Found problems: 3349
2005 China Northern MO, 5
Let $x, y, z$ be positive real numbers such that $x^2 + xy + y^2 = \frac{25}{4}$, $y^2 + yz + z^2 = 36$, and $z^2 + zx + x^2 = \frac{169}{4}$. Find the value of $xy + yz + zx$.
1986 All Soviet Union Mathematical Olympiad, 436
Prove that for every natural $n$ the following inequality is valid $$|\sin 1| + |\sin 2| + |\sin (3n-1)| + |\sin 3n| > \frac{8n}{5}$$
2010 Polish MO Finals, 2
Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$.
2008 Harvard-MIT Mathematics Tournament, 7
Let $ C_1$ and $ C_2$ be externally tangent circles with radius 2 and 3, respectively. Let $ C_3$ be a circle internally tangent to both $ C_1$ and $ C_2$ at points $ A$ and $ B$, respectively. The tangents to $ C_3$ at $ A$ and $ B$ meet at $ T$, and $ TA \equal{} 4$. Determine the radius of $ C_3$.
2008 AMC 12/AHSME, 15
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $ R$ be the region formed by the union of the square and all the triangles, and $ S$ be the smallest convex polygon that contains $ R$. What is the area of the region that is inside $ S$ but outside $ R$?
$ \textbf{(A)} \; \frac{1}{4} \qquad \textbf{(B)} \; \frac{\sqrt{2}}{4} \qquad \textbf{(C)} \; 1 \qquad \textbf{(D)} \; \sqrt{3} \qquad \textbf{(E)} \; 2 \sqrt{3}$
2004 Harvard-MIT Mathematics Tournament, 5
There exists a positive real number $x$ such that $ \cos (\arctan (x)) = x $. Find the value of $x^2$.
2000 Pan African, 1
Solve for $x \in R$:
\[ \sin^3{x}(1+\cot{x})+\cos^3{x}(1+\tan{x})=\cos{2x} \]
2010 Contests, 1
$D, \: E , \: F$ are points on the sides $AB, \: BC, \: CA,$ respectively, of a triangle $ABC$ such that $AD=AF, \: BD=BE,$ and $DE=DF.$ Let $I$ be the incenter of the triangle $ABC,$ and let $K$ be the point of intersection of the line $BI$ and the tangent line through $A$ to the circumcircle of the triangle $ABI.$ Show that $AK=EK$ if $AK=AD.$
PEN R Problems, 3
Prove no three lattice points in the plane form an equilateral triangle.
1963 Putnam, A5
i) Prove that if a function $f$ is continuous on the closed interval $[0, \pi]$ and
$$ \int_{0}^{\pi} f(t) \cos t \; dt= \int_{0}^{\pi} f(t) \sin t \; dt=0,$$
then there exist points $0 < \alpha < \beta < \pi$ such that $f(\alpha) =f(\beta) =0.$
ii) Let $R$ be a bounded, convex, and open region in the Euclidean plane. Prove with the help of i) that the centroid of $R$ bisects at least three different chords of the boundary of $ R.$
2014 Baltic Way, 13
Let $ABCD$ be a square inscribed in a circle $\omega$ and let $P$ be a point on the shorter arc $AB$ of $\omega$. Let $CP\cap BD = R$ and $DP \cap AC = S.$
Show that triangles $ARB$ and $DSR$ have equal areas.
1999 Belarusian National Olympiad, 1
Evaluate the product $\prod_{k=0}^{2^{1999}}(4\sin^2 \frac{k\pi}{2^{2000}}-3)$
1982 Vietnam National Olympiad, 1
Determine a quadric polynomial with intergral coefficients whose roots are $\cos 72^{\circ}$ and $\cos 144^{\circ}.$
1977 IMO Longlists, 58
Prove that for every triangle the following inequality holds:
\[\frac{ab+bc+ca}{4S} \geq \cot \frac{\pi}{6}.\]
where $a, b, c$ are lengths of the sides and $S$ is the area of the triangle.
2008 AMC 12/AHSME, 9
Points $ A$ and $ B$ are on a circle of radius $ 5$ and $ AB\equal{}6$. Point $ C$ is the midpoint of the minor arc $ AB$. What is the length of the line segment $ AC$?
$ \textbf{(A)}\ \sqrt{10} \qquad
\textbf{(B)}\ \frac{7}{2} \qquad
\textbf{(C)}\ \sqrt{14} \qquad
\textbf{(D)}\ \sqrt{15} \qquad
\textbf{(E)}\ 4$
2013 Macedonia National Olympiad, 5
An arbitrary triangle ABC is given. There are 2 lines, p and q, that are not parallel to each other and they are not perpendicular to the sides of the triangle. The perpendicular lines through points A, B and C to line p we denote with $ p_a, p_b, p_c $ and the perpendicular lines to line q we denote with $ q_a, q_b, q_c $. Let the intersection points of the lines $ p_a, q_a, p_b, q_b, p_c $ and $ q_c $ with $ q_b, p_b, q_c, p_c, q_a $ and $ p_a $ are $ K, L, P, Q, N $ and $ M $. Prove that $ KL, MN $ and $ PQ $ intersect in one point.
2011 Kosovo National Mathematical Olympiad, 4
A point $P$ is given in the square $ABCD$ such that $\overline{PA}=3$, $\overline{PB}=7$ and $\overline{PD}=5$. Find the area of the square.
1980 AMC 12/AHSME, 23
Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}2$. The length of the hypotenuse is
$\text{(A)} \ \frac 43 \qquad \text{(B)} \ \frac 32 \qquad \text{(C)} \ \frac{3\sqrt{5}}{5} \qquad \text{(D)} \ \frac{2\sqrt{5}}{3} \qquad \text{(E)} \ \text{not uniquely determined}$
2007 Mediterranean Mathematics Olympiad, 3
In the triangle $ABC$, the angle $\alpha = \angle BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$
2013 Today's Calculation Of Integral, 869
Let $I_n=\frac{1}{n+1}\int_0^{\pi} x(\sin nx+n\pi\cos nx)dx\ \ (n=1,\ 2,\ \cdots).$
Answer the questions below.
(1) Find $I_n.$
(2) Find $\sum_{n=1}^{\infty} I_n.$
2025 Thailand Mathematical Olympiad, 4
Let $D,E$ and $F$ be touch points of the incenter of $\triangle ABC$ at $BC, CA$ and $AB$, respectively. Let $P,Q$ and $R$ be the circumcenter of triangles $AFE, BDF$ and $CED$, respectively. Show that $DP, EQ$ and $FR$ concurrent.
2013 National Olympiad First Round, 27
For how many pairs $(a,b)$ from $(1,2)$, $(3,5)$, $(5,7)$, $(7,11)$, the polynomial $P(x)=x^5+ax^4+bx^3+bx^2+ax+1$ has exactly one real root?
$
\textbf{(A)}\ 4
\qquad\textbf{(B)}\ 3
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 1
\qquad\textbf{(E)}\ 0
$
2014 Bulgaria National Olympiad, 1
Let $k$ be a given circle and $A$ is a fixed point outside $k$. $BC$ is a diameter of $k$. Find the locus of the orthocentre of $\triangle ABC$ when $BC$ varies.
[i]Proposed by T. Vitanov, E. Kolev[/i]
1991 Arnold's Trivium, 10
Investigate the asymptotic behaviour of the solutions $y$ of the equation $x^5 + x^2y^2 = y^6$ that tend to zero as $x\to0$.
2009 Purple Comet Problems, 16
Let the complex number $z = \cos\tfrac{1}{1000} + i \sin\tfrac{1}{1000}.$ Find the smallest positive integer $n$ so that $z^n$ has an imaginary part which exceeds $\tfrac{1}{2}.$