Each of the subsets $ A_1$, $ A_2$, $ \dots,$ $ A_n$ of a 2009-element set $ X$ contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in $ X$ there is a 24-element subset $ B$ containing neither of the sets $ A_1$, $ A_2$, $ \dots,$ $ A_n$.

[b]p17.[/b] Let the roots of the polynomial $f(x) = 3x^3 + 2x^2 + x + 8 = 0$ be $p, q$, and $r$. What is the sum $\frac{1}{p} +\frac{1}{q} +\frac{1}{r}$ ? [b]p18.[/b] Two students are playing a game. They take a deck of five cards numbered $1$ through $5$, shuffle them, and then place them in a stack facedown, turning over the top card next to the stack. They then take turns either drawing the card at the top of the stack into their hand, showing the drawn card to the other player, or drawing the card that is faceup, replacing it with the card on the top of the pile. This is repeated until all cards are drawn, and the player with the largest sum for their cards wins. What is the probability that the player who goes second wins, assuming optimal play? [b]p19.[/b] Compute the sum of all primes $p$ such that $2^p + p^2$ is also prime. [b]p20.[/b] In how many ways can one color the $8$ vertices of an octagon each red, black, and white, such that no two adjacent sides are the same color? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Without using a calculator, prove that $$2^{1995} > 5^{856}$$

If $f(i, j, k) = f(i - 1, j + k , 2i - 1)$ and $f(0, j, k) = j + k$, evaluate $f(n, 0, 0)$.

From an infinite arithmetic progression $ 1,a_1,a_2,\dots$ of real numbers some terms are deleted, obtaining an infinite geometric progression $ 1,b_1,b_2,\dots$ whose ratio is $ q$. Find all the possible values of $ q$.

If $0 \le p \le 1$ and $0 \le q \le 1$, define $F(p, q)$ by \[ F(p, q) = -2pq + 3p(1-q) + 3(1-p)q - 4(1-p)(1-q). \] Define $G(p)$ to be the maximum of $F(p, q)$ over all $q$ (in the interval $0 \le q \le 1$). What is the value of $p$ (in the interval $0 \le p \le 1$) that minimizes $G(p)$?

Determine all pairs $(a,b)$ of real numbers such that $a \lfloor bn \rfloor =b \lfloor an \rfloor $ for all positive integers $n$. (Note that $\lfloor x\rfloor $ denotes the greatest integer less than or equal to $x$.)

A sequence of real numbers $a_0, a_1, \dots$ satisfies the condition\[\sum\limits_{n=0}^{m}a_n\cdot(-1)^n\cdot\dbinom{m}{n}=0\]for all large enough positive integers $m$. Prove that there exists a polynomial $P$ such that $a_n=P(n)$ for all $n\ge0$.

[i]25 problems for 30 minutes[/i] [b]p1.[/b] Compute the sum $2019 + 201 + 20 + 2$. [b]p2.[/b] The sequence $100, 102, 104,..., 996$ and $998$ is the sequence of all three-digit even numbers. How many three digit even numbers are there? [b]p3.[/b] Find the units digit of $25\times 37\times 113\times 22$. [b]p4.[/b] Samuel has a number in his head. He adds $4$ to the number and then divides the result by $2$. After doing this, he ends up with the same number he had originally. What is his original number? [b]p5.[/b] According to Shay's Magazine, every third president is terrible (so the third, sixth, ninth president and so on were all terrible presidents). If there have been $44$ presidents, how many terrible presidents have there been in total? [b]p6.[/b] In the game Tic-Tac-Toe, a player wins by getting three of his or her pieces in the same row, column, or diagonal of a $3\times 3$ square. How many configurations of $3$ pieces are winning? Rotations and reflections are considered distinct. [b]p7.[/b] Eddie is a sad man. Eddie is cursed to break his arm $4$ times every $20$ years. How many times would he break his arm by the time he reaches age $100$? [b]p8. [/b]The figure below is made from $5$ congruent squares. If the figure has perimeter $24$, what is its area? [img]https://cdn.artofproblemsolving.com/attachments/1/9/6295b26b1b09cacf0c32bf9d3ba3ce76ddb658.png[/img] [b]p9.[/b] Sancho Panza loves eating nachos. If he eats $3$ nachos during the first minute, $4$ nachos during the second, $5$ nachos during the third, how many nachos will he have eaten in total after $15$ minutes? [b]p10.[/b] If the day after the day after the day before Wednesday was two days ago, then what day will it be tomorrow? [b]p11.[/b] Neetin the Rabbit and Poonam the Meerkat are in a race. Poonam can run at $10$ miles per hour, while Neetin can only hop at $2$ miles per hour. If Neetin starts the race $2$ miles ahead of Poonam, how many minutes will it take for Poonam to catch up with him? [b]p12.[/b] Dylan has a closet with t-shirts: $3$ gray, $4$ blue, $2$ orange, $7$ pink, and $2$ black. Dylan picks one shirt at random from his closet. What is the probability that Dylan picks a pink or a gray t-shirt? [b]p13.[/b] Serena's brain is $200\%$ the size of Eric's brain, and Eric's brain is $200\%$ the size of Carlson's. The size of Carlson's brain is what percent the size of Serena's? [b]p14.[/b] Find the sum of the coecients of $(2x + 1)^3$ when it is fully expanded. [b]p15. [/b]Antonio loves to cook. However, his pans are weird. Specifically, the pans are rectangular prisms without a top. What is the surface area of the outside of one of Antonio's pans if their volume is $210$, and their length and width are $6$ and $5$, respectively? [b]p16.[/b] A lattice point is a point on the coordinate plane with $2$ integer coordinates. For example, $(3, 4)$ is a lattice point since $3$ and $4$ are both integers, but $(1.5, 2)$ is not since $1.5$ is not an integer. How many lattice points are on the graph of the equation $x^2 + y^2 = 625$? [b]p17.[/b] Jonny has a beaker containing $60$ liters of $50\%$ saltwater ($50\%$ salt and $50\%$ water). Jonny then spills the beaker and $45$ liters pour out. If Jonny adds $45$ liters of pure water back into the beaker, what percent of the new mixture is salt? [b]p18.[/b] There are exactly 25 prime numbers in the set of positive integers between $1$ and $100$, inclusive. If two not necessarily distinct integers are randomly chosen from the set of positive integers from $1$ to $100$, inclusive, what is the probability that at least one of them is prime? [b]p19.[/b] How many consecutive zeroes are at the end of $12!$ when it is expressed in base $6$? [b]p20.[/b] Consider the following figure. How many triangles with vertices and edges from the following figure contain exactly $1$ black triangle? [img]https://cdn.artofproblemsolving.com/attachments/f/2/a1c400ff7d06b583c1906adf8848370e480895.png[/img] [b]p21.[/b] After Akshay got kicked o the school bus for rowdy behavior, he worked out a way to get home from school with his dad. School ends at $2:18$ pm, but since Akshay walks slowly he doesn't get to the front door until $2:30$. His dad doesn't like to waste time, so he leaves home everyday such that he reaches the high school at exactly $2:30$ pm, instantly picks up Akshay and turns around, then drives home. They usually get home at $3:30$ pm. However, one day Akshay left school early at exactly $2:00$ pm because he was expelled. Trying to delay telling his dad for as long as possible, Akshay starts jogging home. His dad left home at the regular time, saw Akshay on the way, picked him up and turned around instantly. They then drove home while Akshay's dad yelled at him for being a disgrace. They reached home at $3:10$ pm. How long had Akshay been walking before his dad picked him up? [b]p22.[/b] In quadrilateral $ABCD$, diagonals $AC$ and $BD$ intersect at $O$. Then $\angle BOC = \angle BCD$, $\angle COD =\angle BAD$, $AB = 4$, $DC = 6$, and $BD = 5$. What is the length of $BO$? [b]p23.[/b] A standard six-sided die is rolled. The number that comes up first determines the number of additional times the die will be rolled (so if the first number is $3$, then the die will be rolled $3$ more times). Each time the die is rolled, its value is recorded. What is the expected value of the sum of all the rolls? [b]p24.[/b] Dora has a peculiar calculator that can only perform $2$ operations: either adding $1$ to the current number or squaring the current number. Each minute, Dora randomly chooses an operation to apply to her number. She starts with $0$. What is the expected number of minutes it takes Dora's number to become greater than or equal to $10$? [b]p25.[/b] Let $\vartriangle ABC$ be such that $AB = 2$, $BC = 1$, and $\angle ACB = 90^o$. Let points $D$ and $E$ be such that $\vartriangle ADE$ is equilateral, $D$ is on segment $\overline{BC}$, and $D$ and $E$ are not on the same side of $\overline{AC}$. Segment $\overline{BE}$ intersects the circumcircle of $\vartriangle ADE$ at a second point $F$. If $BE =\sqrt{6}$, find the length of $\overline{BF}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For each real number $x$, let $\lfloor x \rfloor$ denote the largest integer not exceeding $x$. A sequence $\{a_n \}_{n=1}^{\infty}$ is defined by $a_n = \frac{1}{4^{\lfloor -\log_4 n \rfloor}}, \forall n \geq 1.$ Let $b_n = \frac{1}{n^2} \left( \sum_{k=1}^n a_k - \frac{1}{a_1+a_2} \right), \forall n \geq 1.$ a) Find a polynomial $P(x)$ with real coefficients such that $b_n = P \left( \frac{a_n}{n} \right), \forall n \geq 1$. b) Prove that there exists a strictly increasing sequence $\{n_k \}_{k=1}^{\infty}$ of positive integers such that $$\lim_{k \to \infty} b_{n_k} = \frac{2024}{2025}.$$

Let $x$ and $y$ be real numbers such that $x^2 + y^2 - 1 < xy$. Prove that $x + y - |x - y| < 2$. Slovakia

Given an expression $x^2 + ax + b$ where $a,b$ are integer coefficients. At any step, one can change the expression by adding either $1$ or $-1$ to only one of the two coefficients $a, b$. a) Suppose that the initial expression has $a =-7$ and $b = 19$. Show your modification steps to obtain a new expression that has zero value at some integer value of $x$. b) Starting from the initial expression as above, one gets the expression $x^2 - 17x + 9$ after $m$ modification steps. Prove that at a certain step $k$ with $k < m$, the obtained expression has zero value at some integer value of $x$.

The numbers $1- \sqrt{2}$, $\sqrt{2}$ and $1+\sqrt{2}$ are written on a blackboard. Every minute, if $x, y, z$ are the numbers written, then they are erased and the numbers, $x^2 + xy + y^2$, $y^2 + yz + z^2$ and $z^2 + zx + x^2$ are written. Determine whether it is possible for all written numbers to be rational numbers after a finite number of minutes.

Let $x$ be a real number satisfying \[ \sqrt[3]{125-x^3}-\sqrt[3]{27-x^3}=7. \] Compute $|\sqrt[3]{125-x^3}+\sqrt[3]{27-x^3}|$.

Find the number of counter examples to the statement: \[``\text{If N is an odd positive integer the sum of whose digits is 4 and none of whose digits is 0, then N is prime}."\] $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4 $

A quadruplet of distinct positive integers $(a, b, c, d)$ is called $k$-good if the following conditions hold: 1. Among $a, b, c, d$, no three form an arithmetic progression. 2. Among $a+b, a+c, a+d, b+c, b+d, c+d$, there are $k$ of them, forming an arithmetic progression. $a)$ Find a $4$-good quadruplet. $b)$ What is the maximal $k$, such that there is a $k$-good quadruplet?

Let \(P(x)=x^2+\dfrac x 2 +b\) and \(Q(x)=x^2+cx+d\) be two polynomials with real coefficients such that \(P(x)Q(x)=Q(P(x))\) for all real \(x\). Find all real roots of \(P(Q(x))=0\).

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$ \[f(x + y)=f(f(x)) + y + 2022\]

Solve the equation $ \sqrt{4 \minus{} 3\sqrt{10 \minus{} 3x}} \equal{} x \minus{} 2$.

[b]p19.[/b] $A_1A_2...A_{12}$ is a regular dodecagon with side length $1$ and center at point $O$. What is the area of the region covered by circles $(A_1A_2O)$, $(A_3A_4O)$, $(A_5A_6O)$, $(A_7A_8O)$, $(A_9A_{10}O)$, and $(A_{11}A_{12}O)$? $(ABC)$ denotes the circle passing through points $A,B$, and $C$. [b]p20.[/b] Let $N = 2000... 0x0 ... 00023$ be a $2023$-digit number where the $x$ is the $23$rd digit from the right. If$ N$ is divisible by $13$, compute $x$. [b]p21.[/b] Alice and Bob each visit the dining hall to get a grilled cheese at a uniformly random time between $12$ PM and $1$ PM (their arrival times are independent) and, after arrival, will wait there for a uniformly random amount of time between $0$ and $30$ minutes. What is the probability that they will meet? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For integers $0\le a\le n$, let $f(n,a)$ denote the number of coefficients in the expansion of $(x+1)^a(x+2)^{n-a}$ that is divisible by $3.$ For example, $(x+1)^3(x+2)^1=x^4+5x^3+9x^2+7x+2$, so $f(4,3)=1$. For each positive integer $n$, let $F(n)$ be the minimum of $f(n,0),f(n,1),\ldots ,f(n,n)$. (1) Prove that there exist infinitely many positive integer $n$ such that $F(n)\ge \frac{n-1}{3}$. (2) Prove that for any positive integer $n$, $F(n)\le \frac{n-1}{3}$.

[u]Set 7[/u] [b]p19.[/b] Let circles $\omega_1$ and $\omega_2$, with centers $O_1$ and $O_2$, respectively, intersect at $X$ and $Y$ . A lies on $\omega_1$ and $B$ lies on $\omega_2$ such that $AO_1$ and $BO_2$ are both parallel to $XY$, and $A$ and $B$ lie on the same side of $O_1O_2$. If $XY = 60$, $\angle XAY = 45^o$, and $\angle XBY = 30^o$, then the length of $AB$ can be expressed in the form $\sqrt{a - b\sqrt2 + c\sqrt3}$, where $a, b, c$ are positive integers. Determine $a + b + c$. [b]p20.[/b] If $x$ is a positive real number such that $x^{x^2}= 2^{80}$, find the largest integer not greater than $x^3$. [b]p21.[/b] Justin has a bag containing $750$ balls, each colored red or blue. Sneaky Sam takes out a random number of balls and replaces them all with green balls. Sam notices that of the balls left in the bag, there are $15$ more red balls than blue balls. Justin then takes out $500$ of the balls chosen randomly. If $E$ is the expected number of green balls that Justin takes out, determine the greatest integer less than or equal to $E$. [u]Set 8[/u] These three problems are interdependent; each problem statement in this set will use the answers to the other two problems in this set. As such, let the positive integers $A, B, C$ be the answers to problems $22$, $23$, and $24$, respectively, for this set. [b]p22.[/b] Let $WXYZ$ be a rectangle with $WX =\sqrt{5B}$ and $XY =\sqrt{5C}$. Let the midpoint of $XY$ be $M$ and the midpoint of $YZ$ be $N$. If $XN$ and $W Y$ intersect at $P$, determine the area of $MPNY$ . [b]p23.[/b] Positive integers $x, y, z$ satisfy $$xy \equiv A \,\, (mod 5)$$ $$yz \equiv 2A + C\,\, (mod 7)$$ $$zx \equiv C + 3 \,\, (mod 9).$$ (Here, writing $a \equiv b \,\, (mod m)$ is equivalent to writing $m | a - b$.) Given that $3 \nmid x$, $3 \nmid z$, and $9 | y$, find the minimum possible value of the product $xyz$. [b]p24.[/b] Suppose $x$ and $y$ are real numbers such that $$x + y = A$$ $$xy =\frac{1}{36}B^2.$$ Determine $|x - y|$. [u]Set 9[/u] [b]p25. [/b]The integer $2017$ is a prime which can be uniquely represented as the sum of the squares of two positive integers: $$9^2 + 44^2 = 2017.$$ If $N = 2017 \cdot 128$ can be uniquely represented as the sum of the squares of two positive integers $a^2 +b^2$, determine $a + b$. [b]p26.[/b] Chef Celia is planning to unveil her newest creation: a whole-wheat square pyramid filled with maple syrup. She will use a square flatbread with a one meter diagonal and cut out each of the five polygonal faces of the pyramid individually. If each of the triangular faces of the pyramid are to be equilateral triangles, the largest volume of syrup, in cubic meters, that Celia can enclose in her pyramid can be expressed as $\frac{a-\sqrt{b}}{c}$ where $a, b$ and $c$ are the smallest possible possible positive integers. What is $a + b + c$? [b]p27.[/b] In the Cartesian plane, let $\omega$ be the circle centered at $(24, 7)$ with radius $6$. Points $P, Q$, and $R$ are chosen in the plane such that $P$ lies on $\omega$, $Q$ lies on the line $y = x$, and $R$ lies on the $x$-axis. The minimum possible value of $PQ+QR+RP$ can be expressed in the form $\sqrt{m}$ for some integer $m$. Find m. [u]Set 10[/u] [i]Deja vu?[/i] [b]p28. [/b] Let $ABC$ be a triangle with incircle $\omega$. Let $\omega$ intersect sides $BC$, $CA$, $AB$ at $D, E, F$, respectively. Suppose $AB = 7$, $BC = 12$, and $CA = 13$. If the area of $ABC$ is $K$ and the area of $DEF$ is $\frac{m}{n}\cdot K$, where $m$ and $n$ are relatively prime positive integers, then compute $m + n$. [b]p29.[/b] Sebastian is playing the game Split! again, but this time in a three dimensional coordinate system. He begins the game with one token at $(0, 0, 0)$. For each move, he is allowed to select a token on any point $(x, y, z)$ and take it off, replacing it with three tokens, one at $(x + 1, y, z)$, one at $(x, y + 1, z)$, and one at $(x, y, z + 1)$ At the end of the game, for a token on $(a, b, c)$, it is assigned a score $\frac{1}{2^{a+b+c}}$ . These scores are summed for his total score. If the highest total score Sebastian can get in $100$ moves is $m/n$, then determine $m + n$. [b]p30.[/b] Determine the number of positive $6$ digit integers that satisfy the following properties: $\bullet$ All six of their digits are $1, 5, 7$, or $8$, $\bullet$ The sum of all the digits is a multiple of $5$. [u]Set 11[/u] [b]p31.[/b] The triangular numbers are defined as $T_n =\frac{n(n+1)}{2}$. We also define $S_n =\frac{n(n+2)}{3}$. If the sum $$\sum_{i=16}^{32} \left(\frac{1}{T_i}+\frac{1}{S_i}\right)= \left(\frac{1}{T_{16}}+\frac{1}{S_{16}}\right)+\left(\frac{1}{T_{17}}+\frac{1}{S_{17}}\right)+...+\left(\frac{1}{T_{32}}+\frac{1}{S_{32}}\right)$$ can be written in the form $a/b$ , where $a$ and $b$ are positive integers with $gcd(a, b) = 1$, then find $a + b$. [b]p32.[/b] Farmer Will is considering where to build his house in the Cartesian coordinate plane. He wants to build his house on the line $y = x$, but he also has to minimize his travel time for his daily trip to his barnhouse at $(24, 15)$ and back. From his house, he must first travel to the river at $y = 2$ to fetch water for his animals. Then, he heads for his barnhouse, and promptly leaves for the long strip mall at the line $y =\sqrt3 x$ for groceries, before heading home. If he decides to build his house at $(x_0, y_0)$ such that the distance he must travel is minimized, $x_0$ can be written in the form $\frac{a\sqrt{b}-c}{d}$ , where $a, b, c, d$ are positive integers, $b$ is not divisible by the square of a prime, and $gcd(a, c, d) = 1$. Compute $a+b+c+d$. [b]p33.[/b] Determine the greatest positive integer $n$ such that the following two conditions hold: $\bullet$ $n^2$ is the difference of consecutive perfect cubes; $\bullet$ $2n + 287$ is the square of an integer. [u]Set 12[/u] The answers to these problems are nonnegative integers that may exceed $1000000$. You will be awarded points as described in the problems. [b]p34.[/b] The “Collatz sequence” of a positive integer n is the longest sequence of distinct integers $(x_i)_{i\ge 0}$ with $x_0 = n$ and $$x_{n+1} =\begin{cases} \frac{x_n}{2} & if \,\, x_n \,\, is \,\, even \\ 3x_n + 1 & if \,\, x_n \,\, is \,\, odd \end{cases}.$$ It is conjectured that all Collatz sequences have a finite number of elements, terminating at $1$. This has been confirmed via computer program for all numbers up to $2^{64}$. There is a unique positive integer $n < 10^9$ such that its Collatz sequence is longer than the Collatz sequence of any other positive integer less than $10^9$. What is this integer $n$? An estimate of $e$ gives $\max\{\lfloor 32 - \frac{11}{3}\log_{10}(|n - e| + 1)\rfloor, 0\}$ points. [b]p35.[/b] We define a graph $G$ as a set $V (G)$ of vertices and a set $E(G)$ of distinct edges connecting those vertices. A graph $H$ is a subgraph of $G$ if the vertex set $V (H)$ is a subset of $V (G)$ and the edge set $E(H)$ is a subset of $E(G)$. Let $ex(k, H)$ denote the maximum number of edges in a graph with $k$ vertices without a subgraph of $H$. If $K_i$ denotes a complete graph on $i$ vertices, that is, a graph with $i$ vertices and all ${i \choose 2}$ edges between them present, determine $$n =\sum_{i=2}^{2018} ex(2018, K_i).$$ An estimate of $e$ gives $\max\{\lfloor 32 - 3\log_{10}(|n - e| + 1)\rfloor, 0\}$ points. [b]p36.[/b] Write down an integer between $1$ and $100$, inclusive. This number will be denoted as $n_i$ , where your Team ID is $i$. Let $S$ be the set of Team ID’s for all teams that submitted an answer to this problem. For every ordered triple of distinct Team ID’s $(a, b, c)$ such that a, b, c ∈ S, if all roots of the polynomial $x^3 + n_ax^2 + n_bx + n_c$ are real, then the teams with ID’s $a, b, c$ will each receive one virtual banana. If you receive $v_b$ virtual bananas in total and $|S| \ge 3$ teams submit an answer to this problem, you will be awarded $$\left\lfloor \frac{32v_b}{3(|S| - 1)(|S| - 2)}\right\rfloor$$ points for this problem. If $|S| \le 2$, the team(s) that submitted an answer to this problem will receive $32$ points for this problem. PS. You had better use hide for answers. First sets have been posted [url=https://artofproblemsolving.com/community/c4h2777264p24369138]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $m, n$ be positive integers. Show that the polynomial $$f(x)=x^m(x^2-100)^n-11$$ cannot be expressed as a product of two non-constant polynomials with integral coefficients.

[b]p1.[/b] If $A = 0$, $B = 1$, $C = 2$, $...$, $Z = 25$, then what is the sum of $A + B + M+ C$? [b]p2.[/b] Eric is playing Tetris against Bryan. If Eric wins one-fifth of the games he plays and he plays $15$ games, find the expected number of games Eric will win. [b]p3.[/b] What is the sum of the measures of the exterior angles of a regular $2023$-gon in degrees? [b]p4.[/b] If $N$ is a base $10$ digit of $90N3$, what value of $N$ makes this number divisible by $477$? [b]p5.[/b] What is the rightmost non-zero digit of the decimal expansion of $\frac{1}{2^{2023}}$ ? [b]p6.[/b] if graphs of $y = \frac54 x + m$ and $y = \frac32 x + n$ intersect at $(16, 27)$, what is the value of $m + n$? [b]p7.[/b] Bryan is hitting the alphabet keys on his keyboard at random. If the probability he spells out ABMC at least once after hitting $6$ keys is $\frac{a}{b^c}$ , for positive integers $a$, $b$, $c$ where $b$, $c$ are both as small as possible, find $a+b+c$. Note that the letters ABMC must be adjacent for it to count: AEBMCC should not be considered as correctly spelling out ABMC. [b]p8.[/b] It takes a Daniel twenty minutes to change a light bulb. It takes a Raymond thirty minutes to change a light bulb. It takes a Bryan forty-five minutes to change a light bulb. In the time that it takes two Daniels, three Raymonds, and one and a half Bryans to change $42$ light bulbs, how many light bulbs could half a Raymond change? Assume half a person can work half as productively as a whole person. [b]p9.[/b] Find the value of $5a + 4b + 3c + 2d + e$ given $a, b, c, d, e$ are real numbers satisfying the following equations: $$a^2 = 2e + 23$$ $$b^2 = 10a - 34$$ $$c^2 = 8b - 23$$ $$d^2 = 6c - 14$$ $$e^2 = 4d - 7.$$ [b]p10.[/b] How many integers between $1$ and $1000$ contain exactly two $1$’s when written in base $2$? [b]p11.[/b] Joe has lost his $2$ sets of keys. However, he knows that he placed his keys in one of his $12$ mailboxes, each labeled with a different positive integer from $1$ to $12$. Joe plans on opening the $2$ mailbox labeled $1$ to see if any of his keys are there. However, a strong gust of wind blows by, opening mailboxes $11$ and $12$, revealing that they are empty. If Joe decides to open one of the mailboxes labeled $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$ , or $10$, the probability that he finds at least one of his sets of keys can be expressed as $\frac{a}{b}$, where a and b are relatively prime positive integers. Find the sum $a + b$. Note that a single mailbox can contain $0$, $1$, or $2$ sets of keys, and the mailboxes his sets of keys were placed in are determined independently at random. [b]p12.[/b] As we all know, the top scientists have recently proved that the Earth is a flat disc. Bob is standing on Earth. If he takes the shortest path to the edge, he will fall off after walking $1$ meter. If he instead turns $90$ degrees away from the shortest path and walks towards the edge, he will fall off after $3$ meters. Compute the radius of the Earth. [b]p13.[/b] There are $999$ numbers that are repeating decimals of the form $0.abcabcabc...$ . The sum of all of the numbers of this form that do not have a $1$ or $2$ in their decimal representation can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$. Find $a + b$. [b]p14.[/b] An ant is crawling along the edges of a sugar cube. Every second, it travels along an edge to another adjacent vertex randomly, interested in the sugar it notices. Unfortunately, the cube is about to be added to some scalding coffee! In $10$ seconds, it must return to its initial vertex, so it can get off and escape. If the probability the ant will avoid a tragic doom can be expressed as $\frac{a}{3^{10}}$ , where $a$ is a positive integer, find $a$. Clarification: The ant needs to be on its initial vertex in exactly $10$ seconds, no more or less. [b]p15.[/b] Raymond’s new My Little Pony: Friendship is Magic Collector’s book arrived in the mail! The book’s pages measure $4\sqrt3$ inches by $12$ inches, and are bound on the longer side. If Raymond keeps one corner in the same plane as the book, what is the total area one of the corners can travel without ripping the page? If the desired area in square inches is $a\pi+b\sqrt{c}$ where $a$, $b$, and $c$ are integers and $c$ is squarefree, find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

p1. Let $AB$ be the diameter of the circle and $P$ is a point outside the circle. The lines $PQ$ and $PR$ are tangent to the circles at points $Q$ and $R$. The lines $PH$ is perpendicular on line $AB$ at $H$ . Line $PH$ intersects $AR$ at $S$. If $\angle QPH =40^o$ and $\angle QSA =30^o$, find $\angle RPS$. p2. There is a meeting consisting of $40$ seats attended by $16$ invited guests. If each invited guest must be limited to at least $ 1$ chair, then determine the number of arrangements. p3. In the crossword puzzle, in the following crossword puzzle, each box can only be filled with numbers from $ 1$ to $9$. [img]https://cdn.artofproblemsolving.com/attachments/2/e/224b79c25305e8ed9a8a4da51059f961b9fbf8.png[/img] Across: 1. Composite factor of $1001$ 3. Non-polyndromic numbers 5. $p\times q^3$, with $p\ne q$ and $p,q$ primes Down: 1. $a-1$ and $b+1$ , $a\ne b$ and $p,q$ primes 2. multiple of $9$ 4. $p^3 \times q$, with $p\ne q$ and $p,q$ primes p4. Given a function $f:R \to R$ and a function $g:R \to R$, so that it fulfills the following figure: [img]https://cdn.artofproblemsolving.com/attachments/b/9/fb8e4e08861a3572412ae958828dce1c1e137a.png[/img] Find the number of values ​​of $x$, such that $(f(x))^2-2g(x)-x \in\{-10,-9,-8,…,9,10\}$. p5. In a garden that is rectangular in shape, there is a watchtower in each corner and in the garden there is a monitoring tower. Small areas will be made in the shape of a triangle so that the corner points are towers (free of monitoring and/or supervisory towers). Let $k(m,n)$ be the number of small areas created if there are $m$ control towers and $n$ monitoring towers. a. Find the values ​​of $k(4,1)$, $k(4,2)$, $k(4,3)$, and $k(4,4)$ b. Find the general formula $k(m,n)$ with $m$ and $n$ natural numbers .