least one of $a+b$, $a-c$, $a-d$ is an integer.

1981 Canada National Olympiad, 2

Tags: geometric transformation , reflection , trigonometry , function , calculus , derivative , geometry

Given a circle of radius $r$ and a tangent line $\ell$ to the circle through a given point $P$ on the circle. From a variable point $R$ on the circle, a perpendicular $RQ$ is drawn to $\ell$ with $Q$ on $\ell$. Determine the maximum of the area of triangle $PQR$.

2013 Cono Sur Olympiad, 2

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

In a triangle $ABC$, let $M$ be the midpoint of $BC$ and $I$ the incenter of $ABC$. If $IM$ = $IA$, find the least possible measure of $\angle{AIM}$.

2006 Kyiv Mathematical Festival, 4

Tags: circumcircle , incenter , trigonometry , trig identities , law of sines , geometry

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] Let $O$ be the circumcenter and $H$ be the intersection point of the altitudes of acute triangle $ABC.$ The straight lines $BH$ and $CH$ intersect the segments $CO$ and $BO$ at points $D$ and $E$ respectively. Prove that if triangles $ODH$ and $OEH$ are isosceles then triangle $ABC$ is isosceles too.

2014 Brazil National Olympiad, 1

Tags: geometry , incenter , trigonometry , rhombus , inradius

Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$. The inradii of triangles $ABP$, $BCP$, $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.

2020-2021 OMMC, 9

Tags: trigonometry , trig

The difference between the maximum and minimum values of $$2\cos 2x +7\sin x$$ over the real numbers equals $\frac{p}{q}$ for relatively prime positive integers $p, q.$ Find $p+q.$