For a nonnegative integer $n$, let $s(n)$ be the sum of the digits of the binary representation of $n$. Prove that $$\sum_{n=1}^{2^{2022}-1} \frac{(-1)^{s(n)}}{n+2022}>0.$$

Let $n$ be a positive integer and $\mathcal S$ be the set of points $(x, y)$ with $x, y \in \{1, 2, \ldots , n\}$. Let $\mathcal T$ be the set of all squares with vertices in the set $\mathcal S$. We denote by $a_k$ ($k \geq 0$) the number of (unordered) pairs of points for which there are exactly $k$ squares in $\mathcal T$ having these two points as vertices. Prove that $a_0 = a_2 + 2a_3$. [i]Yugoslavia[/i]

[b]p1.[/b] Find the last digit of $2022^{2022}$. [b]p2.[/b] Let $a_1 < a_2 <... < a_8$ be eight real numbers in an increasing arithmetic progression. If $a_1 + a_3 + a_5 + a_7 = 39$ and $a_2 + a_4 + a_6 + a_8 = 40$, determine the value of $a_1$. [b]p3.[/b] Patrick tries to evaluate the sum of the first $2022$ positive integers, but accidentally omits one of the numbers, $N$, while adding all of them manually, and incorrectly arrives at a multiple of $1000$. If adds correctly otherwise, find the sum of all possible values of $N$. [b]p4.[/b] A machine picks a real number uniformly at random from $[0, 2022]$. Andrew randomly chooses a real number from $[2020, 2022]$. The probability that Andrew’s number is less than the machine’s number is $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [b]p5.[/b] Let $ABCD$ be a square and $P$ be a point inside it such that the distances from $P$ to sides $AB$ and $AD$ respectively are $2$ and $4$, while $PC = 6$. If the side length of the square can be expressed in the form $a +\sqrt{b}$ for positive integers $a, b$, then determine $a + b$. [b]p6.[/b] Positive integers $a_1, a_2, ..., a_{20}$ sum to $57$. Given that $M$ is the minimum possible value of the quantity $a_1!a_2!...a_{20}!$, find the number of positive integer divisors of $M$. [b]p7.[/b] Jessica has $16$ balls in a box, where $15$ of them are red and one is blue. Jessica draws balls out the box three at a time until one of the three is blue. If she ever draws three red marbles, she discards one of them and shuffles the remaining two back into the box. The expected number of draws it takes for Jessica to draw the blue ball can be written as a common fraction $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$. [b]p8.[/b] The Lucas sequence is defined by these conditions: $L_0 = 2$, $L_1 = 1$, and $L_{n+2} =L_{n+1} +L_n$ for all $n \ge 0$. Determine the remainder when $L^2_{2019} +L^2_{2020}$ is divided by $L_{2023}$. [b]p9.[/b] Let $ABCD$ be a parallelogram. Point $P$ is selected in its interior such that the distance from $P$ to $BC$ is exactly $6$ times the distance from $P$ to $AD$, and $\angle APB = \angle CPD = 90^o$. Given that $AP = 2$ and $CP = 9$, the area of $ABCD$ can be expressed as $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$. [b]p10.[/b] Consider the polynomial $P(x) = x^{35} + ... + x + 1$. How many pairs $(i, j)$ of integers are there with $0 \le i < j \le 35$ such that if we flip the signs of the $x^i$ and $x^j$ terms in $P(x)$ to form a new polynomial $Q(x)$, then there exists a nonconstant polynomial $R(x)$ with integer coefficients dividing both $P(x)$ and $Q(x)$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There is a $8*8$ board and the numbers $1,2,3,4,...,63,64$. In all the unit squares of the board, these numbers are places such that only $1$ numbers goes to only one unit square. Prove that there is atleast $4$ $2*2$ squares such that the sum of the numbers in $2*2$ is greater than $100$.

Given the function $$y =|x^2 - 4x + 3|.$$ Study its continuity and differentiability at the point of abscissa $1$. Its graph determines with the $X$ axis a closed figure. Determine the area of said figure.

Examine if we can put the sixteen positive divisors of $2024$ on the cells of the table shown such that the sum of the four numbers of any line or row to be a multiple of $3$. $ \begin{tabular}{ | l | c | c | r| } \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline & & & \\ \hline \end{tabular} $

Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that $$a!b!c!d!=24!$$$\textbf{(A)}~4$ $\textbf{(B)}~4!$ $\textbf{(C)}~4^4$ $\textbf{(D)}~\text{None of the above}$

Let $AB=AC$ in $\triangle ABC$, and let $D$ be a point on segment $AB$. The tangent at $D$ to the circumcircle $\omega$ of $BCD$ hits $AC$ at $E$. The other tangent from $E$ to $\omega$ touches it at $F$, and $G=BF \cap CD$, $H=AG \cap BC$. Prove that $BH=2HC$. [i]Proposed by David Stoner[/i]

There are $n$ girls $G_1,\ldots, G_n$ and $n$ boys $B_1,\ldots,B_n$. A pair $(G_i,B_j)$ is called $\textit{suitable}$ if and only if girl $G_i$ is willing to marry boy $B_j$. Given that there is exactly one way to pair each girl with a distinct boy that she is willing to marry, what is the maximal possible number of suitable pairs?

$N^2$ pieces are placed on an $N \times N$ chessboard. Is it possible to rearrange them in such a way that any two pieces which can capture each other (when considered to be knights) after the rearrangement are on adjacent squares (i.e. squares having at least one common boundary point)? Consider two cases: (a) $N = 3$. (b) $N = 8$ (S Stefanov)

Let $X_1,X_2,\ldots,X_m$ a numbering of the $m=2^n-1$ non-empty subsets of the set $\{1,2,\ldots,n\}$, $n\geq 2$. We consider the matrix $(a_{ij})_{1\leq i,j\leq m}$, where $a_{ij}=0$, if $X_i \cap X_j = \emptyset$, and $a_{ij}=1$ otherwise. Prove that the determinant $d$ of this matrix does not depend on the way the numbering was done and compute $d$.

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $$f(xf(y)-y^2)=(y+1)f(x-y)$$ holds for all $x,y\in\mathbb{R}$.

For how many real numbers $a$ does the quadratic equation $x^2 + ax + 6a=0$ have only integer roots for $x$?

The sum of positive real numbers $ a,b,c,d$ is $ 1$. Prove that: $ \frac{a^2}{a\plus{}b}\plus{}\frac{b^2}{b\plus{}c}\plus{}\frac{c^2}{c\plus{}d}\plus{}\frac{d^2}{d\plus{}a} \ge \frac{1}{2},$ with equality if and only if $ a\equal{}b\equal{}c\equal{}d\equal{}\frac{1}{4}$.

In a scalene triangle $ABC$, $D$ is the foot of the altitude through $A$, $E$ is the intersection of $AC$ with the bisector of $\angle ABC$ and $F$ is a point on $AB$. Let $O$ the circumcenter of $ABC$ and $X=AD\cap BE$, $Y=BE\cap CF$, $Z=CF \cap AD$. If $XYZ$ is an equilateral triangle, prove that one of the triangles $OXY$, $OYZ$, $OZX$ must be equilateral.

Suppose that $n$ and $k$ are positive integers such that \[ 1 = \underbrace{\varphi( \varphi( \dots \varphi(}_{k\ \text{times}} n) \dots )). \] Prove that $n \le 3^k$. Here $\varphi(n)$ denotes Euler's totient function, i.e. $\varphi(n)$ denotes the number of elements of $\{1, \dots, n\}$ which are relatively prime to $n$. In particular, $\varphi(1) = 1$. [i]Proposed by Linus Hamilton[/i]

Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring nof the squares of a $2n$ by $k$ chessboard with $n$ colours, there are $2$ columns and $2$ rows such that the $4$ squares in their intersections have the same colour.

In $\triangle ABC$, $AC = BC$, and point $D$ is on $\overline{BC}$ so that $CD = 3 \cdot BD$. Let $E$ be the midpoint of $\overline{AD}$. Given that $CE = \sqrt{7}$ and $BE = 3$, the area of $\triangle ABC$ can be expressed in the form $m\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m+n$.

You are given $2n$ distinct integers. What's the largest integer $C$ such that you can always form at least $C$ pairs from them, so that no integer is in more than one pair, and the sum of integers in each pair is a composite number? [i](Proposed by Anton Trygub)[/i]

Prove that for every real positive number $a, b, c$ with $1 \le a, b, c \le 8$ the inequality $$\frac{a+b+c}{5}\le \sqrt[3]{abc}$$

$A$ gives $B$ as many cents as $B$ has and $C$ as many cents as $C$ has. Similarly, $B$ then gives $A$ and $C$ as many cents as each then has. $C$, similarly, then gives $A$ and $B$ as many cents as each then has. If each finally has $16$ cents, with how many cents does $A$ start? $\textbf{(A)}\ 24 \qquad \textbf{(B)}\ 26\qquad \textbf{(C)}\ 28 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 32$

For all real number $x$ consider the family $F(x)$ of all sequences $(a_{n})_{n\geq 0}$ satisfying the equation \[a_{n+1}=x-\frac{1}{a_{n}}\quad (n\geq 0).\] A positive integer $p$ is called a [i]minimal period[/i] of the family $F(x)$ if (a) each sequence $\left(a_{n}\right)\in F(x)$ is periodic with the period $p$, (b) for each $0<q<p$ there exists $\left(a_{n}\right)\in F(x)$ such that $q$ is not a period of $\left(a_{n}\right)$. Prove or disprove that for each positive integer $P$ there exists a real number $x=x(P)$ such that the family $F(x)$ has the minimal period $p>P$.

Find all positive integers $k$ such that there exists some permutation of $(1, 2,...,1000)$ namely $(a_1, a_2,..., a_{1000}) $ and satisfy $|a_i - i| = k$ for all $i = 1,1000$.

Let $a$ be an odd natural number and $b$ be a positive integer. We define a sequence of reals $(u_n)$ as follows: $u_0=b$ and, for all $n\in\mathbb N_0$, $u_{n+1}$ is $\frac{u_n}2$ if $u_n$ is even and $a+u_n$ otherwise. (a) Prove that one can find an element of $u_n$ smaller than $a$. (b) Prove that the sequence is eventually periodic.

[b]p1.[/b] Farmer James goes to Kristy’s Krispy Chicken to order a crispy chicken sandwich. He can choose from $3$ types of buns, $2$ types of sauces, $4$ types of vegetables, and $4$ types of cheese. He can only choose one type of bun and cheese, but can choose any nonzero number of sauces, and the same with vegetables. How many different chicken sandwiches can Farmer James order? [b]p2.[/b] A line with slope $2$ and a line with slope $3$ intersect at the point $(m, n)$, where $m, n > 0$. These lines intersect the $x$ axis at points $A$ and $B$, and they intersect the y axis at points $C$ and $D$. If $AB = CD$, find $m/n$. [b]p3.[/b] A multi-set of $11$ positive integers has a median of $10$, a unique mode of $11$, and a mean of $ 12$. What is the largest possible number that can be in this multi-set? (A multi-set is a set that allows repeated elements.) [b]p4.[/b] Farmer James is swimming in the Eggs-Eater River, which flows at a constant rate of $5$ miles per hour, and is recording his time. He swims $ 1$ mile upstream, against the current, and then swims $1$ mile back to his starting point, along with the current. The time he recorded was double the time that he would have recorded if he had swum in still water the entire trip. To the nearest integer, how fast can Farmer James swim in still water, in miles per hour? [b]p5.[/b] $ABCD$ is a square with side length $60$. Point $E$ is on $AD$ and $F$ is on $CD$ such that $\angle BEF = 90^o$. Find the minimum possible length of $CF$. [b]p6.[/b] Farmer James makes a trianglomino by gluing together $5$ equilateral triangles of side length $ 1$, with adjacent triangles sharing an entire edge. Two trianglominoes are considered the same if they can be matched using only translations and rotations (but not reflections). How many distinct trianglominoes can Farmer James make? [b]p7.[/b] Two real numbers $x$ and $y$ satisfy $x^2 - y^2 = 2y - 2x$ , and $x + 6 = y^2 + 2y$. What is the sum of all possible values of$ y$? [b]p8.[/b] Let $N$ be a positive multiple of $840$. When $N$ is written in base $6$, it is of the form $\overline{abcdef}_6$ where $a, b, c, d, e, f$ are distinct base $6$ digits. What is the smallest possible value of $N$, when written in base $6$? [b]p9.[/b] For $S = \{1, 2,..., 12\}$, find the number of functions $f : S \to S$ that satisfy the following $3$ conditions: (a) If $n$ is divisible by $3$, $f(n)$ is not divisible by $3$, (b) If $n$ is not divisible by $3$, $f(n)$ is divisible by $3$, and (c) $f(f(n)) = n$ holds for exactly $8$ distinct values of $n$ in $S$. [b]p10.[/b] Regular pentagon $JAMES$ has area $ 1$. Let $O$ lie on line $EM$ and $N$ lie on line $MA$ so that $E, M, O$ and $M, A, N$ lie on their respective lines in that order. Given that $MO = AN$ and $NO = 11 \cdot ME$, find the area of $NOM$. [b]p11.[/b] Hen Hao is flipping a special coin, which lands on its sunny side and its rainy side each with probability $1/2$. Hen Hao flips her coin ten times. Given that the coin never landed with its rainy side up twice in a row, find the probability that Hen Hao’s last flip had its sunny side up. [b]p12.[/b] Find the product of all integer values of a such that the polynomial $x^4 + 8x^3 + ax^2 + 2x - 1$ can be factored into two non-constant polynomials with integer coefficients. [b]p13.[/b] Isosceles trapezoid $ABCD$ has $AB = CD$ and $AD = 6BC$. Point $X$ is the intersection of the diagonals $AC$ and $BD$. There exist a positive real number $k$ and a point $P$ inside $ABCD$ which satisfy $$[PBC] : [PCD] : [PDA] = 1 : k : 3,$$ where $[XYZ]$ denotes the area of triangle $XYZ$. If $PX \parallel AB$, find the value of $k$. [b]p14.[/b] How many positive integers $n < 1000$ are there such that in base $10$, every digit in $3n$ (that isn’t a leading zero) is greater than the corresponding place value digit (possibly a leading zero) in $n$? For example, $n = 56$, $3n = 168$ satisfies this property as $1 > 0$, $6 > 5$, and $8 > 6$. On the other hand, $n = 506$, $3n = 1518$ does not work because of the hundreds place. [b]p15.[/b] Find the greatest integer that is smaller than $$\frac{2018}{37^2}+\frac{2018}{39^2}+ ... +\frac{2018}{ 107^2}.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].