Prove that the product of 4 consecutive natural numbers cannot be a perfect cube.

Let a [i]triplet [/i] be some set of three distinct pairwise parallel lines. $20$ triplets are drawn on a plane. Find the maximum number of regions these $60$ lines can divide the plane into.

Prove that there exists and infinity of multiples of $1997$ that have $1998$ as first four digits and last four digits.

Let $A$ be a finite set, and $A_1,A_2,\cdots, A_n$ are subsets of $A$ with the following conditions: (1) $|A_1|=|A_2|=\cdots=|A_n|=k$, and $k>\frac{|A|}{2}$; (2) for any $a,b\in A$, there exist $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$ such that $a,b\in A_r\cap A_s\cap A_t$; (3) for any integer $i,j\, (1\leq i<j\leq n)$, $|A_i\cap A_j|\leq 3$. Find all possible value(s) of $n$ when $k$ attains maximum among all possible systems $(A_1,A_2,\cdots, A_n,A)$.

[b]p29.[/b] Consider pentagon $ABCDE$. How many paths are there from vertex $A$ to vertex $E$ where no edge is repeated and does not go through $E$. [b]p30.[/b] Let $a_1, a_2, ...$ be a sequence of positive real numbers such that $\sum^{\infty}_{n=1} a_n = 4$. Compute the maximum possible value of $\sum^{\infty}_{n=1}\frac{\sqrt{a_n}}{2^n}$ (assume this always converges). [b]p31.[/b] Define function $f(x) = x^4 + 4$. Let $$P =\prod^{2021}_{k=1} \frac{f(4k - 1)}{f(4k - 3)}.$$ Find the remainder when $P$ is divided by $1000$. [b]p32.[/b] Reduce the following expression to a simplified rational: $\cos^7 \frac{\pi}{9} + \cos^7 \frac{5\pi}{9}+ \cos^7 \frac{7\pi}{9}$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the smallest positive integer $n$ such that $2016n$ is a perfect cube?

$a$ and $b$ are rationals. Show that if $ax^2 + by^2 = 1$ has a rational solution (in $x$ and $y$), then it must have infinitely many.

Is it possible to divide $5$ apples of the same size equally between six children so that no apple will be cut into more than $3$ pieces? (You are allowed to cut an apple into any number of equal pieces).

Let $p$ be a prime number. For each $k$, $1\le k\le p-1$, there exists a unique integer denoted by $k^{-1}$ such that $1\le k^{-1}\le p-1$ and $k^{-1}\cdot k=1\pmod{p}$. Prove that the sequence \[1^{-1},\quad 1^{-1}+2^{-1},\quad 1^{-1}+2^{-1}+3^{-1},\quad \ldots ,\quad 1^{-1}+2^{-1}+\ldots +(p-1)^{-1} \] (addition modulo $p$) contains at most $\frac{p+1}{2}$ distinct elements.

Prove that there is a number such that its decimal represantation ends with 1994 and it can be written as $1994\cdot 1993^{n}$ ($n\in{Z^{+}}$)

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

Let $P$ be a point of hypotenuse $AB$ (or its extension) of isosceles right triangle $ABC$. Let $s=AP^2+PB^2$. Then: $\textbf{(A) }s<2CP^2\text{ for a finite number of positions of }P$ $\textbf{(B) }s<2CP^2\text{ for an infinite number of positions of }P$ $\textbf{(C) }s=2CP^2\text{ only if }P\text{ is the midpoint of }AB\text{ or an endpoint of }AB$ $\textbf{(D) }s=2CP^2\text{ always}$ $\textbf{(E) }s>2CP^2\text{ if }P\text{ is a trisection point of }AB$

Find all functions $f : R \to R$ such that the inequality $$\sum_{i=1}^{2549} f(x_i + x_{i+1}) + f (\sum_{i=1}^{2550}x_y) \le \sum_{i=1}^{2550}f(2x_i)$$ for all reals $x_1, x_2, . . . , x_{2550}$.

(a) Given a point $O$ inside an equilateral triangle $\vartriangle ABC$. Line $OG$ connects $O$ with the center of mass $G$ of the triangle and intersects the sides of the triangle, or their extensions, at points $A', B', C'$ . Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G} = 3.$$ (b) Point $G$ is the center of the sphere inscribed in a regular tetrahedron $ABCD$. Straight line $OG$ connecting $G$ with a point $O$ inside the tetrahedron intersects the faces at points $A', B', C', D'$. Prove that $$\frac{A'O}{A'G} + \frac{B'O}{B'G} + \frac{C'O}{C'G}+ \frac{D'O}{D'G} = 4.$$

If $x_1 + x_2 + \cdots + x_n = \sum_{i=1}^{n} x_i = \frac{1}{2}$ and $x_i > 0$ ; then prove that: $ \frac{1-x_1}{1+x_1} \cdot \frac{1-x_2}{1+x_2} \cdots \frac{1-x_n}{1+x_n} = \prod_{i=1}^{n} \frac{1-x_i}{1+x_i} \geq \frac{1}{3}$

Solve the equation $$\left(x^2+3x-4\right)^3+\left(2x^2-5x+3\right)^3=\left(3x^2-2x-1\right)^3.$$

There are $2500$ people in Lexington High School, who all start out healthy. After $1$ day, $1$ person becomes infected with coronavirus. Each subsequent day, there are twice as many newly infected people as on the previous day. How many days will it be until over half the school is infected?

A teacher plays the game “Duck-Goose-Goose” with his class. The game is played as follows: All the students stand in a circle and the teacher walks around the circle. As he passes each student, he taps the student on the head and declares her a ‘duck’ or a ‘goose’. Any student named a ‘goose’ leaves the circle immediately. Starting with the first student, the teacher tags students in the pattern: duck, goose, goose, duck, goose, goose, etc., and continues around the circle (re-tagging some former ducks as geese) until only one student remains. This remaining student is the winner. For instance, if there are 8 students, the game proceeds as follows: student 1 (duck), student 2 (goose), student 3 (goose), student 4 (duck), student 5 (goose), student 6 (goose), student 7 (duck), student 8 (goose), student 1 (goose), student 4 (duck), student 7 (goose) and student 4 is the winner. Find, with proof, all values of $n$ with $n>2$ such that if the circle starts with $n$ students, then the $n$th student is the winner.

It is possible to choose $ x > \frac {2}{3}$ in such a way that the value of $ \log_{10}(x^2 \plus{} 3) \minus{} 2 \log_{10}x$ is $ \textbf{(A)}\ \text{negative} \qquad \textbf{(B)}\ \text{zero} \qquad \textbf{(C)}\ \text{one}$ $ \textbf{(D)}\ \text{smaller than any positive number that might be specified}$ $ \textbf{(E)}\ \text{greater than any positive number that might be specified}$

Let $\{a_n\}_{n \in N}$ be a sequence of real numbers with $a_1 = 2$ and $a_n =\frac{n^2 + 1}{\sqrt{n^3 - 2n^2 + n}}$ for all positive integers $n \ge 2$. Let $s_n = a_1 + a_2 + ...+ a_n$ for all positive integers $n$. Prove that $$\frac{1}{s_1s_2}+\frac{1}{s_2s_3}+ ...+\frac{1}{s_ns_{n+1}}<\frac15$$ for all positive integers $n$.

Find all real solutions $(x,y,z)$ of the system $$\begin{cases}\dfrac{4x^2}{1+4x^2}= y\\ \\\dfrac{4y^2}{1+4y^2}= z\\ \\ \dfrac{4z^2}{1+4z^2}= x \end{cases}$$

Let $p$ and $n$ be positive integers such that $p$ is prime and $1 + np$ is a perfect square. Prove that the number $n + 1$ can be expressed as the sum of $p$ perfect squares, where some of them can be equal.

Let $ABC$ be a triangle with $AB = 7, BC = 5,$ and $CA = 6$. Let $D$ be a variable point on segment $BC$, and let the perpendicular bisector of $AD$ meet segments $AC, AB$ at $E, F,$ respectively. It is given that there is a point $P$ inside $\triangle ABC$ such that $\frac{AP}{PC} = \frac{AE}{EC}$ and $\frac{AP}{PB} = \frac{AF}{FB}$. The length of the path traced by $P$ as $D$ varies along segment $BC$ can be expressed as $\sqrt{\frac{m}{n}}\sin^{-1}\left(\sqrt \frac 17\right)$, where $m$ and $n$ are relatively prime positive integers, and angles are measured in radians. Compute $100m + n$. [i]Proposed by Edward Wan[/i]

If $x$, $y$, $z$ are nonnegative integers satisfying the equation below, then compute $x+y+z$. \[\left(\frac{16}{3}\right)^x\times \left(\frac{27}{25}\right)^y\times \left(\frac{5}{4}\right)^z=256.\] [i]Proposed by Jeffrey Shi[/i]

$ ABC$ is a triangle, the bisector of angle $ A$ meets the circumcircle of triangle $ ABC$ in $ A_1$, points $ B_1$ and $ C_1$ are defined similarly. Let $ AA_1$ meet the lines that bisect the two external angles at $ B$ and $ C$ in $ A_0$. Define $ B_0$ and $ C_0$ similarly. Prove that the area of triangle $ A_0B_0C_0 \equal{} 2 \cdot$ area of hexagon $ AC_1BA_1CB_1 \geq 4 \cdot$ area of triangle $ ABC$.