Prove that there is a multiple of $2^{2022}$ that has $2022$ digits, and can be written using digits $1$ and $2$ only.

Let $\Omega_1$ be a circle with centre $O$ and let $AB$ be diameter of $\Omega_1$. Let $P$ be a point on the segment $OB$ different from $O$. Suppose another circle $\Omega_2$ with centre $P$ lies in the interior of $\Omega_1$. Tangents are drawn from $A$ and $B$ to the circle $\Omega_2$ intersecting $\Omega_1$ again at $A_1$ and B1 respectively such that $A_1$ and $B_1$ are on the opposite sides of $AB$. Given that $A_1 B = 5, AB_1 = 15$ and $OP = 10$, find the radius of $\Omega_1$.

Chris has a bag with 4 black socks and 6 red socks (so there are $10$ socks in total). Timothy reaches into the bag and grabs two socks [i]without replacement[/i]. Find the probability that he will not grab two red socks. [i]Proposed by Chris Tong[/i]

[center][u]Guts Round[/u] / [u]Set 7[/u][/center] [b]p19.[/b] Let $N \ge 3$ be the answer to Problem 21. A regular $N$-gon is inscribed in a circle of radius $1$. Let $D$ be the set of diagonals, where we include all sides as diagonals. Then, let $D'$ be the set of all unordered pairs of distinct diagonals in $D$. Compute the sum $$\sum_{\{d,d'\}\in D'} \ell (d)^2 \ell (d')^2,$$where $\ell (d)$ denotes the length of diagonal $d$. [b]p20.[/b] Let $N$ be the answer to Problem $19$, and let $M$ be the last digit of $N$. Let $\omega$ be a primitive $M$th root of unity, and define $P(x)$ such that$$P(x) = \prod^M_{k=1} (x - \omega^{i_k}),$$where the $i_k$ are chosen independently and uniformly at random from the range $\{0, 1, . . . ,M-1\}$. Compute $E \left[P\left(\sqrt{\rfloor \frac{1250}{N} \rfloor } \right)\right].$ [b]p21.[/b] Let $N$ be the answer to Problem $20$. Define the polynomial $f(x) = x^{34} +x^{33} +x^{32} +...+x+1$. Compute the number of primes $p < N$ such that there exists an integer $k$ with $f(k)$ divisible by $p$.

Let $S$ be the sum of all real $x$ such that $4^x = x^4$. Find the nearest integer to $S$.

Let $a,b,c$ be real numbers such that $0 \leq a \leq b \leq c$ and $a+b+c=ab+bc+ca >0.$ Prove that $\sqrt{bc}(a+1) \geq 2$ and determine the equality cases. (Edit: Proposed by sir Leonard Giugiuc, Romania)

Consider the equation $x^4-24x^3+210x^2+mx+n=0$. Given that the roots of this equation are nonnegative reals, find the maximum possible value of a root of this equation across all values of $m$ and $n$. [i]Proposed by Andrew Zhao[/i]

Two identical blue blocks, two identical red blocks, two identical green blocks, and two identical purple blocks are placed next to each other in a row. Find the number of distinct arrangements of these blocks where no blue block is placed next to a red block, and no green block is placed next to a purple block.

Suppose $f_1(x), f_2(x), \dots, f_n(x)$ are functions of $n$ real variables $x = (x_1, \dots, x_n)$ with continuous second-order partial derivatives everywhere on $\mathbb{R}^n$. Suppose further that there are constants $c_{ij}$ such that \[ \frac{\partial f_i}{\partial x_j} - \frac{\partial f_j}{\partial x_i} = c_{ij} \] for all $i$ and $j$, $1\leq i \leq n$, $1 \leq j \leq n$. Prove that there is a function $g(x)$ on $\mathbb{R}^n$ such that $f_i + \partial g/\partial x_i$ is linear for all $i$, $1 \leq i \leq n$. (A linear function is one of the form \( a_0 + a_1 x_1 + a_2 x_2 + \cdots + a_n x_n.) \)

If $ \frac {x^2 \minus{} bx}{ax \minus{} c} \equal{} \frac {m \minus{} 1}{m \plus{} 1}$ has roots which are numerical equal but of opposite signs, the value of $ m$ must be: $ \textbf{(A)}\ \frac {a \minus{} b}{a \plus{} b} \qquad\textbf{(B)}\ \frac {a \plus{} b}{a \minus{} b} \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ \frac {1}{c} \qquad\textbf{(E)}\ 1$

Alice chooses a prime number $p > 2$ and then Bob chooses a positive integer $n_0$. Alice, in the first move, chooses an integer $n_1 > n_0$ and calculates the expression $s_1 = n_0^{n_1} + n_1^{n_0}$; then Bob, in the second move, chooses an integer $n_2 > n_1$ and calculates the expression $s_2 = n_1^{n_2} + n_2^{n_1}$; etc. one by one. (Each player knows the numbers chosen by the other in the previous moves.) The winner is the one who first chooses the number $n_k$ such that $p$ divides $s_k(s_1 + 2s_2 + · · · + ks_k)$. Who has a winning strategy? Proposed by [i]Borche Joshevski, Macedonia[/i]

Two polynomials of the same degree $A(x)=a_nx^n+ \cdots + a_1x+a_0$ and $B(x)=b_nx^n+\cdots+b_1x+b_0$ are called [i]friends[/i] is the coefficients $b_0,b_1, \ldots, b_n$ are a permutation of the coefficients $a_0,a_1, \ldots, a_n$. $P(x)$ and $Q(x)$ be two friendly polynomials with integer coefficients. If $P(16)=3^{2020}$, the smallest possible value of $|Q(3^{2020})|$.

As shown on the figure, square $PQRS$ is inscribed in right triangle $ABC$, whose right angle is at $C$, so that $S$ and $P$ are on sides $BC$ and $CA$, respectively, while $Q$ and $R$ are on side $AB$. Prove that $A B\geq3QR$ and determine when equality occurs. [asy] defaultpen(linewidth(0.7)+fontsize(10)); size(150); real a=8, b=6; real y=a/((a^2+b^2)/(a*b)+1), r=degrees((a,b))+180; pair A=b*dir(-r)*dir(90), B=a*dir(180)*dir(-r), C=origin, S=y*dir(-r)*dir(180), P=(y*b/a)*dir(90-r), Q=foot(P, A, B), R=foot(S, A, B); draw(A--B--C--cycle^^R--S--P--Q); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$S$", S, dir(point--S)); label("$R$", R, dir(270)); label("$Q$", Q, dir(270)); label("$P$", P, dir(point--P));[/asy]

Let $n$ be a positive integer number. The decimal representation of $2^n$ contains $m$ same numbers from the right. What is the largest value of $m$? $\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ \text{None}$

If $f(x) = ax^2 + bx + c$ satisfies the condition $|f(x)| < 1; \forall x \in [-1, 1]$, prove that the equation $f(x) = 2x^2 - 1$ has two real roots.

There exists a unique line tangent to the graph of $y=x^4-20x^3+24x^2-20x+25$ at two distinct points. Compute the product of the $x$-coordinates of the two tangency points.

A given convex pentagon $ ABCDE$ has the property that the area of each of five triangles $ ABC, BCD, CDE, DEA$, and $ EAB$ is unity [i](equal to 1)[/i]. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.

William writes the number $1$ on a blackboard. Every turn, he erases the number $N$ currently on the blackboard and replaces it with either $4N + 1$ or $8N + 1$ until it exceeds $1000$, after which no more moves are made. If the minimum possible value of the final number on the blackboard is $M$, find the remainder when $M$ is divided by $1000$.

A regular $2017$-gon is partitioned into triangles by a set of non-intersecting diagonals. Prove that among those triangles only one is acute-angled.

Let $a, b, c$ be the side-lengths of a triangle, $r$ be the inradius and $r_a, r_b, r_c$ be the corresponding exradius. Show that \[ \frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r} \]

There are some papers of the size $5\times 5$ with two sides which are divided into unit squares for both sides. One uses $n$ colors to paint each cell on the paper, one cell by one color, such that two cells on the same positions for two sides are painted by the same color. Two painted papers are consider as the same if the color of two corresponding cells are the same. Prove that there are no more than $$\frac{1}{8}\left( {{n}^{25}}+4{{n}^{15}}+{{n}^{13}}+2{{n}^{7}} \right)$$ pairwise distinct papers that painted by this way.

Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be two circumferences intersecting at points $A$ and $B$. Let $C$ be a point on line $AB$ such that $B$ lies between $A$ and $C$. Let $P$ and $Q$ be points on $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively such that $CP$ and $CQ$ are tangent to $\mathcal{C}_1$ and $\mathcal{C}_2$ respectively, $P$ is not inside $\mathcal{C}_2$ and $Q$ is not inside $\mathcal{C}_1$. Line $PQ$ cuts $\mathcal{C}_1$ at $R$ and $\mathcal{C}_2$ at $S$, both points different from $P$, $Q$ and $B$. Suppose $CR$ cuts $\mathcal{C}_1$ again at $X$ and $CS$ cuts $\mathcal{C}_2$ again at $Y$. Let $Z$ be a point on line $XY$. Prove $SZ$ is parallel to $QX$ if and only if $PZ$ is parallel to $RX$.

Let $a_1,a_2,\ldots$ be an infinite sequence of integers such that $a_i$ divides $a_{i+1}$ for all $i\geq 1$, and let $b_i$ be the remainder when $a_i$ is divided by $210$. What is the maximal number of distinct terms in the sequence $b_1,b_2,\ldots$?

Find all functions $f:\mathbb{R^+}\to\mathbb {R^+} $ such that for all $x,y\in R^+$ the followings hold: $i) $ $f (x+y)\ge f (x)+y $ $ii) $ $f (f (x))\le x $

In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest possible perimeter of the triangle? $ \textbf{(A) } 43 \qquad \textbf{(B) } 44 \qquad \textbf{(C) } 45 \qquad \textbf{(D) } 46 \qquad \textbf{(E) } 47$