Let $a_0=5/2$ and $a_k=a_{k-1}^2-2$ for $k\ge 1.$ Compute \[\prod_{k=0}^{\infty}\left(1-\frac1{a_k}\right)\] in closed form.

Let $a_1,a_2, a_3,...$ be a sequence of real numbers which satisfy the relation $a_{n+1} =\sqrt{a_n^2 + 1}$ Suppose that there exists a positive integer $n_0$ such that $a_{2n_0} = 3a_{n_0}$ . Find the value of $a_{46}$.

If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$

Let $x, y, z$ be positive real numbers such that $\frac{1}{1 + x}+\frac{1}{1 + y}+\frac{1}{1 + z}= 2$. Prove that $8xyz \le 1$.

Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$ Prove that either $f$ is the identity function or $g$ is periodic. [i]Proposed by Pranjal Srivastava[/i]

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ which satisfy the following conditions: $(\text{i})$ $f(x+f(y))=f(x)f(y)$ for all $x,y>0;$ $(\text{ii})$ there are at most finitely many $x$ with $f(x)=1$.

Let $a,b,c\in R^+$ with $abc=1$. Prove that $$\frac{a^3b^3}{a+b}+\frac{b^3c^3}{b+c}+\frac{c^3c^3}{c+a} \ge \frac12 \left(\frac{1}{a}+ \frac{1}{b}+\frac{1}{c}\right)$$ [i](Zhuge Liang)[/i]

$N$ denotes the set of all natural numbers. Define a function $T: N \to N$ such that $T (2k) = k$ and $T (2k + 1) = 2k + 2$. We write $T^2 (n) = T (T (n))$ and in general $T^k (n) = T^{k-1} (T (n))$ for all $k> 1$. (a) Prove that for every $n \in N$, there exists $k$ such that $T^k (n) = 1$. (b) For $k \in N$, $c_k$ denotes the number of elements in the set $\{n: T^k (n) = 1\}$. Prove that $c_{k + 2} = c_{k + 1} + c_k$, for $1 \le k$.

Find all possible integer values of the sum: $$\frac{a}{b}+ \frac{2023 \times b}{4 \times a},$$ where $a$ and $b$ are positive integers with no prime factors in common.

Find the least positive integer $N$ such that there exist positive real numbers $a_1,a_2,\dots,a_N$ such that \[ \sum_{k=1}^{N}ka_k=1 \quad \text{and} \quad \sum_{k=1}^{N}\frac{a_k^2}{k}\leq \frac{1}{2022^2}. \]

Let $n > 1$ be an odd positive integer. Assume that positive integers $x_1, x_2,..., x_n \ge 0$ satisfy: $$\begin{cases} (x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 \\ (x_3 -x_2)^2 + 2(x_3 +x_2) + 1 = n^2 \\ ...\\ (x_1 - x_n)^2 + 2(x_1 + x_n)+ 1 = n^2 \end {cases}$$ Show that there exists $j, 1 \le j \le n$, such that $x_j = x_{j+1}$. Here $x_{n+1} = x_1$.

Prove that the only function $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying the following is $f(x)=x$. (i) $\forall x \not= 0$, $f(x) = x^2f(\frac{1}{x})$. (ii) $\forall x, y$, $f(x+y) = f(x)+f(y)$. (iii) $f(1)=1$.

Find the largest value of $a^b$ such that the positive integers $a,b>1$ satisfy $$a^bb^a+a^b+b^a=5329$$

[b]7.[/b] Let $V$ be a finite-dimensional subspace of $C[0,1]$ such that every nonzero $f\in V$ attains positive value at some point. Prove that there exists a polynomial $P$ that is strictly positive on $[0,1]$ and orthogonal to $V$, that is, for every $f \in V$, $\int_{0}^{1} f(x)P(x)dx =0$ ([b]F.39[/b]) [A. Pinkus, V. Totik]

Prove that there exist at least six points with rational coordinates on the curve of the equation \[y^3=x^3+x+1370^{1370}\]

Numbers $a,b,c$ are lengths of sides of some triangle. Prove the inequality$$\frac{a}{b+c-a}+\frac{b}{c+a-b}+\frac{c}{a+b-c} \geq \frac{a+b}{2c}+\frac{b+c}{2a}+\frac{c+a}{2b}$$ [i]M. Karpuk[/i]

Georg has a board displaying the numbers from $1$ to $50$. Georg may strike out a number if it can be formed by starting with the number $2$ and doing one or more calculations where he either multiplies by $10$ or subtracts $3$. Which of the board’s numbers may Georg strike out?[img]https://cdn.artofproblemsolving.com/attachments/c/e/1bea13b691d3591d782e698bedee3235f8512f.png[/img] Example: Georg may strike out $26$ because it may, for example, be formed by starting with $2$, multiplying by $10$, subtracting $3$ three times, multiplying by $10$ and subtracting $3$ twenty-eight times.

For a given natural number $n$, consider those sets $A\subseteq \mathbb{Z}_n$ for which the equation $xy=uv$ has no other solution in the residual classes $x,y,u,v\in A$ than the trivial solutions $x=u$, $y=v$ and $x=v$, $y=u$. Let $g(n)$ be the maximum of the size of such sets $A$. Prove that $$\limsup_{n\to\infty}\frac{g(n)}{\sqrt{n}}=1$$

[b]p1.[/b] The remainder of a number when divided by $7$ is $5$. If I multiply the number by $32$ and add $18$ to the product, what is the new remainder when divided by $7$? [b]p2.[/b] If a fair coin is flipped $15$ times, what is the probability that there are more heads than tails? [b]p3.[/b] Let $-\frac{\sqrt{p}}{q}$ be the smallest nonzero real number such that the reciprocal of the number is equal to the number minus the square root of the square of the number, where $p$ and $q$ are positive integers and $p$ is not divisible the square of any prime. Find $p + q$. [b]p4.[/b] Rachel likes to put fertilizers on her grass to help her grass grow. However, she has cows there as well, and they eat $3$ little fertilizer balls on average. If each ball is spherical with a radius of $4$, then the total volume that each cow consumes can be expressed in the form $a\pi$ where $a$ is an integer. What is $a$? [b]p5.[/b] One day, all $30$ students in Precalc class are bored, so they decide to play a game. Everyone enters into their calculators the expression $9 \diamondsuit 9 \diamondsuit 9 ... \diamondsuit 9$, where $9$ appears $2020$ times, and each $\diamondsuit$ is either a multiplication or division sign. Each student chooses the signs randomly, but they each choose one more multiplication sign than division sign. Then all $30$ students calculate their expression and take the class average. Find the expected value of the class average. [b]p6.[/b] NaNoWriMo, or National Novel Writing Month, is an event in November during which aspiring writers attempt to produce novel-length work - formally defined as $50,000$ words or more - within the span of $30$ days. Justin wants to participate in NaNoWriMo, but he's a busy high school student: after accounting for school, meals, showering, and other necessities, Justin only has six hours to do his homework and perhaps participate in NaNoWriMo on weekdays. On weekends, he has twelve hours on Saturday and only nine hours on Sunday, because he goes to church. Suppose Justin spends two hours on homework every single day, including the weekends. On Wednesdays, he has science team, which takes up another hour and a half of his time. On Fridays, he spends three hours in orchestra rehearsal. Assume that he spends all other time on writing. Then, if November $1$st is a Friday, let $w$ be the minimum number of words per minute that Justin must type to finish the novel. Round $w$ to the nearest whole number. [b]p7.[/b] Let positive reals $a$, $b$, $c$ be the side lengths of a triangle with area $2030$. Given $ab + bc + ca = 15000$ and $abc = 350000$, find the sum of the lengths of the altitudes of the triangle. [b]p8.[/b] Find the minimum possible area of a rectangle with integer sides such that a triangle with side lengths $3$, $4$, $5$, a triangle with side lengths $4$, $5$, $6$, and a triangle with side lengths $\frac94$, $4$, $4$ all fit inside the rectangle without overlapping. [b]p9.[/b] The base $16$ number $10111213...99_{16}$, which is a concatenation of all of the (base $10$) $2$-digit numbers, is written on the board. Then, the last $2n$ digits are erased such that the base $10$ value of remaining number is divisible by $51$. Find the smallest possible integer value of $n$. [b]p10.[/b] Consider sequences that consist entirely of $X$'s, $Y$ 's and $Z$'s where runs of consecutive $X$'s, $Y$ 's, and $Z$'s are at most length $3$. How many sequences with these properties of length $8$ are there? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $f(x)$ and $g(x)$ be strictly increasing linear functions from $\mathbb R $ to $\mathbb R $ such that $f(x)$ is an integer if and only if $g(x)$ is an integer. Prove that for any real number $x$, $f(x)-g(x)$ is an integer.

Let $x, y$ be positive real numbers. Find the minimum of $$x^2 + xy +\frac{y^2}{2}+\frac{2^6}{x + y}+\frac{3^4}{x^3}$$

Let $a$ and $b$ be real numbers such that $a \ne 0$. Prove that not all the roots of $ax^4 + bx^3 + x^2 + x + 1 = 0$ can be real.

Solve in $R$ equation $[x] \cdot \{x\} = 2001 x$, where$ [ .]$ and $\{ .\}$ represent respectively the floor and the integer functions.

Let $S=\{1, 2, 3, \dots 2021\}$ and $f:S \to S$ be a function such that $f^{(n)}(n)=n$ for each $n \in S$. Find all possible values for $f(2021)$. (Here, $f^{(n)}(n) = \underbrace{f(f(f\dots f(}_{n \text{ times} }n)))\dots))$.) [i]Authored by Viktor Simjanoski[/i]

Consider sequences that consist entirely of $ A$'s and $ B$'s and that have the property that every run of consecutive $ A$'s has even length, and every run of consecutive $ B$'s has odd length. Examples of such sequences are $ AA$, $ B$, and $ AABAA$, while $ BBAB$ is not such a sequence. How many such sequences have length 14?