Find all integer pairs $(x, y)$ such that $$y^2 = x^3 + 2x^2 + 2x + 1.$$

Quadrilateral $ABCD (\overline{AD} < \overline{BC})$ is inscribed in a circle, and $H(\neq A, B)$ is a point on segment $AB.$ The circumcircle of triangle $BCH$ meets $BD$ at $E(\neq B)$ and line $HE$ meets $AD$ at $F$. The circle passes through $C$ and tangent to line $BD$ at $E$ meets $EF$ at $G(\neq E).$ Prove that $\angle DFG = \angle FCG.$

Positive integers $a_1, a_2, \dots, a_{2023}$ satisfy the following conditions. $a_1 = 5, a_2 = 25$ $a_{n + 2} = 7a_{n+1}-a_n-6$ for each $n = 1, 2, \dots, 2021$ Prove that there exist integers $x, y$ such that $a_{2023} = x^2 + y^2.$

$2023$ players participated in a tennis tournament, and any two players played exactly one match. There was no draw in any match, and no player won all the other players. If a player $A$ satisfies the following condition, let $A$ be "skilled player". (Condition) For each player $B$ who won $A$, there is a player $C$ who won $B$ and lost to $A$. It turned out there are exactly $N(\geq 0)$ skilled player. Find the minimum value of $N$.

For a positive integer $n(\geq 5)$, there are $n$ white stones and $n$ black stones (total $2n$ stones) lined up in a row. The first $n$ stones from the left are white, and the next $n$ stones are black. $$\underbrace{\Circle \Circle \cdots \Circle}_n \underbrace{\CIRCLE \CIRCLE \cdots \CIRCLE}_n $$You can swap the stones by repeating the following operation. (Operation) Choose a positive integer $k (\leq 2n - 5)$, and swap $k$-th stone and $(k+5)$-th stone from the left. Find all positive integers $n$ such that we can make first $n$ stones to be black and the next $n$ stones to be white in finite number of operations.

Find the maximum value of real number $A$ such that $$3x^2 + y^2 + 1 \geq A(x^2 + xy + x)$$for all positive integers $x, y.$

Find the smallest positive integer $N$ such that there are no different sets $A, B$ that satisfy the following conditions. (Here, $N$ is not a power of $2$. That is, $N \neq 1, 2^1, 2^2, \dots$.) $A, B \subseteq \{1, 2^1, 2^2, 2^3, \dots, 2^{2023}\} \cup \{ N \}$ $|A| = |B| \geq 1$ Sum of elements in $A$ and sum of elements in $B$ are equal.

A red equilateral triangle $T$ with side length $1$ is drawn on a plane. For a positive real $c$, we place three blue equilateral triangle shaped paper with side length $c$ on a plane to cover $T$ completely. Find the minimum value of $c$. As shown in the picture, it doesn't matter if the blue papers overlap each other or stick out from $T$. Folding or tearing the paper is not allowed.