Olympiad problems

2005 Today's Calculation Of Integral, 14

Tags: calculus , integration , trigonometry , function , algebra , domain , logarithm

Calculate the following indefinite integrals. [1] $\int \frac{\sin x\cos x}{1+\sin ^ 2 x}dx$ [2] $\int x\log_{10} x dx$ [3] $\int \frac{x}{\sqrt{2x-1}}dx$ [4] $\int (x^2+1)\ln x dx$ [5] $\int e^x\cos x dx$

2018 Ramnicean Hope, 2

Tags: equation , monotone , trigonometry , algebra

Solve in the real numbers the equation $ \arctan\sqrt{3^{1-2x}} +\arctan {3^x} =\frac{7\pi }{12} . $ [i]Ovidiu Țâțan[/i]

2003 Korea - Final Round, 1

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 $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

2012 Today's Calculation Of Integral, 836

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $\int_0^{\pi} e^{\sin x}\cos ^ 2(\sin x )\cos x\ dx$.

2009 Today's Calculation Of Integral, 488

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

For $ 0\leq x <\frac{\pi}{2}$, prove the following inequality. $ x\plus{}\ln (\cos x)\plus{}\int_0^1 \frac{t}{1\plus{}t^2}\ dt\leq \frac{\pi}{4}$

2013 USAJMO, 5

Tags: ratio , trigonometry , geometry , analytic geometry , trapezoid , similar triangles , trig identities

Quadrilateral $XABY$ is inscribed in the semicircle $\omega$ with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Point $Z$ is the foot of the perpendicular from $P$ to line $XY$. Point $C$ lies on $\omega$ such that line $XC$ is perpendicular to line $AZ$. Let $Q$ be the intersection of segments $AY$ and $XC$. Prove that \[\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.\]

2010 Sharygin Geometry Olympiad, 16

Tags: trigonometry , geometry

A circle touches the sides of an angle with vertex $A$ at points $B$ and $C.$ A line passing through $A$ intersects this circle in points $D$ and $E.$ A chord $BX$ is parallel to $DE.$ Prove that $XC$ passes through the midpoint of the segment $DE.$

2013 Stanford Mathematics Tournament, 8

Tags: trigonometry

Rational Man and Irrational Man both buy new cars, and they decide to drive around two racetracks from time $t=0$ to time $t=\infty$. Rational Man drives along the path parametrized by \begin{align*}x&=\cos(t)\\y&=\sin(t)\end{align*} and Irrational Man drives along the path parametrized by \begin{align*}x&=1+4\cos\frac{t}{\sqrt{2}}\\ y&=2\sin\frac{t}{\sqrt{2}}.\end{align*} Find the largest real number $d$ such that at any time $t$, the distance between Rational Man and Irrational Man is not less than $d$.

2014 Cezar Ivănescu, 2

Tags: quadratic , geometry , trigonometry , function , complex number

While there do not exist pairwise distinct real numbers $a,b,c$ satisfying $a^2+b^2+c^2 = ab+bc+ca$, there do exist complex numbers with that property. Let $a,b,c$ be complex numbers such that $a^2+b^2+c^2 = ab+bc+ca$ and $|a+b+c| = 21$. Given that $|a-b| = 2\sqrt{3}$, $|a| = 3\sqrt{3}$, compute $|b|^2+|c|^2$. [hide="Clarifications"] [list] [*] The problem should read $|a+b+c| = 21$. An earlier version of the test read $|a+b+c| = 7$; that value is incorrect. [*] $|b|^2+|c|^2$ should be a positive integer, not a fraction; an earlier version of the test read ``... for relatively prime positive integers $m$ and $n$. Find $m+n$.''[/list][/hide] [i]Ray Li[/i]

2012 Serbia National Math Olympiad, 1

Tags: parallelogram , circumcircle , trigonometry , geometry

Let $ABCD$ be a parallelogram and $P$ be a point on diagonal $BD$ such that $\angle PCB=\angle ACD$. Circumcircle of triangle $ABD$ intersects line $AC$ at points $A$ and $E$. Prove that \[\angle AED=\angle PEB.\]

2010 Brazil National Olympiad, 3

Tags: geometry , 3d geometry , trigonometry , vector , linear algebra , matrix , inequalities

What is the biggest shadow that a cube of side length $1$ can have, with the sun at its peak? Note: "The biggest shadow of a figure with the sun at its peak" is understood to be the biggest possible area of the orthogonal projection of the figure on a plane.

2000 APMO, 3

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

Let $ABC$ be a triangle. Let $M$ and $N$ be the points in which the median and the angle bisector, respectively, at $A$ meet the side $BC$. Let $Q$ and $P$ be the points in which the perpendicular at $N$ to $NA$ meets $MA$ and $BA$, respectively. And $O$ the point in which the perpendicular at $P$ to $BA$ meets $AN$ produced. Prove that $QO$ is perpendicular to $BC$.

2013 Today's Calculation Of Integral, 872

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

Let $n$ be a positive integer. (1) For a positive integer $k$ such that $1\leq k\leq n$, Show that : \[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\] (2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$ If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$ (3) Find $\lim_{n\to\infty} S_n.$

2014 India IMO Training Camp, 1

Tags: geometry , circumcircle , trigonometry , geometric transformation , reflection , dilation , radical axis

In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.

2013 Finnish National High School Mathematics Competition, 3

Tags: trigonometry , geometry

The points $A,B,$ and $C$ lies on the circumference of the unit circle. Furthermore, it is known that $AB$ is a diameter of the circle and \[\frac{|AC|}{|CB|}=\frac{3}{4}.\] The bisector of $ABC$ intersects the circumference at the point $D$. Determine the length of the $AD$.

1966 IMO Shortlist, 9

Tags: trigonometric equation , trigonometry , equation , algebra

Find $x$ such that trigonometric \[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\] where $m$ is a fixed real number.

1952 Miklós Schweitzer, 10

Tags: trigonometry , function , integration , real analysis

Let $ n$ be a positive integer. Prove that, for $ 0<x<\frac{\pi}{n\plus{}1}$, $ \sin{x}\minus{}\frac{\sin{2x}}{2}\plus{}\cdots\plus{}(\minus{}1)^{n\plus{}1}\frac{\sin{nx}}{n}\minus{}\frac{x}{2}$ is positive if $ n$ is odd and negative if $ n$ is even.

2013 Macedonia National Olympiad, 3

Tags: ratio , circumcircle , trigonometry , geometry

Acute angle triangle is given such that $ BC $ is the longest side. Let $ E $ and $ G $ be the intersection points from the altitude from $ A $ to $ BC $ with the circumscribed circle of triangle $ ABC $ and $ BC $ respectively. Let the center $ O $ of this circle is positioned on the perpendicular line from $ A $ to $ BE $. Let $ EM $ be perpendicular to $ AC $ and $ EF $ be perpendicular to $ AB $. Prove that the area of $ FBEG $ is greater than the area of $ MFE $.

2011 Putnam, A3

Tags: limit , integration , trigonometry , calculus , ratio , inequalities

Find a real number $c$ and a positive number $L$ for which \[\lim_{r\to\infty}\frac{r^c\int_0^{\pi/2}x^r\sin x\,dx}{\int_0^{\pi/2}x^r\cos x\,dx}=L.\]

2012 Philippine MO, 3

Tags: trigonometry , inequalities

If $ab>0$ and $\displaystyle 0<x<\frac{\pi}{2}$, prove that \[ \left ( 1+\frac{a^2}{\sin x} \right ) \left ( 1+\frac{b^2}{\cos x} \right ) \geq \frac{(1+\sqrt{2}ab)^2 \sin 2x}{2}. \]

1983 AMC 12/AHSME, 20

Tags: trigonometry , algebra , polynomial

If $\tan{\alpha}$ and $\tan{\beta}$ are the roots of $x^2 - px + q = 0$, and $\cot{\alpha}$ and $\cot{\beta}$ are the roots of $x^2 - rx + s = 0$, then $rs$ is necessarily $\text{(A)} \ pq \qquad \text{(B)} \ \frac{1}{pq} \qquad \text{(C)} \ \frac{p}{q^2} \qquad \text{(D)} \ \frac{q}{p^2} \qquad \text{(E)} \ \frac{p}{q}$

2007 Iran Team Selection Test, 1

Tags: trigonometry , circumcircle , angle bisector , geometry

In triangle $ABC$, $M$ is midpoint of $AC$, and $D$ is a point on $BC$ such that $DB=DM$. We know that $2BC^{2}-AC^{2}=AB.AC$. Prove that \[BD.DC=\frac{AC^{2}.AB}{2(AB+AC)}\]

2002 AMC 12/AHSME, 10

Tags: function , trigonometry

Let $f_n(x)=\sin^n x + \cos^n x$. For how many $x$ in $[0,\pi]$ is it true that \[6f_4(x)-4f_6(x)=2f_2(x)?\] $\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }8\qquad\textbf{(E) }\text{more than 8}$

1983 IMO Shortlist, 25

Tags: geometry , 3d geometry , sphere , trigonometry , combinatorics , partition

Prove that every partition of $3$-dimensional space into three disjoint subsets has the following property: One of these subsets contains all possible distances; i.e., for every $a \in \mathbb R^+$, there are points $M$ and $N$ inside that subset such that distance between $M$ and $N$ is exactly $a.$

2011 Today's Calculation Of Integral, 747

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

Prove that $\int_0^4 \left(1-\cos \frac{x}{2}\right)e^{\sqrt{x}}dx\leq -2e^2+30.$