Each of $K$ friends simultaneously learns one different item of news. They begin to phone one another to tell them their news. Each conversation lasts exactly one hour, during which time it is possible for two friends to tell each other all of their news. What is the minimum number of hours needed in order for all of the friends to know all of the news? Consider in this problem: (a) $K = 64$. (b) $K = 55$. (c) $K = 100$. (A Andjans, Riga) PS. (a) was the junior problem, (a),(b),(c) the senior one

Let $k$ be a positive real number. In the $X-Y$ coordinate plane, let $S$ be the set of all points of the form $(x,x^2+k)$ where $x\in\mathbb{R}$. Let $C$ be the set of all circles whose center lies in $S$, and which are tangent to $X$-axis. Find the minimum value of $k$ such that any two circles in $C$ have at least one point of intersection.

If $-1 < x < 1$ and $-1 < y < 1$, define the "relativistic sum'' $x \oplus y$ to be \[ x \oplus y = \frac{x + y}{1 + xy} \, . \] The operation $\oplus$ is commutative and associative. Let $v$ be the number \[ v = \frac{\sqrt[7]{17} - 1}{\sqrt[7]{17} + 1} \, . \] What is the value of \[ v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \oplus v \, ? \] (In this expression, $\oplus$ appears 13 times.)

The negation of the statement "all men are honest," is: $ \textbf{(A)}\ \text{no men are honest} \qquad\textbf{(B)}\ \text{all men are dishonest} \\ \textbf{(C)}\ \text{some men are dishonest} \qquad\textbf{(D)}\ \text{no men are dishonest} \\ \textbf{(E)}\ \text{some men are honest}$

[b]p1.[/b] Each point in the interior and on the boundary of a square of side $2$ inches is colored either red or blue. Prove that there exists at least one pair of points of the same color whose distance apart is not less than $-\sqrt5$ inches. [b]p2.[/b] $ABC$ is an equilateral triangle of altitude $h$. A circle with center $0$ and radius $h$ is tangent to side $AB$ at $Z$ and intersects side $AC$ in point $X$ and side $BC$ in point $Y$. Prove that the circular arc $XZY$ has measure $60^o$. [img]https://cdn.artofproblemsolving.com/attachments/b/e/ac70942f7a14cd0759ac682c3af3551687dd69.png[/img] [b]p3.[/b] Find all of the real and complex solutions (if any exist) of the equation $x^7 + 7^7 = (x + 7)^7$ [b]p4.[/b] The four points $A, B, C$, and $D$ are not in the same plane. Given that the three angles, angle $ABC$, angle $BCD$, and angle $CDA$, are all right angles, prove that the fourth angle, angle $DAB$, of this skew quadrilateral is acute. [b]p5.[/b] $A, B, C$ and $D$ are four positive whole numbers with the following properties: (i) each is less than the sum of the other three, and (ii) each is a factor of the sum of the other three. Prove that at least two of the numbers must be equal. (An example of four such numbers: $A = 4$, $B = 4$, $C = 2$, $D = 2$.) [b]p6.[/b] $S$ is a set of six points and $L$ is a set of straight line segments connecting certain pairs of points in $S$ so that each point of $S$ is connected with at least four of the other points. Let $A$ and $B$ denote two arbitrary points of $S$. Show that among the triangles having sides in $L$ and vertices in $S$ there are two with the properties: (i) The two triangles have no common vertex. (ii) $A$ is a vertex of one of the triangles, and $B$ is a vertex of the other. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider a triangle $ABC$ . The points $D,E$ are on sides $AB,AC$ such that $BDEC$ is a cyclic quadrilateral. Let $P$ be the intersection of $BE$ and $CD$. $H$ is a point on $AC$ such that $\angle PHA = 90^{\circ}$. Let $M,N$ be the midpoints of $AP,BC$. Prove that: $ ACD \sim MNH $.

Real numbers $x$ and $y$ satisfy the system of equalities $$\begin{cases} \sin x + \cos y = 1 \\ \cos x + \sin y = -1 \end{cases}$$ Prove that $\cos 2x = \cos 2y$.

Let $x, y, z$ be real numbers each of whose absolute value is different from $\frac{1}{\sqrt 3}$ such that $x + y + z = xyz$. Prove that \[\frac{3x - x^3}{1-3x^2} + \frac{3y - y^3}{1-3y^2} + \frac{3z -z^3}{1-3z^2} = \frac{3x - x^3}{1-3x^2} \cdot \frac{3y - y^3}{1-3y^2} \cdot \frac{3z - z^3}{1-3z^2}\]

Let be $a,b,c$ three positive integer.Prove that $4$ divide $a^2+b^2+c^2$ only and only if $a,b,c$ are even.

Find all positive integers $m, n$ such that $n + (n+1) + (n+2) + ...+ (n+m) = 1000$.

Find necessary and sufficient conditions on positive integers $m$ and $n$ so that \[\sum_{i=0}^{mn-1}(-1)^{\lfloor i/m\rfloor+\lfloor i/n\rfloor}=0.\]

Let $A_1, B_1, C_1$, be the feet of the altitudes of $\vartriangle ABC$ drawn from the vertices $A, B, C $ respectively, and let $M$ be the orthocenter (point of intersection of altitudes) of $\vartriangle ABC$. Assume that the orthic triangle (i.e. the triangle whose vertices are the feet of the altitudes of the original triangle) $A_1$,$B_1$,$C_1$ exists. Prove that each of the points $M$, $A$, $B$, and $C$ is the center of a circle tangent to all three sides (extended if necessary) of $\vartriangle A_1B_1C_1$. What is the difference in the behavior of acute and obtuse triangles $ABC$?

King Midas spent $\frac{100}{x}\%$ of his gold deposit yesterday. He is set to earn gold today. What percentage of the amount of gold King Midas currently has would he need to earn today to end up with as much gold as he started?

Find the sum of all possible values of $f(2)$ such that $f(x)f(y)-f(xy) = \frac{y}{x}+\frac{x}{y}$, for every positive real numbers $x,y$ $ \textbf{(A)}\ \frac{5}{2} \qquad\textbf{(B)}\ -\frac{5}{4} \qquad\textbf{(C)}\ \frac{5}{4} \qquad\textbf{(D)}\ \frac{3}{2} \qquad\textbf{(E)}\ \text{None} $

Solve in $ \mathbb{R}^2 $ the following equation. $$ 9^{\sqrt x} +9^{\sqrt{y}} +9^{1/\sqrt{xy}} =\frac{81}{\sqrt{x} +\sqrt{y} +1/\sqrt{xy}} $$ [i]O. Trofin[/i]

A survey of participants was conducted at the Olympiad. $ 90\%$ of the participants liked the first round, $60\%$ of the participants liked the second round, $90\%$ of the participants liked the opening of the Olympiad. Each participant was known to enjoy at least two of these three events. Determine the percentage of participants who rated all three events positively.

For every composite positive integer $n$, define $r(n)$ to be the sum of the factors in the prime factorization of $n$. For example, $r(50)=12$ because the prime factorization of $50$ is $ 2 \cdot 5^2 $, and $ 2 + 5 + 5 = 12 $. What is the range of the function $r$, $ \{ r(n) : n \ \text{is a composite positive integer} \} $? [b](A)[/b] the set of positive integers [b](B)[/b] the set of composite positive integers [b](C)[/b] the set of even positive integers [b](D)[/b] the set of integers greater than 3 [b](E)[/b] the set of integers greater than 4

Let $n \geq 3$ be an integer. At a MEMO-like competition, there are $3n$ participants, there are n languages spoken, and each participant speaks exactly three different languages. Prove that at least $\left\lceil\frac{2n}{9}\right\rceil$ of the spoken languages can be chosen in such a way that no participant speaks more than two of the chosen languages. [b]Note.[/b] $\lceil x\rceil$ is the smallest integer which is greater than or equal to $x$.

The sequence of positive integers $a_1$, $a_2$,$...$ is such that for each $n = 1$,$2$, $...$ the quadratic equation $$a_{n+2}x^2 + a_{n+1}x+ a_n = 0$$ has a real root. Can the sequence consist of (a) $10 $ terms, (b) an infinite number of terms? (A Shapovalov)

How many factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1$, $2$, $3$, $4$, $6$, and $12$.) $\textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 8 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

Given an acute and scalene triangle $ ABC$, let $ H$ be its orthocenter, $ O$ its circumcenter, $ E$ and $ F$ the feet of the altitudes drawn from $ B$ and $ C$, respectively. Line $ AO$ intersects the circumcircle of the triangle again at point $ G$ and segments $ FE$ and $ BC$ at points $ X$ and $ Y$ respectively. Let $ Z$ be the point of intersection of line $ AH$ and the tangent line to the circumcircle at $ G$. Prove that $ HX$ is parallel to $ YZ$.

A game is played on a $1 \times 1000$ board. There are n chips, all of which are initially in a box near the board. Two players move in turn. The first may choose $17$ chips or less, from either on or off the board. She then puts them into unoccupied cells on the board so that there is no more than one chip in each of the cells. The second player may take off the board any number of chips occupying consecutive cells and put them back in the box. The first player wins if she can put all n chips on the board so that they occupy consecutive cells. (a) Show that she can win if $n = 98$. (b) For what maximal value of $n$ can she win? (A Shapovalov)

Let $ x \in \mathbb{Q}^+$. Prove that there exits $\alpha_1,\alpha_2,...,\alpha_k \in \mathbb{N}$ and pairwe distinct such that \[x= \sum_{i=1}^{k} \frac{1}{\alpha_i}\]

Prove that there exists a positive integer $n$, such that the number $k^2+k+n$ does not have a prime divisor less than $2008$ for any integer $k$.

Let $ABCD$ be a convex quadrilateral, and let $P$, $Q$, $R$, and $S$ be points on the sides $AB$, $BC$, $CD$, and $DA$, respectively. Let the line segment $PR$ and $QS$ meet at $O$. Suppose that each of the quadrilaterals $APOS$, $BQOP$, $CROQ$, and $DSOR$ has an incircle. Prove that the lines $AC$, $PQ$, and $RS$ are either concurrent or parallel to each other.