Olympiad problems

2020 Moldova Team Selection Test, 11

Find all functions $f:[-1,1] \rightarrow \mathbb{R},$ which satisfy $$f(\sin{x})+f(\cos{x})=2020$$ for any real number $x.$

2001 China Team Selection Test, 2

Tags: algebra , trigonometry , inequality

Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that: $\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$ [hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]

1998 Irish Math Olympiad, 2

Tags: geometric transformation , rotation , trigonometry , geometry

The distances from a point $ P$ inside an equilateral triangle to the vertices of the triangle are $ 3,4$, and $ 5$. Find the area of the triangle.

Today's calculation of integrals, 883

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

Prove that for each positive integer $n$ \[\frac{4n^2+1}{4n^2-1}\int_0^{\pi} (e^{x}-e^{-x})\cos 2nx\ dx>\frac{e^{\pi}-e^{-\pi}-2}{4}\ln \frac{(2n+1)^2}{(2n-1)(n+3)}.\]

2005 Today's Calculation Of Integral, 69

Tags: calculus , integration , trigonometry , limit , calculus computations

Let $f_1(x)=x,f_n(x)=x+\frac{1}{14}\int_0^\pi xf_{n-1}(t)\cos ^ 3 t\ dt\ (n\geq 2)$. Find $\lim_{n\to\infty} f_n(x)$

2009 Iran Team Selection Test, 1

Tags: incenter , inradius , trigonometry , geometry

Let $ ABC$ be a triangle and $ A'$ , $ B'$ and $ C'$ lie on $ BC$ , $ CA$ and $ AB$ respectively such that the incenter of $ A'B'C'$ and $ ABC$ are coincide and the inradius of $ A'B'C'$ is half of inradius of $ ABC$ . Prove that $ ABC$ is equilateral .

1987 IMO Longlists, 70

Tags: geometry , circumcircle , trigonometry , area of a triangle

In an acute-angled triangle $ABC$ the interior bisector of angle $A$ meets $BC$ at $L$ and meets the circumcircle of $ABC$ again at $N$. From $L$ perpendiculars are drawn to $AB$ and $AC$, with feet $K$ and $M$ respectively. Prove that the quadrilateral $AKNM$ and the triangle $ABC$ have equal areas.[i](IMO Problem 2)[/i] [i]Proposed by Soviet Union.[/i]

Today's calculation of integrals, 878

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

A cubic function $f(x)$ satisfies the equation $\sin 3t=f(\sin t)$ for all real numbers $t$. Evaluate $\int_0^1 f(x)^2\sqrt{1-x^2}\ dx$.

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.

1969 IMO Longlists, 53

Tags: trigonometry , geometry , locus , locus problems

$(POL 2)$ Given two segments $AB$ and $CD$ not in the same plane, find the locus of points $M$ such that $MA^2 +MB^2 = MC^2 +MD^2.$

2020 Ukraine Team Selection Test, 3

Tags: circumcircle , tangent line , angle chasing , trigonometry , geometry

Altitudes $AH1$ and $BH2$ of acute triangle $ABC$ intersect at $H$. Let $w1$ be the circle that goes through $H2$ and touches the line $BC$ at $H1$, and let $w2$ be the circle that goes through $H1$ and touches the line $AC$ at $H2$. Prove, that the intersection point of two other tangent lines $BX$ and $AY$( $X$ and $Y$ are different from $H1$ and $H2$) to circles $w1$ and $w2$ respectively, lies on the circumcircle of triangle $HXY$. Proposed by [i]Danilo Khilko[/i]

1980 IMO Shortlist, 1

Tags: trigonometry , geometry , triangle , perpendicular bisector

Let $\alpha, \beta$ and $\gamma$ denote the angles of the triangle $ABC$. The perpendicular bisector of $AB$ intersects $BC$ at the point $X$, the perpendicular bisector of $AC$ intersects it at $Y$. Prove that $\tan(\beta) \cdot \tan(\gamma) = 3$ implies $BC= XY$ (or in other words: Prove that a sufficient condition for $BC = XY$ is $\tan(\beta) \cdot \tan(\gamma) = 3$). Show that this condition is not necessary, and give a necessary and sufficient condition for $BC = XY$.

2011 Postal Coaching, 1

Tags: ratio , parallelogram , trigonometry , geometry

Let $ABCD$ be a quadrilateral with an inscribed circle, centre $O$. Let \[AO = 5, BO =6, CO = 7, DO = 8.\] If $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$, determine $\frac{OM}{ON}$ .

2015 Mathematical Talent Reward Programme, MCQ: P 12

Tags: trigonometry , algebra , trigonometric equation

Maximum value of $\sin^4\theta +\cos^6\theta $ will be ? [list=1] [*] $\frac{1}{2\sqrt{2}}$ [*] $\frac{1}{2}$ [*] $\frac{1}{\sqrt{2}}$ [*] 1 [/list]

2014 Contests, 1

Tags: trigonometry , modular arithmetic , algebra

Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]

1982 IMO Longlists, 30

Tags: trigonometry , geometry , triangle , midpoint

Let $ABC$ be a triangle, and let $P$ be a point inside it such that $\angle PAC = \angle PBC$. The perpendiculars from $P$ to $BC$ and $CA$ meet these lines at $L$ and $M$, respectively, and $D$ is the midpoint of $AB$. Prove that $DL = DM.$

2024-25 IOQM India, 22

Tags: quadratic formula , trigonometry , geometry

In a triangle $ABC$, $\angle BAC = 90^{\circ}$. Let $D$ be the point on $BC$ such that $AB + BD = AC + CD$. Suppose $BD : DC = 2:1$. if $\frac{AC}{AB} = \frac{m + \sqrt{p}}{n}$, Where $m,n$ are relatively prime positive integers and $p$ is a prime number, determine the value of $m+n+p$.

2002 Kazakhstan National Olympiad, 2

Tags: inequalities , trigonometry , algebra

Let $x_1,x_2,\ldots,x_n$ be arbitrary real numbers. Prove the inequality \[ \frac{x_1}{1+x_1^2} + \frac{x_2}{1+x_1^2 + x_2^2} + \cdots + \frac{x_n}{1 + x_1^2 + \cdots + x_n^2} < \sqrt{n}. \]

2014 ELMO Shortlist, 12

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]

2011 Today's Calculation Of Integral, 719

Tags: calculus , integration , trigonometry , calculus computations

Compute $\int_0^x \sin t\cos t\sin (2\pi\cos t)\ dt$.

2007 Bulgaria Team Selection Test, 1

Tags: circumcircle , trigonometry , inequalities , geometry

Let $ABC$ is a triangle with $\angle BAC=\frac{\pi}{6}$ and the circumradius equal to 1. If $X$ is a point inside or in its boundary let $m(X)=\min(AX,BX,CX).$ Find all the angles of this triangle if $\max(m(X))=\frac{\sqrt{3}}{3}.$

1990 AIME Problems, 7

Tags: ratio , trigonometry , analytic geometry , graphing lines , slope , calculus , integration

A triangle has vertices $P=(-8,5)$, $Q=(-15,-19)$, and $R=(1,-7)$. The equation of the bisector of $\angle P$ can be written in the form $ax+2y+c=0$. Find $a+c$.

1974 IMO Longlists, 41

Tags: circumcircle , euler , trigonometry , geometry

Through the circumcenter $O$ of an arbitrary acute-angled triangle, chords $A_1A_2,B_1B_2, C_1C_2$ are drawn parallel to the sides $BC,CA,AB$ of the triangle respectively. If $R$ is the radius of the circumcircle, prove that \[A_1O \cdot OA_2 + B_1O \cdot OB_2 + C_1O \cdot OC_2 = R^2.\]

1991 AIME Problems, 4

Tags: trigonometry , function , logarithm

How many real numbers $x$ satisfy the equation $\frac{1}{5}\log_2 x = \sin (5\pi x)$?

1969 IMO Shortlist, 29

Tags: trigonometry , algebra , trigonometric equation

$(GDR 1)$ Find all real numbers $\lambda$ such that the equation $\sin^4 x - \cos^4 x = \lambda(\tan^4 x - \cot^4 x)$ $(a)$ has no solution, $(b)$ has exactly one solution, $(c)$ has exactly two solutions, $(d)$ has more than two solutions (in the interval $(0, \frac{\pi}{4}).$