For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Let $A$ be a ring and let $D$ be the set of its non-invertible elements. If $a^2=0$ for any $a \in D,$ prove that: [b]a)[/b] $axa=0$ for all $a \in D$ and $x \in A$; [b]b)[/b] if $D$ is a finite set with at least two elements, then there is $a \in D,$ $a \neq 0,$ such that $ab=ba=0,$ for every $b \in D.$ [i]Ioan Băetu[/i]

In rhombus ABCD, $\angle A = 60^\circ$. Rhombus $BEFG$ is constructed, where $E$ and $G$ are the midpoints of $BC$ and $AB$, respectively. Rhombus $BHIJ$ is constructed, where $H$ and $J$ are the midpoints of $BE$ and $BG$, respectively. This process is repeated forever. If the area of $ABCD$ and the sum of the areas of all of the rhombi are both integers, compute the smallest possible value of $AB$.

Three sequences $(a_0, a_1, \ldots, a_n)$, $(b_0, b_1, \ldots, b_{n})$, $(c_0, c_1, \ldots, c_{2n})$ of non-negative real numbers are given such that for all $0\leq i,j\leq n$ we have $a_ib_j\leq (c_{i+j})^2$. Prove that $$\sum_{i=0}^n a_i\cdot\sum_{j=0}^n b_j\leq \left( \sum_{k=0}^{2n} c_k\right)^2.$$

Is there a convex pentagon in which each diagonal is equal to a side?

For a graph $G$ on $n$ vertices, let $P_G(x)$ be the unique polynomial of degree at most $n$ such that for each $i=0,1,2,\dots,n$, $P_G (i)$ equals the number of ways to color the vertices of the graph $G$ with $i$ distinct colors such that no two vertices connected by an edge have the same color. For each integer $3\le k \le 2017$, define a $k$-[i]tasty[/i] graph to be a connected graph on $2017$ vertices with $2017$ edges and a cycle of length $k$. Let the [i]tastiness[/i] of a $k$-tasty graph $G$ be the number of coefficients in $P_G(x)$ that are odd integers, and let $t$ be the minimal tastiness over all $k$-tasty graphs with $3\le k \le 2017$. Determine the sum of all integers $b$ between $3$ and $2017$ inclusive for which there exists a $b$-tasty graph with tastiness $t$. [i]Proposed by Vincent Huang[/i]

In the triangle $ABC$ it is known that $BC = 5, AC - AB = 3$. Prove that $r <2$ . (here $r$ is the radius of the circle inscribed in the triangle $ABC$). (Mykola Moroz)

Assume that every root of polynomial $P(x) = x^d - a_1x^{d-1} + ... + (-1)^{d-k}a_d$ is in $[0,1]$. Show that for every $k = 1,2,...,d$ the following inequality holds: $ a_k - a_{k+1} + ... + (-1)^{d-k}a_d \geq 0 $

In each cell, a strip of length $100$ is worth a chip. You can change any $2$ neighboring chips and pay $1$ rouble, and you can also swap any $2$ chips for free, between which there are exactly $4$ chips. For what is the smallest amount of rubles you can rearrange chips in reverse order?

For any permutation $p$ of set $\{1, 2, \ldots, n\}$, define $d(p) = |p(1) - 1| + |p(2) - 2| + \ldots + |p(n) - n|$. Denoted by $i(p)$ the number of integer pairs $(i, j)$ in permutation $p$ such that $1 \leqq < j \leq n$ and $p(i) > p(j)$. Find all the real numbers $c$, such that the inequality $i(p) \leq c \cdot d(p)$ holds for any positive integer $n$ and any permutation $p.$

The dates of three Sundays of a month were even numbers. What day of the week was the $20$th of the month?

A rectangular prism with positive integer side lengths formed by stacking unit cubes is called [i]bipartisan[/i] if the same number of unit cubes can be seen on the surface as those which cannot be seen on the surface. How many non-congruent bipartisan rectangular prisms are there? [i]2018 CCA Math Bonanza Team Round #8[/i]

In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis

In triangle $ABC, \angle B = 60^o$, $O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$.The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$.

Find all triples $(a,b,c)$ of positive integers such that the product of any two of them when divided by the third leaves the remainder $1$.

$A$, $B$, $C$, and $D$ are points on a circle, and segments $\overline{AC}$ and $\overline{BD}$ intersect at $P$, such that $AP=8$, $PC=1$, and $BD=6$. Find $BP$, given that $BP<DP$.