(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 $P$?

Moor has $3$ different shirts, labeled $T, E,$ and $A$. Across $5$ days, the only days Moor can wear shirt $T$ are days $2$ and $5$. How many different sequences of shirts can Moor wear across these $5$ days?

Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team?

Right triangle $ ABC,$ with $ \angle A\equal{}90^{\circ},$ is inscribed in circle $ \Gamma.$ Point $ E$ lies on the interior of arc $ {BC}$ (not containing $ A$) with $ EA>EC.$ Point $ F$ lies on ray $ EC$ with $ \angle EAC \equal{} \angle CAF.$ Segment $ BF$ meets $ \Gamma$ again at $ D$ (other than $ B$). Let $ O$ denote the circumcenter of triangle $ DEF.$ Prove that $ A,C,O$ are collinear.

Let $ ABCDE$ be convex pentagon such that $ S(ABC) \equal{} S(BCD) \equal{} S(CDE) \equal{} S(DEA) \equal{} S(EAB)$. Prove that there is a point $ M$ inside pentagon such that $ S(MAB) \equal{} S(MBC) \equal{} S(MCD) \equal{} S(MDE) \equal{} S(MEA)$.

Let $O$ be the circumcircle of the triangle $ABC$. Two circles $O_1,O_2$ are tangent to each of the circle $O$ and the rays $\overrightarrow{AB},\overrightarrow{AC}$, with $O_1$ interior to $O$, $O_2$ exterior to $O$. The common tangent of $O,O_1$ and the common tangent of $O,O_2$ intersect at the point $X$. Let $M$ be the midpoint of the arc $BC$ (not containing the point $A$) on the circle $O$, and the segment $\overline{AA'}$ be the diameter of $O$. Prove that $X,M$, and $A'$ are collinear.

In the plane, there are $n \geqslant 6$ pairwise disjoint disks $D_{1}, D_{2}, \ldots, D_{n}$ with radii $R_{1} \geqslant R_{2} \geqslant \ldots \geqslant R_{n}$. For every $i=1,2, \ldots, n$, a point $P_{i}$ is chosen in disk $D_{i}$. Let $O$ be an arbitrary point in the plane. Prove that \[O P_{1}+O P_{2}+\ldots+O P_{n} \geqslant R_{6}+R_{7}+\ldots+R_{n}.\] (A disk is assumed to contain its boundary.)

Let $P(x)$ be a quadratic polynomial with two distinct real roots. For all real numbers $a$ and $b$ satisfying $|a|,|b| \ge 2017$, we have $P(a^2+b^2) \ge P(2ab)$. Show that at least one of the roots of $P$ is negative.

On the sides $ BC $, $ CA $, $ AB $ of the triangle $ ABC $, the points $ M $, $ N $, $ P $ are taken, respectively, in such a way that $$\frac{BM}{MC} = \frac{CN}{NA} = \frac{AP}{PB} = k, $$ where $ k $ means a given number greater than $ 1 $, then the segments $ AM $, $ BN $, $ CP $ were drawn . Given the area $ S $ of the triangle $ ABC $, calculate the area of the triangle bounded by the lines $ AM $, $ BN $ and $ CP $.

Prove that if you choose $10^{100}$ points on a circle and arrange numbers from $1$ to $10^{100}$ on them in some order, then you can choose $100$ pairwise disjoint chords with ends at the selected points such that the sums of the numbers at the ends of all of them are equal to each other.

How many ways are there to arrange the numbers $1$, $2$, $3$, $4$, $5$, $6$ on the vertices of a regular hexagon such that exactly 3 of the numbers are larger than both of their neighbors? Rotations and reflections are considered the same.

Let $P_1P_2...P_n$ be a regular $n$-gon in the plane and $a_1, . . . , a_n$ be nonnegative integers. It is possible to draw $m$ circles so that for each $1 \le i \le n$, there are exactly $a_i$ circles that contain $P_i$ on their interior. Find, with proof, the minimum possible value of $m$ in terms of the $a_i$. .

Let ${{\left( {{a}_{n}} \right)}_{n\ge 1}}$ an increasing sequence and bounded.Calculate $\underset{n\to \infty }{\mathop{\lim }}\,\left( 2{{a}_{n}}-{{a}_{1}}-{{a}_{2}} \right)\left( 2{{a}_{n}}-{{a}_{2}}-{{a}_{3}} \right)...\left( 2{{a}_{n}}-{{a}_{n-2}}-{{a}_{n-1}} \right)\left( 2{{a}_{n}}-{{a}_{n-1}}-{{a}_{1}} \right).$

Let $ M,N $ be points on the segments $ AB,AC, $ respectively, of the triangle $ ABC. $ Also, let $ P,Q, $ be the midpoints of the segments $ MN,BC, $ respectively. Knowing that $ PQ $ is parallel to the bisector of $ \angle BAC , $ show that $ BM=CN. $ [i]Gheorghe Duță[/i]

A coloring of all plane points with coordinates belonging to the set $S=\{0,1,\ldots,99\}$ into red and white colors is said to be [i]critical[/i] if for each $i,j\in S$ at least one of the four points $(i,j),(i + 1,j),(i,j + 1)$ and $(i + 1, j + 1)$ $(99 + 1\equiv0)$ is colored red. Find the maximal possible number of red points in a critical coloring which loses its property after recoloring of any red point into white.

Given regular $1976$-gon. The midpoints of all the sides and diagonals are marked. What is the greatest number of the marked points lying on one circumference?