Let $ ABC$ be an acute-angled triangle. Let $ L$ be any line in the plane of the triangle $ ABC$. Denote by $ u$, $ v$, $ w$ the lengths of the perpendiculars to $ L$ from $ A$, $ B$, $ C$ respectively. Prove the inequality $ u^2\cdot\tan A \plus{} v^2\cdot\tan B \plus{} w^2\cdot\tan C\geq 2\cdot S$, where $ S$ is the area of the triangle $ ABC$. Determine the lines $ L$ for which equality holds.

[b]p1.[/b] Show that the $n \times n$ determinant $$\begin{vmatrix} 1+x & 1 & 1 & . & . & . & 1 \\ 1 & 1+x & 1 & . & . & . & 1 \\ . & . & . & . & . & . & . \\ . & . & . & . & . & . & . \\ 1 & 1 & . & . & . & . & 1+x \\ \end{vmatrix}$$ has the value zero when $x = -n$ [b]p2.[/b] Let $c > a \ge b$ be the lengths of the sides of an obtuse triangle. Prove that $c^n = a^n + b^n$ for no positive integer $n$. [b]p3.[/b] Suppose that $p_1 = p_2^2+ p_3^2 + p_4^2$ , where $p_1$, $p_2$, $p_3$, and $p_4$ are primes. Prove that at least one of $p_2$, $p_3$, $p_4$ is equal to $3$. [b]p4.[/b] Suppose $X$ and $Y$ are points on tJhe boundary of the triangular region $ABC$ such that the segment $XY$ divides the region into two parts of equal area. If $XY$ is the shortest such segment and $AB = 5$, $BC = 4$, $AC = 3$ calculate the length of $XY$. Hint: Of all triangles having the same area and same vertex angle the one with the shortest base is isosceles. Clearly justify all claims. [b]p5.[/b] Find all solutions of the following system of simultaneous equations $$x + y + z = 7\,\, , \,\, x^2 + y^2 + z^2 = 31\,\,, \,\,x^3 + y^3 + z^3 = 154$$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all surjective functions $f: \mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in \mathbb{N}$ \[f(m)\mid f(n) \mbox{ if and only if }m\mid n.\]

Let $a,b,c$ three positive numbers less than $ 1$. Prove that cannot occur simultaneously these three inequalities: $$a (1- b)>\frac14$$ $$b (1-c)>\frac14 $$ $$c (1-a)>\frac14$$

Let $\triangle ABC$ be such that $AB = 15, BC = 13, CA = 14.$ Let $P$ be the point on the circumcircle of $\triangle ABC$ such that $P$ is on the same side of $AB$ as $C$ and $AP = BP.$ Let $X$ be the foot of the perpendicular from $P$ to $AC.$ Then the length of $AX$ is $\tfrac{m}{n}$ for some relatively prime positive integers $m$ and $n.$ Find $m + n.$

Triangle $ABC$ has perimeter $1$. Its three altitudes form the side lengths of a triangle. Find the set of all possible values of $\min(AB,BC,CA)$.

$64$ numbers (not necessarily distinct) are placed on the squares of a chessboard such that the sum of the numbers in every $2\times2$ square is $7$. What is the sum of the four numbers in the corners of the board?

Solve the equation for real numbers $x,y,z$ $$(x-y+z)^2=x^2-y^2+z^2$$

Isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Point $D$ lies on arc $\frown{BC}$ not containing $A$. Let $E$ be the foot of perpendicular from $A$ to line $CD$. Prove that $BC+DC=2DE$.

Let $I$ be the incircle and $O$ a circumcenter of triangle $ABC$ such that $\angle ACB=30^{\circ}$. On sides $AC$ and $BC$ there are points $E$ and $D$, respectively, such that $EA=AB=BD$. Prove that $DE=IO$ and $DE \perp IO$

Given a pair of concentric circles, chords $AB,BC,CD,\dots$ of the outer circle are drawn such that they all touch the inner circle. If $\angle ABC = 75^{\circ}$, how many chords can be drawn before returning to the starting point ? [img]https://i.imgur.com/Cg37vwa.png[/img]

Find all sequences $(a_n)$ of positive integers satisfying the equality $a_n=a_{a_{n-1}}+a_{a_{n+1}}$ a) for all $n\ge 2$ b) for all $n \ge 3$ (I. Gorodnin)

Let $ABC$ be an acute-angled triangle, and let $F$ be the foot of its altitude from the vertex $C$. Let $M$ be the midpoint of the segment $CA$. Assume that $CF=BM$. Then the angle $MBC$ is equal to angle $FCA$ if and only if the triangle $ABC$ is equilateral.

Find all positive integers $a, b, c$ such that $ab + 1$, $bc + 1$, and $ca + 1$ are all equal to factorials of some positive integers. Proposed by [i]Nikola Velov, Macedonia[/i]

The number of solutions, in real numbers $a$, $b$, and $c$, to the system of equations $$a+bc=1,$$$$b+ac=1,$$$$c+ab=1,$$ is $\text{(A) }3\qquad\text{(B) }4\qquad\text{(C) }5\qquad\text{(D) more than }5\text{, but finitely many}\qquad\text{(E) infinitely many}$

The maximum possible number of knights are placed on a $5 \times 5$ chessboard so that no two attack each other. Prove that there is only one possible placement. (A Kanel)