Any two islands of the Chaos Archipelago are connected by a bridge - a red bridge or a blue bridge. Show that at least one of the following two assertions holds: $\mathcal{A}_{1}$: For any two islands $a$ and $b$, we can reach $b$ from $a$ through at most $3$ red bridges (and no blue bridges). $\mathcal{A}_{2}$: For any two islands $a$ and $b$, we can reach $b$ from $a$ through at most $2$ blue bridges (and no red bridges). [i]Alternative formulation:[/i] Let $G$ be a graph. Prove that the diameter of $G$ is $\leq 3$ or the diameter of the complement of $G$ is $\leq 2$. [i]Note.[/i] This problem is the main Theorem in Frank Harary, Robert W. Robinson, [i]The Diameter of a Graph and its Complement[/i], The American Mathematical Monthly, Vol. 92, No. 3. (Mar., 1985), pp. 211-212. darij

Let $C_1$ and $C_2$ be two circles with centers $O_1$ and $O_2$, respectively, intersecting at $A$ and $B$. Let $l_1$ be the line tangent to $C_1$ passing trough $A$, and $l_2$ the line tangent to $C_2$ passing through $B$. Suppose that $l_1$ and $l_2$ intersect at $P$ and $l_1$ intersects $C_2$ again at $Q$. Show that $PO_1B$ and $PO_2Q$ are similar triangles. [i]Proposed by Pablo Valeriano[/i]

If in $\bigtriangleup ABC$, $AD$ is the altitude and $AE$ is the diameter of the circumcircle through $A$, then prove that $AB\cdot AC = AD \cdot AE$. Use this result to show that if $ABCD$ is a cyclic quadrilateral then show that $AC \cdot (AB \cdot BC + CD \cdot DA) = BD\cdot (DA\cdot AB + BC \cdot CD)$.

If 554 is the base $ b$ representation of the square of the number whose base $ b$ representation is 24, then $ b$, when written in base 10, equals $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 16$

Side $ AC$ of right triangle $ ABC$ is divide into $ 8$ equal parts. Seven line segments parallel to $ BC$ are drawn to $ AB$ from the points of division. If $ BC \equal{} 10$, then the sum of the lengths of the seven line segments: $ \textbf{(A)}\ \text{cannot be found from the given information} \qquad \textbf{(B)}\ \text{is }{33}\qquad \textbf{(C)}\ \text{is }{34}\qquad \textbf{(D)}\ \text{is }{35}\qquad \textbf{(E)}\ \text{is }{45}$

Given is a cube $3 \times 3 \times 3$ with $27$ unit cubes. In each such cube a positive integer is written. Call a $\textit {strip}$ a block $1 \times 1 \times 3$ of $3$ cubes. The numbers are written so that for each cube, its number is the sum of three other numbers, one from each of the three strips it is in. Prove that there are at least $16$ numbers that are at most $60$.

For each subset $A$ of $\{1,2, \dots, n \} $, let $M_A$ be the difference between the largest of the elements of $A$ and the smallest of the elements of $A $. Finds the sum of all values ​​of $M_A$ when all possible subsets $A$ of $\{1,2, \dots, n \} $ are considered.

There are $ n$ students; each student knows exactly $d $ girl students and $d $ boy students ("knowing" is a symmetric relation). Find all pairs $ (n,d) $ of integers .

Let $ABC$ be an acute triangle and let $O$ be its circumcentre. Now, let the diameter $PQ$ of circle $ABC$ intersects sides $AB$ and $AC$ in their interior at$ D$ and $E$, respectively. Now, let $F$ and $G$ be the midpoints of $CD$ and $BE$. Prove that $\angle FOG=\angle BAC$

Prove that there exists a positive constant $c$ such that the following statement is true: Consider an integer $n > 1$, and a set $\mathcal S$ of $n$ points in the plane such that the distance between any two different points in $\mathcal S$ is at least 1. It follows that there is a line $\ell$ separating $\mathcal S$ such that the distance from any point of $\mathcal S$ to $\ell$ is at least $cn^{-1/3}$. (A line $\ell$ separates a set of points S if some segment joining two points in $\mathcal S$ crosses $\ell$.) [i]Note. Weaker results with $cn^{-1/3}$ replaced by $cn^{-\alpha}$ may be awarded points depending on the value of the constant $\alpha > 1/3$.[/i] [i]Proposed by Ting-Feng Lin and Hung-Hsun Hans Yu, Taiwan[/i]

Find the maximum value of natural components of number $96$ that we can seperate such that all of them must be relatively prime number withh each other.

In parallelogram $ABCD$, $\overline{DE}$ is the altitude to the base $\overline{AB}$ and $\overline{DF}$ is the altitude to the base $\overline{BC}$. ['''Note:''' ''Both pictures represent the same parallelogram.''] If $DC=12$, $EB=4$, and $DE=6$, then $DF=$ [asy] unitsize(12); pair A,B,C,D,P,Q,W,X,Y,Z; A = (0,0); B = (12,0); C = (20,6); D = (8,6); W = (18,0); X = (30,0); Y = (38,6); Z = (26,6); draw(A--B--C--D--cycle); draw(W--X--Y--Z--cycle); P = (8,0); Q = (758/25,6/25); dot(A); dot(B); dot(C); dot(D); dot(W); dot(X); dot(Y); dot(Z); dot(P); dot(Q); draw(A--B--C--D--cycle); draw(W--X--Y--Z--cycle); draw(D--P); draw(Z--Q); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,NE); label("$D$",D,NW); label("$E$",P,S); label("$A$",W,SW); label("$B$",X,S); label("$C$",Y,NE); label("$D$",Z,NW); label("$F$",Q,E); [/asy] $\text{(A)}\ 6.4 \qquad \text{(B)}\ 7 \qquad \text{(C)}\ 7.2 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$

Let $\{fn\}$ be the Fibonacci sequence $\{1, 1, 2, 3, 5, \dots.\}. $ (a) Find all pairs $(a, b)$ of real numbers such that for each $n$, $af_n +bf_{n+1}$ is a member of the sequence. (b) Find all pairs $(u, v)$ of positive real numbers such that for each $n$, $uf_n^2 +vf_{n+1}^2$ is a member of the sequence.

Find the real numbers $x, y, z$ such that, $$\frac{1}{x}+\frac{1}{y+z}=\frac{1}{2}, \frac{1}{y}+\frac{1}{z+x}=\frac{1}{3}, \frac{1}{z}+\frac{1}{x+y}=\frac{1}{4}.$$

Prove that there exist an infinite number of triples $n-1 $,$n$,$n + 1$ such that (a) $n$ can be represented as the sum of two squares of natural numbers but neither of $n-1$ and $n+1$ can; (b) each of these three numbers can be represented as the sum of two squares. (V Senderov)

There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference. Note that we count as different ways that one way coincides with another way by rotating or reversing.

By composing a symmetry of axis $r$ with a right angle rotation around from a point $P$ that does not belong to the line, another movement $M$ results. Is $M$ an axis symmetry? Is there any line invariant through $M$?

Find all integers $n \ge 2$ for which there exists $n$ integers $a_1,a_2,\dots,a_n \ge 2$ such that for all indices $i \ne j$, we have $a_i \mid a_j^2+1$.

$ABC$ is an acute triangle and $AD$, $BE$ and $CF$ are the altitudes, with $H$ being the point of intersection of these altitudes. Points $A_1$, $B_1$, $C_1$ are chosen on rays $AD$, $BE$ and $CF$ respectively such that $AA_1 = HD$, $BB_1 = HE$ and $CC_1 =HF$. Let $A_2$, $B_2$ and $C_2$ be midpoints of segments $A_1D$, $B_1E$ and $C_1F$ respectively. Prove that $H$, $A_2$, $B_2$ and $C_2$ are concyclic. (Mykhailo Barkulov)

Consider $ \Lambda = \left\{ \lambda\in\text{Hol} \left[ \mathbb{C}\longrightarrow\mathbb{C} \right] |z\in\mathbb{C}\implies |\lambda (z)|\le e^{|\text{Im}(z)|} \right\} . $ Prove that $ g\in\Lambda $ implies $ g'\in\Lambda . $

Let $ABCD$ be a convex quadrilateral. A circle passing through the points $A$ and $D$ and a circle passing through the points $B$ and $C$ are externally tangent at a point $P$ inside the quadrilateral. Suppose that \[\angle{PAB}+\angle{PDC}\leq 90^\circ\qquad\text{and}\qquad\angle{PBA}+\angle{PCD}\leq 90^\circ.\] Prove that $AB+CD \geq BC+AD$. [i]Proposed by Waldemar Pompe, Poland[/i]

Let $ ABC$ be an acute-angled, non-isosceles triangle with orthocenter $H$, $M$ midpoint of side $AB$ and $w$ bisector of angle $\angle ACB$. Let $S$ be the point of intersection of the perpendicular bisector of side $AB$ with $w$ and $F$ the foot of the perpendicular from $H$ on $w$. Prove that the segments $MS$ and $MF$ are equal. [i](Karl Czakler)[/i]

The dollar is now worth $\frac{1}{980}$ ounce of gold. After the $n^{th}$ 7001 billion dollars bailout package passed by congress, the dollar gains $\frac{1}{2{}^2{}^{n-1}}$ of its $(n-1)^{th}$ value in gold. After four bank bailouts, the dollar is worth $\frac{1}{b}(1-\frac{1}{2^c})$ in gold, where $b, c$ are positive integers. Find $b + c$.

Integers are placed on every cell of an infinite checkerboard. For each cell if it contains integer $a$ then the sum of the numbers in the cell under it and the cell right to it is $2a+1$. Prove that in every infinite diagonal row of direction [i] top-right down-left [/i] all numbers are different.

The diagonals of the inscribed quadrilateral $ABCD$ meet at the point $M$, $\angle AMB = 60^o$. Equilateral triangles $ADK$ and $BCL$ are built outward on sides $AD$ and $BC$. Line $KL$ meets the circle circumscribed ariound $ABCD$ at points $P$ and $Q$. Prove that $PK = LQ$.