In three years, Xingyou’s age in years will be twice his current height in feet. If Xingyou’s current age in years is also his current height in feet, what is Xingyou’s age in years right now?

If $a, b, c$ are non-zero reals, prove that $|\frac{b} {a}-\frac{b} {c}|+|\frac{c} {a}-\frac{c}{b}|+|bc+1|>1$.

Recall that an arithmetic sequence is a sequence of numbers such that the difference between any two consecutive terms is the same. Suppose $x_1$, $x_2$, $x_3$ forms an arithmetic sequence. If $x_2 = 2023$, compute $x_1 + x_2 + x_3$.

Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor \equal{} 4x \plus{} 3$.

If $a+b+c=0$ ,then find the value of $(\frac{a}{b-c}+\frac{b}{c-a}+\frac{c}{a-b})(\frac{b-c}{a}+\frac{c-a}{b}+\frac{a-b}{c})$

Let $S = \{1,2,3,\ldots,n\}$. Consider a function $f\colon S\to S$. A subset $D$ of $S$ is said to be invariant if for all $x\in D$ we have $f(x)\in D$. The empty set and $S$ are also considered as invariant subsets. By $\deg (f)$ we define the number of invariant subsets $D$ of $S$ for the function $f$. [b]i)[/b] Show that there exists a function $f\colon S\to S$ such that $\deg (f)=2$. [b]ii)[/b] Show that for every $1\leq k\leq n$ there exists a function $f\colon S\to S$ such that $\deg (f)=2^{k}$.

Prove that $$ \sin^2 \alpha + \sin^2 \beta \geq \sin \alpha \sin \beta + \sin \alpha + \sin \beta - 1.$$

A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$. [i]S. Tokarev[/i]

Let $a$, $b$, $c$, $d$ be positive real numbers such that $abcd=1$. Prove that $1/[(1/2 +a+ab+abc)^{1/2}]+ 1/[(1/2+b+bc+bcd)^{1/2}] + 1/[(1/2+c+cd+cda)^{1/2}] + 1/[1(1/2+d+da+dab)^{1/2}]$ is greater than or equal to $2^{1/2}$.

Which number is greater, $\frac{1996^{1995}+1}{1996^{1996}+1}$ or $ \frac{1996^{1996}+1}{1996^{1997}+1}$ ?

$A, B, C, D,$ and $E$ are relatively prime integers (i.e., have no single common factor) such that the polynomials $5Ax^4 +4Bx^3 +3Cx^2 +2Dx+E$ and $10Ax^3 +6Bx^2 +3Cx+D$ together have $7$ distinct integer roots. What are all possible values of $A$? [i]Your team has been given a sealed envelope that contains a hint for this problem. If you open the envelope, the value of this problem decreases by 20 points. To get full credit, give the sealed envelope to the judge before presenting your solution.[/i]

Let $a_1,a_2,\ldots,a_n$ and $b_1,b_2,\ldots,b_n$ be two finite sequences consisting of $2n$ real different numbers. Rearranging each of the sequences in increasing order we obtain $a_1',a_2',\ldots,a_n'$ and $b_1',b_2',\ldots,b_n'$. Prove that \[\max_{1\le i\le n}|a_i-b_i|\ge\max_{1\le i\le n}|a_i'-b_i'|.\]

Prove that there is no function $f:(0,\infty )\rightarrow (0,\infty)$ such that \[f(x+y)\ge f(x)+yf(f(x)) \] for every $x,y\in (0,\infty )$.

The roots of the two monic square trinomials are negative integers, and one of these roots is common. Can the values of these trinomials at some positive integer point equal 19 and 98?

Find all functions $ f:\mathbb{Q}\to\mathbb{R}$ that satisfy $ f(x\plus{}y)\equal{}f(x)f(y)\minus{}f(xy)\plus{}1$ for every $x,y\in\mathbb{Q}$.

The sequence $\{x_n\}$ is defined by $x_1=5$ and $x_{k+1}=x_k^2-3x_k+3$ for $k=1,2,3\cdots$. Prove that $x_k>3^{2^{k-1}}$ for any positive integer $k$.

Consider the equation $n^2+(n+1)^4=5(n+2)^3$ (a) Show that any integer of the form $3m+1$ or $3m+2$ can not be a solution of this equation. (b) Does the equation have a solution in positive integers?

Solve $x(8\sqrt{1-x}+\sqrt{1+x}) \le 11\sqrt{1+x}-16\sqrt{1-x}$ when $0<x\le 1$

Let $a$,$b$,$c$ be real numbers different from 0 for which $ab$ + $bc$+ $ca$ = 0 holds a) Prove that ($a$+$b$)($b$+$c$)($c$+$a$)≠ 0 b) Let $X$ = $a$ + $b$ + $c$ and $Y$ = $\frac{1}{a+b}$ + $\frac{1}{b+c}$ + $\frac{1}{c+a}$. Prove that numbers $X$ and $Y$ are both positive or both negative.

Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$ then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.

[u]Round 5[/u] [b]p13.[/b] The expression $\sqrt2 \times \sqrt[3]{3} \times \sqrt[6]{6}$ can be expressed as a single radical in the form $\sqrt[n]{m}$, where $m$ and $n$ are integers, and $n$ is as small as possible. What is the value of $m + n$? [b]p14.[/b] Bertie Bott also produces Bertie Bott’s Every Flavor Pez. Each package contains $6$ peppermint-, $2$ kumquat-, $3$ chili pepper-, and $5$ garlic-flavored candies in a random order. Harold opens a package and slips it into his Dumbledore-shaped Pez dispenser. What is the probability that of the first four candies, at least three are garlic-flavored? [b]p15.[/b] Quadrilateral $ABCD$ with $AB = BC = 1$ and $CD = DA = 2$ is circumscribed around and inscribed in two different circles. What is the area of the region between these circles? [u] Round 6[/u] [b]p16.[/b] Find all values of x that satisfy $\sqrt[3]{x^7} + \sqrt[3]{x^4} = \sqrt[3]{x}$. [b]p17.[/b] An octagon has vertices at $(2, 1)$, $(1, 2)$, $(-1, 2)$, $(-2, 1)$, $(-2, -1)$, $(-1, -2)$, $(1, -2)$, and $(2, -1)$. What is the minimum total area that must be cut off of the octagon so that the remaining figure is a regular octagon? [b]p18.[/b] Ron writes a $4$ digit number with no zeros. He tells Ronny that when he sums up all the two-digit numbers that are made by taking 2 consecutive digits of the number, he gets 99. He also reveals that his number is divisible by 8. What is the smallest possible number Ron could have written? [u]Round 7[/u] [b]p19.[/b] In a certain summer school, 30 kids enjoy geometry, 40 kids enjoy number theory, and 50 kids enjoy algebra. Also, the number of kids who only enjoy geometry is equal to the number of kids who only enjoy number theory and also equal to the number of kids who only enjoy algebra. What is the difference between the maximum and minimum possible numbers of kids who only enjoy geometry and algebra? [b]p20.[/b] A mouse is trying to run from the origin to a piece of cheese, located at $(4, 6)$, by traveling the shortest path possible along the lattice grid. However, on a lattice point within the region $\{0 \le x \le 4, 0 \le y \le 6$, $(x, y) \ne (0, 0),(4, 6)\}$ lies a rock through which the mouse cannot travel. The number of paths from which the mouse can choose depends on where the rock is placed. What is the difference between the maximum possible number of paths and the minimum possible number of paths available to the mouse? [b]p21.[/b] The nine points $(x, y)$ with $x, y \in \{-1, 0, 1\}$ are connected with horizontal and vertical segments to their nearest neighbors. Vikas starts at $(0, 1)$ and must travel to $(1, 0)$, $(0, -1)$, and $(-1, 0)$ in any order before returning to $(0, 1)$. However, he cannot travel to the origin $4$ times. If he wishes to travel the least distance possible throughout his journey, then how many possible paths can he take? [u]Round 8[/u] [b]p22.[/b] Let $g(x) = x^3 - x^2- 5x + 2$. If a, b, and c are the roots of g(x), then find the value of $((a + b)(b + c)(c + a))^3$. [b]p23.[/b] A regular octahedron composed of equilateral triangles of side length $1$ is contained within a larger tetrahedron such that the four faces of the tetrahedron coincide with four of the octahedron’s faces, none of which share an edge. What is the ratio of the volume of the octahedron to the volume of the tetrahedron? [b]p24.[/b] You are the lone soul at the south-west corner of a square within Elysium. Every turn, you have a $\frac13$ chance of remaining at your corner and a $\frac13$ chance of moving to each of the two closest corners. What is the probability that after four turns, you will have visited every corner at least once? PS. You should use hide for answers.Rounds 1-4 are [url=https://artofproblemsolving.com/community/c3h3134177p28401527]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3134489p28406583]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For how many integers $n$ between 1 and 100 does $x^2+x-n$ factor into the product of two linear factors with integer coefficients? $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 9 \qquad \text{(E)} \ 10$

Let $\mathbb{Z}_{\ge 0}$ be the set of all nonnegative integers. Find all the functions $f: \mathbb{Z}_{\ge 0} \rightarrow \mathbb{Z}_{\ge 0} $ satisfying the relation \[ f(f(f(n))) = f(n+1 ) +1 \] for all $ n\in \mathbb{Z}_{\ge 0}$.

Determine all functions $f : R \to R$ such that for all $x, y \in R$ holds $$f(xf(x) + f(y)) = y + f(x)^2$$

Let $a_1,a_2,\cdots,a_{n+1}>0$. Prove that$$\sum_{i-1}^{n}a_i\sum_{i=1}^{n}a_{i+1}\geq \sum_{i=1}^{n}\frac{a_i a_{i+1}}{a_i+a_{i+1}}\cdot \sum_{i=1}^{n}(a_i+a_{i+1})$$