Let $S(n)=1\varphi(1)+2\varphi(2) \ldots +n\varphi(n)$, where $\varphi(n)$ is the number of positive integers less than or equal to $n$ that are relatively prime to $n$. (For instance $\varphi(12)=4$ and $\varphi(20)=8$.) Prove that for all $n \geq 2018$, the following inequality holds: $$0.17n^3 \leq S(n) \leq 0.23n^3$$

When $a_0, a_1, \dots , a_{1000}$ denote digits, can the sum of the $1001$-digit numbers $a_0a_1\cdots a_{1000}$ and $a_{1000}a_{999}\cdots a_0$ have odd digits only?

[u]Set 1[/u] [b]p1.[/b] Farmer John has $4000$ gallons of milk in a bucket. On the first day, he withdraws $10\%$ of the milk in the bucket for his cows. On each following day, he withdraws a percentage of the remaining milk that is $10\%$ more than the percentage he withdrew on the previous day. For example, he withdraws $20\%$ of the remaining milk on the second day. How much milk, in gallons, is left after the tenth day? [b]p2.[/b] Will multiplies the first four positive composite numbers to get an answer of $w$. Jeremy multiplies the first four positive prime numbers to get an answer of $j$. What is the positive difference between $w$ and $j$? [b]p3.[/b] In Nathan’s math class of $60$ students, $75\%$ of the students like dogs and $60\%$ of the students like cats. What is the positive difference between the maximum possible and minimum possible number of students who like both dogs and cats? [u]Set 2[/u] [b]p4.[/b] For how many integers $x$ is $x^4 - 1$ prime? [b]p5.[/b] Right triangle $\vartriangle ABC$ satisfies $\angle BAC = 90^o$. Let $D$ be the foot of the altitude from $A$ to $BC$. If $AD = 60$ and $AB = 65$, find the area of $\vartriangle ABC$. [b]p6.[/b] Define $n! = n \times (n - 1) \times ... \times 1$. Given that $3! + 4! + 5! = a^2 + b^2 + c^2$ for distinct positive integers $a, b, c$, find $a + b + c$. [u]Set 3[/u] [b]p7.[/b] Max nails a unit square to the plane. Let M be the number of ways to place a regular hexagon (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Vincent, on the other hand, nails a regular unit hexagon to the plane. Let $V$ be the number of ways to place a square (of any size) in the same plane such that the square and hexagon share at least $2$ vertices. Find the nonnegative difference between $M$ and $V$ . [b]p8.[/b] Let a be the answer to this question, and suppose $a > 0$. Find $\sqrt{a +\sqrt{a +\sqrt{a +...}}}$ . [b]p9.[/b] How many ordered pairs of integers $(x, y)$ are there such that $x^2 - y^2 = 2019$? [u]Set 4[/u] [b]p10.[/b] Compute $\frac{p^3 + q^3 + r^3 - 3pqr}{p + q + r}$ where $p = 17$, $q = 7$, and $r = 8$. [b]p11.[/b] The unit squares of a $3 \times 3$ grid are colored black and white. Call a coloring good if in each of the four $2 \times 2$ squares in the $3 \times 3$ grid, there is either exactly one black square or exactly one white square. How many good colorings are there? Consider rotations and reflections of the same pattern distinct colorings. [b]p12.[/b] Define a $k$-[i]respecting [/i]string as a sequence of $k$ consecutive positive integers $a_1$, $a_2$, $...$ , $a_k$ such that $a_i$ is divisible by $i$ for each $1 \le i \le k$. For example, $7$, $8$, $9$ is a $3$-respecting string because $7$ is divisible by $1$, $8$ is divisible by $2$, and $9$ is divisible by $3$. Let $S_7$ be the set of the first terms of all $7$-respecting strings. Find the sum of the three smallest elements in $S_7$. [u]Set 5[/u] [b]p13.[/b] A triangle and a quadrilateral are situated in the plane such that they have a finite number of intersection points $I$. Find the sum of all possible values of $I$. [b]p14.[/b] Mr. DoBa continuously chooses a positive integer at random such that he picks the positive integer $N$ with probability $2^{-N}$ , and he wins when he picks a multiple of 10. What is the expected number of times Mr. DoBa will pick a number in this game until he wins? [b]p15.[/b] If $a, b, c, d$ are all positive integers less than $5$, not necessarily distinct, find the number of ordered quadruples $(a, b, c, d)$ such that $a^b - c^d$ is divisible by $5$. PS. You had better use hide for answers. Last 4 sets have been posted [url=https://artofproblemsolving.com/community/c4h2777362p24370554]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that in a group of $50$ people there are always two who have an even number (possibly zero) of common acquaintances within the group. (SI Tokarev)

Define sequence $a_{0}, a_{1}, a_{2}, \ldots, a_{2018}, a_{2019}$ as below: $ a_{0}=1 $ $a_{n+1}=a_{n}-\frac{a_{n}^{2}}{2019}$, $n=0,1,2, \ldots, 2018$ Prove $a_{2019} < \frac{1}{2} < a_{2018}$

Given is a directed graph with $28$ vertices, such that there do not exist vertices $u, v$, such that $u \rightarrow v$ and $v \rightarrow u$. Every $16$ vertices form a directed cycle. Prove that among any $17$ vertices, we can choose $15$ which form a directed cycle.

Let $P =\{(a, b) / a, b \in \{1, 2, ..., n\}, n \in N\}$ be a set of point of the Cartesian plane and draw horizontal, vertical, or diagonal segments, of length $1$ or $\sqrt 2$, so that both ends of the segment are in $P$ and do not intersect each other. Furthermore, for each point $(a, b)$ it is true that i) if $a + b$ is a multiple of $3$, then it is an endpoint of exactly $3$ segments. ii) if $a + b$ is an even not multiple of $3$, then it is an endpoint of exactly $2$ segments. iii) if $a + b$ is an odd not multiple of $3$, then it is endpoint of exactly $1$ segment. a) Check that with $n = 6$ it is possible to satisfy all the conditions. b) Show that with $n = 2015$ it is not possible to satisfy all the conditions.

Let $I{}$ be the incenter of triangle $ABC{}.$ Let $N{}$ be the foot of the bisector of angle $B{}.$ The tangent line to the circumcircle of triangle $AIN$ at $A{}$ and the tangent line to the circumcircle of triangle $CIN{}$ at $C{}$ intersect at $D{}.$ Prove that lines $AC{}$ and $DI$ are perpendicular. [i]Mikhail Evdokimov[/i]

$n$ is a positive integer. Let $a(n)$ be the smallest number for which $n\mid a(n)!$ Find all solutions of:$$\frac{a(n)}{n}=\frac{2}{3}$$

In an acute angled triangle $ABC$, let $H$ be the orthocenter, and let $D,E,F$ be the feet of altitudes from $A,B,C$ to the opposite sides, respectively. Let $L,M,N$ be the midpoints of the segments $AH, EF, BC$ respectively. Let $X,Y$ be the feet of altitudes from $L,N$ on to the line $DF$ respectively. Prove that $XM$ is perpendicular to $MY$.

Let $T_a, T_b$ and $T_c$ be the tangent points of the incircle $\Omega$ of the triangle $ABC$ with the sides $BC, CA$ and $AB{}$ respectively. Let $X, Y$ and $Z{}$ be points on the circle $\Omega$ such that $A{}$ lies on the ray $YX$, $B{}$ lies on the ray $ZY$ and $C{}$ lies on the ray $XZ$. Let $P{}$ be the intersection point of the segments $ZX$ and $T_bT_c$, and similarly $Q=XY \cap T_cT_a$ and $R=YZ\cap T_aT_b$. Prove that the points $A, B$ and $C{}$ lie on the lines $RP, PQ$ and $QR{}$, respectively. [i]Proposed by L. Shatunov (11th grade student)[/i]

Solve the following equation over the integers $$ 25x^2y^2+10x^2y+25xy^2+x^2+30xy+2y^2+5x+7y+6= 0.$$

The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is $ 20$ cents. If she had one more quarter, the average value would be $ 21$ cents. How many dimes does she have in her purse? $ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 3\qquad \textbf{(E)}\ 4$

Let $a, b$ and $c$ be positive real numbers such that $a+b+c = abc$. Prove that at least one of $a, b$ or $c$ is greater than $\frac{17}{10}$ .

Bahman wants to build an area next to his garden's wall for keeping his poultry. He has three fences each of length $10$ meters. Using the garden's wall, which is straight and long, as well as the three pieces of fence, what is the largest area Bahman can enclose in meters squared? $\textbf{(A)}\ 100 \qquad\textbf{(B)}\ 50+25 \sqrt 3\qquad\textbf{(C)}\ 50 + 50\sqrt 2\qquad\textbf{(D)}\ 75 \sqrt 3 \qquad\textbf{(E)}\ 300$

Define the function $f_1$ on the positive integers by setting $f_1(1)=1$ and if $n=p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the prime factorization of $n>1$, then \[f_1(n)=(p_1+1)^{e_1-1}(p_2+1)^{e_2-1}\cdots (p_k+1)^{e_k-1}.\] For every $m \ge 2$, let $f_m(n)=f_1(f_{m-1}(n))$. For how many $N$ in the range $1 \le N \le 400$ is the sequence $(f_1(N), f_2(N), f_3(N),...)$ unbounded? [b]Note:[/b] a sequence of positive numbers is unbounded if for every integer $B$, there is a member of the sequence greater than $B$. $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 16 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 18\qquad\textbf{(E)}\ 19 $

What is the last digit of $2021^{2021}$? [i]Proposed by Yifan Kang[/i]