Can the graphs of a polynomial of degree 20 and the function $\displaystyle y={1\over x^{40}}$ have exactly 30 points of intersection? [i]Proposed by K. Kokhas[/i]

Find the sum of all integers $x$ for which there is an integer $y$, such that $x^3-y^3=xy+61$.

Let $A_1A_2\cdots A_n$ be a regular polygon with $n$ sides, $n \geqslant 3$. Initially, there are three ants standing at vertex $A_1$. Every minute, two ants simultaneously move to an adjacent vertex, but in different directions (one clockwise and the other counterclockwise), and the third stays at its current vertex. Determine all the values of $n$ for which, after some time, the three ants can meet at the same vertex of the polygon, different from $A_1$.

In $\triangle ABC$ with $AC>AB$, let $D$ be the foot of the altitude from $A$ to side $\overline{BC}$, and let $M$ be the midpoint of side $\overline{AC}$. Let lines $AB$ and $DM$ intersect at a point $E$. If $AC=8$, $AE=5$, and $EM=6$, find the square of the area of $\triangle ABC$. [i]Proposed by [b]DeToasty3[/b][/i]

Let $\mathcal{S}$ be a set of $2023$ points in a plane, and it is known that the distances of any two different points in $S$ are all distinct. Ivan colors the points with $k$ colors such that for every point $P \in \mathcal{S}$, the closest and the furthest point from $P$ in $\mathcal{S}$ also have the same color as $P$. What is the maximum possible value of $k$? [i]Proposed by Ivan Chan Kai Chin[/i]

Let $p$ and $q$ be different prime numbers. We are given an infinite decreasing arithmetic progression in which each of the numbers $p^{23}, p^{24}, q^{23}$ and $q^{24}$ occurs. Show that the numbers $p$ and $q$ also occur in this progression. [i]Proposed by A. Kuznetsov[/i]

For each positive integer $ n$, let $ f(n)$ denote the number of ways of representing $ n$ as a sum of powers of 2 with nonnegative integer exponents. Representations which differ only in the ordering of their summands are considered to be the same. For instance, $ f(4) \equal{} 4$, because the number 4 can be represented in the following four ways: 4; 2+2; 2+1+1; 1+1+1+1. Prove that, for any integer $ n \geq 3$ we have $ 2^{\frac {n^2}{4}} < f(2^n) < 2^{\frac {n^2}2}$.

Danial went to a fruit stall that sells apples, mangoes, and papayas. Each apple costs $3$ RM ,each mango costs $4$ RM , and each papaya costs $5$ RM . He bought at least one of each fruit, and paid exactly $50$ RM. What is the maximum number of fruits that he could have bought?

In the figures, the vertices are marked with a circle. The segments that join vertices are called paths. Non-negative integers are distributed to the vertices and, to the paths, the differences between the numbers at their ends. [img]https://cdn.artofproblemsolving.com/attachments/d/6/e6fce93719a5b35dbf34d58652b01a8631de57.gif[/img] We will say that a distribution of numbers is [i]graceful [/i] if all the numbers from $1$ to $n$ appear in the paths, where $n$ is the number of paths. The following is an example of graceful distribution: [img]https://cdn.artofproblemsolving.com/attachments/1/1/a8c2b4fde673ca902b655804c4f5321f9666e9.gif[/img] Give -if possible- a graceful distribution for the following figures. If you can't do it, show why.

Let $A$ be an non-invertible of order $n$, $n>1$, with the elements in the set of complex numbers, with all the elements having the module equal with 1 a)Prove that, for $n=3$, two rows or two columns of the $A$ matrix are proportional b)Does the conclusion from the previous exercise remains true for $n=4$?

Describe the amount of points $P(x, y)$ that are twice as far apart $A(3, 0)$ as to $0(0, 0)$.

Draw two tangents to the circle from point $P$ outside a circle, touching the circle at $A$ and $B$, then draw a secant line passes $P$, intersecting the circle at points $C$ and $D$ ($C$ is between $P$ and $D$). $Q$ is a point on the chord $CD$ such that $\angle DAQ=\angle PBC$. Prove that $\angle DBQ=\angle PAC$.

The acute angles of a right triangle are $a^{\circ}$ and $b^{\circ}$, where $a>b$ and both $a$ and $b$ are prime numbers. What is the least possible value of $b$? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }5\qquad\textbf{(D) }7\qquad\textbf{(E) }11$

Let $Q(x)$ be a cubic polynomial with integer coefficients. Suppose that a prime $p$ divides $Q(x_j)$ for $j = 1$ ,$2$ ,$3$ ,$4$ , where $x_1 , x_2 , x_3 , x_4$ are distinct integers from the set $\{0,1,\cdots, p-1\}$. Prove that $p$ divides all the coefficients of $Q(x)$.

The sum of $k$ consecutive integers is $90$. Then the sum of all possible values of $k$ is? (A)$89$ (B)$179$ (C)$168$ (D)$119$

Let $ABC$ be a triangle such that $AB = 7$, and let the angle bisector of $\angle BAC$ intersect line $BC$ at $D$. If there exist points $E$ and $F$ on sides $AC$ and $BC$, respectively, such that lines $AD$ and $EF$ are parallel and divide triangle $ABC$ into three parts of equal area, determine the number of possible integral values for $BC$.

Let $ABC$ be a triangle with $AB = 5, BC = 13,$ and $AC = 12$. Let $D$ be a point on minor arc $AC$ of the circumcircle of $ABC$ (endpoints excluded) and $P$ on $\overline{BC}$. Let $B_1, C_1$ be the feet of perpendiculars from $P$ onto $CD, AB$ respectively and let $BB_1, CC_1$ hit $(ABC)$ again at $B_2, C_2$ respectively. Suppose that $D$ is chosen uniformly at random and $AD, BC, B_2C_2$ concur at a single point. Compute the expected value of $BP/PC$. [i]Proposed by Grant Yu[/i]

The sum of two prime numbers is $85$. What is the product of these two prime numbers? $\textbf{(A) }85\qquad\textbf{(B) }91\qquad\textbf{(C) }115\qquad\textbf{(D) }133\qquad \textbf{(E) }166$

Each of the small squares of a $50\times 50$ table is coloured in red or blue. Initially all squares are red. A [i]step[/i] means changing the colour of all squares on a row or on a column. a) Prove that there exists no sequence of steps, such that at the end there are exactly $2011$ blue squares. b) Describe a sequence of steps, such that at the end exactly $2010$ squares are blue. [i]Adriana & Lucian Dragomir[/i]

Three circles are constructed for the triangle $ABC $: the circle ${{w} _ {A}} $ passes through the vertices $B $ and $C $ and intersects the sides $AB $ and $ AC $ at points ${{A} _ {1}} $ and ${{A} _ {2}} $ respectively, the circle ${{w} _ {B}} $ passes through the vertices $A $ and $C $ and intersects the sides $BA $ and $BC $ at the points ${{B} _ {1}} $ and ${{B} _ {2}} $, ${{w} _ {C}} $ passes through the vertices $A $ and $B $ and intersects the sides $CA $ and $CB $ at the points ${{C} _ {1}} $ and ${{C} _ {2}} $. Let ${{A} _ {1}} {{A} _ {2}} \cap {{B} _ {1}} {{B} _ {2}} = {C} '$, ${{A} _ {1}} {{A} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {B} '$ ta ${ {B} _ {1}} {{B} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {A} '$ is Prove that the perpendiculars, which are omitted from the points ${A} ', \, \, {B}', \, \, {C} '$ to the lines $BC $, $CA $ and $AB $ respectively intersect at one point. (Rudenko Alexander)

There are $n \geq 3$ islands in a city. Initially, the ferry company offers some routes between some pairs of islands so that it is impossible to divide the islands into two groups such that no two islands in different groups are connected by a ferry route. After each year, the ferry company will close a ferry route between some two islands $X$ and $Y$. At the same time, in order to maintain its service, the company will open new routes according to the following rule: for any island which is connected to a ferry route to exactly one of $X$ and $Y$, a new route between this island and the other of $X$ and $Y$ is added. Suppose at any moment, if we partition all islands into two nonempty groups in any way, then it is known that the ferry company will close a certain route connecting two islands from the two groups after some years. Prove that after some years there will be an island which is connected to all other islands by ferry routes.

In the beginning, there are $100$ cards on the table, and each card has a positive integer written on it. An odd number is written on exactly $43$ cards. Every minute, the following operation is performed: for all possible sets of $3$ cards on the table, the product of the numbers on these three cards is calculated, all the obtained results are summed, and this sum is written on a new card and placed on the table. A day later, it turns out that there is a card on the table, the number written on this card is divisible by $2^{2018}.$ Prove that one hour after the start of the process, there was a card on the table that the number written on that card is divisible by $2^{2018}.$

The figure shows a triangle, a circle circumscribed around it and the center of its inscribed circle. Using only one ruler (one-sided, without divisions), construct the center of the circumscribed circle.

In the triangle $ABC$ the angle bisector $AD$ is drawn, $E$ is the point of tangency of the inscribed circle to the side $BC$, $I$ is the center of the inscribed circle $\Delta ABC$. The point ${{A} _ {1}}$ on the circumscribed circle $\Delta ABC$ is such that $A {{A} _ {1}} || BC$. Denote by $T$ - the second point of intersection of the line $E {{A} _ {1}}$ and the circumscribed circle $\Delta AED$. Prove that $IT = IA$.

At a party, each man danced with exactly three women and each woman danced with exactly two men. Twelve men attended the party. How many women attended the party? $ \textbf{(A)}\ 8\qquad \textbf{(B)}\ 12\qquad \textbf{(C)}\ 16\qquad \textbf{(D)}\ 18\qquad \textbf{(E)}\ 24$