Olympiad problems

2010 Today's Calculation Of Integral, 632

Tags: calculus , integration , limit , trigonometry , floor function , ceiling function , calculus computations

Find $\lim_{n\to\infty} \int_0^1 |\sin nx|^3dx\ (n=1,\ 2,\ \cdots).$ [i]2010 Kyoto Institute of Technology entrance exam/Textile, 2nd exam[/i]

2013 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: algebra , equation , trigonometry

Let $a$ and $b$ be real numbers from interval $\left[0,\frac{\pi}{2}\right]$. Prove that $$\sin^6 {a}+3\sin^2 {a}\cos^2 {b}+\cos^6 {b}=1$$ if and only if $a=b$

2013 ELMO Shortlist, 10

Tags: circumcircle , ratio , symmetry , projective geometry , trigonometry , geometry

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

2004 Nicolae Coculescu, 2

Tags: equation , monotone , quadratic , algebra , trigonometry , logarithm

Solve in the real numbers the equation: $$ \cos^2 \frac{(x-2)\pi }{4} +\cos\frac{(x-2)\pi }{3} =\log_3 (x^2-4x+6) $$ [i]Gheorghe Mihai[/i]

2023 CMIMC Team, 7

Tags: team , function , trigonometry

Compute the value of $$\sin^2\left(\frac{\pi}{7}\right) + \sin^2\left(\frac{3\pi}{7}\right) + \sin^2\left(\frac{5\pi}{7}\right).$$ Your answer should not involve any trigonometric functions. [i]Proposed by Howard Halim[/i]

1992 IMO Longlists, 68

Tags: trigonometry , number theory , relatively prime , algebra , equation

Show that the numbers $\tan \left(\frac{r \pi }{15}\right)$, where $r$ is a positive integer less than $15$ and relatively prime to $15$, satisfy \[x^8 - 92x^6 + 134x^4 - 28x^2 + 1 = 0.\]

2009 China Northern MO, 2

Tags: geometry , angle bisector , right angle , trigonometry

In an acute triangle $ABC$ , $AB>AC$ , $ \cos B+ \cos C=1$ , $E,F$ are on the extend line of $AB,AC$ such that $\angle ABF = \angle ACE = 90$ . (1) Prove :$BE+CF=EF$ ; (2) Assume the bisector of $\angle EBC$ meet $EF$ at $P$ , prove that $CP$ is the bisector of $\angle BCF$. [img]https://cdn.artofproblemsolving.com/attachments/a/2/c554c2bc0b4e044c45f88138568f5234d544a8.png[/img]

Ukrainian TYM Qualifying - geometry, V.8

Tags: geometry , trigonometry , geometric inequality , equilateral

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

2002 All-Russian Olympiad, 3

Tags: trigonometry , function , inequalities

Prove that if $0<x<\frac{\pi}{2}$ and $n>m$, where $n$,$m$ are natural numbers, \[ 2 \left| \sin^n x - \cos^n x \right| \le 3 \left| \sin^m x - \cos^m x \right|.\]

2020 LIMIT Category 2, 10

Tags: limit , trigonometry , geometry

In a triangle $\triangle XYZ$, $\tan(x)\tan(z)=2$, $\tan(y)\tan(z)=18$. Then what is $\tan^2(z)$?

2011 Today's Calculation Of Integral, 764

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

Let $f(x)$ be a continuous function defined on $0\leq x\leq \pi$ and satisfies $f(0)=1$ and \[\left\{\int_0^{\pi} (\sin x+\cos x)f(x)dx\right\}^2=\pi \int_0^{\pi}\{f(x)\}^2dx.\] Evaluate $\int_0^{\pi} \{f(x)\}^3dx.$

2001 Vietnam Team Selection Test, 2

Tags: trigonometry , geometric transformation , reflection , a-hm point , geometry

In the plane let two circles be given which intersect at two points $A, B$; Let $PT$ be one of the two common tangent line of these circles ($P, T$ are points of tangency). Tangents at $P$ and $T$ of the circumcircle of triangle $APT$ meet each other at $S$. Let $H$ be a point symmetric to $B$ under $PT$. Show that $A, S, H$ are collinear.

2021 Bangladeshi National Mathematical Olympiad, 8

Tags: geometry , euclidean geometry , angle bisector , trigonometry

Let $ABC$ be an acute-angled triangle. The external bisector of $\angle{BAC}$ meets the line $BC$ at point $N$. Let $M$ be the midpoint of $BC$. $P$ and $Q$ are two points on line $AN$ such that, $\angle{PMN}=\angle{MQN}=90^{\circ}$. If $PN=5$ and $BC=3$, then the length of $QA$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are coprime positive integers. What is the value of $(a+b)$?

2002 Romania Team Selection Test, 1

Tags: symmetry , trigonometry , geometry

Let $ABCDE$ be a cyclic pentagon inscribed in a circle of centre $O$ which has angles $\angle B=120^{\circ},\angle C=120^{\circ},$ $\angle D=130^{\circ},\angle E=100^{\circ}$. Show that the diagonals $BD$ and $CE$ meet at a point belonging to the diameter $AO$. [i]Dinu Șerbănescu[/i]

2011 China Team Selection Test, 1

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

Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.

2006 AIME Problems, 1

Tags: geometry , perimeter , trigonometry , pythagorean theorem

In quadrilateral $ABCD, \angle B$ is a right angle, diagonal $\overline{AC}$ is perpendicular to $\overline{CD},$ $AB=18, BC=21,$ and $CD=14.$ Find the perimeter of $ABCD$.

1985 All Soviet Union Mathematical Olympiad, 403

Tags: trigonometry , inequalities , algebra

Find all the pairs $(x,y)$ such that $|\sin x-\sin y| + \sin x \sin y \le 0$.

1989 IberoAmerican, 2

Tags: trigonometry , inequalities

Let $x,y,z$ be real numbers such that $0\le x,y,z\le\frac{\pi}{2}$. Prove the inequality \[\frac{\pi}{2}+2\sin x\cos y+2\sin y\cos z\ge\sin 2x+\sin 2y+\sin 2z.\]

1999 Mediterranean Mathematics Olympiad, 4

Tags: trigonometry , geometry

In triangle $\triangle ABC$ we have $BC=a,CA=b,AB=c$ and $\angle B=4\angle A$ Show that \[ab^2c^3=(b^2-a^2-ac)((a^2-b^2)^2-a^2c^2)\]

IV Soros Olympiad 1997 - 98 (Russia), 11.8

Tags: trigonometry , algebra , polynomial

Sum of all roots of the equation $$cos^{100} x + a_1 cos^{99} x + a_2cos^{98} x +... + a_99 cos x+ a_{100} = 0$$, in interval $\left[\pi, \frac{3\pi}{2} \right]$, is equal to $21\pi$, and the sum of all roots of the equation $$sin^{100} x + a_1 sin^{99} x + a_2sin ^{98} x +... + a_99sin x+ a_{100} = 0$$, in the same interval, is equal to $24\pi $. How many roots does the first equation have on the segment $\left[ \frac{\pi}{2}, \pi\right]$?

Estonia Open Senior - geometry, 2007.2.5

Tags: circumcircle , inradius , trigonometry , geometry

Consider triangles whose each side length squared is a rational number. Is it true that (a) the square of the circumradius of every such triangle is rational; (b) the square of the inradius of every such triangle is rational?

2000 India National Olympiad, 4

Tags: trigonometry , pythagorean theorem , geometry

In a convex quadrilateral $PQRS$, $PQ =RS$, $(\sqrt{3} +1 )QR = SP$ and $\angle RSP - \angle SQP = 30^{\circ}$. Prove that $\angle PQR - \angle QRS = 90^{\circ}.$

2003 IberoAmerican, 2

Tags: trigonometry , geometry , trapezoid , circumcircle , radical axis , cyclic quadrilateral , power of a point

Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.

2004 France Team Selection Test, 2

Tags: inequalities , geometry , incenter , trigonometry , inradius , perimeter , cyclic quadrilateral

Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

1966 IMO Shortlist, 50

Tags: trigonometric equation , parameterization , trigonometry , parabola

Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac 1p$ where $p$ is a real parameter. Discuss for which values of $p$ the equation has at least one real solution and determine the number of solutions in $[0, 2\pi)$ for a given $p.$