Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$

A unit square is divided into polygons, so that all sides of a polygon are parallel to sides of the given square. If the total length of the segments inside the square (without the square) is $2n$ (where $n$ is a positive real number), prove that there exists a polygon whose area is greater than $\frac{1}{(n+1)^2}$.

Let's construct a family $\{K_n\}$ of geometric figures following the pattern shown in pictures: [center][img]https://cdn.artofproblemsolving.com/attachments/4/1/76d6cf2b7ec3bd69de7bf33e2a382885f744a0.png[/img][/center] where each hexagon (like the starting one) is constructed by cutting the two corners tops of a square, in such a way that the two figures removed are identical isosceles triangles, and the three resulting upper sides have the same length. Continuing like this, a pattern is produced with which we can build the figures $K_n$, for integer $n \ge 0$ . Then, we denote by $P_n$ and $A_n$ the perimeter and area of the figure $K_n$, respectively. If the side of square to build $K_0$ measures $x$: (a) Calculate $P_0$ and $A_0$ (in terms of the length $x$). (b) Find an explicit formula for $P_n$, and for $A_n$, in terms of $x$ and of $n$. Simplify your answers. (c) If $P_{2022} = A_{2022}$, find the measure of the six sides of the figure $K_0$, in its simplest form.

For the real pairs $(x,y)$ satisfying the equation $x^2 + y^2 + 2x - 6y = 6$, which of the following cannot be equal to $(x-1)^2 + (y-2)^2$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 23 \qquad\textbf{(E)}\ 30 $

Let $ABC$ be a triangle and $M$ be the midpoint of side $BC$. The perpendicular bisector of $BC$ intersects the circumcircle of $\triangle ABC$ at points $K$ and $L$ ($K$ and $A$ lie on the opposite sides of $BC$). A circle passing through $L$ and $M$ intersects $AK$ at points $P$ and $Q$ ($P$ lies on the line segment $AQ$). $LQ$ intersects the circumcircle of $\triangle KMQ$ again at $R$. Prove that $BPCR$ is a cyclic quadrilateral.

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

$6$ people stand in a circle with water guns. Each person randomly selects another person to shoot. What is the probability that no pair of people shoots at each other?

Let $ABC$ be a triangle with $AC > AB$. Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$. Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$. Let $Z$ be the intersection point of $XY$ and $BC$. Determine the value of $\frac{BZ}{ZC}$.

If the line $L$ in the $xy$-plane has half the slope and twice the y-intercept of the line $y = \frac{2}{3} x + 4$, then an equation for $L$ is: $ \textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad \textbf{(B)}\ y = \frac{4}{3} x + 2 \qquad \textbf{(C)}\ y = \frac{1}{3} x + 4 \qquad\\ \textbf{(D)}\ y = \frac{4}{3} x + 4 \qquad \textbf{(E)}\ y = \frac{1}{3} x + 2 $

Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying \[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \] for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)

A domino is a $1 \times 2$ (or 2 $\times 1$) rectangular piece; namely, made up of two squares. There is an $8 \times 8$ board such that each domino can be cover exactly two of its squares. John places $n$ dominoes on the board, so that each one covers exactly two squares of the board and it is no longer possible to place a piece more without overlapping with any of those already placed. Determine the smallest value of $n$ for which the described situation is possible.

Let $p\ge 5$ be a prime number. Consider the function given by the formula $$f (x_1,..., x_p) = x_1 + 2x_2 +... + px_p.$$ Let $A_k$ denote the set of all these permutations $(a_1,..., a_p)$ of the set $\{1,..., p\}$, for integer number $f (a_1,..., a_p) - k$ is divisible by $p$ and $a_i \ne i$ for all $i \in \{1,..., p\}$. Prove that the sets $A_1$ and $A_4$ have the same number of elements.

Characterize all increasing sequences $(s_n)$ of positive reals for which there exists a set $A\subset \mathbb{R}$ with positive measure such that $\lambda(A\cap I)<\frac{s_n}{n}$ holds for every interval $I$ with length $1/n$, where $\lambda$ denotes the Lebesgue measure.

Let $H$ be the orthocenter of triangle $ABC$, and $D$ be the foot of $A$ onto $BC$. Given that $DB=3$, $DH=2$, and $DC=6$, calculate $HA$.

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect in point $M$. The angle bisector of $\angle ACD$ intersects the ray $\overrightarrow{BA}$ in point $K$. If $MA.MC+MA.CD=MB.MD$, prove that $\angle BKC=\angle CDB$.

The Fibonacci numbers are a sequence of numbers defined recursively as follows: $F_1=1$, $F_2=1$, and $F_n=F_{n-1}+F_{n-2}$. Using this definition, compute the sum $$\sum_{k=1}^{10}\frac{F_k}{F_{k+1}F_{k+2}}.$$

Let $ABCD$ be a trapezoid with $AB\parallel CD$. The bisectors of $\angle CDA$ and $\angle DAB$ meet at $E$, the bisectors of $\angle ABC$ and $\angle BCD$ meet at $F$, the bisectors of $\angle BCD$ and $\angle CDA$ meet at $G$, and the bisectors of $\angle DAB$ and $\angle ABC$ meet at $H$. Quadrilaterals $EABF$ and $EDCF$ have areas $24$ and $36$, respectively, and triangle $ABH$ has area $25$. Find the area of triangle $CDG$.

Diameter $ACE$ is divided at $C$ in the ratio $2:3$. The two semicircles, $ABC$ and $CDE$, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is [asy]pair A,B,C,D,EE; A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0); fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray); draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)); draw(circle((5,0),5)); dot(A); dot(B); dot(C); dot(D); dot(EE); label("$A$",A,W); label("$B$",B,N); label("$C$",C,E); label("$D$",D,N); label("$E$",EE,W); [/asy] $\textbf{(A)}\ 2:3 \qquad \textbf{(B)}\ 1:1 \qquad \textbf{(C)}\ 3:2 \qquad \textbf{(D)}\ 9:4 \qquad \textbf{(E)}\ 5:2$

The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star. [img]https://1.bp.blogspot.com/-Rso2aWGUq_k/XzcAm4BkAvI/AAAAAAAAMW0/277afcqTfCgZOHshf_6ce2XpinWWR4SZACLcBGAsYHQ/s0/2006%2BMohr%2Bp1.png[/img]

Billiam is distributing his ample supply of balls among an ample supply of boxes. He distributes the balls as follows: he places a ball in the first empty box, and then for the greatest positive integer n such that all $n$ boxes from box $1$ to box $n$ have at least one ball, he takes all of the balls in those $n$ boxes and puts them into box $n +1$. He then repeats this process indefinitely. Find the number of repetitions of this process it takes for one box to have at least $2022$ balls.

Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot 7^y = z^3$

Naomi has a class of $100$ students who will compete with each other in five teams. Once the teams are made, each student will shake hands with every other student, except the students in his or her own team. Naomi chooses to partition the students into teams so as to maximize the number of handshakes. How many handshakes will there be?

Let $n, m, k$ be natural numbers, with $m > n$. Which of the numbers is greater: $$\sqrt{n+\sqrt{m+\sqrt{n+...}}}\,\,\, or \,\,\,\, \sqrt{m+\sqrt{n+\sqrt{m+...}}}\,\, ?$$ Note: Each of the expressions contains $k$ square root signs; $n, m$ alternate within each expression. (N. Kurlandchik)

Let $O$ be a point of three-dimensional space and let $l_1, l_2, l_3$ be mutually perpendicular straight lines passing through $O$. Let $S$ denote the sphere with center $O$ and radius $R$, and for every point $M$ of $S$, let $S_M$ denote the sphere with center $M$ and radius $R$. We denote by $P_1, P_2, P_3$ the intersection of $S_M$ with the straight lines $l_1, l_2, l_3$, respectively, where we put $P_i \neq O$ if $l_i$ meets $S_M$ at two distinct points and $P_i = O$ otherwise ($i = 1, 2, 3$). What is the set of centers of gravity of the (possibly degenerate) triangles $P_1P_2P_3$ as $M$ runs through the points of $S$?