The transportation ministry has just decided to pay $80$ companies to repair $2400$ roads. These roads connects $100$ cities. Each road is between two cities and there is at most one road between any two cities. Each company must repair exactly $30$ roads, and each road is repaired by exactly one company. For a company to repair a road, it is necessary for the company to set up stations at the both cities on its endpoints. Prove that there are at least $8$ companies stationed at one city.

Let be a natural number $ n\ge 4, $ and a group $ G $ for which the applications $ \iota ,\eta : G\longrightarrow G $ defined by $ \iota (g) =g^n ,\eta (g) =g^{2n} $ are endomorphisms. Prove that $ G $ is commutative if $ \iota $ is injective or surjective. [i]Gh. Andrei[/i]

Find the maximum number of planes in the space, such there are $ 6$ points, that satisfy to the following conditions: [b]1.[/b]Each plane contains at least $ 4$ of them [b]2.[/b]No four points are collinear.

Let $ABCD$ be a cyclic quadrilateral wih both diagonals perpendicular to each other and intersecting at point $O$. Let $E,F,G,H$ be the orthogonal projections of $O$ on sides $AB,BC,CD,DA$ respectively. a. Prove that $\angle EFG + \angle GHE = 180^o$ b. Prove that $OE$ bisects angle $\angle FEH$ .

The numbers $1,2,\dots,9$ are randomly placed into the $9$ squares of a $3 \times 3$ grid. Each square gets one number, and each of the numbers is used once. What is the probability that the sum of the numbers in each row and each column is odd? $\textbf{(A) }1/21\qquad\textbf{(B) }1/14\qquad\textbf{(C) }5/63\qquad\textbf{(D) }2/21\qquad\textbf{(E) } 1/7$

Let $\mathcal{F}$ be a family of $4$-element subsets of a set of size $5^m$, where $m$ is a fixed positive integer. If the intersection of any two sets in $\mathcal{F}$ does not have size exactly $2$, find the maximal value of $|\mathcal{F}|$.

The set $N$ is partitioned into three subsets $A_1,A_2,A_3$. Prove that at least one of them has the following property: There exists a positive number $m$ such that for any $k$ one can find numbers $a_1 < a_2 < ... < a_k$ in that subset satisfying $a_{j+1} -a_j \le m$ for $j = 1,...,k -1$.

A certain state issues license plates consisting of six digits (from 0 to 9). The state requires that any two license plates differ in at least two places. (For instance, the numbers 027592 and 020592 cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.

Show that for any $n \not \equiv 0 \pmod{10}$ there exists a multiple of $n$ not containing the digit $0$ in its decimal expansion.

\[\int_1^\infty \frac{\lfloor x \rfloor}{9x^3-x}\mathrm dx\] [i]Proposed by Robert Trosten and Connor Gordon[/i]

In cyclic quadrilateral $ABCD$ with circumcenter $O$ and circumradius $R$, define $X=\overline{AB}\cap\overline{CD}$, $Y=\overline{AC}\cap \overline{BD}$, and $Z=\overline{AD}\cap\overline{BC}$. Prove that \[OX^2+OY^2+OZ^2\ge 2R^2+2[ABCD].\] [i]Rohan Bodke[/i]

Sam colors each tile in a 4 by 4 grid white or black. A coloring is called [i]rotationally symmetric[/i] if the grid can be rotated 90, 180, or 270 degrees to achieve the same pattern. Two colorings are called [i]rotationally distinct[/i] if neither can be rotated to match the other. How many rotationally distinct ways are there for Sam to color the grid such that the colorings are [i]not[/i] rotationally symmetric? [i]Proposed by Gabriel Wu[/i]

Consider a rectangle $R$ partitioned into $2016$ smaller rectangles such that the sides of each smaller rectangle is parallel to one of the sides of the original rectangle. Call the corners of each rectangle a vertex. For any segment joining two vertices, call it basic if no other vertex lie on it. (The segments must be part of the partitioning.) Find the maximum/minimum possible number of basic segments over all possible partitions of $R$.

Let all vertices of a convex quadrilateral $ABCD$ lie on the circumference of a circle with center $O$. Let $F$ be the second intersection point of the circumcircles of the triangles $ABO$ and $CDO$. Prove that the circle passing through the points $A, F$ and $D$ also passes through the intersection point of the segments $AC$ and $BD$. (A Zaslavskiy)

Two circles intersect at $M$ and $N$. A line through $M$ meets the circles at $A$ and $B$, with $M$ between $A$ and $B$. Let $C$ and $D$ be the midpoints of the arcs $AN$ and $BN$ not containing $M$, respectively, and $K$ and $L$ be the midpoints of $AB$ and $CD$, respectively. Prove that $CL=KL$.

Let $a_1, a_2, ... , a_{1000}$ be a sequence of integers such that $a_1=3, a_2=7$ and for all $n=2, 3, ... , 999$ $a_{n+1}-a_n=4(a_1+a_2)(a_2+a_3) ... (a_{n-1}+a_n)$. Find the number of indices $1\leq n\leq 1000$ for which $a_n+2018$ is a perfect square.

Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained? [i](A. Golovanov)[/i]

Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?

Square $ABCD$ has side length 2. A semicircle with diameter $AB$ is constructed inside the square, and the tangent to the semicircle from $C$ intersects side $AD$ at $E$. What is the length of $CE$? [asy]defaultpen(linewidth(0.8)); pair A=origin, B=(1,0), C=(1,1), D=(0,1), X=tangent(C, (0.5,0), 0.5, 1), F=C+2*dir(C--X), E=intersectionpoint(C--F, A--D); draw(C--D--A--B--C--E); draw(Arc((0.5,0), 0.5, 0, 180)); pair point=(0.5,0.5); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E));[/asy] $ \textbf{(A)}\frac{2+\sqrt5}2\qquad \textbf{(B)}\sqrt5\qquad \textbf{(C)}\sqrt6\qquad \textbf{(D)}\frac52\qquad \textbf{(E)}5-\sqrt5 $

An integer is [i]balance[/i] if the number of digit in its decimal representation is equal to the number of its distinct prime factors (For example, 15 is [i]balanced[/i], but not 49). Prove that there are [b]finite[/b] [i]balanced[/i] number.

Find all functions $f : R \to R$ such that $f( 2x^3 + f (y)) = y + 2x^2 f (x)$ for all real numbers $x, y$.

$ABCD$ is a convex quadrilateral with $AB=36$, $CD=9$, $DA=39$, and $BD=15$. Given that $\angle{C}$ is right, compute the area of $ABCD$. [i]2018 CCA Math Bonanza Team Round #4[/i]

Let $p$ be an odd prime number. Show that the smallest positive quadratic nonresidue of $p$ is smaller than $\sqrt{p}+1$.

Which of the following changes will result in an [i]increase[/i] in the period of a simple pendulum? (A) Decrease the length of the pendulum (B) Increase the mass of the pendulum (C) Increase the amplitude of the pendulum swing (D) Operate the pendulum in an elevator that is accelerating upward (E) Operate the pendulum in an elevator that is moving downward at constant speed.

Find the smallest integer $n > 3$ such that, for each partition of $\{3, 4,..., n\}$ in two sets, at least one of these sets contains three (not necessarily distinct) numbers $ a, b, c$ for which $ab = c$. (Alberto Alfarano)