Let $A_1A_2...A_{101}$ be a regular $101$-gon, and colour every vertex red or blue. Let $N$ be the number of obtuse triangles satisfying the following: The three vertices of the triangle must be vertices of the $101$-gon, both the vertices with acute angles have the same colour, and the vertex with obtuse angle have different colour. $(1)$ Find the largest possible value of $N$. $(2)$ Find the number of ways to colour the vertices such that maximum $N$ is acheived. (Two colourings a different if for some $A_i$ the colours are different on the two colouring schemes).

A cylinder is inscribed within a sphere of radius 10 such that its volume is [i]almost-half[/i] that of the sphere. If [i]almost-half[/i] is defined such that the cylinder has volume $\frac12+\frac{1}{250}$ times the sphere’s volume, find the sum of all possible heights for the cylinder.

In a triangle $ABC ~(\overline{AB} < \overline{AC})$, points $D (\neq A, B)$ and $E (\neq A, C)$ lies on side $AB$ and $AC$ respectively. Point $P$ satisfies $\overline{PB}=\overline{PD}, \overline{PC}=\overline{PE}$. $X (\neq A, C)$ is on the arc $AC$ of the circumcircle of triangle $ABC$ not including $B$. Let $Y (\neq A)$ be the intersection of circumcircle of triangle $ADE$ and line $XA$. Prove that $\overline{PX} = \overline{PY}$.

Let $\frac{35x-29}{x^2-3x+2}=\frac{N_1}{x-1}+\frac{N_2}{x-2}$ be an identity in $x$. The numerical value of $N_1N_2$ is: $\text{(A)} \ -246 \qquad \text{(B)} \ -210 \qquad \text{(C)} \ -29 \qquad \text{(D)} \ 210 \qquad \text{(E)} \ 246$

Let $A$ and $B$ be two non-infinite sets of natural numbers, each of which contains at least 3 elements. Two numbers $a\in A$ and $b\in B$ are called [i]"harmonious"[/i], if they are not coprime. It is known that each element from $A$ is not [i]harmonious[/i] with at least one element from $B$ and each element from $B$ is harmonious with at least one from $A$. Prove that there exist $a_1,a_2\in A$ and $b_1,b_2\in B$ such that $(a_1,b_1)$ and $(a_2,b_2)$ are [i]harmonious[/i] but $(a_1,b_2)$ and $(a_2,b_1)$ are not.

Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?

Let $a,b,c$ be real numbers, which are not all equal, such that $$a+b+c=\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3.$$ Prove that at least one of $a, b, c$ is negative.

Given a square whose side measures $a$, consider the set of all points of its plane through which passes a circumference of radius whose circle contains to the quoted square. You are asked to prove that the contour of the figure formed by the points with this property is formed by arcs of circumference, and determine the positions, their centers, their radii and their lengths.

Determine the number of positive integers $N = \overline{abcd}$, with $a, b, c, d$ nonzero digits, which satisfy $(2a -1) (2b -1) (2c- 1) (2d - 1) = 2abcd -1$.

Let $\mathfrak F=\{A_1,A_2,\ldots,A_n\}$ be a collection of subsets of the set $S=\{1,2,\ldots,n\}$ satisfying the following conditions: (a) Any two distinct sets from $\mathfrak F$ have exactly one element in common; (b) each element of $S$ is contained in exactly $k$ of the sets in $\mathfrak F$. Can $n$ be equal to $1996$?

From an $n \times n $ square, $n \ge 2,$ the unit squares situated on both odd numbered rows and odd numbers columns are removed. Determine the minimum number of rectangular tiles needed to cover the remaining surface.

Find the least positive integer ending in $7$ with exactly $12$ positive divisors.

The set $M = \{1, 2, . . . , 2n\}$ is partitioned into $k$ nonintersecting subsets $M_1,M_2, \dots, M_k,$ where $n \ge k^3 + k.$ Prove that there exist even numbers $2j_1, 2j_2, \dots, 2j_{k+1}$ in $M$ that are in one and the same subset $M_i$ $(1 \le i \le k)$ such that the numbers $2j_1 - 1, 2j_2 - 1, \dots, 2j_{k+1} - 1$ are also in one and the same subset $M_j (1 \le j \le k).$

Prove that the modulus of an integer root of a polynomial with integer coefficients cannot exceed the maximum of the moduli of the coefficients.

Rectangle $ABCD$ has side lengths $AB = 3$ and $BC = 7$. Let $E$ be a point on $BC$, and let $F$ be the intersection of $DE$ and $AC$. Given that $[CDF] = 4$, find $\frac{DF}{FE}$ .

Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

In the figure, $ ABCD$ is a $ 2\times 2$ square, $ E$ is the midpoint of $ \overline{AD}$, and $ F$ is on $ \overline{BE}$. If $ \overline{CF}$ is perpendicular to $ \overline{BE}$, then the area of quadrilateral $ CDEF$ is [asy]defaultpen(linewidth(.8pt)); dotfactor=4; pair A = (0,2); pair B = origin; pair C = (2,0); pair D = (2,2); pair E = midpoint(A--D); pair F = foot(C,B,E); dot(A);dot(B);dot(C);dot(D);dot(E);dot(F); label("$A$",A,N);label("$B$",B,S);label("$C$",C,S);label("$D$",D,N);label("$E$",E,N);label("$F$",F,NW); draw(A--B--C--D--cycle); draw(B--E); draw(C--F); draw(rightanglemark(B,F,C,4));[/asy]$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 3 \minus{} \frac {\sqrt {3}}{2}\qquad \textbf{(C)}\ \frac {11}{5}\qquad \textbf{(D)}\ \sqrt {5}\qquad \textbf{(E)}\ \frac {9}{4}$

Find the least positive integer, which may not be represented as ${2^a-2^b\over 2^c-2^d}$, where $a,\,b,\,c,\,d$ are positive integers.

Find all functions $f: R\to R$, such that $f (xf (y) + f (x)) = xy$ for all $x, y \in R $.

Point $A,B,C,D,$ and $E$ are equally spaced on a minor arc of a circle. Points $E,F,G,H,I$ and $A$ are equally spaced on a minor arc of a second circle with center $C$ as shown in the figure below. The angle $\angle ABD$ exceeds $\angle AHG$ by $12^\circ$. Find the degree measure of $\angle BAG$.[asy] pair A,B,C,D,E,F,G,H,I,O; O=(0,0); C=dir(90); B=dir(70); A=dir(50); D=dir(110); E=dir(130); draw(arc(O,1,50,130)); real x=2*sin(20*pi/180); F=x*dir(228)+C; G=x*dir(256)+C; H=x*dir(284)+C; I=x*dir(312)+C; draw(arc(C,x,200,340)); label("$A$",A,dir(0)); label("$B$",B,dir(75)); label("$C$",C,dir(90)); label("$D$",D,dir(105)); label("$E$",E,dir(180)); label("$F$",F,dir(225)); label("$G$",G,dir(260)); label("$H$",H,dir(280)); label("$I$",I,dir(315)); [/asy]

A box $2$ centimeters high, $3$ centimeters wide, and $5$ centimeters long can hold $40$ grams of clay. A second box with twice the height, three times the width, and the same length as the first box can hold $n$ grams of clay. What is $n$? $\textbf{(A)}\ 120\qquad\textbf{(B)}\ 160\qquad\textbf{(C)}\ 200\qquad\textbf{(D)}\ 240\qquad\textbf{(E)}\ 280$

Let $ABC$ be a triangle with $AB<AC$ and with its incircle touching the sides $AB$ and $BC$ at $M$ and $J$ respectively. A point $D$ lies on the extension of $AB$ beyond $B$ such that $AD=AC$. Let $O$ be the midpoint of $CD$. Prove that the points $J$, $O$, $M$ are collinear. [i](Proposed by Tan Rui Xuen)[/i]

What is the arithmetic mean of all values of the expression $ | a_1-a_2 | + | a_3-a_4 | $ Where $ a_1, a_2, a_3, a_4 $ is a permutation of the elements of the set {$ 1,2,3,4 $}?

Determine strictly positive real numbers $ a_{1},a_{2},...,a_{n}$ if for any $ n\in N^*$ takes place equality: $ a_{1}^2\plus{}a_{2}^2\plus{}...\plus{}a_{n}^2\equal{}a_{1}\plus{}a_{2}\plus{}...\plus{}a_{n}\plus{}\frac{n(n^2\plus{}6n\plus{}11)}{3}$

Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.