The equation $ x^{n+1} +x=0 $ admits $ 0 $ and $ 1 $ as its unique solutions in a ring of order $ n\ge 2. $ Prove that this ring is a skew field.

Prove that $ \prod_{i\equal{}1}^{n}(1\plus{}x_{1}\plus{}x_{2}\plus{}...\plus{}x_{i})\geq\sqrt{(n\plus{}1)^{n\plus{}1}x_{1}x_{2}...x_{n}}\forall x_{1},...,x_{n}> 0$.

Let $S = \{ a_0, a_1, a_2, a_3, \dots \}$ be a set of positive integers with $1 = a_0 < a_1 < a_2 < a_3 < \dots$. For a subset $T$ of $S$, let $\sigma(T)$ be the sum of the elements of $T$. For instance, $\sigma(\{1, 2, 3\}) = 6$. By convention, $\sigma(\emptyset) = 0$, where $\emptyset$ denotes an empty set. Call a number $n$ representable if there exists a subset $T$ of $S$ such that $\sigma(T) = n$. We aim to prove for any set $S$ satisfying $a_{k+1} \le 2a_k$ for every $k \ge 0$, that all non-negative integers are representable. (a) Prove there is a unique value of $a_1$, and find this value. Use this to determine, with proof, all possible sets $\{a_0, a_1, a_2, a_3 \}$. (Hint: there are 7 possible sets.) [Not for credit] I recommend that you show that for all 7 sets in part (a), every integer between $0$ and $a_3 - 1$ is representable. (Note that this does not depend on the values of $a_4, a_5, a_6, \dots$.) (b) Show that if $a_k \le n \le a_{k+1} - 1$, then $0 \le n - a_k \le a_k - 1$. (c) Prove that any non-negative integer is representable.

Define the numbers $a_0, a_1, \ldots, a_n$ in the following way: \[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \] Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]

As shown in the following figure, there is a line segment consisting of five line segments $AB, BC, CD, DE, and EA$ and $10$ intersection points of these five line segments. Find the number of ways to write $1$ or $2$ at each of the $10$ vertices so that the following conditions are satisfied. $\bigstar$ The sum of the four numbers written on each line segment $AB, BC, CD, DE, and EA$ is the same.

Prove the inequality $$\sum _{k = 1} ^n (x_k - x_{k-1})^2 \geq 4 \sin ^2 \frac{\pi}{2n} \cdot \sum ^n _{k = 0} x_k ^2$$ for any sequence of real numbers $x_0, x_1, ..., x_n$ for which $x_0 = x_n = 0.$

Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.

A rectangular piece of of paper measures 4 units by 5 units. Several lines are drawn parallel to the edges of the paper. A rectangle determined by the intersections of some of these lines is called [i]basic [/i]if (i) all four sides of the rectangle are segments of drawn line segments, and (ii) no segments of drawn lines lie inside the rectangle. Given that the total length of all lines drawn is exactly 2007 units, let $N$ be the maximum possible number of basic rectangles determined. Find the remainder when $N$ is divided by 1000.

A square and a rectangle of the same perimeter have a common corner. Prove that the intersection point of the diagonals of the rectangle lies on the diagonal of the square. (Yu. Blinkov)

Points $P$, $Q$, and $M$ lie on a circle $\omega$ such that $M$ is the midpoint of minor arc $PQ$ and $MP=MQ=3$. Point $X$ varies on major arc $PQ$, $MX$ meets segment $PQ$ at $R$, the line through $R$ perpendicular to $MX$ meets minor arc $PQ$ at $S$, $MS$ meets line $PQ$ at $T$. If $TX=5$ when $MS$ is minimized, what is the minimum value of $MS$? [i]2019 CCA Math Bonanza Team Round #9[/i]