Given a finite nonempty set $X$ with real values, let $f(X) = \frac{1}{|X|} \displaystyle\sum\limits_{a\in X} a$, where $\left\lvert X \right\rvert$ denotes the cardinality of $X$. For ordered pairs of sets $(A,B)$ such that $A\cup B = \{1, 2, \dots , 100\}$ and $A\cap B = \emptyset$ where $1\leq |A| \leq 98$, select some $p\in B$, and let $A_{p} = A\cup \{p\}$ and $B_{p} = B - \{p\}.$ Over all such $(A,B)$ and $p\in B$ determine the maximum possible value of $(f(A_{p})-f(A))(f(B_{p})-f(B)).$

[b]p1.[/b] Each box represents $1$ square unit. Find the area of the shaded region. [img]https://cdn.artofproblemsolving.com/attachments/0/0/f8f8ad6d771f3bbbc59b374a309017cecdce5a.png[/img] [b]p2.[/b] Evaluate $(3^3)\sqrt{5^2-2^4} -5 \cdot 9$. [b]p3.[/b] Find the last two digits of $21^3$. [b]p4.[/b] Let $L$, $M$, and $T$ be distinct prime numbers. Find the least possible odd value of$ L+M +T$ . [b]p5.[/b]Two circles have areas that sum to $20\pi$ and diameters that sum to $12$. Find the radius of the smaller circle. [b]p6.[/b] Zach and Evin each independently choose a date in the year $2022$, uniformly and randomly. The probability that at least one of the chosen dates is December $17$, $2022$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $A$. [b]p7.[/b] Let $L$ be a list of $2023$ real numbers with medianm. When any two numbers are removed from $L$, its median is still $m$. Find the greatest possible number of distinct values in $L$. [b]p8.[/b] Some children and adults are eating a delicious pile of sand. Children comprise $20\%$ of the group and combined, they consume $80\%$ of the sand. Given that on average, each child consumes $N$ pounds of sand and on average, each adult consumes $M$ pounds of sand, find $\frac{N}{M}$. [b]p9.[/b] An integer $N$ is chosen uniformly and randomly from the set of positive integers less than $100$. The expectedm number of digits in the base-$10$-representation of $N$ can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p10.[/b] Dunan is taking a calculus course in which the final exam counts for $15\%$ of the total grade. Dunan wishes to have an $A$ in the course, which is defined as a grade of $93\%$ or above. When counting everything but the final exam, he currently has a $92\%$ in the course. What is the minimum integer grade Dunan must get on the final exam in order to get an $A$ in the course? [b]p11.[/b] Norbert, Eorbert, Sorbert, andWorbert start at the origin of the Cartesian Plane and walk in the positive $y$, positive $x$, negative $y$, and negative $x$ directions respectively at speeds of $1$, $2$, $3$, and $4$ units per second respectively. After how many seconds will the quadrilateral with a vertex at each person’s location have area $300$? [b]p12.[/b] Find the sum of the unique prime factors of $1020201$. [b]p13.[/b] HacoobaMatata rewrites the base-$10$ integers from $0$ to $30$ inclusive in base $3$. How many times does he write the digit $1$? [b]p14.[/b] The fractional part of $x$ is $\frac17$. The greatest possible fractional part of $x^2$ can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p15.[/b] For howmany integers $x$ is $-2x^2 +8 \ge x^2 -3x +2$? [b]p16.[/b] In the figure below, circle $\omega$ is inscribed in square $EFGH$, which is inscribed in unit square $ABCD$ such that $\overline{EB} = 2\overline{AE}$. If the minimum distance from a point on $\omega$ to $ABCD$ can be written as $\frac{P-\sqrt{Q}}{R}$ with $Q$ square-free, find $10000P +100Q +R$. [img]https://cdn.artofproblemsolving.com/attachments/a/1/c6e5400bc508ab14f34987c9f5f4039daaa4d6.png[/img] [b]p17.[/b] There are two base number systems in use in the LHS Math Team. One member writes “$13$ people usemy base, while $23$ people use the other, base $12$.” Another member writes “out of the $34$ people in the club, $10$ use both bases while $9$ use neither.” Find the sum of all possible numbers ofMath Team members, as a regular decimal number. [b]p18.[/b] Sam is taking a test with $100$ problems. On this test the questions gradually get harder in such a way that for question $i$ , Sam has a $\frac{(101-i)^2}{ 100} \%$ chance to get the question correct. Suppose the expected number of questions Sam gets correct can be written as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p19.[/b] In an ordered $25$-tuple, each component is an integer chosen uniformly and randomly from $\{1,2,3,4,5\}$. Ephram and Zach both copy this tuple into a $5\times 5$ grid, both starting from the top-left corner. Ephram writes five components from left to right to fill one row before continuing down to the next row. Zach writes five components from top to bottom to fill one column before continuing right to the next column. Find the expected number of spaces on their grids where Zach and Ephram have the same integer written. [b]p20.[/b] In $\vartriangle ABC$ with circumcenter $O$ and circumradius $8$, $BC = 10$. Let $r$ be the radius of the circle that passes through $O$ and is tangent to $BC$ at $C$. The value of $r^2$ can be written as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $1000m+n$. [b]p21.[/b] Find the number of integer values of $n$ between $1$ and $100$ inclusive such that the sum of the positive divisors of $2n$ is at least $220\%$ of the sum of the divisors of $n$. [b]p22.[/b] Twenty urns containing one ball each are arranged in a circle. Ernie then moves each ball either $1$, $2$ or $3$ urns clockwise, chosen independently, uniformly, and randomly. The expected number of empty urns after this process is complete can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p23.[/b] Hannah the cat begins at $0$ on a number line. Every second, Hannah jumps $1$ unit in the positive or negative direction, chosen uniformly at random. After $7$ seconds,Hannah‘s expected distance from $0$, in units, can be expressed as $\frac{A}{B}$ for relatively prime positive integers $A$ and $B$. Find $1000A+B$. [b]p24.[/b] Find the product of all primes $p < 30$ for which there exists an integer $n$ such that $p$ divides $n +(n +1)^{-1}\,\, (mod \,\,p)$. [b]p25.[/b] In quadrilateral $ABCD$, $\angle ABD = \angle CBD = \angle C AD$, $AB = 9$, $BC = 6$, and $AC = 10$. The area of $ABCD$ can be expressed as $\frac{P\sqrt{Q}}{R}$ with $Q$ squarefree and $P$ and $R$ relatively prime. Find $10000P +100Q +R$. [img]https://cdn.artofproblemsolving.com/attachments/4/8/28569605b262c8f26e685e27f5f261c70a396c.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[u]Round 4[/u] [b]p13.[/b] What is the units digit of the number $(2^1 + 1)(2^2 - 1)(2^3 + 1)(2^4 - 1)...(2^{2010} - 1)$? [b]p14.[/b] Mr. Fat noted that on January $2$, $2010$, the display of the day is $01/02/2010$, and the sequence $01022010$ is a palindrome (a number that reads the same forwards and backwards). How many days does Mr. Fat need to wait between this palindrome day and the last palindrome day of this decade? [b]p15.[/b] Farmer Tim has a $30$-meter by $30$-meter by $30\sqrt2$-meter triangular barn. He ties his goat to the corner where the two shorter sides meet with a 60-meter rope. What is the area, in square meters, of the land where the goat can graze, given that it cannot get inside the barn? [b]p16.[/b] In triangle $ABC$, $AB = 3$, $BC = 4$, and $CA = 5$. Point $P$ lies inside the triangle and the distances from $P$ to two of the sides of the triangle are $ 1$ and $2$. What is the maximum distance from $P$ to the third side of the triangle? [u]Round 5[/u] [b]p17.[/b] Let $Z$ be the answer to the third question on this guts quadruplet. If $x^2 - 2x = Z - 1$, find the positive value of $x$. [b]p18.[/b] Let $X$ be the answer to the first question on this guts quadruplet. To make a FATRON2012, a cubical steel body as large as possible is cut out from a solid sphere of diameter $X$. A TAFTRON2013 is created by cutting a FATRON2012 into $27$ identical cubes, with no material wasted. What is the length of one edge of a TAFTRON2013? [b]p19.[/b] Let $Y$ be the smallest integer greater than the answer to the second question on this guts quadruplet. Fred posts two distinguishable sheets on the wall. Then, $Y$ people walk into the room. Each of the Y people signs up on $0, 1$, or $2$ of the sheets. Given that there are at least two people in the room other than Fred, how many possible pairs of lists can Fred have? [b]p20.[/b] Let $A, B, C$, be the respective answers to the first, second, and third questions on this guts quadruplet. At the Robot Design Convention and Showcase, a series of robots are programmed such that each robot shakes hands exactly once with every other robot of the same height. If the heights of the $16$ robots are $4$, $4$, $4$, $5$, $5$, $7$, $17$, $17$, $17$, $34$, $34$, $42$, $100$, $A$, $B$, and $C$ feet, how many handshakes will take place? [u]Round 6[/u] [b]p21.[/b] Determine the number of ordered triples $(p, q, r)$ of primes with $1 < p < q < r < 100$ such that $q - p = r - q$. [b]p22.[/b] For numbers $a, b, c, d$ such that $0 \le a, b, c, d \le 10$, find the minimum value of $ab + bc + cd + da - 5a - 5b - 5c - 5d$. [b]p23.[/b] Daniel has a task to measure $1$ gram, $2$ grams, $3$ grams, $4$ grams , ... , all the way up to $n$ grams. He goes into a store and buys a scale and six weights of his choosing (so that he knows the value for each weight that he buys). If he can place the weights on either side of the scale, what is the maximum value of $n$? [b]p24.[/b] Given a Rubik’s cube, what is the probability that at least one face will remain unchanged after a random sequence of three moves? (A Rubik’s cube is a $3$ by $3$ by $3$ cube with each face starting as a different color. The faces ($3$ by $3$) can be freely turned. A move is defined in this problem as a $90$ degree rotation of one face either clockwise or counter-clockwise. The center square on each face–six in total–is fixed.) PS. You should use hide for answers. First rounds have been posted [url=https://artofproblemsolving.com/community/c4h2766534p24230616]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $n\ge 2$ be a positive integer. Find the number of functions $f:\{1,2,\ldots ,n\}\rightarrow\{1,2,3,4,5 \}$ which have the following property: $|f(k+1)-f(k)|\ge 3$, for any $k=1,2,\ldots n-1$. [i]Vasile Pop[/i]

A function $g: \mathbb{Z} \to \mathbb{Z}$ is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+\dots+f(30)$ is minimized. Find the smallest possible value of $f(25)$.

Let $P(x)$ be any polynomial of degree at most $3.$ It can be shown that there are numbers $x_1$ and $x_2$ such that $\textstyle\int_{-1}^1 P(x) \ dx = P(x_1) + P(x_2),$ where $x_1$ and $x_2$ are independent of the polynomial $P.$ (a) Show that $x_1=-x_2.$ (b) Find $x_1$ and $x_2.$

Find all functions $f:\mathbb R^+\to \mathbb R^+$ such that $$f(x+y)f(f(x))=f(1+yf(x))$$ for all $x,y\in \mathbb R^+.$ [i]Proposed by Ming Hsiao[/i]

Find all functions $f : \mathbb{N}\rightarrow{\mathbb{N}}$ such that for all positive integers $m$ and $n$ the number $f(m)+n-m$ is divisible by $f(n)$.

Let $y = f (x)$ be a convex function defined on $[0,1]$, $f (0) = 0,$ $f (1) = 0$. It is also known that the area of ​​the segment bounded by this function and the segment $[0, 1]$ is equal to $1$. Find and draw the set of points of the coordinate plane through which the graph of such a function can pass. (A function is called convex if all points of the line segment connecting any two points on its graph are located no higher than the graph of this function.)

Let $P(x) = x^2 + ax + b$ be a quadratic polynomial with real coefficients. Suppose there are real numbers $ s \neq t$ such that $P(s) = t$ and $P(t) = s$. Prove that $b-st$ is a root of $x^2 + ax + b - st$.

Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$? [i]E. Malinnikova[/i]

Let $a, b, c$ be positive real numbers such that $ab +be + ba \le 3abc$. Prove that $$a^3+b^3+c^3 \ge a+b+c.$$

Find positive integers $a$, $b$, and $c$ such that \[\sqrt{a}+\sqrt{b}+\sqrt{c}=\sqrt{219+\sqrt{10080}+\sqrt{12600}+\sqrt{35280}}.\] Prove that your solution is correct. (Warning: numerical approximations of the values do not constitute a proof.)

The sequence $a_1, a_2,...$ is defined by the equalities $a_1 = 2, a_2 = 12$ and $a_{n+1} = 6a_n-a_{n-1}$ for every positive integer $n \ge 2$. Prove that no member of this sequence is equal to a perfect power (greater than one) of a positive integer.

Which of the followings is congruent (in $\bmod{25}$) to the sum in of integers $0\leq x < 25$ such that $x^3+3x^2-2x+4 \equiv 0 \pmod{25}$? $ \textbf{(A)}\ 3 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 17 \qquad\textbf{(D)}\ 22 \qquad\textbf{(E)}\ \text{None of the preceding} $

Given the Fibonacci sequence with $f_0=f_1=1$and for $n\geq 1, f_{n+1}=f_n+f_{n-1}$, find all real solutions to the equation: $$x^{2024}=f_{2023}x+f_{2022}.$$

Prove that for every natural number $k$ ($k \geq 2$) there exists an irrational number $r$ such that for every natural number $m$, \[[r^m] \equiv -1 \pmod k .\] [i]Remark.[/i] An easier variant: Find $r$ as a root of a polynomial of second degree with integer coefficients. [i]Proposed by Yugoslavia.[/i]

Find all the pairs of integers $ (m, n)$ such that $ \sqrt {n +\sqrt {2016}} +\sqrt {m-\sqrt {2016}} \in \mathbb {Q}.$

$A, B, C$ can walk at $5$ km/hr. They have a car that can accomodate any two of them whch travels at $50$ km/hr. Can they reach a point $62$ km away in less than $3$ hrs?

Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a bijection and let $k$ be a positive integer such that $|f(x+1)-f(x)| \leq k$ for all positive integers $x$. Show that there exists an integer $d$, such that $f(x)=x+d$ for infinitely many positive integers $x$.

Suppose that $n\ge2$ and $a_1,a_2,...,a_n$ are natural numbers that $ (a_1,a_2,...,a_n)=1$. Find all strictly increasing function $f: \mathbb{Z} \to \mathbb{R} $ that: $$ \forall x_1,x_2,...,x_n \in \mathbb{Z} : f(\sum_{i=1}^{n} {x_ia_i}) = \sum_{i=1}^{n} {f(x_ia_i})$$ [i]Proposed by Navid Safaei and Ali Mirzaei [/i]

Let $n$ be a positive integer. Positive numbers $a$, $b$, $c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Find the greatest possible value of $$E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}}$$

Consider a $ 7\times 7$ numbers table $ a_{ij} \equal{} (i^2 \plus{} j)(i \plus{} j^2), 1\le i,j\le 7.$ When we add arbitrarily each term of an arithmetical progression consisting of $ 7$ integers to corresponding to term of certain row (or column) in turn, call it an operation. Determine whether such that each row of numbers table is an arithmetical progression, after a finite number of operations.

For all triplets $a,b,c$ of (pairwise) distinct real numbers, prove the inequality $$ \left | \dfrac {a} {b-c} \right | + \left | \dfrac {b} {c-a} \right | + \left | \dfrac {c} {a-b} \right | \geq 2$$ and determine all cases of equality. Prove that if we also impose $a,b,c$ positive, then all equality cases disappear, but the value $2$ remains the best constant possible. ([i]Dan Schwarz[/i])

Find all triplets of real $(a,b,c)$ that solve the equation $a(a-b-c)+(b^2+c^2-bc)=4c^2\left(abc-\frac{a^2}{4}-b^2c^2\right)$