In a simple graph with 300 vertices no two vertices of the same degree are adjacent (boo hoo hoo). What is the maximal possible number of edges in such a graph?

Consider an integer $n\geq 2$ and an odd prime $p$. Let $U$ be the set of all positive integers $($strictly$)$ less than $p^n$ that are not divisible by $p$, and let $N$ be the number of elements of $U$. Does there exist permutation $a_1,a_2,\cdots a_N$ of the numbers in $U$ such that the sum $\sum_{k=1}^N a_ka_{k+1}$,where $a_{N+1}=a_1$, be divisible by $p^{n-1}$ but not by $p^n$? $Alexander \ Ivanov \, Bulgaria$

Let $ ABCD$ be a rhombus with $ \angle BAD \equal{} 60^{\circ}$. Points $ S$ and $ R$ are chosen inside the triangles $ ABD$ and $ DBC$, respectively, such that $ \angle SBR \equal{} \angle RDS \equal{} 60^{\circ}$. Prove that $ SR^2\geq AS\cdot CR$.

Given a triangle $ABC$ and a point $O$ on a plane. Let $\Gamma$ be the circumcircle of $ABC$. Suppose that $CO$ intersects with $AB$ at $D$, and $BO$ and $CA$ intersect at $E$. Moreover, suppose that $AO$ intersects with $\Gamma$ at $A,F$. Let $I$ be the other intersection of $\Gamma$ and the circumcircle of $ADE$, and $Y$ be the other intersection of $BE$ and the circumcircle of $CEI$, and $Z$ be the other intersection of $CD$ and the circumcircle of $BDI$. Let $T$ be the intersection of the two tangents of $\Gamma$ at $B,C$, respectively. Lastly, suppose that $TF$ intersects with $\Gamma$ again at $U$, and the reflection of $U$ w.r.t. $BC$ is $G$. Show that $F,I,G,O,Y,Z$ are concyclic.

Consider the following $2 \times 3$ arrangement of pegs on a board. Jane places three rubber bands on the pegs on the board such that the following conditions are satisfied: $~$ [center] (I) No two rubber bands cross each other. (II) Each peg has a rubber band wrapped around it [/center]$~$ How many distinct arrangements could Jane create exist? One acceptable arrangement is shown below. [asy] size(100); filldraw(circle((0,0),0.2),black); filldraw(circle((1,0),0.2),black); filldraw(circle((2,0),0.2),black); filldraw(circle((0,1),0.2),black); filldraw(circle((1,1),0.2),black); filldraw(circle((2,1),0.2),black); draw((0,1.2)--(1,1.2)); draw((0,0.8)--(1,0.8)); draw((1,0.2)--(2,0.2)); draw((1,-0.2)--(2,-0.2)); draw((0,0.2)--(2,1.2)); draw((0,-0.2)--(2,0.8)); [/asy] $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }8$

Point $D$ is selected inside acute $\triangle ABC$ so that $\angle DAC = \angle ACB$ and $\angle BDC = 90^{\circ} + \angle BAC$. Point $E$ is chosen on ray $BD$ so that $AE = EC$. Let $M$ be the midpoint of $BC$. Show that line $AB$ is tangent to the circumcircle of triangle $BEM$. [i]Proposed by Anton Trygub[/i]

Let $ABC$ be a triangle with $\angle A = 90^o$ and $\angle B < \angle C$. The tangent at $A$ to the circumcircle $k$ of $\vartriangle ABC$ intersects line $BC$ at $D$. Let $E$ be the reflection of $A$ in $BC$. Also, let $X$ be the feet of the perpendicular from $A$ to $BE$ and let $Y$ be the midpoint of $AX$. Line $BY$ meets $k$ again at $Z$. Prove that line $BD$ is tangent to the circumcircle of $\vartriangle ADZ$.

Let $S$ is a finite set with $n$ elements. We divided $AS$ to $m$ disjoint parts such that if $A$, $B$, $A \cup B$ are in the same part, then $A=B.$ Find the minimum value of $m$.

As in the picture below, the rectangle on the left hand side has been divided into four parts by line segments which are parallel to a side of the rectangle. The areas of the small rectangles are $A,B,C$ and $D$. Similarly, the small rectangles on the right hand side have areas $A^\prime,B^\prime,C^\prime$ and $D^\prime$. It is known that $A\leq A^\prime$, $B\leq B^\prime$, $C\leq C^\prime$ but $D\leq B^\prime$. [asy] import graph; size(12cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.3,xmax=12.32,ymin=-10.68,ymax=6.3; draw((0,3)--(0,0)); draw((3,0)--(0,0)); draw((3,0)--(3,3)); draw((0,3)--(3,3)); draw((2,0)--(2,3)); draw((0,2)--(3,2)); label("$A$",(0.86,2.72),SE*lsf); label("$B$",(2.38,2.7),SE*lsf); label("$C$",(2.3,1.1),SE*lsf); label("$D$",(0.82,1.14),SE*lsf); draw((5,2)--(11,2)); draw((5,2)--(5,0)); draw((11,0)--(5,0)); draw((11,2)--(11,0)); draw((8,0)--(8,2)); draw((5,1)--(11,1)); label("$A'$",(6.28,1.8),SE*lsf); label("$B'$",(9.44,1.82),SE*lsf); label("$C'$",(9.4,0.8),SE*lsf); label("$D'$",(6.3,0.86),SE*lsf); dot((0,3),linewidth(1pt)+ds); dot((0,0),linewidth(1pt)+ds); dot((3,0),linewidth(1pt)+ds); dot((3,3),linewidth(1pt)+ds); dot((2,0),linewidth(1pt)+ds); dot((2,3),linewidth(1pt)+ds); dot((0,2),linewidth(1pt)+ds); dot((3,2),linewidth(1pt)+ds); dot((5,0),linewidth(1pt)+ds); dot((5,2),linewidth(1pt)+ds); dot((11,0),linewidth(1pt)+ds); dot((11,2),linewidth(1pt)+ds); dot((8,0),linewidth(1pt)+ds); dot((8,2),linewidth(1pt)+ds); dot((5,1),linewidth(1pt)+ds); dot((11,1),linewidth(1pt)+ds); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy] Prove that the big rectangle on the left hand side has area smaller or equal to the area of the big rectangle on the right hand side, i.e. $A+B+C+D\leq A^\prime+B^\prime+C^\prime+D^\prime$.

There are $n$ instruments in the laboratory, each two of them can be connected with a wire. Moreover, if four devices $A, B, C, D$, are such that wires of $AB$, $BC$ and $CD$ are connected but there is no connected pair between $CA$, $AD$ and $DB$, a collapse occurs. A professor invented a wiring diagram that does not collapse. Coming to the laboratory, he found that the collapse has not yet occurred, but the devices are connected not according to his scheme. Prove that he can implement his scheme, each time connecting or disconnecting a pair of devices, so that the collapse won’t happen anytime.

The figure shows $9$ circles connected by $12$ lines. Georg must colour each circle either red or blue. He gets one point for each line connecting circles with different colours. How many points can he at most achieve? [img]https://cdn.artofproblemsolving.com/attachments/3/9/983d3c5755547246899891db141fe2383f3dc1.png[/img]

It is known that $x$ and $y$ are reals satisfying \[ 5x^2 + 4xy + 11y^2 = 3. \] Without using calculus (differentials/integrals), determine the maximum value of $xy - 2x + 5y$.

Show that the inequality \[\prod_{k=2}^n\left(1-\frac{1}{k^3}\right)>\frac12\] holds for every positive integer $n>1.$

Let $m$, $n$ be positive integers. Prove that, for some positive integer $a$, each of $\phi(a)$, $\phi(a+1)$, $\cdots$, $\phi(a+n)$ is a multiple of $m$.

Solve in $\mathbb{R}$ the equation $$\frac{1}{x}+\frac{1}{\lfloor x\rfloor} + \frac{1}{\{x\}} = 0.$$ [i]Mathematical Gazette[/i]

Solve in integers the following equation $\frac{1}{2}(x + y)(y + z)(z + x) + (x + y + z)^3 = 1 - xyz$.

Fix an integer $ b \geq 2$. Let $ f(1) \equal{} 1$, $ f(2) \equal{} 2$, and for each $ n \geq 3$, define $ f(n) \equal{} n f(d)$, where $ d$ is the number of base-$ b$ digits of $ n$. For which values of $ b$ does \[ \sum_{n\equal{}1}^\infty \frac{1}{f(n)} \] converge?

Find all functions $ f : R \to R$ that satisfies $$xf(y) - yf(x)= f\left(\frac{y}{x}\right)$$ for all $x, y \in R$.

Let $ABC$ be an acute-angled triangle with orthocenter $H$. Let $D$ be the foot of the altitude from $A$ to the line $BC$. Let $T$ be a point on the circle with diameter $AH$ such that this circle is internally tangent to the circumcircle of triangle $BDT$. Let $N$ be the midpoint of segment $AH$. Prove that $BT \perp CN$.

$1956$ points are chosen in a cube with edge $13$. Is it possible to fit inside the cube a cube with edge $1$ that would not contain any of the selected points?

Consider any complex units $Z,W$ with $\text{Im}\ Z\ge0,\text{Re}\,W\ge 0.$ Determine and draw the locus of all possible sums $S=Z+W$ in the complex plane.

In $\triangle{ABC}$, $\angle{C} = 90^{\circ}$ and $AB = 12$. Squares $ABXY$ and $ACWZ$ are constructed outside of the triangle. The points $X, Y, Z$, and $W$ lie on a circle. What is the perimeter of the triangle? $ \textbf{(A)}\ 12+9\sqrt{3}\qquad\textbf{(B)}\ 18+6\sqrt{3}\qquad\textbf{(C)}\ 12+12\sqrt{2}\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 32 $

Given two equilateral triangles $A_1B_1C_1$ and $A_2B_2C_2$ in the plane. (The vertices are mentioned counterclockwise.) We draw vectors $\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}$, from the arbitrary point $O$, equal to $\overrightarrow{A_1A_2}, \overrightarrow{B_1B_2}, \overrightarrow{C_1C_2}$ respectively. Prove that the triangle $ABC$ is equilateral.

On the side $AB$ of the parallelogram $ABCD$ we take the points $X$ and $Y$ such that the points $A$, $X$, $Y$ and $B$ appear in this order. The lines $DX$ and $CY$ intersect at the point $Z$. Suppose that the area of the triangle $\triangle XYZ$ is equal to the sum of the areas of the triangles $\triangle AXD$ and $\triangle CYB$. Prove that the area of the quadrilateral $XYCD$ is equal to $3$ times the area of the triangle $\triangle XYZ$.

Let $ABC$ be an isosceles triangle with base $AB$ and $D$ be a point on side $AB$ such that the incircle of triangle $ACD$ is congruent to the excircle of triangle $DCB$ across $C$. Prove that the diameter of each of these circles equals half the altitude of $\vartriangle ABC$ from $A$