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

2010 Today's Calculation Of Integral, 601
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate $\int_0^{\frac{\pi}{4}} (\tan x)^{\frac{3}{2}}dx$. created by kunny

2004 Moldova Team Selection Test, 7
Tags: inequalities , circumcircle , inradius , function , trigonometry , geometric transformation , geometry
Let $ABC$ be a triangle, let $O$ be its circumcenter, and let $H$ be its orthocenter. Let $P$ be a point on the segment $OH$. Prove that $6r\leq PA+PB+PC\leq 3R$, where $r$ is the inradius and $R$ the circumradius of triangle $ABC$. [b]Moderator edit:[/b] This is true only if the point $P$ lies inside the triangle $ABC$. (Of course, this is always fulfilled if triangle $ABC$ is acute-angled, since in this case the segment $OH$ completely lies inside the triangle $ABC$; but if triangle $ABC$ is obtuse-angled, then the condition about $P$ lying inside the triangle $ABC$ is really necessary.)

1983 IMO Longlists, 29
Tags: trigonometry , geometry
Let $O$ be a point outside a given circle. Two lines $OAB, OCD$ through $O$ meet the circle at $A,B,C,D$, where $A,C$ are the midpoints of $OB,OD$, respectively. Additionally, the acute angle $\theta$ between the lines is equal to the acute angle at which each line cuts the circle. Find $\cos \theta$ and show that the tangents at $A,D$ to the circle meet on the line $BC.$

2011 Serbia JBMO TST, 3
Tags: geometry , trigonometry
Let $\triangle ABC$ be a right-angled triangle and $BC > AC$. $M$ is a point on $BC$ such that $BM = AC$ and $N$ is a point on $AC$ such that $AN = CM$. Find the angle between $BN$ and $AM$.

1989 Vietnam National Olympiad, 1
Tags: trigonometry , inequalities
Let $ n$ and $ N$ be natural number. Prove that for any $ \alpha$ , $ 0\le\alpha\le N$, and any real $ x$, it holds that \[{ |\sum_ {k=0}^n}\frac{\sin((\alpha+k)x)}{N+k}|\le\min\{(n+1)|x|, \frac{1}{N|\sin\frac{x}{2}|}\}\]

2013 Tuymaada Olympiad, 8
Tags: inequalities , perimeter , analytic geometry , trigonometry , geometry
The point $A_1$ on the perimeter of a convex quadrilateral $ABCD$ is such that the line $AA_1$ divides the quadrilateral into two parts of equal area. The points $B_1$, $C_1$, $D_1$ are defined similarly. Prove that the area of the quadrilateral $A_1B_1C_1D_1$ is greater than a quarter of the area of $ABCD$. [i]L. Emelyanov [/i]

1996 ITAMO, 1
Tags: trigonometry , geometry
Among all the triangles which have a fixed side $l$ and a fixed area $S$, determine for which triangles the product of the altitudes is maximum.

1990 IMO Longlists, 70
Tags: trigonometry , geometry theorems , geometry
$BC$ is a segment, $M$ is point on $BC$, $A$ is a point such that $A, B, C$ are non-collinear. (i) Prove that if $M$ is the midpoint of $BC$, then $AB^2 + AC^2 = 2(AM^2 + BM^2).$ (ii) If there exists another point (except $M$) on segment $BC$ satisfying (i), find the region of point $A$ might occupy.

2011 Graduate School Of Mathematical Sciences, The Master Cource, The University Of Tokyo, 3
Tags: calculus , integration , trigonometry , real analysis
Let $a$ be a positive real number. Evaluate $I=\int_0^{+\infty} \frac{\sin x\cos x}{x(x^2+a^2)}dx.$

1992 Dutch Mathematical Olympiad, 3
Tags: geometry , trigonometry , geometric transformation , reflection
Consider the configuration of six squares as shown on the picture. Prove that the sum of the area of the three outer squares ($ I,II$ and $ III$) equals three times the sum of the areas of the three inner squares ($ IV,V$ and $ VI$).

2020 Flanders Math Olympiad, 1
Tags: trigonometry
Let $x$ be an angle between $0^o$ and $90^o$ so that $$\frac{\sin^4 x}{9}+\frac{\cos^4 x}{16 }=\frac{1}{25} .$$ Then what is $\tan x$?

2011 Today's Calculation Of Integral, 728
Tags: calculus , integration , trigonometry , calculus computations , logarithm
Evaluate \[\int_{\frac {\pi}{12}}^{\frac{\pi}{6}} \frac{\sin x-\cos x-x(\sin x+\cos x)+1}{x^2-x(\sin x+\cos x)+\sin x\cos x}\ dx.\]

2014 Contests, 3
Tags: trigonometry , geometry , angle bisector
Let $ABC$ be an acute-angled triangle in which $\angle ABC$ is the largest angle. Let $O$ be its circumcentre. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The internal angle bisectors of $\angle AXB$ and $\angle BYC$ meet $AB$ and $BC$ at $D$ and $E$ respectively. Prove that $BO$ is perpendicular to $AC$ if $DE$ is parallel to $AC$.

2006 Estonia National Olympiad, 4
Tags: trigonometry , geometry
Triangle $ ABC$ is isosceles with $ AC \equal{} BC$ and $ \angle{C} \equal{} 120^o$. Points $ D$ and $ E$ are chosen on segment $ AB$ so that $ |AD| \equal{} |DE| \equal{} |EB|$. Find the sizes of the angles of triangle $ CDE$.

1986 China National Olympiad, 2
Tags: trigonometry , trig identities , law of sines , geometry
In $\triangle ABC$, the length of altitude $AD$ is $12$, and the bisector $AE$ of $\angle A$ is $13$. Denote by $m$ the length of median $AF$. Find the range of $m$ when $\angle A$ is acute, orthogonal and obtuse respectively.

2006 Moldova National Olympiad, 10.1
Tags: trigonometry , function , inequalities , geometry
Let $a,b$ be the smaller sides of a right triangle. Let $c$ be the hypothenuse and $h$ be the altitude from the right angle. Fint the maximal value of $\frac{c+h}{a+b}$.

2003 Romania Team Selection Test, 11
Tags: trigonometry , pigeonhole principle , area of a triangle , geometry
In a square of side 6 the points $A,B,C,D$ are given such that the distance between any two of the four points is at least 5. Prove that $A,B,C,D$ form a convex quadrilateral and its area is greater than 21. [i]Laurentiu Panaitopol[/i]

2007 Today's Calculation Of Integral, 217
Tags: calculus , integration , function , geometry , trigonometry , calculus computations , logarithm
Evaluate $ \int_{0}^{1}e^{\sqrt{e^{x}}}\ dx\plus{}2\int_{e}^{e^{\sqrt{e}}}\ln (\ln x)\ dx$.

1942 Putnam, B5
Tags: trigonometry , integration
Sketch the curve $$y= \frac{x}{1+x^6 (\sin x)^{2}},$$ and show that $$ \int_{0}^{\infty} \frac{x}{1+x^6 (\sin x)^{2}}\; dx$$ exists.

2014 National Olympiad First Round, 17
Tags: geometry , ratio , trigonometry
Let $E$ be the midpoint of side $[AB]$ of square $ABCD$. Let the circle through $B$ with center $A$ and segment $[EC]$ meet at $F$. What is $|EF|/|FC|$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ \dfrac{3}{2} \qquad\textbf{(C)}\ \sqrt{5}-1 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \sqrt{3} $

2013 AIME Problems, 8
Tags: algebra , function , domain , modular arithmetic , trigonometry , logarithm
The domain of the function $f(x) = \text{arcsin}(\log_{m}(nx))$ is a closed interval of length $\frac{1}{2013}$, where $m$ and $n$ are positive integers and $m > 1$. Find the remainder when the smallest possible sum $m+n$ is divided by $1000$.

2004 France Team Selection Test, 2
Tags: trigonometry , angle bisector , geometry
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$. [i]Alternative formulation.[/i] Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

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 $|AB|=|CE|$.

1986 IMO Longlists, 11
Tags: geometry , 3d geometry , tetrahedron , trigonometry , congruent triangles
Prove that the sum of the face angles at each vertex of a tetrahedron is a straight angle if and only if the faces are congruent triangles.

2009 Today's Calculation Of Integral, 405
Tags: calculus , integration , trigonometry , calculus computations
Calculate $ \displaystyle \left|\frac {\int_0^{\frac {\pi}{2}} (x\cos x + 1)e^{\sin x}\ dx}{\int_0^{\frac {\pi}{2}} (x\sin x - 1)e^{\cos x}\ dx}\right|$.