Find the sum of the answers to all even numbered Short Answer problems, with the exception of #26, rounded to the nearest tenth. [i](.7 points)[/i]

Find all integers $n$ such that if \[ 1 = d_1 < d_2 < \cdots < d_{k-1} < d_k = n \] are the divisors of $n$, then the sequence \[ d_2 - d_1,\, d_3 - d_2,\, \ldots,\, d_k - d_{k-1} \] forms a permutation of an arithmetic progression. [i](Kritesh Dhakal, Nepal)[/i]

Suppose $n \ge 3$ is a positive integer. Let $a_1 < a_2 < ... < a_n$ be an increasing sequence of positive real numbers, and let $a_{n+1} = a_1$. Prove that $$\sum_{k=1}^{n}\frac{a_k}{a_{k+1}}>\sum_{k=1}^{n}\frac{a_{k+1}}{a_k}$$

How many ordered pairs of positive integers $(x, y)$ satisfy $yx^y = y^{2017}$?

For which positive integers $n$ can the polynomial $p(x) = 1 + x^n + x^{2n}$ is written as a product of two polynomials with integer coefficients (of degree $\ge 1$)?

Find all real functions $f$ defined on $ \mathbb R$, such that \[f(f(x)+y) = f(f(x)-y)+4f(x)y ,\] for all real numbers $x,y$.

Find all polynomials whose coefficients are all $\pm1$ and whose roots are all real.

In the plane of a triangle $ABC,$ in its exterior$,$ we draw the triangles $ABR, BCP, CAQ$ so that $\angle PBC = \angle CAQ = 45^{\circ}$, $\angle BCP = \angle QCA = 30^{\circ}$, $\angle ABR = \angle RAB = 15^{\circ}$. Prove that [b]a.)[/b] $\angle QRP = 90\,^{\circ},$ and [b]b.)[/b] $QR = RP.$

Let \[M=\{1, 2, 3, \ldots, 2022\}\] Determine the least positive integer $k$, such that for every $k$ subsets of $M$ with the cardinality of each subset equal to $3$, there are two of these subsets with exactly one common element.

Let $ABCD$ be a cyclic quadrilateral with circumcircle $\omega$ and let $AC$ and $BD$ intersect at $X$. Let the line through $A$ parallel to $BD$ intersect line $CD$ at $E$ and $\omega$ at $Y \ne A$. If $AB = 10, AD = 24, XA = 17$, and $XB = 21$, then the area of $\vartriangle DEY$ can be written in simplest form as $\frac{m}{n}$ . Find $m + n$.

Find the smallest positive constant $c$ satisfying: For any simple graph $G=G(V,E)$, if $|E|\geq c|V|$, then $G$ contains $2$ cycles with no common vertex, and one of them contains a chord. Note: The cycle of graph $G(V,E)$ is a set of distinct vertices ${v_1,v_2...,v_n}\subseteq V$, $v_iv_{i+1}\in E$ for all $1\leq i\leq n$ $(n\geq 3, v_{n+1}=v_1)$; a cycle containing a chord is the cycle ${v_1,v_2...,v_n}$, such that there exist $i,j, 1< i-j< n-1$, satisfying $v_iv_j\in E$.

Let $p$ be a prime, and $n > p$ be an integer. Prove that \[ \binom{n+p-1}{p} - \binom{n}{p} \] is divisible by $n$.

Let $ n$ be an even positive integer. Let $ A_1, A_2, \ldots, A_{n \plus{} 1}$ be sets having $ n$ elements each such that any two of them have exactly one element in common while every element of their union belongs to at least two of the given sets. For which $ n$ can one assign to every element of the union one of the numbers 0 and 1 in such a manner that each of the sets has exactly $ \frac {n}{2}$ zeros?

How many integer pairs $(a, b)$ with $1 < a, b \le 2015$ are there such that $\log_a b$ is an integer?

The perimeter of a square is equal in value to its area. Determine the length of one of its sides.

If $a,b,c$ are the sides of a triangle whose perimeter is equal to 2 then prove that: a) $abc+\frac{28}{27}\geq ab+bc+ac$; b) $abc+1<ab+bc+ac$ See also [url]http://www.mathlinks.ro/Forum/viewtopic.php?t=47939&view=next[/url] (problem 1) :)

Let $ x$, $ y$, $ z$ be real numbers satisfying $ x^2\plus{}y^2\plus{}z^2\plus{}9\equal{}4(x\plus{}y\plus{}z)$. Prove that \[ x^4\plus{}y^4\plus{}z^4\plus{}16(x^2\plus{}y^2\plus{}z^2) \ge 8(x^3\plus{}y^3\plus{}z^3)\plus{}27\] and determine when equality holds.

In the segment $A C$ a point $B$ is taken. Construct circles $T_1, T_2$ and $T_3$ of diameters $A B, BC$ and $AC$ respectively. A line that passes through $B$ cuts $T_3$ in the points $P$ and $Q$, and the circles $T_1$ and $T_2$ respectively at points $R$ and $S$. Prove that $PR = Q S$.

Show that if $k\leq \frac n2$ and $\mathcal F$ is a family $k\times k$ submatrices of an $n\times n$ matrix such that any two intersect then $$|\mathcal F|\leq \binom{n-1}{k-1}^2$$ [Gy. Katona]

The points $P$ and $Q$ lie on the sides $BC$ and $CD$ of the parallelogram $ABCD$ so that $BP = QD$. Show that the intersection point between the lines $BQ$ and $DP$ lies on the line bisecting $\angle BAD$.

What are the last $2$ digits of the number $2018^{2018}$ when written in base $7$?

$ S$ be a set of $ n$ points in the plane. No three points of $ S$ are collinear. Prove that there exists a set $ P$ containing $ 2n \minus{} 5$ points satisfying the following condition: In the interior of every triangle whose three vertices are elements of $ S$ lies a point that is an element of $ P.$

Let $ \xi_1,\xi_2,...$ be independent random variables such that $ E\xi_n=m>0$ and $ \textrm{Var}(\xi_n)=\sigma^2 < \infty \;(n=1,2,...)\ .$ Let $ \{a_n \}$ be a sequence of positive numbers such that $ a_n\rightarrow 0$ and $ \sum_{n=1}^{\infty} a_n= \infty$. Prove that \[ P \left( \lim_{n\rightarrow \infty} %Error. "diaplaymath" is a bad command. \sum_{k=1}^n a_k \xi_k =\infty \right)=1.\] [i]P. Revesz[/i]

An empty $2020 \times 2020 \times 2020$ cube is given, and a $2020 \times 2020$ grid of square unit cells is drawn on each of its six faces. A [i]beam[/i] is a $1 \times 1 \times 2020$ rectangular prism. Several beams are placed inside the cube subject to the following conditions: [list=] [*]The two $1 \times 1$ faces of each beam coincide with unit cells lying on opposite faces of the cube. (Hence, there are $3 \cdot {2020}^2$ possible positions for a beam.) [*]No two beams have intersecting interiors. [*]The interiors of each of the four $1 \times 2020$ faces of each beam touch either a face of the cube or the interior of the face of another beam. [/list] What is the smallest positive number of beams that can be placed to satisfy these conditions? [i]Proposed by Alex Zhai[/i]

$\textbf{Juan's Old Stamping Grounds}$ Juan organizes the stamps in his collection by country and by the decade in which they were issued. The prices he paid for them at a stamp shop were: Brazil and France, 6 cents each, Peru 4 cents each, and Spain 5 cents each. (Brazil and Peru are South American countries and France and Spain are in Europe.) [asy] /* AMC8 2002 #8, 9, 10 Problem */ size(3inch, 1.5inch); for ( int y = 0; y <= 5; ++y ) { draw((0,y)--(18,y)); } draw((0,0)--(0,5)); draw((6,0)--(6,5)); draw((9,0)--(9,5)); draw((12,0)--(12,5)); draw((15,0)--(15,5)); draw((18,0)--(18,5)); draw(scale(0.8)*"50s", (7.5,4.5)); draw(scale(0.8)*"4", (7.5,3.5)); draw(scale(0.8)*"8", (7.5,2.5)); draw(scale(0.8)*"6", (7.5,1.5)); draw(scale(0.8)*"3", (7.5,0.5)); draw(scale(0.8)*"60s", (10.5,4.5)); draw(scale(0.8)*"7", (10.5,3.5)); draw(scale(0.8)*"4", (10.5,2.5)); draw(scale(0.8)*"4", (10.5,1.5)); draw(scale(0.8)*"9", (10.5,0.5)); draw(scale(0.8)*"70s", (13.5,4.5)); draw(scale(0.8)*"12", (13.5,3.5)); draw(scale(0.8)*"12", (13.5,2.5)); draw(scale(0.8)*"6", (13.5,1.5)); draw(scale(0.8)*"13", (13.5,0.5)); draw(scale(0.8)*"80s", (16.5,4.5)); draw(scale(0.8)*"8", (16.5,3.5)); draw(scale(0.8)*"15", (16.5,2.5)); draw(scale(0.8)*"10", (16.5,1.5)); draw(scale(0.8)*"9", (16.5,0.5)); label(scale(0.8)*"Country", (3,4.5)); label(scale(0.8)*"Brazil", (3,3.5)); label(scale(0.8)*"France", (3,2.5)); label(scale(0.8)*"Peru", (3,1.5)); label(scale(0.8)*"Spain", (3,0.5)); label(scale(0.9)*"Juan's Stamp Collection", (9,0), S); label(scale(0.9)*"Number of Stamps by Decade", (9,5), N); [/asy] How many of his European stamps were issued in the '80s? $\text{(A)}\ 9 \qquad \text{(B)}\ 15 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 24 \qquad \text{(E)}\ 42$