Olympiad problems

2011 IberoAmerican, 1

Tags: vector , parallelogram , circumcircle , rhombus , trigonometry , perpendicular bisector , geometry

Let $ABC$ be an acute-angled triangle, with $AC \neq BC$ and let $O$ be its circumcenter. Let $P$ and $Q$ be points such that $BOAP$ and $COPQ$ are parallelograms. Show that $Q$ is the orthocenter of $ABC$.

2014 Harvard-MIT Mathematics Tournament, 6

Tags: geometry , ratio , trigonometry , area of a triangle , heron's formula , trig identities , law of cosines

In quadrilateral $ABCD$, we have $AB = 5$, $BC = 6$, $CD = 5$, $DA = 4$, and $\angle ABC = 90^\circ$. Let $AC$ and $BD$ meet at $E$. Compute $\dfrac{BE}{ED}$.

2004 Germany Team Selection Test, 1

Tags: parallelogram , trigonometry , angle bisector , geometry

The $A$-excircle of a triangle $ABC$ touches the side $BC$ at the point $K$ and the extended side $AB$ at the point $L$. The $B$-excircle touches the lines $BA$ and $BC$ at the points $M$ and $N$, respectively. The lines $KL$ and $MN$ meet at the point $X$. Show that the line $CX$ bisects the angle $ACN$.

2007 Today's Calculation Of Integral, 247

Tags: calculus , integration , trigonometry , calculus computations

Evaluate $ \int_{\frac{\pi}{8}}^{\frac{3}{8}\pi} \frac{11\plus{}4\cos 2x \plus{}\cos 4x}{1\minus{}\cos 4x}\ dx.$

2010 Romanian Master of Mathematics, 3

Tags: geometric transformation , trigonometry , geometry , ratio , homothety , projective geometry , trig identities

Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent. [i]Pavel Kozhevnikov, Russia[/i]

2014 Dutch IMO TST, 2

Tags: trigonometry , geometry , geometric transformation , reflection , projective geometry , trig identities , law of sines

Let $\triangle ABC$ be a triangle. Let $M$ be the midpoint of $BC$ and let $D$ be a point on the interior of side $AB$. The intersection of $AM$ and $CD$ is called $E$. Suppose that $|AD|=|DE|$. Prove that $|AB|=|CE|$.

2003 CHKMO, 1

Tags: trigonometry , geometry

Two circles meet at points $A$ and $B$. A line through $B$ intersects the first circle again at $K$ and the second circle at $M$. A line parallel to $AM$ is tangent to the first circle at $Q$. The line $AQ$ intersects the second circle again at $R$. $(a)$ Prove that the tangent to the second circle at $R$ is parallel to $AK$. $(b)$ Prove that these two tangents meet on $KM$.

2000 Moldova Team Selection Test, 3

Tags: function , algebra , polynomial , integration , trigonometry , calculus , probability

For each positive integer $ n$, evaluate the sum \[ \sum_{k\equal{}0}^{2n}(\minus{}1)^{k}\frac{\binom{4n}{2k}}{\binom{2n}{k}}\]

2020 AMC 12/AHSME, 25

Tags: trigonometry , probability

For each real number $a$ with $0 \leq a \leq 1$, let numbers $x$ and $y$ be chosen independently at random from the intervals $[0, a]$ and $[0, 1]$, respectively, and let $P(a)$ be the probability that $$\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1.$$ What is the maximum value of $P(a)?$ $\textbf{(A)}\ \frac{7}{12} \qquad\textbf{(B)}\ 2 - \sqrt{2} \qquad\textbf{(C)}\ \frac{1+\sqrt{2}}{4} \qquad\textbf{(D)}\ \frac{\sqrt{5}-1}{2} \qquad\textbf{(E)}\ \frac{5}{8}$

1998 Hungary-Israel Binational, 2

Tags: inequalities , trigonometry , circumcircle , geometry

A triangle ABC is inscribed in a circle with center $ O$ and radius $ R$. If the inradii of the triangles $ OBC, OCA, OAB$ are $ r_{1}, r_{2}, r_{3}$ , respectively, prove that $ \frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\geq\frac{4\sqrt{3}+6}{R}.$

2012 AMC 12/AHSME, 17

Tags: analytic geometry , graphing lines , slope , trigonometry , geometric transformation , dilation , geometry

Square $PQRS$ lies in the first quadrant. Points $(3,0), (5,0), (7,0),$ and $(13,0)$ lie on lines $SP, RQ, PQ$, and $SR$, respectively. What is the sum of the coordinates of the center of the square $PQRS$? $ \textbf{(A)}\ 6\qquad\textbf{(B)}\ 6.2\qquad\textbf{(C)}\ 6.4\qquad\textbf{(D)}\ 6.6\qquad\textbf{(E)}\ 6.8 $

2008 AIME Problems, 4

Tags: trigonometry , quadratic , special factorizations , diophantine equation

There exist unique positive integers $ x$ and $ y$ that satisfy the equation $ x^2 \plus{} 84x \plus{} 2008 \equal{} y^2$. Find $ x \plus{} y$.

2008 Harvard-MIT Mathematics Tournament, 7

Tags: trigonometry

Given that $ x \plus{} \sin y \equal{} 2008$ and $ x \plus{} 2008 \cos y \equal{} 2007$, where $ 0 \leq y \leq \pi/2$, find the value of $ x \plus{} y$.

2010 Sharygin Geometry Olympiad, 21

Tags: geometric transformation , reflection , trigonometry , geometry

A given convex quadrilateral $ABCD$ is such that $\angle ABD + \angle ACD > \angle BAC + \angle BDC.$ Prove that \[S_{ABD}+S_{ACD} > S_{BAC}+S_{BDC}.\]

1971 IMO Longlists, 28

Tags: 3d geometry , tetrahedron , trigonometry , distance , geometry

All faces of the tetrahedron $ABCD$ are acute-angled. Take a point $X$ in the interior of the segment $AB$, and similarly $Y$ in $BC, Z$ in $CD$ and $T$ in $AD$. [b]a.)[/b] If $\angle DAB+\angle BCD\ne\angle CDA+\angle ABC$, then prove none of the closed paths $XYZTX$ has minimal length; [b]b.)[/b] If $\angle DAB+\angle BCD=\angle CDA+\angle ABC$, then there are infinitely many shortest paths $XYZTX$, each with length $2AC\sin k$, where $2k=\angle BAC+\angle CAD+\angle DAB$.

1998 India National Olympiad, 4

Tags: trigonometry , rectangle , cyclic quadrilateral , geometric inequality , geometry

Suppose $ABCD$ is a cyclic quadrilateral inscribed in a circle of radius one unit. If $AB \cdot BC \cdot CD \cdot DA \geq 4$, prove that $ABCD$ is a square.

2011 Morocco National Olympiad, 1

Tags: trigonometry , inequalities

Let $x$, $y$, and $z$ be three real positive numbers such that $x^{2}+y^{2}+z^{2}+2xyz=1$. Prove that $2(x+y+z)\leq 3$.

2007 Today's Calculation Of Integral, 183

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

Let $n\geq 2$ be integer. On a plane there are $n+2$ points $O,\ P_{0},\ P_{1},\ \cdots P_{n}$ which satisfy the following conditions as follows. [1] $\angle{P_{k-1}OP_{k}}=\frac{\pi}{n}\ (1\leq k\leq n),\ \angle{OP_{k-1}P_{k}}=\angle{OP_{0}P_{1}}\ (2\leq k\leq n).$ [2] $\overline{OP_{0}}=1,\ \overline{OP_{1}}=1+\frac{1}{n}.$ Find $\lim_{n\to\infty}\sum_{k=1}^{n}\overline{P_{k-1}P_{k}}.$

2010 Harvard-MIT Mathematics Tournament, 9

Tags: calculus , trigonometry

Let $x(t)$ be a solution to the differential equation \[\left(x+x^\prime\right)^2+x\cdot x^{\prime\prime}=\cos t\] with $x(0)=x^\prime(0)=\sqrt{\frac{2}{5}}$. Compute $x\left(\dfrac{\pi}{4}\right)$.

1991 Arnold's Trivium, 17

Tags: geometry , 3d geometry , sphere , integration , calculus , trigonometry , function

Find the distance of the centre of gravity of a uniform $100$-dimensional solid hemisphere of radius $1$ from the centre of the sphere with $10\%$ relative error.

2010 Contests, 3

Tags: geometric transformation , trigonometry , ratio , homothety , projective geometry , trig identities , geometry

Let $A_1A_2A_3A_4$ be a quadrilateral with no pair of parallel sides. For each $i=1, 2, 3, 4$, define $\omega_1$ to be the circle touching the quadrilateral externally, and which is tangent to the lines $A_{i-1}A_i, A_iA_{i+1}$ and $A_{i+1}A_{i+2}$ (indices are considered modulo $4$ so $A_0=A_4, A_5=A_1$ and $A_6=A_2$). Let $T_i$ be the point of tangency of $\omega_i$ with the side $A_iA_{i+1}$. Prove that the lines $A_1A_2, A_3A_4$ and $T_2T_4$ are concurrent if and only if the lines $A_2A_3, A_4A_1$ and $T_1T_3$ are concurrent. [i]Pavel Kozhevnikov, Russia[/i]

2013 Stanford Mathematics Tournament, 10

Tags: calculus , limit , integration , trigonometry

Evaluate $\lim_{n\to\infty}\left[\left(\prod_{k=1}^{n}\frac{2k}{2k-1}\right)\int_{-1}^{\infty}\frac{(\cos x)^{2n}}{2^x} \, dx\right]$.

2007 Today's Calculation Of Integral, 251

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

Evaluate $ \int_0^{n\pi} e^x\sin ^ 4 x\ dx\ (n\equal{}1,\ 2,\ \cdots).$

2003 USA Team Selection Test, 5

Tags: trigonometry , function , inequalities

Let $A, B, C$ be real numbers in the interval $\left(0,\frac{\pi}{2}\right)$. Let \begin{align*} X &= \frac{\sin A\sin (A-B)\sin (A-C)}{\sin (B+C)} \\ Y &= \frac{\sin B\sin(B-C)\sin (B-A)}{\sin (C+A)} \\ Z &= \frac{\sin C\sin (C-A)\sin (C-B)}{\sin (A+B)} . \end{align*} Prove that $X+Y+Z \geq 0$.

2013 Albania Team Selection Test, 4

Tags: circumcircle , trigonometry , geometry

It is given a triangle $ABC$ whose circumcenter is $O$ and orthocenter $H$. If $AO=AH$ find the angle $\hat{BAC}$ of that triangle.