This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 3349

2001 Junior Balkan MO, 4
Tags: geometry , perimeter , trigonometry , combinatorics
Let $N$ be a convex polygon with 1415 vertices and perimeter 2001. Prove that we can find 3 vertices of $N$ which form a triangle of area smaller than 1.

2000 Pan African, 2
Tags: geometry , trigonometry
Let $\gamma$ be circle and let $P$ be a point outside $\gamma$. Let $PA$ and $PB$ be the tangents from $P$ to $\gamma$ (where $A, B \in \gamma$). A line passing through $P$ intersects $\gamma$ at points $Q$ and $R$. Let $S$ be a point on $\gamma$ such that $BS \parallel QR$. Prove that $SA$ bisects $QR$.

2010 Estonia Team Selection Test, 5
Tags: polynomial , algebra , trigonometry
Let $P(x, y)$ be a non-constant homogeneous polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for every real number $t$. Prove that there exists a positive integer $k$ such that $P(x, y) = (x^2 + y^2)^k$.

2005 QEDMO 1st, 5 (G1)
Tags: circumcircle , trigonometry , geometry
Let $ABC$ be a triangle, and let $C^{\prime}$ and $A^{\prime}$ be the feet of its altitudes issuing from the vertices $C$ and $A$, respectively. Denote by $P$ the midpoint of the segment $C^{\prime}A^{\prime}$. The circumcircles of triangles $AC^{\prime}P$ and $CA^{\prime}P$ have a common point apart from $P$; denote this common point by $Q$. Prove that: [b](a)[/b] The point $Q$ lies on the circumcircle of the triangle $ABC$. [b](b)[/b] The line $PQ$ passes through the point $B$. [b](c)[/b] We have $\frac{AQ}{CQ}=\frac{AB}{CB}$. Darij

2011 China Team Selection Test, 1
Tags: circumcircle , trigonometry , trapezoid , geometric transformation , reflection , parallelogram , geometry
Let $H$ be the orthocenter of an acute trangle $ABC$ with circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) of $\Gamma$, and let $M$ be a point on the arc $CA$ (not containing $B$) of $\Gamma$ such that $H$ lies on the segment $PM$. Let $K$ be another point on $\Gamma$ such that $KM$ is parallel to the Simson line of $P$ with respect to triangle $ABC$. Let $Q$ be another point on $\Gamma$ such that $PQ \parallel BC$. Segments $BC$ and $KQ$ intersect at a point $J$. Prove that $\triangle KJM$ is an isosceles triangle.

2004 India IMO Training Camp, 1
Tags: geometry , circumcircle , trigonometry
Let $ABC$ be an acute-angled triangle and $\Gamma$ be a circle with $AB$ as diameter intersecting $BC$ and $CA$ at $F ( \not= B)$ and $E (\not= A)$ respectively. Tangents are drawn at $E$ and $F$ to $\Gamma$ intersect at $P$. Show that the ratio of the circumcentre of triangle $ABC$ to that if $EFP$ is a rational number.

1991 India National Olympiad, 2
Tags: geometry , trigonometry
Given an acute-angled triangle $ABC$, let points $A' , B' , C'$ be located as follows: $A'$ is the point where altitude from $A$ on $BC$ meets the outwards-facing semicircle on $BC$ as diameter. Points $B', C'$ are located similarly. Prove that $A[BCA']^2 + A[CAB']^2 + A[ABC']^2 = A[ABC]^2$ where $A[ABC]$ is the area of triangle $ABC$.

1999 National Olympiad First Round, 9
Tags: geometry , trigonometry , rectangle
Find the area of inscribed convex octagon, if the length of four sides is $2$, and length of other four sides is $ 6\sqrt {2}$. $\textbf{(A)}\ 120 \qquad\textbf{(B)}\ 24 \plus{} 68\sqrt {2} \qquad\textbf{(C)}\ 88\sqrt {2} \qquad\textbf{(D)}\ 124 \qquad\textbf{(E)}\ 72\sqrt {3}$

2003 China Team Selection Test, 1
Tags: trigonometry , calculus , geometry
Let $ ABCD$ be a quadrilateral which has an incircle centered at $ O$. Prove that \[ OA\cdot OC\plus{}OB\cdot OD\equal{}\sqrt{AB\cdot BC\cdot CD\cdot DA}\]

1992 IMO Longlists, 19
Tags: trigonometry , algebra , sequence , counting , recurrence relation
Denote by $a_n$ the greatest number that is not divisible by $3$ and that divides $n$. Consider the sequence $s_0 = 0, s_n = a_1 +a_2+\cdots+a_n, n \in \mathbb N$. Denote by $A(n)$ the number of all sums $s_k \ (0 \leq k \leq 3^n, k \in \mathbb N_0)$ that are divisible by $3$. Prove the formula \[A(n) = 3^{n-1} + 2 \cdot 3^{(n/2)-1} \cos \left(\frac{n\pi}{6}\right), \qquad n\in \mathbb N_0.\]

1992 India National Olympiad, 1
Tags: trigonometry , algebra , trig identities
In a triangle $ABC$, $\angle A = 2 \cdot \angle B$. Prove that $a^2 = b (b+c)$.

2015 CCA Math Bonanza, T10
Tags: trigonometry
If $\cos 2^{\circ} - \sin 4^{\circ} -\cos 6^{\circ} + \sin 8^{\circ} \ldots + \sin 88^{\circ}=\sec \theta - \tan \theta$, compute $\theta$ in degrees. [i]2015 CCA Math Bonanza Team Round #10[/i]

1990 Irish Math Olympiad, 3
Tags: trigonometry
Let $t$ be a real number, and let $$a_n=2\cos \left(\frac{t}{2^n}\right)-1,\quad n=1,2,3,\dots$$ Let $b_n$ be the product $a_1a_2a_3\cdots a_n$. Find a formula for $b_n$ that does not involve a product of $n$ terms, and deduce that $$\lim_{n\to \infty}b_n=\frac{2\cos t+1}{3}$$

2009 Today's Calculation Of Integral, 404
Tags: calculus , integration , trigonometry , calculus computations
Evaluate $ \int_{ \minus{} \pi}^{\pi} \frac {\sin nx}{(1 \plus{} 2009^x)\sin x}\ dx\ (n\equal{}0,\ 1,\ 2,\ \cdots)$.

2010 Today's Calculation Of Integral, 622
Tags: calculus , integration , trigonometry , geometry , analytic geometry , function , calculus computations
For $0<k<2$, consider two curves $C_1: y=\sin 2x\ (0\leq x\leq \pi),\ C_2: y=k\cos x\ (0\leqq x\leqq \pi).$ Denote by $S(k)$ the sum of the areas of four parts enclosed by $C_1,\ C_2$ and two lines $x=0,\ x=\pi$. Find the minimum value of $S(k).$ [i]2010 Nagoya Institute of Technology entrance exam[/i]

2008 Baltic Way, 18
Tags: trigonometry , circumcircle , radical axis , geometry
Let $ AB$ be a diameter of a circle $ S$, and let $ L$ be the tangent at $ A$. Furthermore, let $ c$ be a fixed, positive real, and consider all pairs of points $ X$ and $ Y$ lying on $ L$, on opposite sides of $ A$, such that $ |AX|\cdot |AY| \equal{} c$. The lines $ BX$ and $ BY$ intersect $ S$ at points $ P$ and $ Q$, respectively. Show that all the lines $ PQ$ pass through a common point.

2001 China Western Mathematical Olympiad, 2
Tags: rectangle , trigonometry , geometry
$ ABCD$ is a rectangle of area 2. $ P$ is a point on side $ CD$ and $ Q$ is the point where the incircle of $ \triangle PAB$ touches the side $ AB$. The product $ PA \cdot PB$ varies as $ ABCD$ and $ P$ vary. When $ PA \cdot PB$ attains its minimum value, a) Prove that $ AB \geq 2BC$, b) Find the value of $ AQ \cdot BQ$.

1989 Federal Competition For Advanced Students, P2, 4
Tags: trigonometry , geometry
We are given a circle $ k$ and nonparallel tangents $ t_1,t_2$ at points $ P_1,P_2$ on $ k$, respectively. Lines $ t_1$ and $ t_2$ meet at $ A_0$. For a point $ A_3$ on the smaller arc $ P_1 P_2,$ the tangent $ t_3$ to $ k$ at $ P_3$ meets $ t_1$ at $ A_1$ and $ t_2$ at $ A_2$. How must $ P_3$ be chosen so that the triangle $ A_0 A_1 A_2$ has maximum area?

1985 IMO Longlists, 38
Tags: geometry , circumcircle , trigonometry , triangle
The tangents at $B$ and $C$ to the circumcircle of the acute-angled triangle $ABC$ meet at $X$. Let $M$ be the midpoint of $BC$. Prove that [i](a)[/i] $\angle BAM = \angle CAX$, and [i](b)[/i] $\frac{AM}{AX} = \cos\angle BAC.$

1961 IMO Shortlist, 2
Tags: inequalities , trigonometry , geometry , area of a triangle , heron's formula
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?

2014 AIME Problems, 7
Tags: sum , trigonometry , logarithm
Let $f(x) = (x^2+3x+2)^{\cos(\pi x)}$. Find the sum of all positive integers $n$ for which \[\left| \sum_{k=1}^n \log_{10} f(k) \right| = 1.\]

1989 Poland - Second Round, 1
Tags: algebra , trigonometry
Solve the equation $$ tg 7x - \sin 6x=\cos 4x - ctg 7x.$$

2008 Sharygin Geometry Olympiad, 3
Tags: inequalities , inradius , trigonometry , geometry
(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$.

2002 Moldova National Olympiad, 1
Tags: trigonometry
Solve in $ \mathbb R$ the equation $ \sqrt{1\minus{}x}\equal{}2x^2\minus{}1\plus{}2x\sqrt{1\minus{}x^2}$.

1974 IMO Longlists, 37
Tags: trigonometry
Let $a, b$, and $c$ denote the three sides of a billiard table in the shape of an equilateral triangle. A ball is placed at the midpoint of side $a$ and then propelled toward side $b$ with direction defined by the angle $\theta$. For what values of $\theta$ will the ball strike the sides $b, c, a$ in that order?