Solve in positive integers $y^2 = x^2 + x + 1$
Solve in integers $\frac{1}{x}+\frac{1}{y} + \frac{1}{z} = \frac{3}{5}$
There are $7$ boys each of which has at least $3$ brothers among the other $6$ boys. Are these $7$ boys necessarily all brothers? Explain.
$\textbf{Lily Pads}$
There are $24$ lily pads shown below. A toad can jump from one pad to an adjacent one either horizontally or vertically, but not diagonally. Can this toad visit all the pads without stopping at a pad for more than once? It can choose any pad to start its journey.
Let $x$ and $y$ be two positive real numbers. Find the maximum value of $\frac{(3x+4y)^2}{x^2 + y^2}$.
$N$ delegates attend a round-table meeting, where $N$ is an even number. After a break, these delegates randomly pick a seat to sit down again to continue the meeting. Prove that there must exist two delegates so that the number of people sitting between them is the same before and after the break.
Find the largest multiple of 99 among the nine-digit integers, whose digits are all distinct.
Prove there is no integer solutions to $x^2 = y^5 - 4$.
Find all positive integer solutions to: $x^2 + 3y^2 = 1998x$.
A code consists of four different digits from $1$ to $9$, inclusive. What is the probability of selection a code that consists of four consecutive digits but not necessarily in order? Express your answer as a common fraction.
If $x$ and $y$ are positive integer solutions to the equation $x^2 - 2y^2 = 1$, then $6\mid xy$.
Let $\alpha$ and $\beta$ be two real roots of the equation $x^2 + x - 4=0$. Find the value of $\alpha^2 - 5\beta + 10$ without computing the value of $\alpha$ and $\beta$.
Prove that there exist infinite many triples of consecutive integers each of which is a sum of two squares. For example: $8 = 2^2 + 2^2$, $9 = 3^2 + 0^2$, and $10=3^1 + 1^2$
Find all triangles whose sides are consecutive integers and areas are also integers.
Find all positive integers $k$, $m$ such that $k < m$ and
$$1+ 2 +\cdots+ k = (k +1) + (k + 2) +\cdots+ m$$
Prove that there are infinitely many positive integers $n$ such that $(n^2+1)$ divides $n!$.
Let $a_1, a_2, \cdots, a_n$ be $n > 2$ real numbers. Show that it is possible to select $\epsilon_1, \epsilon_2, \cdots, \epsilon_n \in \{1, -1\}$ such that $$(\sum_{i=1}^na_i)^2 + (\sum_{i=1}^n\epsilon_ia_i)^2 \le (n+1)(\sum_{i=1}^na_i^2)$$
What is the maximum number of acute triangles 2n + 1 lines can create?
There are $100$ tigers, $100$ foxes, and $100$ monkeys in the animal kingdom. Tigers always tell the truth; Foxes always tell lies; and monkeys sometimes tell the truth but sometimes not. These $300$ animals are divided into $100$ groups, each of which has exactly two of the same kind and one of another kind. Now comes the Kong Fu Panda. He asks every animal: "is there a tiger in your group?" and receives $138$ "yes" answers. Then he asks everyone: "is there a fox in your group?" and receives $188$ positive answer this time. Find the number of monkeys who tell the truth both time.
How many different ways to express $13$ as the sum of several positive odd numbers? Order
matters. For example, $1 + 1 + 3 + 3 + 5$ is treated as a different expression as 1 + 3 + 1 + 3 + 5
Let $x$ be a real number between 0 and 1. Find the maximum value of $x(1-x^4)$.
Prove: any convex pentagon must have three vertices $A$, $B$, and $C$ satisfying $\angle{ABC} \le 36^\circ$.
Given any four points on a plane, prove the ratio of the farthest distance between any two points over the shortest distance must be at least $\sqrt{2}$. What if the number of points is 5?
There are $99$ points on a plane. Among any three points, at least two of them are not more than $1$ unit length apart. Prove: it is possible to cover $50$ of these points using a unit circle.
Let three non-zero numbers $a$, $b$, and $c$ satisfy $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{a+b+c}$. Prove at least two of these three numbers are opposite numbers