Olympiad problems

1969 Swedish Mathematical Competition, 2

Show that $\tan \frac{\pi}{3n}$ is irrational for all positive integers $n$.

2013 USA TSTST, 9

Tags: rotation , invariant , niven theorem , geocombontturnedintoalg , trigonometry , geometry , inequalities

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.

2006 Iran Team Selection Test, 5

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

Let $ABC$ be an acute angle triangle. Suppose that $D,E,F$ are the feet of perpendicluar lines from $A,B,C$ to $BC,CA,AB$. Let $P,Q,R$ be the feet of perpendicular lines from $A,B,C$ to $EF,FD,DE$. Prove that \[ 2(PQ+QR+RP)\geq DE+EF+FD \]

2016 Indonesia TST, 3

Tags: algebra , trigonometry

Let $n$ be a positive integer greater than $1$. Evaluate the following summation: \[ \sum_{k=0}^{n-1} \frac{1}{1 + 8 \sin^2 \left( \frac{k \pi}{n} \right)}. \]

1969 Kurschak Competition, 2

Tags: trigonometry , equilateral , geometry

A triangle has side lengths $a, b, c$ and angles $A, B, C$ as usual (with $b$ opposite $B$ etc). Show that if $$a(1 - 2 \cos A) + b(1 - 2 \cos B) + c(1 - 2 \cos C) = 0$$ then the triangle is equilateral.

2013 AIME Problems, 14

Tags: trig identities , number theory , relatively prime , geometric series , quadratic , ratio , trigonometry

For $\pi\leq\theta<2\pi$, let \[ P=\dfrac12\cos\theta-\dfrac14\sin2\theta-\dfrac18\cos3\theta+\dfrac1{16}\sin4\theta+\dfrac1{32}\cos5\theta-\dfrac1{64}\sin6\theta-\dfrac1{128}\cos7\theta+\ldots \] and \[ Q=1-\dfrac12\sin\theta-\dfrac14\cos2\theta+\dfrac1{8}\sin3\theta+\dfrac1{16}\cos4\theta-\dfrac1{32}\sin5\theta-\dfrac1{64}\cos6\theta+\dfrac1{128}\sin7\theta +\ldots \] so that $\tfrac PQ = \tfrac{2\sqrt2}7$. Then $\sin\theta = -\tfrac mn$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2013 Tuymaada Olympiad, 8

Tags: trigonometry , inequalities , analytic geometry , perimeter , geometry

The point $A_1$ on the perimeter of a convex quadrilateral $ABCD$ is such that the line $AA_1$ divides the quadrilateral into two parts of equal area. The points $B_1$, $C_1$, $D_1$ are defined similarly. Prove that the area of the quadrilateral $A_1B_1C_1D_1$ is greater than a quarter of the area of $ABCD$. [i]L. Emelyanov [/i]

2008 Alexandru Myller, 2

Tags: trigonometry , binomial coefficient , complex number , binomial theorem , number theory

There are no integers $ a,b,c $ that satisfy $ \left( a+b\sqrt{-3}\right)^{17}=c+\sqrt{-3} . $ [i]Dorin Andrica, Mihai Piticari[/i]

2006 International Zhautykov Olympiad, 2

Tags: trapezoid , parallelogram , analytic geometry , function , ratio , geometry , trigonometry

Let $ ABC$ be a triangle and $ K$ and $ L$ be two points on $ (AB)$, $ (AC)$ such that $ BK \equal{} CL$ and let $ P \equal{} CK\cap BL$. Let the parallel through $ P$ to the interior angle bisector of $ \angle BAC$ intersect $ AC$ in $ M$. Prove that $ CM \equal{} AB$.

1966 IMO Shortlist, 59

Tags: trigonometric equation , geometry , trigonometry

Let $a,b,c$ be the lengths of the sides of a triangle, and $\alpha, \beta, \gamma$ respectively, the angles opposite these sides. Prove that if \[ a+b=\tan{\frac{\gamma}{2}}(a\tan{\alpha}+b\tan{\beta}) \] the triangle is isosceles.

2010 Kurschak Competition, 2

Tags: trigonometry , geometry

Consider a triangle $ABC$, with the points $A_1$, $A_2$ on side $BC$, $B_1,B_2\in\overline{AC}$, $C_1,C_2\in\overline{AB}$ such that $AC_1<AC_2$, $BA_1<BA_2$, $CB_1<CB_2$. Let the circles $AB_1C_1$ and $AB_2C_2$ meet at $A$ and $A^*$. Similarly, let the circles $BC_1A_1$ and $BC_2A_2$ intersect at $B^*\neq B$, let $CA_1B_1$ and $CA_2B_2$ intersect at $C^*\neq C$. Prove that the lines $AA^*$, $BB^*$, $CC^*$ are concurrent.

2023 UMD Math Competition Part II, 4

Tags: ratio , trigonometry , similar triangles , geometry

Assume every side length of a triangle $ABC$ is more than $2$ and two of its angles are given by $\angle ABC = 57^\circ$ and $ACB = 63^\circ$. Point $P$ is chosen on side $BC$ with $BP:PC = 2:1$. Points $M,N$ are chosen on sides $AB$ and $AC$, respectively so that $BM = 2$ and $CN = 1$. Let $Q$ be the point on segment $MN$ for which $MQ:QN = 2:1$. Find the value of $PQ$. Your answer must be in simplest form.

1996 Poland - Second Round, 2

Tags: geometry , trigonometry

A circle with center O inscribed in a convex quadrilateral ABCD is tangent to the lines AB, BC, CD, DA at points K, L, M, N respectively. Assume that the lines KL and MN are not parallel and intersect at the point S. Prove that BD is perpendicular OS. I think it is very good and beautiful problem. I solved it without help. I'm wondering is it a well known theorem? Also I'm interested who is the creator of this problem? I'll be glad to see simple solution of this problem.

2011 District Olympiad, 2

Tags: geometry , absolute value , complex number , rectangle , trigonometry

[b]a)[/b] Show that if four distinct complex numbers have the same absolute value and their sum vanishes, then they represent a rectangle. [b]b)[/b] Let $ x,y,z,t $ be four real numbers, and $ k $ be an integer. Prove the following implication: $$ \sum_{j\in\{ x,y,z,t\}} \sin j = 0 = \sum_{j\in\{ x,y,z,t\}} \cos j\implies \sum_{j\in\{ x,y,z,t\}} \sin (1+2n)j. $$

2009 Today's Calculation Of Integral, 431

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

Consider the function $ f(\theta) \equal{} \int_0^1 |\sqrt {1 \minus{} x^2} \minus{} \sin \theta|dx$ in the interval of $ 0\leq \theta \leq \frac {\pi}{2}$. (1) Find the maximum and minimum values of $ f(\theta)$. (2) Evaluate $ \int_0^{\frac {\pi}{2}} f(\theta)\ d\theta$.

2014 Contests, 3

Tags: parallelogram , trapezoid , geometric transformation , logarithm , dilation , trigonometry , geometry

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

1995 Greece National Olympiad, 2

Tags: geometry , trigonometry

Let $ABC$ be a triangle with $AB = AC$ and let $D$ be a point on $BC$ such that the incircle of $ABD$ and the excircle of $ADC$ corresponding to $A$ have the same radius. Prove that this radius is equal to one quarter of the altitude from $B$ of triangle $ABC$.

1983 AMC 12/AHSME, 27

Tags: trig identities , trigonometry , power of a point , 3d geometry , sphere , geometry

A large sphere is on a horizontal field on a sunny day. At a certain time the shadow of the sphere reaches out a distance of $10$ m from the point where the sphere touches the ground. At the same instant a meter stick (held vertically with one end on the ground) casts a shadow of length $2$ m. What is the radius of the sphere in meters? (Assume the sun's rays are parallel and the meter stick is a line segment.) $ \textbf{(A)}\ \frac{5}{2}\qquad\textbf{(B)}\ 9 - 4\sqrt{5}\qquad\textbf{(C)}\ 8\sqrt{10} - 23\qquad\textbf{(D)}\ 6 - \sqrt{15}\qquad\textbf{(E)}\ 10\sqrt{5} - 20 $

2008 Iran MO (2nd Round), 2

Tags: incenter , trigonometry , radical axis , power of a point , geometry

Let $I_a$ be the $A$-excenter of $\Delta ABC$ and the $A$-excircle of $\Delta ABC$ be tangent to the lines $AB,AC$ at $B',C'$, respectively. $ I_aB,I_aC$ meet $B'C'$ at $P,Q$, respectively. $M$ is the meet point of $BQ,CP$. Prove that the length of the perpendicular from $M$ to $BC$ is equal to $r$ where $r$ is the radius of incircle of $\Delta ABC$.

2009 Today's Calculation Of Integral, 513

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

Find the constants $ a,\ b,\ c$ such that a function $ f(x)\equal{}a\sin x\plus{}b\cos x\plus{}c$ satisfies the following equation for any real numbers $ x$. \[ 5\sin x\plus{}3\cos x\plus{}1\plus{}\int_0^{\frac{\pi}{2}} (\sin x\plus{}\cos t)f(t)\ dt\equal{}f(x).\]

2022 Moldova EGMO TST, 10

Tags: trigonometry

Compute $$\frac{\sum_{k=0}^{2022}\sin\frac{k\pi}{3033}}{\sum_{k=0}^{2022}\cos\frac{k\pi}{3033}}.$$

1986 China Team Selection Test, 1

Tags: angle bisector , geometry , congruent triangles , perimeter , trigonometry

Given a square $ABCD$ whose side length is $1$, $P$ and $Q$ are points on the sides $AB$ and $AD$. If the perimeter of $APQ$ is $2$ find the angle $PCQ$.

2014 NIMO Problems, 7

Tags: trigonometry , geometry

Let $ABC$ be a triangle and let $Q$ be a point such that $\overline{AB} \perp \overline{QB}$ and $\overline{AC} \perp \overline{QC}$. A circle with center $I$ is inscribed in $\triangle ABC$, and is tangent to $\overline{BC}$, $\overline{CA}$ and $\overline{AB}$ at points $D$, $E$, and $F$, respectively. If ray $QI$ intersects $\overline{EF}$ at $P$, prove that $\overline{DP} \perp \overline{EF}$. [i]Proposed by Aaron Lin[/i]

2012 National Olympiad First Round, 33

Tags: geometry , 3d geometry , prism , trigonometry , rectangle

Let $ABCDA'B'C'D'$ be a rectangular prism with $|AB|=2|BC|$. $E$ is a point on the edge $[BB']$ satisfying $|EB'|=6|EB|$. Let $F$ and $F'$ be the feet of the perpendiculars from $E$ at $\triangle AEC$ and $\triangle A'EC'$, respectively. If $m(\widehat{FEF'})=60^{\circ}$, then $|BC|/|BE| = ? $ $ \textbf{(A)}\ \sqrt\frac53 \qquad \textbf{(B)}\ \sqrt\frac{15}2 \qquad \textbf{(C)}\ \frac32\sqrt{15} \qquad \textbf{(D)}\ 5\sqrt\frac53 \qquad \textbf{(E)}\ \text{None}$

I Soros Olympiad 1994-95 (Rus + Ukr), 10.2

Tags: algebra , trigonometry

Find the smallest positive number $a$ for which $$\sin a^o = \sin a$$ (on the left ($a^o$) is an angle of $a$ degrees, on the right is an angle in $a$ radians).