Olympiad problems

2002 India IMO Training Camp, 17

Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.

1997 Baltic Way, 14

Tags: inequalities , geometry , trigonometry

In the triangle $ABC$, $AC^2$ is the arithmetic mean of $BC^2$ and $AB^2$. Show that $\cot^2B\ge \cot A\cdot\cot C$.

III Soros Olympiad 1996 - 97 (Russia), 11.1

Tags: algebra , fractional part , trigonometry

Find the smallest positive root of the equation $$\{tg x\}=\sin x. $$ ($\{a\}$ is the fractional part of $a$, $\{a\}$ is equal to the difference between $ a$ and the largest integer not exceeding $a$.)

1992 Balkan MO, 3

Tags: inequalities , circumcircle , trigonometry , geometry

Let $D$, $E$, $F$ be points on the sides $BC$, $CA$, $AB$ respectively of a triangle $ABC$ (distinct from the vertices). If the quadrilateral $AFDE$ is cyclic, prove that \[ \frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 . \] [i]Greece[/i]

1971 Canada National Olympiad, 8

Tags: geometry , trigonometry

A regular pentagon is inscribed in a circle of radius $r$. $P$ is any point inside the pentagon. Perpendiculars are dropped from $P$ to the sides, or the sides produced, of the pentagon. a) Prove that the sum of the lengths of these perpendiculars is constant. b) Express this constant in terms of the radius $r$.

1998 Iran MO (3rd Round), 2

Tags: isogonal conjugate , geometry , reflection , trigonometry , circumcircle

Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that \[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC. \] Prove that \[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1. \]

1982 All Soviet Union Mathematical Olympiad, 335

Tags: algebra , compare , angle , trigonometry

Three numbers $a,b,c$ belong to $[0,\pi /2]$ interval with $$\cos a = a, \sin(\cos b) = b, \cos(\sin c ) = c$$ Sort those numbers in increasing order.

1965 Putnam, B1

Tags: limit , integration , calculus , trigonometry

Evaluate $ \lim_{n\to\infty} \int_0^1 \int_0^1 \cdots \int_0^1 \cos ^ 2 \left\{\frac{\pi}{2n}(x_1\plus{}x_2\plus{}\cdots \plus{}x_n)\right\} dx_1dx_2\cdots dx_n.$

1994 IberoAmerican, 2

Tags: perimeter , rectangle , trigonometry , geometry , ratio , inradius , incenter

Let $ ABCD$ a cuadrilateral inscribed in a circumference. Suppose that there is a semicircle with its center on $ AB$, that is tangent to the other three sides of the cuadrilateral. (i) Show that $ AB \equal{} AD \plus{} BC$. (ii) Calculate, in term of $ x \equal{} AB$ and $ y \equal{} CD$, the maximal area that can be reached for such quadrilateral.

2009 Tuymaada Olympiad, 3

Tags: geometry , trigonometry

In a cyclic quadrilateral $ ABCD$ the sides $ AB$ and $ AD$ are equal, $ CD>AB\plus{}BC$. Prove that $ \angle ABC>120^\circ$.

2012 Mexico National Olympiad, 1

Tags: geometry , trigonometry

Let $\mathcal{C}_1$ be a circumference with center $O$, $P$ a point on it and $\ell$ the line tangent to $\mathcal{C}_1$ at $P$. Consider a point $Q$ on $\ell$ different from $P$, and let $\mathcal{C}_2$ be the circumference passing through $O$, $P$ and $Q$. Segment $OQ$ cuts $\mathcal{C}_1$ at $S$ and line $PS$ cuts $\mathcal{C}_2$ at a point $R$ diffferent from $P$. If $r_1$ and $r_2$ are the radii of $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively, Prove \[\frac{PS}{SR} = \frac{r_1}{r_2}.\]

2011 Middle European Mathematical Olympiad, 6

Tags: trig identities , law of cosines , pythagorean theorem , geometry , trigonometry , circumcircle

Let $ABC$ be an acute triangle. Denote by $B_0$ and $C_0$ the feet of the altitudes from vertices $B$ and $C$, respectively. Let $X$ be a point inside the triangle $ABC$ such that the line $BX$ is tangent to the circumcircle of the triangle $AXC_0$ and the line $CX$ is tangent to the circumcircle of the triangle $AXB_0$. Show that the line $AX$ is perpendicular to $BC$.

2011 Today's Calculation Of Integral, 745

Tags: calculus , integration , calculus computations , trigonometry

When real numbers $a,\ b$ move satisfying $\int_0^{\pi} (a\cos x+b\sin x)^2dx=1$, find the maximum value of $\int_0^{\pi} (e^x-a\cos x-b\sin x)^2dx.$

2011 Spain Mathematical Olympiad, 1

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

In triangle $ABC$, $\angle B=2\angle C$ and $\angle A>90^\circ$. Let $D$ be the point on the line $AB$ such that $CD$ is perpendicular to $AC$, and let $M$ be the midpoint of $BC$. Prove that $\angle AMB=\angle DMC$.

1988 IMO Shortlist, 27

Tags: function , area of a triangle , geometric inequality , trigonometry , geometry

Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.

2005 Today's Calculation Of Integral, 45

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

Find the function $f(x)$ which satisfies the following integral equation. \[f(x)=\int_0^x t(\sin t-\cos t)dt+\int_0^{\frac{\pi}{2}} e^t f(t)dt\]

2001 AIME Problems, 4

Tags: trig identities , law of sines , trigonometry , geometry

In triangle $ABC$, angles $A$ and $B$ measure 60 degrees and 45 degrees, respectively. The bisector of angle $A$ intersects $\overline{BC}$ at $T$, and $AT=24.$ The area of triangle $ABC$ can be written in the form $a+b\sqrt{c},$ where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Find $a+b+c.$

2009 All-Russian Olympiad, 7

Tags: function , parallelogram , geometry , circumcircle , trigonometry

Let be given a parallelogram $ ABCD$ and two points $ A_1$, $ C_1$ on its sides $ AB$, $ BC$, respectively. Lines $ AC_1$ and $ CA_1$ meet at $ P$. Assume that the circumcircles of triangles $ AA_1P$ and $ CC_1P$ intersect at the second point $ Q$ inside triangle $ ACD$. Prove that $ \angle PDA \equal{} \angle QBA$.

2010 Today's Calculation Of Integral, 618

Tags: calculus , calculus computations , integration , trigonometry

Find the minimu value of $\frac{1}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \{x\cos t+(1-x)\sin t\}^2dt.$ [i]2010 Ibaraki University entrance exam/Science[/i]

1986 IMO Longlists, 59

Tags: geometry , circumcircle , trigonometry , ratio

Let $ABCD$ be a convex quadrilateral whose vertices do not lie on a circle. Let $A'B'C'D'$ be a quadrangle such that $A',B', C',D'$ are the centers of the circumcircles of triangles $BCD,ACD,ABD$, and $ABC$. We write $T (ABCD) = A'B'C'D'$. Let us define $A''B''C''D'' = T (A'B'C'D') = T (T (ABCD)).$ [b](a)[/b] Prove that $ABCD$ and $A''B''C''D''$ are similar. [b](b) [/b]The ratio of similitude depends on the size of the angles of $ABCD$. Determine this ratio.

1999 Romania Team Selection Test, 12

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

Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.

2008 Junior Balkan MO, 2

Tags: perpendicular bisector , angle bisector , power of a point , geometry , trigonometry

The vertices $ A$ and $ B$ of an equilateral triangle $ ABC$ lie on a circle $k$ of radius $1$, and the vertex $ C$ is in the interior of the circle $ k$. A point $ D$, different from $ B$, lies on $ k$ so that $ AD\equal{}AB$. The line $ DC$ intersects $ k$ for the second time at point $ E$. Find the length of the line segment $ CE$.

2003 AMC 10, 17

Tags: perimeter , law of sines , trig identities , circumcircle , trigonometry , perpendicular bisector , geometry

The number of inches in the perimeter of an equilateral triangle equals the number of square inches in the area of its circumscribed circle. What is the radius, in inches, of the circle? $ \textbf{(A)}\ \frac{3\sqrt2}{\pi} \qquad \textbf{(B)}\ \frac{3\sqrt3}{\pi} \qquad \textbf{(C)}\ \sqrt3 \qquad \textbf{(D)}\ \frac{6}{\pi} \qquad \textbf{(E)}\ \sqrt3\pi$

2006 Stanford Mathematics Tournament, 13

Tags: graphing lines , conic , hyperbola , slope , analytic geometry , trigonometry

A ray is drawn from the origin tangent to the graph of the upper part of the hyperbola $y^2=x^2-x+1$ in the first quadrant. This ray makes an angle of $\theta$ with the positive $x$-axis. Compute $\cos\theta$.

2004 Austrian-Polish Competition, 2

Tags: geometry , trigonometry , inequalities , angle bisector

In a triangle $ABC$ let $D$ be the intersection of the angle bisector of $\gamma$, angle at $C$, with the side $AB.$ And let $F$ be the area of the triangle $ABC.$ Prove the following inequality: \[2 \cdot \ F \cdot \left( \frac{1}{AD} -\frac{1}{BD} \right) \leq AB.\]