Simplify $$\dbinom{2020}{1010}\dbinom{1010}{1010}+\dbinom{2019}{1010}\dbinom{1011}{1010}+\cdots+\dbinom{1011}{1010}\dbinom{2019}{1010} + \dbinom{1010}{1010}\dbinom{2020}{1010}.$$

For a positive integer $N$, let $f_N$ be the function defined by \[ f_N (x)=\sum_{n=0}^N \frac{N+1/2-n}{(N+1)(2n+1)} \sin\left((2n+1)x \right). \] Determine the smallest constant $M$ such that $f_N (x)\le M$ for all $N$ and all real $x$.

Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indenitely. If Nathaniel goes first, determine the probability that he ends up winning.

Let $ S \equal{} \{0,1,2,\ldots,1999\}$ and $ T \equal{} \{0,1,2,\ldots \}.$ Find all functions $ f: T \mapsto S$ such that [b](i)[/b] $ f(s) \equal{} s \quad \forall s \in S.$ [b](ii)[/b] $ f(m\plus{}n) \equal{} f(f(m)\plus{}f(n)) \quad \forall m,n \in T.$

At the ends of a checkered strip measuring $1 \times 101$ squares there are two chips: on the left is the chip of the first player, on the right is the second. Per turn dares to move his piece in the direction of the opposite edge of the strip by 1, 2, 3 or 4 cells. In this case, you are allowed to jump over opponent's chip, but it is forbidden to place your chip on the same square with her. The first one to reach the opposite edge of the strip wins. Who wins if the game is played correctly: the one who goes first, or him rival?

Find the maximum value of the real constant $C$ such that $x^{2}+y^{2}+1\geq C(x+y)$, and $ x^{2}+y^{2}+xy+1\geq C(x+y)$ for all reals $x,y$.

Find the maximum value of an integer $B$ such that for every 9 distinct natural number with the sum of $2023$, there must exist a sum of 4 of the number that is greater than or equal to $B$

Let $f(x)=x^2-3x+1,$ and let $\alpha_1, \alpha_2, \alpha_3, \alpha_4$ be the $4$ roots of $f(f(x))=x.$ Evaluate $\lfloor 10\alpha_1\rfloor+ $ $\lfloor 10\alpha_2\rfloor$ $+$ $\lfloor 10\alpha_3\rfloor+\lfloor 10\alpha_4\rfloor.$

There is a set of cards with numbers from $1$ to $30$ (which may be repeated) . Each student takes one such card. The teacher can perform the following operation: He reads a list of such numbers (possibly only one) and then asks the students to raise an arm if their number was in this list. How many times must he perform such an operation in order to determine the number on each student 's card? (Indicate the number of operations and prove that it is minimal . Note that there are not necessarily 30 students.)

Let $0<x_{1},x_{2},x_{3},x_{4}\leq\frac{1}{2}$ are real numbers. Prove that $\frac{x_{1}x_{2}x_{3}x_{4}}{(1-x_{1})(1-x_{2})(1-x_{3})(1-x_{4})}\leq\frac{x_{1}^{4}+x_{2}^{4}+x_{3}^{4}+x_{4}^{4}}{(1-x_{1})^{4}+(1-x_{2})^{4}+(1-x_{3})^{4}+(1-x_{4})^{4}}$.

There are $ 11$ coins, which are indistinguishable by sight. Nevertheless, among them there are $ 10$ geniune coins ( of weight $ 20$ g each) and one counterfeit (of weight $ 21$ g). You have a two-pan scale which is blanced when the weight in the left-hand pan is twice as much as the weight in the right-hand one. Using this scale only, find the false coin by three weighings.

Given a set of $4n$ positive numbers such that any distinct choice of ordered foursomes of these numbers constitutes a geometric progression. Prove that at least $4$ numbers of the set are identical.

In a long line of people, the 1013th person from the left is also the 1010th person from the right. How many people are in the line? $ \textbf{(A) }2021 \qquad \textbf{(B) }2022 \qquad \textbf{(C) }2023 \qquad \textbf{(D) }2024 \qquad \textbf{(E) }2025 \qquad $

There are $2022$ signs arranged in a straight line. Mark tasks Auto to color each sign with either red or blue with the following condition: for any given sequence of length $1011$ whose each term is either red or blue, Auto can always remove $1011$ signs from the line so that the remaining $1011$ signs match the given color sequence without changing the order. Determine the number of ways Auto can color the signs to satisfy Mark's condition.

Let $f(x)=e^{-x}\ \forall\ x\geq 0$ and let $g$ be a function defined as for every integer $k \ge 0$, a straight line joining $(k,f(k))$ and $(k+1,f(k+1))$ . Find the area between the graphs of $f$ and $g$.

Let $P$ be a polynomial with integer coefficients. There exist integers $a$ and $b$ such that $P(a) \cdot P(b)=-(a-b)^2$. Prove that $P(a)+P(b)=0$.

A [i]word[/i] is defined as any finite string of letters. A word is a [i]palindrome[/i] if it reads the same backwards and forwards. Let a sequence of words $W_0, W_1, W_2,...$ be defined as follows: $W_0 = a, W_1 = b$, and for $n \ge 2$, $W_n$ is the word formed by writing $W_{n-2}$ followed by $W_{n-1}$. Prove that for any $n \ge 1$, the word formed by writing $W_1, W_2, W_3,..., W_n$ in succession is a palindrome.

Let $ABC$ be an acute triangle and let $M$ be the midpoint of $AC$. A circle $\omega$ passing through $B$ and $M$ meets the sides $AB$ and $BC$ at points $P$ and $Q$ respectively. Let $T$ be the point such that $BPTQ$ is a parallelogram. Suppose that $T$ lies on the circumcircle of $ABC$. Determine all possible values of $\frac{BT}{BM}$.

Which of the following describes the graph of the equation $ (x \plus{} y)^2 \equal{} x^2 \plus{} y^2$? $ \textbf{(A)}\text{ the empty set}\qquad \textbf{(B)}\text{ one point}\qquad \textbf{(C)}\text{ two lines}$ $\textbf{(D)}\text{ a circle}\qquad \textbf{(E)}\text{ the entire plane}$

Given a circumscribed quadrilateral $ABCD$. Prove that its inradius is smaller than the sum of the inradii of triangles $ABC$ and $ACD$.

Let $a,b,c$ be real numbers $a + b +c = 0$. Show that [list=a] [*] $\displaystyle \frac{a^2 + b^2 + c^2}{2} \cdot \frac{a^3 + b^3 + c^3}{3} = \frac{a^5 + b^5 + c^5}{5}$. [*] $\displaystyle \frac{a^2 + b^2 + c^2}{2} \cdot \frac{a^5 + b^5 + c^5}{5} = \frac{a^7 + b^7 + c^7}{7}$. [/list] [I]Folklore[/I]

We look at an octahedron, a regular octahedron, having painted one of the side surfaces red and the other seven surfaces blue. We throw the octahedron like a die. The surface that comes up is painted: if it is red it is painted blue and if it is blue it is painted red. Then we throw the octahedron again and paint it again according to the above rule. In total we throw the octahedron $10$ times. How many different octahedra can we get after finishing the $10$th time? [i]Two octahedra are different if they cannot be converted into each other by rotation.[/i]

Let $ABCD$ be a convex pentagon such that \[\angle DCB = \angle DEA = 90^\circ, \ \text{and} \ DC=DE.\] Let $F$ be a point on AB such that $AF:BF=AE:BC$. Show that \[\angle FEC= \angle BDC, \ \text{and} \ \angle FCE= \angle ADE.\]

An (ordered) triple $(x_1,x_2,x_3)$ of positive [i]irrational[/i] numbers with $x_1+x_2+x_3=1$ is called [b]balanced[/b] if each $x_i< 1/2.$ If a triple is not balanced, say if $x_j>1/2$, one performs the following [b] balancing act[/b] $$B(x_1,x_2,x_3)=(x'_1,x'_2,x'_3),$$ where $x'_i=2x_i$ if $i\neq j$ and $x'_j=2x_j-1.$ If the new triple is not balanced, one performs the balancing act on it. Does the continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act?

The Judgment of the Council of Sages proceeds as follows: the king arranges the sages in a line and places either a white hat, black hat or a red hat on each sage's head. Each sage can see the color of the hats of the sages in front of him, but not of his own hat or of the hats of the sages behind him. Then one by one (in an order of their choosing), each sage guesses a color. Afterward, the king executes those sages who did not correctly guess the color of their own hat. The day before, the Council meets and decides to minimize the number of executions. What is the smallest number of sages guaranteed to survive in this case? [i]K. Knop[/i] P.S. Of course, the sages hear the previous guesses. See also [url]http://www.artofproblemsolving.com/Forum/viewtopic.php?f=42&t=530552[/url]