Let $R$ be a commutative ring with $1$ such that the number of elements of $R$ is equal to $p^3$ where $p$ is a prime number. Prove that if the number of elements of $\text{zd}(R)$ be in the form of $p^n$ ($n \in \mathbb{N^*}$) where $\text{zd}(R) = \{a \in R \mid \exists 0 \neq b \in R, ab = 0\}$, then $R$ has exactly one maximal ideal.

Four circles $\omega,$ $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$ with the same radius are drawn in the interior of triangle $ABC$ such that $\omega_{A}$ is tangent to sides $AB$ and $AC$, $\omega_{B}$ to $BC$ and $BA$, $\omega_{C}$ to $CA$ and $CB$, and $\omega$ is externally tangent to $\omega_{A},$ $\omega_{B},$ and $\omega_{C}$. If the sides of triangle $ABC$ are $13,$ $14,$ and $15,$ the radius of $\omega$ can be represented in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

The incircle of triangle $ABC$ touches $BC$ at $D$ and $AC$ at $E$. Let $K$ be the point on $CB$ with $CK=BD$, and $L$ be the point on $CA$ with $AE=CL$. Lines $AK$ and $BL$ meet at $P$. If $Q$ is the midpoint of $BC$, $I$ the incenter, and $G$ the centroid of $\triangle ABC$, show that: $(a)$ $IQ$ and $AK$ are parallel, $(b)$ the triangles $AIG$ and $QPG$ have equal area.

The Fibonacci sequence is given by $x_1 = x_2 = 1$ and $x_{k+2} = x_{k+1} + x_k$ for each $k \in N$. (a) Prove that there are Fibonacci numbes that end in a $9$ in the decimal system. (b) Determine for which $n$ can a Fibonacci number end in $n$ $9$-s in the decimal system.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

Prove that exists a infinity of triplets $a, b, c\in\mathbb{R}$ satisfying simultaneously the relations $a+b+c=0$ and $a^4+b^4+c^4=50$. Moldova National Math Olympiad 2010, 12th grade

A rectangular box with side lengths $1$, $2$, and $16$ is cut into two congruent smaller boxes with integer side lengths. Compute the square of the largest possible length of the space diagonal of one of the smaller boxes. [i]2020 CCA Math Bonanza Lightning Round #2.2[/i]

[i]Nocycleland[/i] is a country with $500$ cities and $2013$ two-way roads, each one of them connecting two cities. A city $A$ [i]neighbors[/i] $B$ if there is one road that connects them, and a city $A$ [i]quasi-neighbors[/i] $B$ if there is a city $C$ such that $A$ neighbors $C$ and $C$ neighbors $B$. It is known that in Nocycleland, there are no pair of cities connected directly with more than one road, and there are no four cities $A$, $B$, $C$ and $D$ such that $A$ neighbors $B$, $B$ neighbors $C$, $C$ neighbors $D$, and $D$ neighbors $A$. Show that there is at least one city that quasi-neighbors at least $57$ other cities.

Prove that for any positive integer $n\ge $, $2 \cdot \sqrt3 \cdot \sqrt[3]{4} ...\sqrt[n-1]{n} > n$

The point P lies on the side $CD$ of the parallelogram $ABCD$ with $\angle DBA = \angle CBP$. Point $O$ is the center of the circle passing through the points $D$ and $P$ and tangent to the straight line $AD$ at point $D$. Prove that $AO = OC$.

Draw a tangent with positive slope to a parabola $y=x^2+1$. Find the $x$-coordinate such that the area of the figure bounded by the parabola, the tangent and the coordinate axisis is $\frac{11}{3}.$

In a fish shop with 28 kinds of fish, there are 28 fish sellers. In every seller, there exists only one type of each fish kind, depending on where it comes, Mediterranean or Black Sea. Each of the $k$ people gets exactly one fish from each seller and exactly one fish of each kind. For any two people, there exists a fish kind which they have different types of it (one Mediterranean, one Black Sea). What is the maximum possible number of $k$?

Let $S$ be a set of $\displaystyle { 2n \choose n } + 1$ real numbers, where $n$ is an positive integer. Prove that there exists a monotone sequence $\{a_i\}_{1\leq i \leq n+2} \subset S$ such that \[ |x_{i+1} - x_1 | \geq 2 | x_i - x_1 | , \] for all $i=2,3,\ldots, n+1$.

Let $D_{1}, D_{2}, . . . , D_{n}$ be rectangular parallelepipeds in space, with edges parallel to the $x, y, z$ axes. For each $D_{i}$, let $x_{i} , y_{i} , z_{i}$ be the lengths of its projections onto the $x, y, z$ axes, respectively. Suppose that for all pairs $D_{i}$ , $D_{j}$, if at least one of $x_{i} < x_{j}$ , $y_{i} < y_{j}$, $z_{i} < z_{j}$ holds, then $x_{i} \leq x_{j}$ , $y_{i} \leq y_{j}$, and $z_{i} < z_{j}$ . If the volume of the region $\bigcup^{n}_{i=1}{D_{i}}$ equals 1997, prove that there is a subset $\{D_{i_{1}}, D_{i_{2}}, . . . , D_{i_{m}}\}$ of the set $\{D_{1}, . . . , D_{n}\}$ such that $(i)$ if $k \not= l $ then $D_{i_{k}} \cap D_{i_{l}} = \emptyset $, and $(ii)$ the volume of $\bigcup^{m}_{k=1}{D_{i_{k}}}$ is at least 73.

Let $ n\ge 3 $ be a natural number and $ A_0A_1...A_{n-1} $ a regular polygon. Consider $ B_0 $ on the segment $ A_0A_1 $ such that $ A_0B_0<\frac{1}{2}A_0A_1; B_1 $ on $ A_1A_2 $ so that $ A_1B_1<\frac{1}{2} A_1A_2; $ etc.; $ B_{n-2} $ on $ A_{n-2}A_{n-1} $ so that $ A_{n-2}B_{n-2} <\frac{1}{2} A_{n-2}A_{n-1} , $ and $ B_{n-1} $ on $ A_{n-1}A_0 $ with $ A_{n-1}B_{n-1} <\frac{1}{2} A_{n-1}A_{0} . $ Show that the perimeter of any ploygon that has its vertices on the segments $ A_1B_1,A_2B_2,...,A_{n-1}B_{n-1}, $ is equal or greater than the perimeter of $ B_0B_1...B_{n-1} . $

Find the sum of all integers $n$ with this property: both $n$ and $n + 2021$ are perfect squares.

There are four points $A, B, C, D$ on a circle. Circles are drawn through each pair of neighboring points. Denote the intersection points of neighboring circles by $A_1, B_1, C_1, D_1$. (Some of these points may coincide with previously given ones.) Prove that points $A_1, B_1, C_1, D_1$ lie on one circle.

Each square of a $7 \times 8$ board is painted black or white, in such a way that each $3 \times 3$ subboard has at least two black squares that are neighboring. What is the least number of black squares that can be on the entire board? Clarification: Two squares are [i]neighbors [/i] if they have a common side.

Given a positive integer $N$. Prove that: there are infinitely elements of the [i]Fibonacci[/i] sequence that are divisible by $N$.

Given real numbers $b_0,b_1,\ldots, b_{2019}$ with $b_{2019}\neq 0$, let $z_1,z_2,\ldots, z_{2019}$ be the roots in the complex plane of the polynomial \[ P(z) = \sum_{k=0}^{2019}b_kz^k. \] Let $\mu = (|z_1|+ \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots, z_{2019}$ to the origin.  Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots, b_{2019}$ that satisfy \[ 1\leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019. \]

Find the sum of the solutions to $\dfrac{1}{1+\dfrac{1}{|x-25|}}=\frac{49}{50}$.

The figure below shows a smaller square within a larger square. Both squares have integer side lengths. The region inside the larger square but outside the smaller square has area $52$. Find the area of the larger square. [img]https://cdn.artofproblemsolving.com/attachments/a/f/2cb8c70109196bf30f88aef0c53bbac07d6cc3.png[/img]

Given a convex quadrilateral $ABCD$. The points $P$ and $Q$ are the midpoints of the diagonals $AC$ and $BD$ respectively. The line $PQ$ intersects the lines $AB$ and $CD$ at $N$ and $M$ respectively. Prove that the circumcircles of triangles $NAP$, $NBQ$, $MQD$, and $MPC$ have a common point.

The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught $n$ fish for various values of $n$. \[ \begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ \hline \text{number of contestants who caught} \ n \ \text{fish} & 9 & 5 & 7 & 23 & \dots & 5 & 2 & 1 \\ \hline \end{array} \] In the newspaper story covering the event, it was reported that (a) the winner caught 15 fish; (b) those who caught 3 or more fish averaged 6 fish each; (c) those who caught 12 or fewer fish averaged 5 fish each. What was the total number of fish caught during the festival?

Let $a_1, a_2, \ldots, a_8$ be real numbers. Prove that $$\sum_{i=1}^{8} \left( a_i^2 + a_i a_{i+2} \right) \geq \sum_{i=1}^{8} \left( a_i a_{i+1} + a_i a_{i+3} \right),$$ where the indices are taken modulo 8, i.e., $a_9 = a_1$, $a_{10} = a_2$, and $a_{11} = a_3$. In which cases does equality hold? [i]Proposed by Vukašin Pantelić and Andrija Živadinović[/i]