Let $f=f_0+f_1z+f_2z^2+\ldots+f_{2n}z^{2n}$ and $f_k=f_{2n-k}$ for each $k$. Prove that $f(z)=z^ng(z+z^{-1})$, where $g$ is a polynomial of degree $n$.

Give a square of side $1$. Show that for each finite set of points of the sides of the square you can find a vertex of the square with the following property: the arithmetic mean of the squares of the distances from this vertex to the points of the set is greater than or equal to $3/4$.

Let $A_0,A_1, \ldots , A_n$ be points in a plane such that (i) $A_0A_1 \leq \frac{1}{ 2} A_1A_2 \leq \cdots \leq \frac{1}{2^{n-1} } A_{n-1}A_n$ and (ii) $0 < \measuredangle A_0A_1A_2 < \measuredangle A_1A_2A_3 < \cdots < \measuredangle A_{n-2}A_{n-1}A_n < 180^\circ,$ where all these angles have the same orientation. Prove that the segments $A_kA_{k+1},A_mA_{m+1}$ do not intersect for each $k$ and $n$ such that $0 \leq k \leq m - 2 < n- 2.$

[b]p1.[/b] Al usually arrives at the train station on the commuter train at $6:00$, where his wife Jane meets him and drives him home. Today Al caught the early train and arrived at $5:00$. Rather than waiting for Jane, he decided to jog along the route he knew Jane would take and hail her when he saw her. As a result, Al and Jane arrived home $12$ minutes earlier than usual. If Al was jogging at a constant speed of $5$ miles per hour, and Jane always drives at the constant speed that would put her at the station at $6:00$, what was her speed, in miles per hour? [b]p2.[/b] In the figure, points $M$ and $N$ are the respective midpoints of the sides $AB$ and $CD$ of quadrilateral $ABCD$. Diagonal $AC$ meets segment $MN$ at $P$, which is the midpoint of $MN$, and $AP$ is twice as long as $PC$. The area of triangle $ABC$ is $6$ square feet. (a) Find, with proof, the area of triangle $AMP$. (b) Find, with proof, the area of triangle $CNP$. (c) Find, with proof, the area of quadrilateral $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/a/c/4bdcd8390bae26bc90fc7eae398ace06900a67.png[/img] [b]p3.[/b] (a) Show that there is a triangle whose angles have measure $\tan^{-1}1$, $\tan^{-1}2$ and $\tan^{-1}3$. (b) Find all values of $k$ for which there is a triangle whose angles have measure $\tan^{-1}\left(\frac12 \right)$, $\tan^{-1}\left(\frac12 +k\right)$, and $\tan^{-1}\left(\frac12 +2k\right)$ [b]p4.[/b] (a) Find $19$ consecutive integers whose sum is as close to $1000$ as possible. (b) Find the longest possible sequence of consecutive odd integers whose sum is exactly $1000$, and prove that your sequence is the longest. [b]p5.[/b] Let $AB$ and $CD$ be chords of a circle which meet at a point $X$ inside the circle. (a) Suppose that $\frac{AX}{BX}=\frac{CX}{DX}$. Prove that $|AB|=|CD|$. (b) Suppose that $\frac{AX}{BX}>\frac{CX}{DX}>1$. Prove that $|AB|>|CD|$. ($|PQ|$ means the length of the segment $PQ$.) PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A, B, C, D$ be points that lie on the same circle . Let $F$ be such that the arc $AF$ is congruent with the arc $BF$. Let $P$ be the intersection point of the segments $DF$ and $AC$. Let $Q$ be intersection point of the $CF$ and $BD$ segments. Prove that $PQ \parallel AB$.

Forty-nine students solve a set of 3 problems. The score for each problem is a whole number of points from 0 to 7. Prove that there exist two students $ A$ and $ B$ such that, for each problem, $ A$ will score at least as many points as $ B.$

Given is a simple connected graph with $2n$ vertices. Prove that its vertices can be colored with two colors so that if there are $k$ edges connecting vertices with different colors and $m$ edges connecting vertices with the same color, then $k-m \geq n$.

Find the maximum value of $\frac{xyz}{(1 + 5x)(4x + 3y)(5y + 6z)(z + 18)}$ as $x, y$ and $z$ range over the set of all positive real numbers. Justify your answer.

The volume of a certain rectangular solid is $ 8 \text{ cm}^3$, its total surface area is $ 32 \text{ cm}^3$, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is $ \textbf{(A)}\ 28 \qquad \textbf{(B)}\ 32 \qquad \textbf{(C)}\ 36 \qquad \textbf{(D)}\ 40 \qquad \textbf{(E)}\ 44$

How maay zeros are there at the end of the number $$1000! = 1 \cdot 2 \cdot 3 \cdot ... \cdot 999 \cdot 1000?$$

A circle is drawn on the board, and $2n$ points are marked on it, dividing it into $2n$ equal arcs. Petya and Vasya are playing the following game. Petya chooses a positive integer $d \leqslant n$ and announces this number to Vasya. To win the game, Vasya needs to color all marked points using $n$ colors, such that each color is assigned to exactly two points, and for each pair of same-colored points, one of the arcs between them contains exactly $(d - 1)$ marked points. Find all $n$ for which Petya will be able to prevent Vasya from winning.

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

We color all points of a plane using $3$ colors. Prove that there are at least two points of the plane having same colours and with distance among them $1$.

Find all four-element sets of positive integers $\{w,x,y,z\}$ such that $w^x+w^y=w^z$.

Find all continuous functions $f : \mathbb{R}\to\mathbb{R}$ such that for all $x \in\mathbb{ R}$, \[f (f (x)) = f (x)+x.\]

Find all functions $f: \mathbb R \to \mathbb R$ such that $$f(x+f(x+y))=x+f(f(x)+y)$$ holds for all real numbers $x$ and $y$.

Let $ 2n$ distinct points on a circle be given. Arrange them into disjoint pairs in an arbitrary way and join the couples by chords. Determine the probability that no two of these $ n$ chords intersect. (All possible arrangement into pairs are supposed to have the same probability.)

A positive integer is called [i]hypotenuse[/i] if it can be represented as a sum of two squares of non-negative integers. Prove that any natural number greater than $10$ is the difference of two hypotenuse numbers.

[i]20 problems for 25 minutes.[/i] [b]p1.[/b] Evaluate $20 \times 21 + 2021$. [b]p2.[/b] Let points $A$, $B$, $C$, and $D$ lie on a line in that order. Given that $AB = 5CD$ and $BD = 2BC$, compute $\frac{AC}{BD}$. [b]p3.[/b] There are $18$ students in Vincent the Bug's math class. Given that $11$ of the students take U.S. History, $15$ of the students take English, and $2$ of the students take neither, how many students take both U.S. History and English? [b]p4.[/b] Among all pairs of positive integers $(x, y)$ such that $xy = 12$, what is the least possible value of $x + y$? [b]p5.[/b] What is the smallest positive integer $n$ such that $n! + 1$ is composite? [b]p6.[/b] How many ordered triples of positive integers $(a, b,c)$ are there such that $a + b + c = 6$? [b]p7.[/b] Thomas orders some pizzas and splits each into $8$ slices. Hungry Yunseo eats one slice and then finds that she is able to distribute all the remaining slices equally among the $29$ other math club students. What is the fewest number of pizzas that Thomas could have ordered? [b]p8.[/b] Stephanie has two distinct prime numbers $a$ and $b$ such that $a^2-9b^2$ is also a prime. Compute $a + b$. [b]p9.[/b] Let $ABCD$ be a unit square and $E$ be a point on diagonal $AC$ such that $AE = 1$. Compute $\angle BED$, in degrees. [b]p10.[/b] Sheldon wants to trace each edge of a cube exactly once with a pen. What is the fewest number of continuous strokes that he needs to make? A continuous stroke is one that goes along the edges and does not leave the surface of the cube. [b]p11.[/b] In base $b$, $130_b$ is equal to $3n$ in base ten, and $1300_b$ is equal to $n^2$ in base ten. What is the value of $n$, expressed in base ten? [b]p12.[/b] Lin is writing a book with $n$ pages, numbered $1,2,..., n$. Given that $n > 20$, what is the least value of $n$ such that the average number of digits of the page numbers is an integer? [b]p13.[/b] Max is playing bingo on a $5\times 5$ board. He needs to fill in four of the twelve rows, columns, and main diagonals of his bingo board to win. What is the minimum number of boxes he needs to fill in to win? [b]p14.[/b] Given that $x$ and $y$ are distinct real numbers such that $x^2 + y = y^2 + x = 211$, compute the value of $|x - y|$. [b]p15.[/b] How many ways are there to place 8 indistinguishable pieces on a $4\times 4$ checkerboard such that there are two pieces in each row and two pieces in each column? [b]p16.[/b] The Manhattan distance between two points $(a, b)$ and $(c, d)$ in the plane is defined to be $|a - c| + |b - d|$. Suppose Neil, Neel, and Nail are at the points $(5, 3)$, $(-2,-2)$ and $(6, 0)$, respectively, and wish to meet at a point $(x, y)$ such that their Manhattan distances to$ (x, y)$ are equal. Find $10x + y$. [b]p17.[/b] How many positive integers that have a composite number of divisors are there between $1$ and $100$, inclusive? [b]p18.[/b] Find the number of distinct roots of the polynomial $$(x - 1)(x - 2) ... (x - 90)(x^2 - 1)(x^2 - 2) ... (x^2 - 90)(x^4 - 1)(x^4 - 2)...(x^4 - 90)$$. [b]p19.[/b] In triangle $ABC$, let $D$ be the foot of the altitude from $ A$ to $BC$. Let $P,Q$ be points on $AB$, $AC$, respectively, such that $PQ$ is parallel to $BC$ and $\angle PDQ = 90^o$. Given that $AD = 25$, $BD = 9$, and $CD = 16$, compute $111 \times PQ$. [b]p20.[/b] The simplified fraction with numerator less than $1000$ that is closest but not equal to $\frac{47}{18}$ is $\frac{p}{q}$ , where $p$ and $q$ are relatively prime positive integers. Compute $p$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let be $f: (0,+\infty)\rightarrow \mathbb{R}$ monoton-decreasing . If $f(2a^2+a+1)<f(3a^2-4a+1)$ find interval of $a$ .

Consider the equation $x^4 + ax^3 + bx^2 + ax + 1 = 0$ with real coefficients $a, b$. Determine the number of distinct real roots and their multiplicities for various values of $a$ and $b$. Display your result graphically in the $(a, b)$ plane.

$a$ is a real number. Find the all $(x,y)$ real number pairs satisfy;$$y^2=x^3+(a-1)x^2+a^2x$$$$x^2=y^3+(a-1)y^2+a^2y$$

Let $n$ be an arbitrary positive integer. Calculate $S_n = \sum_{r=0}^n 2^{r-2n} \binom{2n-r}{n}.$

For arbitrary real number $x$, the function $f : \mathbb R \to \mathbb R$ satisfies $f(f(x))-x^2+x+3=0$. Show that the function $f$ does not exist.

Assume $c$ is a real number. If there exists $x\in[1,2]$ such that $\max\left\{\left |x+\frac cx\right |, \left |x+\frac cx + 2\right |\right\}\geq 5$, please find the value range of $c$.