Let $ABCD$ be a rectangle, and let $\omega$ be a circular arc passing through the points $A$ and $C$. Let $\omega_{1}$ be the circle tangent to the lines $CD$ and $DA$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Similiarly let $\omega_{2}$ be the circle tangent to the lines $AB$ and $BC$ and to the circle $\omega$, and lying completely inside the rectangle $ABCD$. Denote by $r_{1}$ and $r_{2}$ the radii of the circles $\omega_{1}$ and $\omega_{2}$, respectively, and by $r$ the inradius of triangle $ABC$. [b](a)[/b] Prove that $r_{1}+r_{2}=2r$. [b](b)[/b] Prove that one of the two common internal tangents of the two circles $\omega_{1}$ and $\omega_{2}$ is parallel to the line $AC$ and has the length $\left|AB-AC\right|$.

Define the function $m$ of the three real variables $x$, $y$, $z$ by $m$($x$,$y$,$z$) = max($x^2$,$y^2$,$z^2$), $x$, $y$, $z$ ∈ $R$. Determine, with proof, the minimum value of $m$ if $x$,$y$,$z$ vary in $R$ subject to the following restrictions: $x$ + $y$ + $z$ = 0, $x^2$ + $y^2$ + $z^2$ = 1.

Let $a_1, a_2, \ldots, a_{2024}$ be positive integers such that $a_{i+1}+1$ is a multiple of $a_i$ for all $i = 1, 2, \ldots , 2024$, with indices taken modulo $2024$. Find the maximum possible value of $a_1 + a_2 + \ldots + a_{2024}$. [i](Proposed by Ivan Chan Guan Yu)[/i]

Let $ABCDE$ be a regular pentagon, and $F$ some point on the arc $AB$ of the circumcircle of $ABCDE$. Show that \[ \frac{FD}{FE + FC} = \frac{FB + FA}{FD} = \frac{-1 + \sqrt{5}}{2}, \] and that $FD + FB + FA = FE + FC$.

Find the value of $x$ that satisfies $\log_{3}(\log_9x)=\log_9(\log_3x)$

Let $P$ be a plane. Prove that there is no function $f :P\rightarrow P$ where, for any convex quadrilateral $ABCD$, the points $f(A)$, $f(B)$, $f(C)$, $f (D)$ are the vertices of a concave quadrilateral. [i](tatari/nightmare)[/i]

Observe that the number $4$ is such that $4 \choose k$ $= \frac{4!}{k!(4-k)!}$ divisible by $k + 1$ for $k = 0,1,2,3$. Find all the natural numbers $n$ between $50$ and $90$ such that $n \choose k$ is divisible by $k + 1$ for $k = 0,1,2,..., n - 1$. Justify your answers.

Let $\Gamma = \{\varepsilon,0,00,\ldots\}$ be the set of all finite strings consisting of only zeroes. We consider $\textit{six-state unary DFAs}$ $D = (F,q_0,\delta)$ where $F$ is a subset of $Q = \{1,2,3,4,5,6\}$, not necessarily strict and possibly empty; $q_0\in Q$ is some $\textit{start state}$; and $\delta: Q\rightarrow Q$ is the $\textit{transition function}$. For each such DFA $D$, we associate a set $F_D\subseteq\Gamma$ as the set of all strings $w\in\Gamma$ such that \[\underbrace{\delta(\cdots(\delta(q_0))\cdots)}_{|w|\text{ applications}}\in F,\] We say a set $\mathcal D$ of DFAs is $\textit{diverse}$ if for all $D_1,D_2\in\mathcal D$ we have $F_{D_1}\neq F_{D_2}$. What is the maximum size of a diverse set?

Prove that for a triangle $ ABC $ with $ \angle BAC \ge 90^{\circ } , $ having circumradius $ R $ and inradius $ r, $ the following inequality holds: $$ R\sin A>2r $$ [i]Romeo Ilie[/i]

Let $ABC$ be a right triangle with a right angle at $C.$ Two lines, one parallel to $AC$ and the other parallel to $BC,$ intersect on the hypotenuse $AB.$ The lines split the triangle into two triangles and a rectangle. The two triangles have areas $512$ and $32.$ What is the area of the rectangle? [i]Author: Ray Li[/i]

Let $a_0, a_1, a_2, ... , a_n$ be real numbers, which fulfill the following two conditions: a) $0 = a_0 \leq a_1 \leq a_2 \leq ... \leq a_n$. b) For all $0 \leq i < j \leq n$ holds: $a_j - a_i \leq j-i$. Prove that $$\left( \displaystyle \sum_{i=0}^n a_i \right)^2 \geq \sum_{i=0}^n a_i^3.$$

The formula which expresses the relationship between $x$ and $y$ as shown in the accompanying table is: \[ \begin{tabular}[t]{|c|c|c|c|c|c|}\hline x&0&1&2&3&4\\\hline y&100&90&70&40&0\\\hline \end{tabular}\] $\textbf{(A)}\ y=100-10x \qquad \textbf{(B)}\ y=100-5x^2 \qquad \textbf{(C)}\ y=100-5x-5x^2 \qquad\\ \textbf{(D)}\ y=20-x-x^2 \qquad \textbf{(E)}\ \text{None of these}$

A positive integer is called lonely if the sum of the reciprocals of its positive divisors (including 1 and itself) is different from the sum of the reciprocals of the positive divisors of any positive integer. a) Prove that every prime number is lonely. b) Prove that there are infinitely many positive integers that are not lonely.

Let $ ABCDEF$ be a convex hexagon such that $ AB$ is parallel to $ DE$, $ BC$ is parallel to $ EF$, and $ CD$ is parallel to $ FA$. Let $ R_{A},R_{C},R_{E}$ denote the circumradii of triangles $ FAB,BCD,DEF$, respectively, and let $ P$ denote the perimeter of the hexagon. Prove that \[ R_{A} \plus{} R_{C} \plus{} R_{E}\geq \frac {P}{2}. \]

Consider three positive real numbers $a,b$ and $c$. Show that there cannot exist two distinct positive integers $m$ and $n$ such that both $a^m+b^m=c^m$ and $a^n+b^n=c^n$ hold.

We define $N$ as the set of natural numbers $n<10^6$ with the following property: There exists an integer exponent $k$ with $1\le k \le 43$, such that $2012|n^k-1$. Find $|N|$.

Let $n$ be the answer to this problem. In acute triangle $ABC$, point $D$ is located on side $BC$ so that $\angle BAD = \angle DAC$ and point $E$ is located on $AC$ so that $BE \perp AC$. Segments $BE$ and $AD$ intersect at $X$ such that $\angle BXD = n^o$: Given that $\angle XBA = 16^o$, find the measure of $\angle BCA$.

Determine all pairs $(k, n)$ of positive integers that satisfy $$1! + 2! + ... + k! = 1 + 2 + ... + n.$$

A $k$-coloring of a graph $G$ is a coloring of its vertices using $k$ possible colors such that the end points of any edge have different colors. We say a graph $G$ is uniquely $k$-colorable if one hand it has a $k$-coloring, on the other hand there do not exist vertices $u$ and $v$ such that $u$ and $v$ have the same color in one $k$-coloring and $u$ and $v$ have different colors in another $k$-coloring. Prove that if a graph $G$ with $n$ vertices $(n \ge 3)$ is uniquely $3$-colorable, then it has at least $2n-3$ edges.

Let $P$ be a point inside triangle $ABC$. Let $AP$ meet $BC$ at $A_1$, let $BP$ meet $CA$ at $B_1$, and let $CP$ meet $AB$ at $C_1$. Let $A_2$ be the point such that $A_1$ is the midpoint of $PA_2$, let $B_2$ be the point such that $B_1$ is the midpoint of $PB_2$, and let $C_2$ be the point such that $C_1$ is the midpoint of $PC_2$. Prove that points $A_2, B_2$, and $C_2$ cannot all lie strictly inside the circumcircle of triangle $ABC$. (Australia)

Prove that none of the numbers $2^{2^n}+ 1$, $n = 0, 1, 2, \dots$ is a perfect cube.

One point of the plane is called $rational$ if both coordinates are rational and $irrational$ if both coordinates are irrational. Check whether the following statements are true or false: [b]a)[/b] Every point of the plane is in a line that can be defined by $2$ rational points. [b]b)[/b] Every point of the plane is in a line that can be defined by $2$ irrational points. This maybe is not algebra so sorry if I putted it in the wrong category!

Let $M$ be an interior point of the tetrahedron $ABCD$. Prove that \[ \begin{array}{c}\ \stackrel{\longrightarrow }{MA} \text{vol}(MBCD) +\stackrel{\longrightarrow }{MB} \text{vol}(MACD) +\stackrel{\longrightarrow }{MC} \text{vol}(MABD) + \stackrel{\longrightarrow }{MD} \text{vol}(MABC) = 0 \end{array}\] ($\text{vol}(PQRS)$ denotes the volume of the tetrahedron $PQRS$).

Find all positive integers $n$ such that for any integer $k$ there exists an integer $a$ for which $a^3+a-k$ is divisible by $n$. [i]Warut Suksompong, Thailand[/i]

Let $AB$ be a segment of length $1.$ Let $\odot A, \odot B$ be circles with radius $\overline{AB}$ centered at $A, B.$ Denote their intersection points $C, D.$ Draw circles $\odot C, \odot D$ with radius $\overline{CD}.$ Denote their intersection points $C, D.$ Draw circles $\odot C, \odot D$ with radius $\overline{CD}.$ Denote the intersection points of $\odot C$ and $\odot D$ by $E, F.$ Draw circles $\odot E, \odot F$ with radius $\overline{EF}$ and denote their intersection points $G, H.$ Compute the area of the pentagon $ACFHE.$