Let $a,b,c>0$ and $a+b+c=6$ . Prove that $$ \frac{1}{a^2b+16}+\frac{1}{b^2c+16}+\frac{1}{c^2a+16} \ge \frac{1}{8}.$$

Given that $a, b$, and $c$ are distinct positive integers such that $a \cdot b \cdot c = 2020$, the minimum possible positive value of $\frac{1}{a}-\frac{1}{b}-\frac{1}{c}$, is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

What is the remainder, in base $10$, when $24_7 + 364_7 + 43_7 + 12_7 + 3_7 + 1_7$ is divided by $6$?

Fix a positive integer $n\ge 2$. For any cyclic $2n$-gon $P_1 P_2\cdots P_{2n}$ in this order, define its score as the maximal possible value of $$\angle P_iXP_{i+1} + \angle P_{i+n}XP_{i+n+1}$$ across all $1\le i\le n$ (indices modulo $n$), and over all points $X$ inside the $2n$-gon including its boundary. Prove that there exist a real number $r$ such that a cyclic $2n$-gon is regular if and only if it has score $r$. [i]Proposed by Wong Jer Ren[/i]

A square is given. Lines divide it into $n$ polygons. What is he the largest possible sum of the internal angles of all polygons?

Let $f(x)$ be a nonconstant monic polynomial of degree $n$ with rational coefficents that is irreducible, meaning it cannot be factored into two nonconstant rational polynomials. Find and prove a formula for the number of monic complex polynomials that divide $f$.

A sequence whose first term is positive has the property that any given term is the area of an equilateral triangle whose perimeter is the preceding term. If the first three terms form an arithmetic progression, determine all possible values of the first term.

Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$.

For a set $S$ of at least $3$ points in the plane, let $d_{\text{min}}$ denote the minimal distance between two different points in $S$ and $d_{\text{max}}$ the maximal distance between two different points in $S$. For a real $c>0$, a set $S$ will be called $c$-[i]balanced[/i] if \[\frac{d_{\text{max}}}{d_{\text{min}}}\leq c|S|\] Prove that there exists a real $c>0$ so that for every $c$-balanced set of points $S$, there exists a triangle with vertices in $S$ that contains at least $\sqrt{|S|}$ elements of $S$ in its interior or on its boundary.

Alice, Bob, and Chebyshev play a game. Alice puts six red chips into a bag, Bob puts seven blue chips into the bag, and Chebyshev puts eight green chips into the bag. Then, the almighty Zan randomly removes chips from the bag one at a time and gives them back to the corresponding player. The winner of the game is the first player to get all of their chips back. Find, with proof, the probability that Bob wins the game.

Determine the lowest positive integer n such that following statement is true: If polynomial with integer coefficients gets value 2 for n different integers, then it can't take value 4 for any integer.

Call a positive integer [i]descending[/i] if, reading left to right, each of its digits (other than its leftmost) is less than or equal to the previous digit. For example, $4221$ and $751$ are descending while $476$ and $455$ are not descending. Determine whether there exists a positive integer $n$ for which $16^n$ is descending.

In Hidro planet were living $2008^2$ hydras some time ago. One of them had 1 tentacle, other 2 and so on to the last with $2008^2$ tentacles. Let $t(H)$ be the number of tentacles of hydra $H$. The pairing of $H_1$ and $H_2$ (where $t(H_1) < t(H_2)$) is a new hydra with $t(H_2)-t(H_1)+8$ tentacles, in case of $t(H_1)=t(H_2)$ they die. An expedition found in Hidro the last hydra with 23 tentacles. That could be true ?

For real numbers $a$, $b$, $c$ and $d$ holds: $$ a+b+c+d=0$$ $$a^3+b^3+c^3+d^3=0$$ Prove that sum of some two numbers $a$, $b$, $c$ and $d$ is equal to zero

Let $a$, $b$, $c$, $d$, $e$ be integers such that $1 \le a < b < c < d < e$. Prove that \[\frac{1}{[a,b]} + \frac{1}{[b,c]} + \frac{1}{[c,d]} + \frac{1}{[d,e]} \le \frac{15}{16},\] where $[m,n]$ denotes the least common multiple of $m$ and $n$ (e.g. $[4,6] = 12$).

Let $M$ be a set of points in a plane with at least two elements. Prove that if $M$ has two axes of symmetry $g_1$ and $g_2$ intersecting at an angle $\alpha = q\pi$, where $q$ is irrational, then $M$ must be infinite.

Let $ k$ be a positive integer. Set $ A \subseteq \mathbb{Z}$ is called a $ \textbf{k \minus{} set}$ if there exists $ x_1, x_2, \cdots, x_k \in \mathbb{Z}$ such that for any $ i \neq j$, $ (x_i \plus{} A) \cap (x_j \plus{} A) \equal{} \emptyset$, where $ x \plus{} A \equal{} \{ x \plus{} a \mid a \in A \}$. Prove that if $ A_i$ is $ \textbf{k}_i\textbf{ \minus{} set}$($ i \equal{} 1,2, \cdots, t$), and $ A_1 \cup A_2 \cup \cdots \cup A_t \equal{} \mathbb{Z}$, then $ \displaystyle \frac {1}{k_1} \plus{} \frac {1}{k_2} \plus{} \cdots \plus{} \frac {1}{k_t} \geq 1$.

In an apartment block there live only couples of parents with children. It is known that every couple has at least one child, that every child has exactly two parents, that every little boy in this building has a sister, and that among the children there are more boys than girls. You may also assume that there are no grandparents living in the building. Is it possible that there are more parents than children in the building? Explain your reasoning.

Let $a$ and $c$ be fixed positive numbers. For each real number $t$ let $(x_t, y_t)$ be the vertex of the parabola $y = ax^2+bx+c$. If the set of vertices $(x_t, y_t)$ for all real values of $t$ is graphed in the plane, the graph is A. a straight line B. a parabola C. part, but not all, of a parabola D. one branch of a hyperbola E. None of these

Let $ x_n \equal{} \int_0^{\frac {\pi}{2}} \sin ^ n \theta \ d\theta \ (n \equal{} 0,\ 1,\ 2,\ \cdots)$. (1) Show that $ x_n \equal{} \frac {n \minus{} 1}{n}x_{n \minus{} 2}$. (2) Find the value of $ nx_nx_{n \minus{} 1}$. (3) Show that a sequence $ \{x_n\}$ is monotone decreasing. (4) Find $ \lim_{n\to\infty} nx_n^2$.

[b]p1.[/b] There is a string of numbers $1234567891023...910134 ...91012...$ that concatenates the numbers $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, then $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, then $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$, $1$, $2$, and so on. After $10$, $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, the string will be concatenated with $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, $10$ again. What is the $2021$st digit? [b]p2.[/b] Bob really likes eating rice. Bob starts eating at the rate of $1$ bowl of rice per minute. Every minute, the number of bowls of rice Bob eats per minute increases by $1$. Given there are $78$ bowls of rice, find number of minutes Bob needs to finish all the rice. [b]p3.[/b] Suppose John has $4$ fair coins, one red, one blue, one yellow, one green. If John flips all $4$ coins at once, the probability he will land exactly $3$ heads and land heads on both the blue and red coins can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$. [b]p4.[/b] Three of the sides of an isosceles trapezoid have lengths $1$, $10$, $20$ Find the sum of all possible values of the fourth side. [b]p5.[/b] An number two-three-delightful if and only if it can be expressed as the product of $2$ consecutive integers larger than $1$ and as the product of $3$ consecutive integers larger than $1$. What is the smallest two-three-delightful number? [b]p6.[/b] There are $3$ students total in Justin's online chemistry class. On a $100$ point test, Justin's two classmates scored $4$ and $7$ points. The teacher notices that the class median score is equal to $gcd(x, 42)$, where the positive integer $x$ is Justin's score. Find the sum of all possible values of Justin's score. [b]p7.[/b] Eddie's gym class of $10$ students decides to play ping pong. However, there are only $4$ tables and only $2$ people can play at a table. If $8$ students are randomly selected to play and randomly assigned a partner to play against at a table, the probability that Eddie plays against Allen is $\frac{a}{b}$ for relatively prime positive integers $a$, $b$, Find $a + b$. [b]p8.[/b] Let $S$ be the set of integers $k$ consisting of nonzero digits, such that $300 < k < 400$ and $k - 300$ is not divisible by $11$. For each $k$ in $S$, let $A(k)$ denote the set of integers in $S$ not equal to $k$ that can be formed by permuting the digits of $k$. Find the number of integers $k$ in $S$ such that $k$ is relatively prime to all elements of $A(k)$. [b]p9.[/b] In $\vartriangle ABC$, $AB = 6$ and $BC = 5$. Point $D$ is on side $AC$ such that $BD$ bisects angle $\angle ABC$. Let $E$ be the foot of the altitude from $D$ to $AB$. Given $BE = 4$, find $AC^2$. [b]p10.[/b] For each integer $1 \le n \le 10$, Abe writes the number $2^n + 1$ on a blackboard. Each minute, he takes two numbers $a$ and $b$, erases them, and writes $\frac{ab-1}{a+b-2}$ instead. After $9$ minutes, there is one number $C$ left on the board. The minimum possible value of $C$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$. [b]p11.[/b] Estimation (Tiebreaker) Let $A$ and $B$ be the proportions of contestants that correctly answered Questions $9$ and $10$ of this round, respectively. Estimate $\left \lfloor \dfrac{1}{(AB)^2} \right \rfloor$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integer $n$ such that $$\frac{\sin{n\theta}}{\sin{\theta}} - \frac{\cos{n\theta}}{\cos{\theta}} = n-1$$ holds for all $\theta$ which are not integral multiples of $\frac{\pi}{2}$

Let $\triangle ABC$ be acute with $\angle BAC = 45^{\circ}$. Let $\overline{AD}$ be an altitude of $\triangle ABC$, let $E$ be the midpoint of $\overline{BC}$, and let $F$ be the midpoint of $\overline{AD}$. Let $O$ be the center of the circumcircle of $\triangle ABC$, let $K$ be the intersection of lines $DO$ and $EF$, and let $L$ be the foot of the perpendicular from $O$ to line $AK$. If $BL = 6$ and $CL = 8$, find $AL^2$. [i]Proposed by [b]Awesome_guy[/b][/i]

Let $ABC$ be a right triangle with $\angle ACB = 90^{\circ}$ and centroid $G$. The circumcircle $k_1$ of triangle $AGC$ and the circumcircle $k_2$ of triangle $BGC$ intersect $AB$ at $P$ and $Q$, respectively. The perpendiculars from $P$ and $Q$ respectively to $AC$ and $BC$ intersect $k_1$ and $k_2$ at $X$ and $Y$. Determine the value of $\frac{CX \cdot CY}{AB^2}$.

Let $ (a_n)^{\infty}_{n\equal{}1}$ be a sequence of integers with $ a_{n} < a_{n\plus{}1}, \quad \forall n \geq 1.$ For all quadruple $ (i,j,k,l)$ of indices such that $ 1 \leq i < j \leq k < l$ and $ i \plus{} l \equal{} j \plus{} k$ we have the inequality $ a_{i} \plus{} a_{l} > a_{j} \plus{} a_{k}.$ Determine the least possible value of $ a_{2008}.$