Olympiad problems

2010 Contests, 3

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

2005 Danube Mathematical Olympiad, 1

Tags: trigonometry , number theory

Prove that the equation $4x^3-3x+1=2y^2$ has at least $31$ solutions in positive integers $x$ and $y$ with $x\leq 2005$.

2009 Stanford Mathematics Tournament, 5

Tags: geometry , trigonometry , college , trig identities , law of cosines

In the 2009 Stanford Olympics, Willy and Sammy are two bikers. The circular race track has two lanes, the inner lane with radius 11, and the outer with radius 12. Willy will start on the inner lane, and Sammy on the outer. They will race for one complete lap, measured by the inner track. What is the square of the distance between Willy and Sammy's starting positions so that they will both race the same distance? Assume that they are of point size and ride perfectly along their respective lanes

2007 Romania Team Selection Test, 3

Tags: geometry , trigonometry

Let $ABCDE$ be a convex pentagon, such that $AB=BC$, $CD=DE$, $\angle B+\angle D=180^{\circ}$, and it's area is $\sqrt2$. a) If $\angle B=135^{\circ}$, find the length of $[BD]$. b) Find the minimum of the length of $[BD]$.

2005 Today's Calculation Of Integral, 47

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

Find the condition of $a,b$ for which the function $f(x)\ (0\leq x\leq 2\pi)$ satisfying the following equality can be determined uniquely,then determine $f(x)$, assuming that $f(x) $ is a continuous function at $0\leq x\leq 2\pi$. \[f(x)=\frac{a}{2\pi}\int_0^{2\pi} \sin (x+y)f(y)dy+\frac{b}{2\pi}\int_0^{2\pi} \cos (x-y)f(y)dy+\sin x+\cos x\]

2001 Austrian-Polish Competition, 3

Tags: trigonometry , inequalities , triangle inequality

Let $a,b,c$ be sides of a triangle. Prove that \[ 2 < \frac{a+b}{c} + \frac{b+c}{a} + \frac{c+a}{b} - \frac{a^3+b^3+c^3}{abc}\leq 3 \]

2013 Harvard-MIT Mathematics Tournament, 14

Tags: ratio , trigonometry , hmmt , angle bisector , geometry

Consider triangle $ABC$ with $\angle A=2\angle B$. The angle bisectors from $A$ and $C$ intersect at $D$, and the angle bisector from $C$ intersects $\overline{AB}$ at $E$. If $\dfrac{DE}{DC}=\dfrac13$, compute $\dfrac{AB}{AC}$.

2004 India IMO Training Camp, 2

Tags: trigonometry , number theory

Show that the only solutions of te equation \[ p^{k} + 1 = q^{m} \], in positive integers $k,q,m > 1$ and prime $p$ are (i) $(p,k,q,m) = (2,3,3,2)$ (ii) $k=1 , q=2,$and $p$ is a prime of the form $2^{m} -1$, $m > 1 \in \mathbb{N}$

2014 PUMaC Team, 14

Tags: function , trigonometry

Define the function $f_k(x)$ (where $k$ is a positive integer) as follows: \[f_k(x)=(\cos kx)(\cos x)^k+(\sin kx)(\sin x)^k-(\cos 2x)^k.\] Find the sum of all distinct value(s) of $k$ such that $f_k(x)$ is a constant function.

2007 IberoAmerican, 2

Tags: circumcircle , incenter , trig identities , law of sines , geometry , inradius , trigonometry

Let $ ABC$ be a triangle with incenter $ I$ and let $ \Gamma$ be a circle centered at $ I$, whose radius is greater than the inradius and does not pass through any vertex. Let $ X_{1}$ be the intersection point of $ \Gamma$ and line $ AB$, closer to $ B$; $ X_{2}$, $ X_{3}$ the points of intersection of $ \Gamma$ and line $ BC$, with $ X_{2}$ closer to $ B$; and let $ X_{4}$ be the point of intersection of $ \Gamma$ with line $ CA$ closer to $ C$. Let $ K$ be the intersection point of lines $ X_{1}X_{2}$ and $ X_{3}X_{4}$. Prove that $ AK$ bisects segment $ X_{2}X_{3}$.

2012 Math Prize For Girls Problems, 12

Tags: trigonometry

What is the sum of all positive integer values of $n$ that satisfy the equation \[ \cos \Bigl( \frac{\pi}{n} \Bigr) \cos \Bigl( \frac{2\pi}{n} \Bigr) \cos \Bigl( \frac{4\pi}{n} \Bigr) \cos \Bigl( \frac{8\pi}{n} \Bigr) \cos \Bigl( \frac{16\pi}{n} \Bigr) = \frac{1}{32} \, ? \]

2007 USA Team Selection Test, 3

Tags: algebra , polynomial , algorithm , greatest common divisor , trigonometry , induction , number theory

Let $ \theta$ be an angle in the interval $ (0,\pi/2)$. Given that $ \cos \theta$ is irrational, and that $ \cos k \theta$ and $ \cos[(k \plus{} 1)\theta ]$ are both rational for some positive integer $ k$, show that $ \theta \equal{} \pi/6$.

2012 IMO Shortlist, G4

Tags: geometry , circumcircle , trigonometry , triangle , reflection

Let $ABC$ be a triangle with $AB \neq AC$ and circumcenter $O$. The bisector of $\angle BAC$ intersects $BC$ at $D$. Let $E$ be the reflection of $D$ with respect to the midpoint of $BC$. The lines through $D$ and $E$ perpendicular to $BC$ intersect the lines $AO$ and $AD$ at $X$ and $Y$ respectively. Prove that the quadrilateral $BXCY$ is cyclic.

2000 Spain Mathematical Olympiad, 2

Tags: trigonometry , combinatorics

Four points are given inside or on the boundary of a unit square. Prove that at least two of these points are on a mutual distance at most $1.$

1961 IMO Shortlist, 2

Tags: inequalities , trigonometry , geometry , heron's formula , area of a triangle

Let $ a$, $ b$, $ c$ be the sides of a triangle, and $ S$ its area. Prove: \[ a^{2} \plus{} b^{2} \plus{} c^{2}\geq 4S \sqrt {3} \] In what case does equality hold?

2000 Baltic Way, 1

Tags: parallelogram , similar triangles , trigonometry , geometry

Let $ K$ be a point inside the triangle $ ABC$. Let $ M$ and $ N$ be points such that $ M$ and $ K$ are on opposite sides of the line $ AB$, and $ N$ and $ K$ are on opposite sides of the line $ BC$. Assume that $ \angle MAB \equal{} \angle MBA \equal{} \angle NBC \equal{} \angle NCB \equal{} \angle KAC \equal{} \angle KCA$. Show that $ MBNK$ is a parallelogram.

2010 Today's Calculation Of Integral, 537

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

Evaluate $ \int_0^{\frac{\pi}{6}} \frac{\sqrt{1\plus{}\sin x}}{\cos x}\ dx$.

III Soros Olympiad 1996 - 97 (Russia), 9.8

Tags: algebra , trigonometry , geometric inequality

The two sides of the triangle are equal to $1$ and $x$, and $ x \ge 1$. The values $a$ and $b$ are the largest and smallest angles of this triangle, respectively. Find the greatest value of $\cos a$ and the smallest value of $\cos b$.

2000 Brazil Team Selection Test, Problem 1

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

Consider a triangle $ABC$ and $I$ its incenter. The line $(AI)$ meets the circumcircle of $ABC$ in $D$. Let $E$ and $F$ be the orthogonal projections of $I$ on $(BD)$ and $(CD)$ respectively. Assume that $IE+IF=\frac{1}{2}AD$. Calculate $\angle{BAC}$. [color=red][Moderator edited: Also discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=5088 .][/color]

1993 India National Olympiad, 4

Tags: symmetry , circumcircle , trigonometry , geometry

Let $ABC$ be a triangle in a plane $\pi$. Find the set of all points $P$ (distinct from $A,B,C$ ) in the plane $\pi$ such that the circumcircles of triangles $ABP$, $BCP$, $CAP$ have the same radii.

1971 IMO Longlists, 9

Tags: trigonometry , geometry , 3d geometry , prism

The base of an inclined prism is a triangle $ABC$. The perpendicular projection of $B_1$, one of the top vertices, is the midpoint of $BC$. The dihedral angle between the lateral faces through $BC$ and $AB$ is $\alpha$, and the lateral edges of the prism make an angle $\beta$ with the base. If $r_1, r_2, r_3$ are exradii of a perpendicular section of the prism, assuming that in $ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C,$ and $BC = a$, calculate $r_1r_2 + r_1r_3 + r_2r_3.$

1990 IMO Longlists, 66

Tags: algebra , limit , trigonometry , function

Find all the continuous bounded functions $f: \mathbb R \to \mathbb R$ such that \[(f(x))^2 -(f(y))^2 = f(x + y)f(x - y) \text{ for all } x, y \in \mathbb R.\]

2008 Sharygin Geometry Olympiad, 3

Tags: geometry , inradius , trigonometry , inequalities

(R.Pirkuliev) Prove the inequality \[ \frac1{\sqrt {2\sin A}} \plus{} \frac1{\sqrt {2\sin B}} \plus{} \frac1{\sqrt {2\sin C}}\leq\sqrt {\frac {p}{r}}, \] where $ p$ and $ r$ are the semiperimeter and the inradius of triangle $ ABC$.

2005 Today's Calculation Of Integral, 14

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

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] $\int e^x\cos x dx$

2006 Pre-Preparation Course Examination, 2

Tags: algebra , trigonometry , polynomial , quadratic

a) Show that you can divide an angle $\theta$ to three equal parts using compass and ruler if and only if the polynomial $4t^3-3t-\cos (\theta)$ is reducible over $\mathbb{Q}(\cos (\theta))$. b) Is it always possible to divide an angle into five equal parts?