The area of a square inscribed in a semicircle is to the area of the square inscribed in the entire circle as: $ \textbf{(A)}\ 1: 2 \qquad\textbf{(B)}\ 2: 3 \qquad\textbf{(C)}\ 2: 5 \qquad\textbf{(D)}\ 3: 4 \qquad\textbf{(E)}\ 3: 5$

Determine the maximum value of the sum \[ S=\sum_{n=1}^{\infty}\frac{n}{2^n}(a_1 a_2 \dots a_n)^{\frac{1}{n}} \] over all sequences $a_1,a_2,a_3,\dots$ of nonnegative real numbers satisfying \[ \sum_{k=1}^{\infty}a_k=1. \]

Let $n\geq 2$ be a fixed positive integer. Let $\{a_1,a_2,...,a_n\}$ be fixed positive integers whose sum is $2n-1$. Denote by $S_{\mathbb{A}}$ the sum of elements of a set $A$. Find the minimal and maximal value of $S_{\mathbb{X}}\cdot S_{\mathbb{Y}}$ where $\mathbb{X}$ and $\mathbb{Y}$ are two sets with the property that $\mathbb{X}\cup \mathbb{Y}=\{a_1,a_2,...,a_n\}$ and $\mathbb{X}\cap \mathbb{Y}=\emptyset.$ [i]Note: $\mathbb{X}$ and $\mathbb{Y}$ can have multiple equal elements. For example, when $n=5$ and $a_1=...=a_4=1$ and $a_5=5$, we can consider $\mathbb{X}=\{1,1,1\}$ and $\mathbb{Y}=\{1,5\}$. Moreover, in this case, $S_\mathbb{X}=3$ and $S_{\mathbb{Y}}=6.$[/i]

The sides of a convex regular polygon of $L + M + N$ sides are to be given draw in three colors: $L$ of them with a red stroke, $M$ with a yellow stroke, and $N$ with a blue. Express, through inequalities, the necessary and sufficient conditions so that there is a solution (several, in general) to the problem of doing it without leaving two adjacent sides drawn with the same color.

Find all real numbers $a$ for which there exists a function $f$ defined on the set of all real numbers which takes as its values all real numbers exactly once and satisfies the equality $$ f(f(x))=x^2f(x)+ax^2 $$ for all real $x$.

Let $a_n = n(2n+1)$. Evaluate \[ \biggl | \sum_{1 \le j < k \le 36} \sin\bigl( \frac{\pi}{6}(a_k-a_j) \bigr) \biggr |. \]

Let $ ABCD$ be a regular tetrahedron, and let $ O$ be the centroid of triangle $ BCD$. Consider the point $ P$ on $ AO$ such that $ P$ minimizes $ PA \plus{} 2(PB \plus{} PC \plus{} PD)$. Find $ \sin \angle PBO$.

There are 2018 cards numbered from 1 to 2018. The numbers of the cards are visible at all times. Tito and Pepe play a game. Starting with Tito, they take turns picking cards until they're finished. Then each player sums the numbers on his cards and whoever has an even sum wins. Determine which player has a winning strategy and describe it. P.S. Proposed by yours truly :-D

Let $\phi(n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Evaluate \[\lim_{m \rightarrow \infty}\frac{\sum_{n = 1}^m \phi(60n)}{\sum_{n = 1}^m \phi(n)}\]

A whole number is written on the board. Its last digit is remembered is then erased and multiplied by $5$ added to the number that remained on the board after erasing. The number was originally written $7^{1998}$. After applying several such operations, can one get the number $1998^7$?

We draw a triangle inside of a circle with one vertex at the center of the circle and the other two vertices on the circumference of the circle. The angle at the center of the circle measures $75$ degrees. We draw a second triangle, congruent to the first, also with one vertex at the center of the circle and the other vertices on the circumference of the circle rotated $75$ degrees clockwise from the first triangle so that it shares a side with the first triangle. We draw a third, fourth, and fifth such triangle each rotated $75$ degrees clockwise from the previous triangle. The base of the fifth triangle will intersect the base of the first triangle. What is the degree measure of the obtuse angle formed by the intersection?

In triangle $ ABC$, $ AB\equal{}AC$, $ \angle A\equal{}40^\circ$. Point $ O$ is within the triangle with $ \angle OBC \cong \angle OCA$. The number of degrees in angle $ BOC$ is: $ \textbf{(A)}\ 110 \qquad \textbf{(B)}\ 35 \qquad \textbf{(C)}\ 140 \qquad \textbf{(D)}\ 55 \qquad \textbf{(E)}\ 70$

Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition: \[f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.\]

$2.$Let $ABC$ be an acute-angeled triangle , non-isosceles and with barycentre $G$ (which is , in fact , the intersection of the medians).Let $M$ be the midpoint of $BC$ , and let Ω be the circle with centre $G$ and radius $GM$ , and let $N$ be the point of intersection between Ω and $BC$ that is distinct from $M$.Let $S$ be the symmetric point of $A$ with respect to $N$ , that is , the point on the line $AN$ such that $AN=NS$. Prove that $GS$ is perpendicular to $BC$

Given the monic polynomial \begin{align*} P(x) = x^N +a_{N-1}x^{N-1} + \ldots + a_1 x + a_0 \in \mathbb{R}[x] \end{align*} of even degree $N$ $=$ $2n$ and having all real positive roots $x_i$, for $1 \le i \le N$. Prove, for any $c$ $\in$ $[0, \underset{1 \le i \le N}{\min} \{x_i \} )$, the following inequality \begin{align*} c + \sqrt[N]{P(c)} \le \sqrt[N]{a_0} \end{align*}

Prove that if $X{}$ is an infinite set of cardinality $\kappa$ then there is a collection $\mathcal{F}$ of subsets of $X$ such that[list] [*]For any $A\subseteq X$ with cardinality $\kappa$ there exists $F\in\mathcal{F}$ for which $A\cap F$ has cardinality $\kappa,$ and [*]$X$ cannot be written as the union of less than $\kappa$ sets from $\mathcal{F}$ and a single set of cardinality less than $\kappa$. [/list]

The sequence of nonnegative integers $F_0, F_1, F_2, \dots$ is defined recursively as $F_0 = 0$, $F_1 = 1$, and $F_{n+2}= F_{n+1} + F_{n}$ for all integers $n \geq 0$. Let $d$ be the largest positive integer such that, for all integers $n\geq 0$, $d$ divides $F_{n+2020}-F_n$. Compute the remainder when $d$ is divided by $1001$. [i]Proposed by Ankit Bisain[/i]

[b]p1.[/b] Lisa is playing the piano at a tempo of $80$ beats per minute. If four beats make one measure of her rhythm, how many seconds are in one measure? [b]p2.[/b] Compute the smallest integer $n > 1$ whose base-$2$ and base-$3$ representations both do not contain the digit $0$. [b]p3.[/b] In a room of $24$ people, $5/6$ of the people are old, and $5/8$ of the people are male. At least how many people are both old and male? [b]p4.[/b] Juan chooses a random even integer from $1$ to $15$ inclusive, and Gina chooses a random odd integer from $1$ to $15$ inclusive. What is the probability that Juan’s number is larger than Gina’s number? (They choose all possible integers with equal probability.) [b]p5.[/b] Set $S$ consists of all positive integers less than or equal to $ 2016$. Let $A$ be the subset of $S$ consisting of all multiples of $6$. Let $B$ be the subset of $S$ consisting of all multiples of $7$. Compute the ratio of the number of positive integers in $A$ but not $B$ to the number of integers in $B$ but not $A$. [b]p6.[/b] Three peas form a unit equilateral triangle on a flat table. Sebastian moves one of the peas a distance $d$ along the table to form a right triangle. Determine the minimum possible value of $d$. [b]p7.[/b] Oumar is four times as old as Marta. In $m$ years, Oumar will be three times as old as Marta will be. In another $n$ years after that, Oumar will be twice as old as Marta will be. Compute the ratio $m/n$. [b]p8.[/b] Compute the area of the smallest square in which one can inscribe two non-overlapping equilateral triangles with side length $ 1$. [b]p9.[/b] Teemu, Marcus, and Sander are signing documents. If they all work together, they would finish in $6$ hours. If only Teemu and Sander work together, the work would be finished in 8 hours. If only Marcus and Sander work together, the work would be finished in $10$ hours. How many hours would Sander take to finish signing if he worked alone? [b]p10.[/b]Triangle $ABC$ has a right angle at $B$. A circle centered at $B$ with radius $BA$ intersects side $AC$ at a point $D$ different from $A$. Given that $AD = 20$ and $DC = 16$, find the length of $BA$. [b]p11.[/b] A regular hexagon $H$ with side length $20$ is divided completely into equilateral triangles with side length $ 1$. How many regular hexagons with sides parallel to the sides of $H$ are formed by lines in the grid? [b]p12[/b]. In convex pentagon $PEARL$, quadrilateral $PERL$ is a trapezoid with side $PL$ parallel to side $ER$. The areas of triangle $ERA$, triangle $LAP$, and trapezoid $PERL$ are all equal. Compute the ratio $\frac{PL}{ER}$. [b]p13.[/b] Let $m$ and $n$ be positive integers with $m < n$. The first two digits after the decimal point in the decimal representation of the fraction $m/n$ are $74$. What is the smallest possible value of $n$? [b]p14.[/b] Define functions $f(x, y) = \frac{x + y}{2} - \sqrt{xy}$ and $g(x, y) = \frac{x + y}{2} + \sqrt{xy}$. Compute $g (g (f (1, 3), f (5, 7)), g (f (3, 5), f (7, 9)))$. [b]p15.[/b] Natalia plants two gardens in a $5 \times 5$ grid of points. Each garden is the interior of a rectangle with vertices on grid points and sides parallel to the sides of the grid. How many unordered pairs of two non-overlapping rectangles can Nataliia choose as gardens? (The two rectangles may share an edge or part of an edge but should not share an interior point.) PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The $7$-digit numbers $\underline{7}$ $ \underline{4}$ $ \underline{A}$ $ \underline{5}$ $ \underline{2}$ $ \underline{B}$ $ \underline{1}$ and $\underline{3}$ $ \underline{2}$ $ \underline{6}$ $ \underline{A}$ $ \underline{B}$ $ \underline{4}$ $ \underline{C}$ are each multiples of $3$. Which of the following could be the value of $C$? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }5\qquad \textbf{(E) }8$

Three different points $A$, $B$ and $C$ are placed in the plane. Construct a line $m$ through $C$ so that the product of the distances from $A$ and $B$ to $m$ has the maximal value. Is $m$ unique for every triple $A$, $B$ and $C$? (NB Vassiliev)

Prove that there exist at least six points with rational coordinates on the curve of the equation \[y^3=x^3+x+1370^{1370}\]

Let $ABC$ be a triangle with circumcircle $\omega$, incenter $I$, and $A$-excenter $I_A$. Let the incircle and the $A$-excircle hit $BC$ at $D$ and $E$, respectively, and let $M$ be the midpoint of arc $BC$ without $A$. Consider the circle tangent to $BC$ at $D$ and arc $BAC$ at $T$. If $TI$ intersects $\omega$ again at $S$, prove that $SI_A$ and $ME$ meet on $\omega$. [i]Amol Aggarwal.[/i]

Fifteen pairwise coprime positive integers chosen so that each of them less than 2010. Show that at least one of them is prime.

Let $a, b, c$ be positive real numbers such that $a^3 + b^3 + c^3 = 5abc$. Show that \[ \left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9. \]

Consider the solid with 4 triangles and 4 regular hexagons as faces, where each triangle borders 3 hexagons, and all the sides are of length 1. Compute the [i]square[/i] of the volume of the solid. Express your result in reduced fraction and concatenate the numerator with the denominator (e.g., if you think that the square is $\frac{1734}{274}$, then you would submit 1734274).