Three pairwairs distinct positive integers $a,b,c,$ with $gcd(a,b,c)=1$, satisfy \[a|(b-c)^2 ,b|(a-c)^2 , c|(a-b)^2\] Prove that there doesnt exist a non-degenerate triangle with side lengths $a,b,c.$

Find all triangles $ABC$ such that $\frac{a cos A + b cos B + c cos C}{a sin A + b sin B + c sin C} =\frac{a + b + c}{9R}$, where, as usual, $a, b, c$ are the lengths of sides $BC, CA, AB$ and $R$ is the circumradius.

Let $ f_i (x), i \equal{} 1,2,3 \cdots$ be defined by $ f_1 (x) \equal{} \frac{1}{1 \minus{} x}$ and $ f_{i\plus{}1} (x) \equal{} f_i (f_1 (x))$. Then $ f_{1998} (1998)$ equals A. 0 B. 1998 C. -1/1997 D. 1997/1998 E. None of these

Matrices $A$, $B$ are given as follows. \[A=\begin{pmatrix} 2 & 1 & 0 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}, \quad B = \begin{pmatrix} 4 & 2 & 0 \\ 2 & 4 & 0 \\ 0 & 0 & 12\end{pmatrix}\] Find volume of $V=\{\mathbf{x}\in\mathbb{R}^3 : \mathbf{x}\cdot A\mathbf{x} \leq 1 < \mathbf{x}\cdot B\mathbf{x} \}$.

[b]8.[/b] Prove the following generalization of the well-known Chinese remainder theorem: Let $R$ be a ring with unit element and let $A_{1},A_{2},\dots . A_{n} (n\geqslant 2)$ be pairwise relative prime ideals of $R$. Then, for arbitrary elements $c_{1},c_{2}, \dots , c_{n}$ of $R$, there exists an element $x\in R$ such that $x-c_{k} \in A_{k} (k= 1,2, \dots , n)$. [b](A. 17)[/b]

Find the sum of all real numbers $x$ such that $5x^4-10x^3+10x^2-5x-11=0$.

Let $ABC$ be an acute triangle with orthocenter $H$ and circumcircle $\Omega$. Let $M$ be the midpoint of side $BC$. Point $D$ is chosen from the minor arc $BC$ on $\Gamma$ such that $\angle BAD = \angle MAC$. Let $E$ be a point on $\Gamma$ such that $DE$ is perpendicular to $AM$, and $F$ be a point on line $BC$ such that $DF$ is perpendicular to $BC$. Lines $HF$ and $AM$ intersect at point $N$, and point $R$ is the reflection point of $H$ with respect to $N$. Prove that $\angle AER + \angle DFR = 180^\circ$. [i]Proposed by Li4.[/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]

We consider positive integers written down in the (usual) decimal system. Within such an integer, we number the positions of the digits from left to right, so the leftmost digit (which is never a $0$) is at position $1$. An integer is called [i]even-steven[/i] if each digit at an even position (if there is one) is greater than or equal to its neighbouring digits (if these exist). An integer is called [i]oddball[/i] if each digit at an odd position is greater than or equal to its neighbouring digits (if these exist). For example, $3122$ is [i]oddball[/i] but not [i]even-steven[/i], $7$ is both [i]even-steven[/i] and [i]oddball[/i], and $123$ is neither [i]even-steven[/i] nor [i]oddball[/i]. (a) Prove: every oddball integer greater than $9$ can be obtained by adding two [i]oddball [/i] integers. (b) Prove: there exists an oddball integer greater than $9$ that cannot be obtained by adding two [i]even-steven[/i] integers.

$ABCDEFG$ is a convex polygon with area 1. Points $X,Y,Z,U,V$ are arbitrary points on $AB, BC, CD, EF, FG$. Let $M, I, N, K, S$ be the midpoints of $EZ, BU, AV, FX, TE$. Find the largest and smallest possible values of the area of $AKBSCMDEIFNG$.

Let $a$, $b$, $c$ and $d$ be real numbers such that no two of them are equal, \[\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}=4\] and $ac=bd$. Find the maximum possible value of \[\frac{a}{c}+\frac{b}{d}+\frac{c}{a}+\frac{d}{b}.\]

For each positive integer $n$, let an be the smallest nonnegative integer such that there is only one positive integer at most $n$ that is relatively prime to all of $n$, $n + 1$, .. , $n + a_n$. If $n < 100$, compute the largest possible value of $n - a_n$.

a) Prove that the equation $2^x + 21^x = y^3$ has no solution in the set of natural numbers. b) Solve the equation $2^x + 21^y = z^2y$ in the set of non-negative integer numbers.

In triangle $ABC$, let $\omega$ be the excircle opposite to $A$. Let $D, E$ and $F$ be the points where $\omega$ is tangent to $BC, CA$, and $AB$, respectively. The circle $AEF$ intersects line $BC$ at $P$ and $Q$. Let $M$ be the midpoint of $AD$. Prove that the circle $MPQ$ is tangent to $\omega$.

Fames is playing a computer game with falling two-dimensional blocks. The playing field is $7$ units wide and infinitely tall with a bottom border. Initially the entire field is empty. Each turn, the computer gives Fames a $1\times 3$ solid rectangular piece of three unit squares. Fames must decide whether to orient the piece horizontally or vertically and which column(s) the piece should occupy ($3$ consecutive columns for horizontal pieces, $1$ column for vertical pieces). Once he confirms his choice, the piece is dropped straight down into the playing field in the selected columns, stopping all three of the piece's squares as soon as the piece hits either the bottom of the playing field or any square from another piece. All of the pieces must be contained completely inside the playing field after dropping and cannot partially occupy columns. If at any time a row of $7$ spaces is all filled with squares, Fames scores a point. Unfortunately, Fames is playing in [i]invisible mode[/i], which prevents him from seeing the state of the playing field or how many points he has, and he has already arbitrarily dropped some number of pieces without remembering what he did with them or how many there were. For partial credit, find a strategy that will allow Fames to eventually earn at least one more point. For full credit, find a strategy for which Fames can correctly announce "I have earned at least one more point" and know that he is correct.

The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$. What is $|r-s|$? $\text{A) } \frac{1}{4} \qquad \text{B) } \frac{1}{2} \qquad \text{C) } \frac{3}{4} \qquad \text{D) } \frac{2}{3} \qquad \text{E) } 1$

There are 100 houses in the street, divided into 50 pairs. In each pair houses are right across the street one from another. On the right side of the street the houses have even numbers, while the houses on the left side have odd numbers. On both sides of the street the numbers increase from the beginning to the end of the street, but are not necessarily consecutive (some numbers may be omitted). For each house on the right side of the street, the difference between its number and the number of the opposite house was computed, and it turned out that all these values were different. Let $n$ be the greatest number of a house on this street. Find the smallest possible value of $n$. (8 points) Maxim Didin

The sides of an equilateral triangle $ABC$ are divided into $n$ equal parts $(n \geq 2) .$ For each point on a side, we draw the lines parallel to other sides of the triangle $ABC,$ e.g. for $n=3$ we have the following diagram: [asy] unitsize(150); defaultpen(linewidth(0.7)); int n = 3; /* # of vertical lines, including AB */ pair A = (0,0), B = dir(-30), C = dir(30); draw(A--B--C--cycle,linewidth(2)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0)); label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE); for(int i = 1; i < n; ++i) { draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n); draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n); draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n); } [/asy] For each $n \geq 2,$ find the number of existing parallelograms.

Evaluate \[\sum_{n=0}^{2006}\int_{0}^{1}\frac{dx}{2(x+n+1)\sqrt{(x+n)(x+n+1)}}\]

Jesse cuts a circular paper disk of radius $12$ along two radii to form two sectors, the smaller having a central angle of $120$ degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger? $ \textbf{(A)}\ \frac{1}{8} \qquad\textbf{(B)}\ \frac{1}{4} \qquad\textbf{(C)}\ \frac{\sqrt{10}}{10} \qquad\textbf{(D)}\ \frac{\sqrt{5}}{6} \qquad\textbf{(E)}\ \frac{\sqrt{10}}{5} $

In a class there are n students with unequal heights. $\textbf{(a)}$ Find the number of orderings of the students such that the shortest person is not at the front and the tallest person is not at the end. $\textbf{(b)}$ Define the [i]badness[/i] of an ordering as the maximum number $k$ such that there are $k$ many people with height greater than in front of a person. For example: the sequence $66, 61, 65, 64, 62, 70$ has [i]badness [/i] $3$ since there are $3$ numbers greater than $62$ in front of it. Let $f_k(n)$ denote the number of orderings of $n$ with [i]badness[/i] $k$. Find $f_k(n)$. [hide=hint](Hint: Consider $g_k(n)$ as the number of orderings of n with [i]badness [/i]less than or equal to $k$)[/hide]

Let $P_1(x), P_2(x), \ldots, P_k(x)$ be monic polynomials of degree $13$ with integer coefficients. Suppose there are pairwise distinct positive integers $n_1, n_2, \ldots, n_k$ for which, for all positive integers $i$ and $j$ less than or equal to $k,$ the statement "$n_i$ divides $P_j(m)$ for every integer $m$" holds if and only if $i=j.$ Compute the largest possible value of $k.$

Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies \[ f(g(x)y + f(x)) = (y+2015)f(x) \] for every $x,y \in \mathbb{R^+} $ a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$ b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$

Let $S(r)$ denote the sum of the infinite geometric series $17 + 17r + 17r^2 +17r^3 + . . . $for $-1 < r < 1$. If $S(a) \times S(-a) = 2023$, find $S(a) + S(-a)$.

Let $f(x) = \frac{x+1}{x-1}$. Then for $x^2 \neq 1$, $f(-x)$ is $ \textbf{(A)}\ \frac{1}{f(x)}\qquad\textbf{(B)}\ -f(x)\qquad\textbf{(C)}\ \frac{1}{f(-x)}\qquad\textbf{(D)}\ -f(-x)\qquad\textbf{(E)}\ f(x) $