Let $ABC$ be an acute triangle with the circumcircle $k$ and incenter $I$. The perpendicular through $I$ in $CI$ intersects segment $[BC]$ in $U$ and $k$ in $V$. In particular $V$ and $A$ are on different sides of $BC$. The parallel line through $U$ to $AI$ intersects $AV$ in $X$. Prove: If $XI$ and $AI$ are perpendicular to each other, then $XI$ intersects segment $[AC]$ in its midpoint $M$. [i](Notation: $[\cdot]$ denotes the line segment.)[/i]

The circle with radius $4$ passing through $A$ and $B$ is tangent to the circle with radius $3$ passing through $A$ and $C$, where $|AB|=2$. If the line $BC$ is tangent to the second circle, what is $|BC|$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 2 + \dfrac{\sqrt{43}}2 \qquad\textbf{(C)}\ \dfrac 52 \qquad\textbf{(D)}\ 4 + \sqrt 9 \qquad\textbf{(E)}\ \sqrt 7 $

[u]Set 2[/u] [b]p4.[/b] $\vartriangle ABC$ is an isosceles triangle with $AB = BC$. Additionally, there is $D$ on $BC$ with $AC = DA = BD = 1$. Find the perimeter of $\vartriangle ABC$. [b]p5[/b]. Let $r$ be the positive solution to the equation $100r^2 + 2r - 1 = 0$. For an appropriate $A$, the infinite series $Ar + Ar^2 + Ar^3 + Ar^4 +...$ has sum $1$. Find $A$. [b]p6.[/b] Let $N(k)$ denote the number of real solutions to the equation $x^4 -x^2 = k$. As $k$ ranges from $-\infty$ to $\infty$, the value of $N(k)$ changes only a finite number of times. Write the sequence of values of $N(k)$ as an ordered tuple (i.e. if $N(k)$ went from $1$ to $3$ to $2$, you would write $(1, 3, 2)$). PS. You should use hide for answers.

Find all positive integers $n$ for which all positive divisors of $n$ can be put into the cells of a rectangular table under the following constraints: [list] [*]each cell contains a distinct divisor; [*]the sums of all rows are equal; and [*]the sums of all columns are equal. [/list]

The value of $ x$ at the intersection of $ y\equal{}\frac{8}{x^2\plus{}4}$ and $ x\plus{}y\equal{}2$ is: $ \textbf{(A)}\ \minus{}2\plus{}\sqrt{5} \qquad\textbf{(B)}\ \minus{}2\minus{}\sqrt{5} \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{none of these}$

Find all integers $k\ge 3$ with the following property: There exist integers $m,n$ such that $1<m<k$, $1<n<k$, $\gcd (m,k)=\gcd (n,k) =1$, $m+n>k$ and $k\mid (m-1)(n-1)$.

There are $4n$ pebbles of weights $1, 2, 3, \dots, 4n.$ Each pebble is coloured in one of $n$ colours and there are four pebbles of each colour. Show that we can arrange the pebbles into two piles so that the following two conditions are both satisfied: [list] [*]The total weights of both piles are the same. [*] Each pile contains two pebbles of each colour. [/list] [i]Proposed by Milan Haiman, Hungary and Carl Schildkraut, USA[/i]

Find all natural numbers $n$ such that $n$ equals the cube of the sum of its digits.

Find the positive integer n such that $10^n$ cubic centimeters is the same as 1 cubic kilometer.

Find all polynomials of the form $$P_n(x)=n!x^n+a_{n-1}x^{n-1}+\dots+a_1x+(-1)^n(n+1)$$ with integer coefficients, having $n$ real roots $x_1,\dots,x_n$ satisfying $k \leq x_k \leq k+1$ for $k=1, \dots,n$.

1.14. Let P and Q be interior points of triangle ABC such that \ACP = \BCQ and \CAP = \BAQ. Denote by D;E and F the feet of the perpendiculars from P to the lines BC, CA and AB, respectively. Prove that if \DEF = 90, then Q is the orthocenter of triangle BDF.

Let $a,b,c$ be distinct positive real numbers such that $\frac{1}{1+a}+\frac{1}{1+b}+\frac{1}{1+c}\leq 1$. Prove that $$2\left(\sqrt{\frac{a+b}{ac}}+\sqrt{\frac{b+c}{ba}}+\sqrt{\frac{c+a}{cb}}\right)<\frac{a^3}{(a-b)(a-c)}+\frac{b^3}{(b-c)(b-a)}+\frac{c^3}{(c-a)(c-b)}.$$

A non-isosceles trapezoid $ABCD$ ($AB // CD$) is given. An arbitrary circle passing through points $A$ and $B$ intersects the sides of the trapezoid at points $P$ and $Q$, and the intersect the diagonals at points $M$ and $N$. Prove that the lines $PQ, MN$ and $CD$ are concurrent.

$ABCD$ is a four digit number ($A\neq0$) such that both $ABC$ and $BCD$ are divisible by $9$ ($ABCD$ is not necessarily divisible by $9$, and $B,C,D$ may be $0$). Compute the number of four digit numbers satisfying this property.

Find all the pairs of natural numbers $(a, b),$ such that \[a!+1=(a+1)^{(2^b)}\]

Let $h$ be a semicircle with diameter $AB$. The two circles $k_1$ and $k_2$, $k_1 \ne k_2$, touch the segment $AB$ at the points $C$ and $D$, respectively, and the semicircle $h$ fom the inside at the points $E$ and $F$, respectively. Prove that the four points $C$, $D$, $E$ and $F$ lie on a circle. [i](Walther Janous)[/i]