The Devil and the Man play a game. Initially, the Man pays some cash $s$ to the Devil. Then he lists some $97$ triples $\{i,j,k\}$ consisting of positive integers not exceeding $100$. After that, the Devil draws some convex polygon $A_1A_2...A_{100}$ with area $100$ and pays to the Man, the sum of areas of all triangles $A_iA_jA_k$. Determine the maximal value of $s$ which guarantees that the Man receives at least as much cash as he paid. [i]Proposed by Nikolai Beluhov, Bulgaria[/i]

Alice wrote a sequence of $n > 2$ nonzero nonequal numbers such that each is greater than the previous one by the same amount. Bob wrote the inverses of those n numbers in some order. It so happened that each number in his row also is greater than the previous one by the same amount, possibly not the same as in Alice’s sequence. What are the possible values of $n{}$? [i]Alexey Zaslavsky[/i]

In triangle $ABC$, the medians and bisectors corresponding to sides $BC$, $CA$, $AB$ are $m_a$, $m_b$, $m_c$ and $w_a$, $w_b$, $w_c$ respectively. $P=w_a \cap m_b$, $Q=w_b \cap m_c$, $R=w_c \cap m_a$. Denote the areas of triangle $ABC$ and $PQR$ by $F_1$ and $F_2$ respectively. Find the least positive constant $m$ such that $\frac{F_1}{F_2}<m$ holds for any $\triangle{ABC}$.

Let $ABC$ be a scalene triangle with angle bisectors $AD$, $BE$, and $CF$ so that $D$, $E$, and $F$ lie on segments $BC$, $CA$, and $AB$ respectively. Let $M$ and $N$ be the midpoints of $BC$ and $EF$ respectively. Prove that line $AN$ and the line through $M$ parallel to $AD$ intersect on the circumcircle of $ABC$ if and only if $DE=DF$. [i]Proposed by Michael Ren.[/i]

Is there a triangle with angles in ratio of $ 1: 2: 4$ and the length of its sides are integers with at least one of them is a prime number? [i]Nanang Susyanto, Jogjakarta[/i]

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

[b]Q[/b] On a \(10 \times 10\) chess board whose colors of square are green and blue in an arbitrary way and we are simultaneously allowed to switch all the colors of all squares in any \((2 \times 2)\) and \((5\times 5)\) region. Can we transform any coloring of the board into one where all squares are blue ? Give a proper explanation of your answer. Note. that if a unit square is part of both the $2\times 2$ and $5\times 5$ region,then its color switched is twice(i.e switching is additive) [i]Proposed by Aritra12[/i]

(a) Given a positive integer $k$, prove that there do not exist two distinct integers in the open interval $(k^2, (k + 1)^2)$ whose product is a perfect square. (b) Given an integer $n > 2$, prove that there exist $n$ distinct integers in the open interval $(k^n, (k + 1)^n)$ whose product is the $n$-th power of an integer, for all but a finite number of positive integers $k$. [i]AMM Magazine[/i]

Let $f$ be a real-valued function of real and positive argument such that $f(x) + 3xf(\tfrac1x) = 2(x + 1)$ for all real numbers $x > 0$. Find $f(2003)$.

Let $ABCDE$ be a convex pentagon in which $\angle{A}=\angle{B}=\angle{C}=\angle{D}=120^{\circ}$ and the side lengths are five [i]consecutive integers[/i] in some order. Find all possible values of $AB+BC+CD$.

Consider a circle $K$ and an inscribed quadrilateral $ABCD$, such that the diagonal $BD$ is not the diameter of the circle. Prove that the intersection of the lines tangent to $K$ through the points $B$ and $D$ lies on the line $AC$ if and only if $AB \cdot CD = AD \cdot BC$.

A square with the dimension $ 1 \times1 $ has been removed from a square board $ 3 ^n \times 3 ^n $ ($ n \in \mathbb {N}, $ $ n> 1 $). a) Prove that any defective board with the dimension $ 3 ^ n \times3 ^ n $ can be covered with shaped figures of shape 1 (the 3 squares' one) and of shape 2 (the 5 squares' one). Figures covering the board must not overlap each other and must not cross the edge of the board. Also the squares removed from the board must not be covered. (b) How many small figures in shape 2 must be used to cover the board? [img]https://cdn.artofproblemsolving.com/attachments/4/7/e970fadd7acc7fd6f5897f1766a84787f37acc.png[/img]

Let $A$ be the set of the $16$ first positive integers. Find the least positive integer $k$ satisfying the condition: In every $k$-subset of $A$, there exist two distinct $a, b \in A$ such that $a^2 + b^2$ is prime.

The product of three positive numbers equals to one, their sum is strictly greater than the sum of the inverse numbers. Prove that one and only one of them is greater than one.

Find $\lim_{n\to\infty} \frac{1}{(\ln n)^2}\sum_{k=3}^n \frac{\ln k}{k}.$

(a) Let there be two given points $A,B$ in the plane. i. Find the triangles $MAB$ with the given area and the minimal perimeter. ii. Find the triangles $MAB$ with a given perimeter and the maximal area. (b) In a tetrahedron of volume $V$, let $a,b,c,d$ be the lengths of its four edges, no three of which are coplanar, and let $L=a+b+c+d$. Determine the maximum value of $\frac V{L^3}$.

Find all triples $(a,b,c)$ of distinct positive integers so that there exists a subset $S$ of the positive integers for which for all positive integers $n$ exactly one element of the triple $(an,bn,cn)$ is in $S$. Proposed by Carl Schildkraut, MIT

Find a 3rd degree polynomial whose roots are $r_a$, $r_b$ and $r_c$ where $r_a$ is the radius of the outer inscribed circle of $ABC$ with respect to $A$.

Given three non-collinear points $A,B,C$, construct a circle with centre $C$ such that the tangents from $A$ and $B$ are parallel.

A triangle with sides a, b, c has area S. The distances of its centroid from the vertices are x, y, z. Show that: if (x + y + z)^2 ≤ (a^2 + b^2 + c^2)/2 + 2S√3, then the triangle is equilateral.

On a blackboard, there are $17$ integers not divisible by $17$. Alice and Bob play a game. Alice starts and they alternately play the following moves: $\bullet$ Alice chooses a number $a$ on the blackboard and replaces it with $a^2$ $\bullet$ Bob chooses a number $b$ on the blackboard and replaces it with $b^3$. Alice wins if the sum of the numbers on the blackboard is a multiple of $17$ after a finite number of steps. Prove that Alice has a winning strategy. (Daniel Holmes)

Q. For integers $m,n\geq 1$, Let $A_{m,n}$ , $B_{m,n}$ and $C_{m,n}$ denote the following sets: $A_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m) \colon 1\leq \alpha _1\leq \alpha_2 \leq \ldots \leq \alpha_m\leq n\}$ given that $\alpha _i \in \mathbb{Z}$ for all $i$ $B_{m,n}=\{(\alpha _1,\alpha _2,\ldots ,\alpha _m) \colon \alpha _1+\alpha _2+\ldots + \alpha _m=n\}$ given that $\alpha _i \geq 0$ and $\alpha_ i\in \mathbb{Z}$ for all $i$ $C_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m)\colon 1\leq \alpha _1< \alpha_2 < \ldots< \alpha_m\leq n\}$ given that $\alpha _i \in \mathbb{Z}$ for all $i$ $(a)$ Define a one-one onto map from $A_{m,n}$ onto $B_{m+1,n-1}$. $(b)$ Define a one-one onto map from $A_{m,n}$ onto $C_{m,n+m-1}$. $(c)$ Find the number of elements of the sets $A_{m,n}$ and $B_{m,n}$.

Find all polynomials $p_n(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0$ ($n\geq 2$) with real and non-zero coeficients s.t. $p_n(x)-p_1(x)p_2(x)...p_{n-1}(x)$ be a constant polynomial. ;)

Let $a,\ b$ be given positive constants. Evaluate \[\int_0^1 \frac{\ln\ (x+a)^{x+a}(x+b)^{x+b}}{(x+a)(x+b)}dx.\] Own

Let $S$ be the set of points $A$ in the xy-plane such that the four points $A$, $(2, 3)$, $(-1, 0)$, and $(0, 6)$ form the vertices of a parallelogram. Let $P$ be the convex polygon whose vertices are the points in $S$. What is the area of $P$?