For how many positive integers $x$ is $\log_{10}{(x-40)} + \log_{10}{(60-x)} < 2$? ${ \textbf{(A)}\ 10\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 19\qquad\textbf{(D)}}\ 20\qquad\textbf{(E)}\ \text{infinitely many} $

For positive numbers $x$, $y$, and $z$, we know that $x + y^2 + z^3 = 1$. Prove that $$\frac{x}{2 + xy} + \frac{y}{2 + yz} + \frac{z}{2 + zx} > \frac{1}{2} .$$

[b]p1.[/b] There are $2024$ apples in a very large basket. First, Julie takes away half of the apples in the basket; then, Diane takes away $202$ apples from the remaining bunch. How many apples remain in the basket? [b]p2.[/b] The set of all permutations (different arrangements) of the letters in ”ABMC” are listed in alphabetical order. The first item on the list is numbered $1$, the second item is numbered $2$, and in general, the kth item on the list is numbered $k$. What number is given to ”ABMC”? [b]p3.[/b] Daniel has a water bottle that is three-quarters full. After drinking $3$ ounces of water, the water bottle is three-fifths full. The density of water is $1$ gram per milliliter, and there are around $28$ grams per ounce. How many milliliters of water could the bottle fit at full capacity? [b]p4.[/b] How many ways can four distinct $2$-by-$1$ rectangles fit on a $2$-by-$4$ board such that each rectangle is fully on the board? [b]p5.[/b] Iris and Ivy start reading a $240$ page textbook with $120$ left-hand pages and $120$ right-hand pages. Iris takes $4$ minutes to read each page, while Ivy takes $5$ minutes to read a left-hand page and $3$ minutes to read a right-hand page. Iris and Ivy move onto the next page only when both sisters have completed reading. If a sister finishes reading a page first, the other sister will start reading three times as fast until she completes the page. How many minutes after they start reading will both sisters finish the textbook? [b]p6.[/b] Let $\vartriangle ABC$ be an equilateral triangle with side length $24$. Then, let $M$ be the midpoint of $BC$. Define $P$ to be the set of all points $P$ such that $2PM = BC$. The minimum value of $AP$ can be expressed as $\sqrt{a}- b$, where $a$ and $b$ are positive integers. Find $a + b$. [b]p7.[/b] Jonathan has $10$ songs in his playlist: $4$ rap songs and $6$ pop songs. He will select three unique songs to listen to while he studies. Let $p$ be the probability that at least two songs are rap, and let $q$ be the probability that none of them are rap. Find $\frac{p}{q}$ . [b]p8.[/b] A number $K$ is called $6,8$-similar if $K$ written in base $6$ and $K$ written in base $8$ have the same number of digits. Find the number of $6,8$-similar values between $1$ and $1000$, inclusive. [b]p9.[/b] Quadrilateral $ABCD$ has $\angle ABC = 90^o$, $\angle ADC = 120^o$, $AB = 5$, $BC = 18$, and $CD = 3$. Find $AD^2$. [b]p10.[/b] Bob, Eric, and Raymond are playing a game. Each player rolls a fair $6$-sided die, and whoever has the highest roll wins. If players are tied for the highest roll, the ones that are tied reroll until one wins. At the start, Bob rolls a $4$. The probability that Eric wins the game can be expressed as $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$. [b]p11.[/b] Define the following infinite sequence $s$: $$s = \left\{\frac92,\frac{99}{2^2},\frac{999}{2^3} , ... , \overbrace{\frac{999...999}{2^k}}^{k\,\,nines}, ...\right\}$$ The sum of the first $2024$ terms in $s$, denoted $S$, can be expressed as $$S =\frac{5^a - b}{4}+\frac{1}{2^c},$$ where $a, b$, and $c$ are positive integers. Find $a + b + c$. [b]p12.[/b] Andy is adding numbers in base $5$. However, he accidentally forgets to write the units digit of each number. If he writes all the consecutive integers starting at $0$ and ending at $50$ (base $10$) and adds them together, what is the difference between Andy’s sum and the correct sum? (Express your answer in base-$10$.) [b]p13.[/b] Let $n$ be the positive real number such that the system of equations $$y =\frac{1}{\sqrt{2024 - x^2}}$$ $$y =\sqrt{x^2 - n}$$ has exactly two real solutions for $(x, y)$: $(a, b)$ and $(-a, b)$. Then, $|a|$ can be expressed as $j\sqrt{k}$, where $j$ and $k$ are integers such that $k$ is not divisible by any perfect square other than $1$. Find $j · k$. [b]p14.[/b] Nakio is playing a game with three fair $4$-sided dice. But being the cheater he is, he has secretly replaced one of the three die with his own $4$-sided die, such that there is a $1/2$ chance of rolling a $4$, and a $1/6$ chance to roll each number from $1$ to $3$. To play, a random die is chosen with equal probability and rolled. If Nakio guesses the number that is on the die, he wins. Unfortunately for him, Nakio’s friends have an anti-cheating mechanism in place: when the die is picked, they will roll it three times. If each roll lands on the same number, that die is thrown out and one of the two unused dice is chosen instead with equal probability. If Nakio always guesses $4$, the probability that he wins the game can be expressed as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime. Find $m + n$. [b]p15.[/b] A particle starts in the center of a $2$m-by-$2$m square. It moves in a random direction such that the angle between its direction and a side of the square is a multiple of $30^o$. It travels in that direction at $1$ m/s, bouncing off of the walls of the square. After a minute, the position of the particle is recorded. The expected distance from this point to the start point can be written as $$\frac{1}{a}\left(b - c\sqrt{d}\right),$$ where $a$ and $b$ are relatively prime, and d is not divisible by any perfect square. Find $a + b + c + d$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ b_1, b_2, \ldots, b_{1989}$ be positive real numbers such that the equations \[ x_{r\minus{}1} \minus{} 2x_r \plus{} x_{r\plus{}1} \plus{} b_rx_r \equal{} 0 \quad (1 \leq r \leq 1989)\] have a solution with $ x_0 \equal{} x_{1989} \equal{} 0$ but not all of $ x_1, \ldots, x_{1989}$ are equal to zero. Prove that \[ \sum^{1989}_{k\equal{}1} b_k \geq \frac{2}{995}.\]

Rational numbers are written in the following sequence: $\frac{1}{1},\frac{2}{1},\frac{1}{2},\frac{3}{1},\frac{2}{2},\frac{1}{3},\frac{4}{1},\frac{3}{2},\frac{2}{3},\frac{1}{4}, . . .$ In which position of this sequence is $\frac{2005}{2004}$ ?

William was bored at the math lesson, so he drew a circle and $n\ge3$ empty cells around the circumference. In every cell he wrote a positive number. Later on he erased the numbers and in every cell wrote the geometric mean of the numbers previously written in the two neighboring cells. Show that there exists a cell whose number was not replaced by a larger number.

[b]p1.[/b] While computing $7 - 2002 \cdot x$, John accidentally evaluates from left to right $((7 - 2002) \cdot x)$ instead of correctly using order of operations $(7 - (2002 \cdot x))$. If he gets the correct answer anyway, what is $x$? [b]p2.[/b] Given that $$x^2 + y^2 + z^2 = 6$$ $$ \left( \frac{x}{y} + \frac{y}{x} \right)^2 + \left( \frac{y}{z} + \frac{z}{y} \right)^2 + \left( \frac{z}{x} + \frac{x}{z} \right)^2 = 16.5,$$ what is $\frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2}$ ? [b]p3.[/b] Evaluate $$\frac{tan \frac{\pi}{4}}{4}+\frac{tan \frac{3\pi}{4}}{8}+\frac{tan \frac{5\pi}{4}}{16}+\frac{tan \frac{7\pi}{4}}{32}+ ...$$ [b]p4.[/b] Note that $2002 = 22 \cdot 91$, and so $2002$ is a multiple of the number obtained by removing its middle $2$ digits. Generalizing this, how many $4$-digit palindromes, $abba$, are divisible by the $2$-digit palindrome, $aa$? [b]p5.[/b] Let $ABCDE$ be a pyramid such that $BCDE$ is a square with side length $2$, and $A$ is $2$ units above the center of $BCDE$. If $F$ is the midpoint of $\overline{DE}$ and $G$ is the midpoint of $\overline{AC}$, what is the length of $\overline{DE}$? [b]p6.[/b] Suppose $a_1, a_2,..., a_{100}$ are real numbers with the property that $$i(a_1 + a_2 +... + a_i) = 1 + (a_{i+1} + a_{i+2} + ... + a_{100})$$ for all $i$. Compute $a_{10}$. [b]p7.[/b] A bug is sitting on one corner of a $3' \times 4' \times 5'$ block of wood. What is the minimum distance nit needs to travel along the block’s surface to reach the opposite corner? [b]p8.[/b] In the number game, a pair of positive integers $(n,m)$ is written on a blackboard. Two players then take turns doing the following: 1. If $n \ge m$, the player chooses a positive integer $c$ such that $n - cm \ge 0$, and replaces $(n,m)$ with $(n - cm,m)$. 2. If $m > n$, the player chooses a positive integer $c$ such that $m - cn \ge 0$, and replaces $(n,m)$ with $(n,m - cn)$. If $m$ or $n$ ever become $0$, the game ends, and the last player to have moved is declared the winner. If $(n,m)$ are originally $(20021000, 2002)$, what choices of $c$ are winning moves for the first player? PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $a_1,a_2,\ldots,a_n$ ($n\geq 2$) be nonnegative real numbers whose sum is $\frac{n}{2}$. For every $i=1,\ldots,n$ define $$b_i=a_i+a_ia_{i+1}+a_ia_{i+1}a_{i+2}+\cdots+ a_ia_{i+1}\cdots a_{i+n-2}+2a_ia_{i+1}\cdots a_{i+n-1}$$ where $a_{j+n}=a_j$ for every $j$. Prove that $b_i\geq 1$ holds for at least one index $i$.

Sequences $x_1,x_2,\dots,$ and $y_1,y_2,\dots,$ are defined with $x_1=\dfrac{1}{8}$, $y_1=\dfrac{1}{10}$ and $x_{n+1}=x_n+x_n^2$, $y_{n+1}=y_n+y_n^2$. Prove that $x_m\neq y_n$ for all $m,n\in\mathbb{Z}^{+}$. [I]Proposed by A. Golovanov[/i]

Given any four distinct positive real numbers, show that one can choose three numbers $A,B,C$ from among them, such that all three quadratic equations \begin{eqnarray*} Bx^2 + x + C &=& 0\\ Cx^2 + x + A &=& 0 \\ Ax^2 + x +B &=& 0 \end{eqnarray*} have only real roots, or all three equations have only imaginary roots.

Find the number of polynomials $P(x)$ of degree $6$ whose coefficients are in the set $\{1,2,\ldots,1999\}$ and which are divisible by $x^3+x^2+x+1$.

Suppose in a given collection of $2016$ integer, the sum of any $1008$ integers is positive. Show that sum of all $2016$ integers is positive.

For any positive integer $n$, define $$c_n=\min_{(z_1,z_2,...,z_n)\in\{-1,1\}^n} |z_1\cdot 1^{2018} + z_2\cdot 2^{2018} + ... + z_n\cdot n^{2018}|.$$ Is the sequence $(c_n)_{n\in\mathbb{Z}^+}$ bounded?

Let $n $ be a positive integer. Let $f $ be a function from nonnegative integers to themselves. Let $f (0,i)=f (i,0)=0$, $f (1, 1)=n $, and $ f(i, j)= [\frac {f(i-1,j)}{2}]+ [\frac {f(i, j-1)}{2}] $ for positive integers $i, j$ such that $i*j>1$. Find the number of pairs $(i,j) $ such that $f (i, j) $ is an odd number.( $[x]$ is the floor function).

Ana, Beto, Carlos, Diana, Elena and Fabian are in a circle, located in that order. Ana, Beto, Carlos, Diana, Elena and Fabian each have a piece of paper, where are written the real numbers $a,b,c,d,e,f$ respectively. At the end of each minute, all the people simultaneously replace the number on their paper by the sum of three numbers; the number that was at the beginning of the minute on his paper and on the papers of his two neighbors. At the end of the minute $2022, 2022$ replacements have been made and each person have in his paper it´s initial number. Find all the posible values of $abc+def$. $\textbf{Note:}$ [i]If at the beginning of the minute $N$ Ana, Beto, Carlos have the numbers $x,y,z$, respectively, then at the end of the minute $N$, Beto is going to have the number $x+y+z$[/i].

What is the largest possible value of $|a_1 - 1| + |a_2-2|+...+ |a_n- n|$ where $a_1, a_2,..., a_n$ is a permutation of $1,2,..., n$?

Determine all $ f:R\rightarrow R $ such that $$ f(xf(y)+y^3)=yf(x)+f(y)^3 $$

Consider a function $f$ on nonnegative integers such that $f(0)=1, f(1)=0$ and $f(n)+f(n-1)=nf(n-1)+(n-1)f(n-2)$ for $n \ge 2$. Show that \[\frac{f(n)}{n!}=\sum_{k=0}^n \frac{(-1)^k}{k!}\]

For x > 1, let $f(x) = log_2(x + log_2(x + log_2(x +...)))$. Compute $\Sigma_{k=2}^{10} f^{-1}(k)$

Prove for all $a_1, ..., a_n > 0$ the following inequality and determine all cases in where the equaloty holds: $$\sum_{k=1}^{n}ka_k\le {n \choose 2}+\sum_{k=1}^{n}a_k^k.$$

Let $f(x) = x^2 + 6x + 7$. Determine the smallest possible value of $f(f(f(f(x))))$ over all real numbers $x.$

Let $P(x) \in \mathbb{R}[x]$ be a monic, non-constant polynomial. Determine all continuous functions $f: \mathbb{R} \to \mathbb{R}$ such that $$f(f(P(x))+y+2023f(y))=P(x)+2024f(y),$$ for all reals $x,y$.

For what values of the variable $x$ does the following inequality hold: \[ \dfrac{4x^2}{(1-\sqrt{2x+1})^2}<2x+9 \ ? \]

Let $f:\mathbb{R}_{+}^{*}$$\rightarrow$$\mathbb{R}_{+}^{*}$ such that $f'''(x)>0$ for all $x$ $\in$ $\mathbb{R}_{+}^{*}$. Prove that: $f(a^{2}+b^{2}+c^{2})+2f(ab+bc+ac)$ $\geq$ $f(a^{2}+2bc)+f(b^{2}+2ca)+f(c^{2}+2ab)$, for all $a,b,c$ $\in$ $\mathbb{R}_{+}^{*}$.

Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $f(x^3)-f(y^3)=(x^2+xy+y^2)(f(x)-f(y))$.