Let $P$, $Q$, and $R$ be the points where the incircle of a triangle $ABC$ touches the sides $AB$, $BC$, and $CA$, respectively. Prove the inequality $\frac{BC} {PQ} + \frac{CA} {QR} + \frac{AB} {RP} \geq 6$.

If $n\in\mathbb{Z_+}$, positive real numbers $a+b=2$, then the minumum value of $\frac{1}{1+a^n}+\frac{1}{1+b^n}$ is________.

x,y,z positive real numbers such that $x^2+y^2+z^2=25$ Find the min price of $A=\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}$

Let $ \alpha_1$, $ \alpha_2$, $ \ldots$, $ \alpha_{2008}$ be real numbers. Find the maximum value of \[ \sin\alpha_1\cos\alpha_2 \plus{} \sin\alpha_2\cos\alpha_3 \plus{} \cdots \plus{} \sin\alpha_{2007}\cos\alpha_{2008} \plus{} \sin\alpha_{2008}\cos\alpha_1\]

A circle meets the three sides $BC,CA,AB$ of a triangle $ABC$ at points $D_1,D_2;E_1,E_2; F_1,F_2$ respectively. Furthermore, line segments $D_1E_1$ and $D_2F_2$ intersect at point $L$, line segments $E_1F_1$ and $E_2D_2$ intersect at point $M$, line segments $F_1D_1$ and $F_2E_2$ intersect at point $N$. Prove that the lines $AL,BM,CN$ are concurrent.

Let $\alpha , \beta ,\gamma$ be the angles of a triangle $ABC$ of perimeter $ 2p $ and $R$ is the radius of its circumscribed circle. $(a)$ Prove that \[\cot^{2}\alpha +\cot^{2}\beta+\cot^{2}\gamma\geq 3\left(9\cdot \frac{R^{2}}{p^{2}} - 1\right).\] $(b)$ When do we have equality?

Let $a, b$ and $c$ be real numbers such that $ab + bc + ca> a + b + c> 0$. Show then that $a+b+c>3$

Let $a, b, c, d$ be positive real numbers so that $abc+bcd+cda+dab = 4$. Prove that $a^2 + b^2 + c^2 + d^2 \ge 4$

Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.

Let $ ABCD$ be a rhombus with $ AC\equal{}16$ and $ BD\equal{}30$. Let $ N$ be a point on $ \overline{AB}$, and let $ P$ and $ Q$ be the feet of the perpendiculars from $ N$ to $ \overline{AC}$ and $ \overline{BD}$, respectively. Which of the following is closest to the minimum possible value of $ PQ$? [asy]unitsize(2.5cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); pair D=(0,0), C=dir(0), A=dir(aSin(240/289)), B=shift(A)*C; pair Np=waypoint(B--A,0.6), P=foot(Np,A,C), Q=foot(Np,B,D); draw(A--B--C--D--cycle); draw(A--C); draw(B--D); draw(Np--Q); draw(Np--P); label("$D$",D,SW); label("$C$",C,SE); label("$B$",B,NE); label("$A$",A,NW); label("$N$",Np,N); label("$P$",P,SW); label("$Q$",Q,SSE); draw(rightanglemark(Np,P,C,2)); draw(rightanglemark(Np,Q,D,2));[/asy]$ \textbf{(A)}\ 6.5 \qquad \textbf{(B)}\ 6.75 \qquad \textbf{(C)}\ 7 \qquad \textbf{(D)}\ 7.25 \qquad \textbf{(E)}\ 7.5$

Let $n\geq 2$ be an integer. Let $a_{ij}, \ i,j=1,2,\ldots,n$ be $n^2$ positive real numbers satisfying the following conditions: [list=1] [*]For all $i=1,\ldots,n$ we have $a_{ii}=1$ and, [*]For all $j=2,\ldots,n$ the numbers $a_{ij}, \ i=1,\ldots, j-1$ form a permutation of $1/a_{ji}, \ i=1,\ldots, j-1.$ [/list] Given that $S_i=a_{i1}+\cdots+a_{in}$, determine the maximum value of the sum $1/S_1+\cdots+1/S_n.$

Find all the real numbers $x,y,z$ which satisfy the following conditions: $$ \begin{cases} 3(x^2+y^2+z^2)=1\\ x^2y^2+y^2z^2+z^2x^2=xyz(x+y+z)^3\\ \end{cases} $$

Let $ABC$ be a triangle such that \[ \left( \cot \dfrac{A}{2} \right)^2 + \left( 2\cot \dfrac{B}{2} \right)^2 + \left( 3\cot \dfrac{C}{2} \right)^2 = \left( \dfrac{6s}{7r} \right)^2, \] where $s$ and $r$ denote its semiperimeter and its inradius, respectively. Prove that triangle $ABC$ is similar to a triangle $T$ whose side lengths are all positive integers with no common divisors and determine these integers.

Let $a_1,a_2,\dots,a_n$ be positive real numbers whose product is $1$. Show that the sum \[\textstyle\frac{a_1}{1+a_1}+\frac{a_2}{(1+a_1)(1+a_2)}+\frac{a_3}{(1+a_1)(1+a_2)(1+a_3)}+\cdots+\frac{a_n}{(1+a_1)(1+a_2)\cdots(1+a_n)}\] is greater than or equal to $\frac{2^n-1}{2^n}$.

Determine the set of all points $(x,y)$ in two-dimensional cartesian coordinate system such that \begin{align*}0\le &\,x\le\frac{\pi}{2}, \\ \sqrt{1-\sin 2x}-\sqrt{1+\sin 2x}\le &\,y\le\sqrt{1-\cos2x}-\sqrt{1+\cos2x}.\end{align*} Draw a picture of the set.

Let $a_1, a_2, ..., a_n$ be more than one real numbers, such that $0\leq a_i\leq \frac{\pi}{2}$. Prove that $$\Bigg(\frac{1}{n}\sum_{i=1}^{n}\frac{1}{1+\sin a_i}\Bigg)\Bigg(1+\prod_{i=1}^{n}(\sin a_i)^{\frac{1}{n}}\Bigg)\leq1.$$

$D$ is an arbitrary point inside triangle $ABC$, and $E$ is inside triangle $BDC$. Prove that \[\frac{S_{DBC}}{(P_{DBC})^{2}}\geq\frac{S_{EBC}}{(P_{EBC})^{2}}\]

Suppose $x$, $y$, $z$ are positive numbers such that $x+y+z=1$. Prove that \[ \frac{(1+xy+yz+zx)(1+3x^3 + 3y^3 + 3z^3)}{9(x+y)(y+z)(z+x)} \ge \left( \frac{x \sqrt{1+x} }{\sqrt[4]{3+9x^2}} + \frac{y \sqrt{1+y} }{\sqrt[4]{3+9y^2}} + \frac{z \sqrt{1+z}}{\sqrt[4]{3+9z^2}} \right)^2. \]

Let $x, y, z$ be positive real numbers. Prove that $\frac{y^2 + z^2}{x}+\frac{z^2 + x^2}{y}+\frac{x^2 + y^2}{z}\ge 2(x + y + z)$.

Let $ I_n\equal{}\int_0^{\sqrt{3}} \frac{1}{1\plus{}x^{n}}\ dx\ (n\equal{}1,\ 2,\ \cdots)$. (1) Find $ I_1,\ I_2$. (2) Find $ \lim_{n\to\infty} I_n$.

If the function $f\colon [1,2]\to R$ is such that $\int_{1}^{2}f(x) dx=\frac{73}{24}$, then show that there exists a $x_{0}\in (1;2)$ such that \[x_{0}^{2}<f(x_{0})<x_{0}^{3}\] [Edit: $f$ is continuous]

Prove that if $a,b,c,d$ are lengths of the successive sides of a quadrangle (not necessarily convex) with the area equal to $S$, then the following inequality holds \[S \leq \frac{1}{2}(ac+bd).\] For which quadrangles does the inequality become equality?

We are given $ 3^{2k}$ apparently identical coins,one of which is fake,being lighter than the others. We also dispose of three apparently identical balances without weights, one of which is broken (and yields outcomes unrelated to the actual situations). How can we find the fake coin in $ 3k\plus{}1$ weighings?

Find the largest $c$ such that for any $\lambda \ge 1$ there is an a that satisfies the inequality $$\sin a + \sin (a\lambda ) \ge c.$$

In the adjoining figure, two circles of radii 6 and 8 are drawn with their centers 12 units apart. At $P$, one of the points of intersection, a line is drawn in such a way that the chords $QP$ and $PR$ have equal length. Find the square of the length of $QP$. [asy]unitsize(2.5mm); defaultpen(linewidth(.8pt)+fontsize(12pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=intersectionpoints(C1,C2)[0]; path C3=Circle(P,sqrt(130)); pair Q=intersectionpoints(C3,C1)[0]; pair R=intersectionpoints(C3,C2)[1]; draw(C1); draw(C2); //draw(O2--O1); //dot(O1); //dot(O2); draw(Q--R); label("$Q$",Q,N); label("$P$",P,dir(80)); label("$R$",R,E); //label("12",waypoint(O1--O2,0.4),S);[/asy]