Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let $$b_k=\sum_{i=1}^n a_i \omega^{ki}$$ for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.

Determine the smallest and the greatest possible values of the expression $$\left( \frac{1}{a^2+1}+\frac{1}{b^2+1}+\frac{1}{c^2+1}\right)\left( \frac{a^2}{a^2+1}+\frac{b^2}{b^2+1}+\frac{c^2}{c^2+1}\right)$$ provided $a,b$ and $c$ are non-negative real numbers satisfying $ab+bc+ca=1$. [i]Proposed by Walther Janous, Austria [/i]

In $\triangle ABC$ let point $D$ be the foot of the altitude from $A$ to $BC.$ Suppose that $\angle A = 90^{\circ}, AB - AC = 5,$ and $BD - CD = 7.$ Find the area of $\triangle ABC.$

The exam has $25$ topics, each of which has $8$ questions. On a test, there are $4$ questions of different topics. Is it possible to make $50$ tests so that each question was asked exactly once, and for any two topics there is a test where are questions of both topics?

Given are $m\ge 3$ points in the plane. Prove that you can always choose three of these points $A,B,C$ such that $$\angle ABC \le \frac{180^o}{m}.$$

In the sequence of digits $2,0,2,9,3,...$ any digit it equal to the last digit in the decimal representation of the sum of four previous digits. Do the four numbers $2,0,1,5$ in that order occur in the sequence? Folklore

Right triangle $ABC$ with $\angle ABC = 90^o$ is inscribed in a circle $\omega_1$ with radius $3$. A circle $\omega_2$ tangent to $AB$, $BC$, and $\omega_1$ has radius $2$. Compute the area of $\vartriangle ABC$.

Two circles $ \omega_1$ and $ \omega_2$ tangents internally in point $ P$. On their common tangent points $ A$, $ B$ are chosen such that $ P$ lies between $ A$ and $ B$. Let $ C$ and $ D$ be the intersection points of tangent from $ A$ to $ \omega_1$, tangent from $ B$ to $ \omega_2$ and tangent from $ A$ to $ \omega_2$, tangent from $ B$ to $ \omega_1$, respectively. Prove that $ CA \plus{} CB \equal{} DA \plus{} DB$.

Find all functions $f: \mathbb R \to \mathbb R$ such that \[ f( xf(x) + f(y) ) = f^2(x) + y \] for all $x,y\in \mathbb R$.

Petya chooses $100$ pairwise distinct positive numbers less than $1$ and arranges them in a circle. In one operation, he may take three consecutive numbers \( a, b, c \) (in this order) and replace \( b \) with \( a - b + c \). What is the greatest value of \( k \) such that Petya could initially choose the numbers and perform several operations so that \( k \) of the resulting numbers are integers? \\

Consider the quadratic equation \(x^2 + a_0 x + b_0\) for some real numbers \((a_0, b_0)\). Repeat the following procedure as many times as possible: Let \(c_i = \min \{r_i, s_i\}\), with \(r_i, s_i\) being the roots of the equation \(x^2 + a_i x + b_i\). The new equation is written as \(x^2 + b_i x + c_i\). That is, for the next iteration of the procedure, \(a_{i+1} = b_i\) and \(b_{i+1} = c_i\). We say that \((a_0, b_0)\) is an $\textit{interesting}$ pair if, after a finite number of steps, the equation we obtain after one step is the same, so that \((a_i, b_i) = (a_{i+1}, b_{i+1})\). Find all $\textit{interesting}$ pairs.

For a positive integer $n,$ let $P_n$ be the set of sequences of $2n$ elements, each $0$ or $1,$ where there are exactly $n$ $1$’s and $n$ $0$’s. I choose a sequence uniformly at random from $P_n.$ Then, I partition this sequence into maximal blocks of consecutive $0$’s and $1$’s. Define $f(n)$ to be the expected value of the sum of squares of the block lengths of this uniformly random sequence. What is the largest integer value that $f(n)$ can take on?

Find the number of 4 digit positive integers '$n$' that follow these. 1) the number of digit $ \le $ 6 2) $ 3 \mid n$, but $ 6 \nmid n $

In terms of $n\ge2$, find the largest constant $c$ such that for all nonnegative $a_1,a_2,\ldots,a_n$ satisfying $a_1+a_2+\cdots+a_n=n$, the following inequality holds: \[\frac1{n+ca_1^2}+\frac1{n+ca_2^2}+\cdots+\frac1{n+ca_n^2}\le \frac{n}{n+c}.\] [i]Calvin Deng.[/i]

A solution of $\ce{KNO3}$ in water is prepared for which the following data have been obtained: \[ \text{masses of solute and solvent} \] \[ \text{molar masses of solute and solvent} \] Which of these quantitative descriptions of the solution can be determined? \[ \text{I. molarity II. molality III. density of solution} \] $ \textbf{(A) }\text{I. only}\qquad\textbf{(B) }\text{II. only}\qquad\textbf{(C) }\text{I. and II. only}\qquad\textbf{(D) }\text{I., II. and III.}\qquad$

Solve the equation $ \frac{\sqrt{2+x} +\sqrt{2-x}}{\sqrt{2+x} -\sqrt{2-x}} =\sqrt 3. $

Let $n \ge 1$ and $x_1, \ldots, x_n \ge 0$. Prove that $$ (x_1 + \frac{x_2}{2} + \ldots + \frac{x_n}{n}) (x_1 + 2x_2 + \ldots + nx_n) \le \frac{(n+1)^2}{4n} (x_1 + x_2 + \ldots + x_n)^2 .$$

In the triangle $ABC$ let $AD$ be the interior angle bisector of $\angle ACB$, where $D\in AB$. The circumcenter of the triangle $ABC$ coincides with the incenter of the triangle $BCD$. Prove that $AC^2 = AD\cdot AB$.

One side of a square sheet of paper is colored red, the other - in blue. On both sides, the sheet is divided into $n^2$ identical square cells. In each of these $2n^2$ cells is written a number from $1$ to $k$. Find the smallest $k$,for which the following properties hold simultaneously: (i) on the red side, any two numbers in different rows are distinct; (ii) on the blue side, any two numbers in different columns are different; (iii) for each of the $n^2$ squares of the partition, the number on the blue side is not equal to the number on the red side.

The positive integers $x_1$, $x_2$, $\ldots$, $x_5$, $x_6 = 144$ and $x_7$ are such that $x_{n+3} = x_{n+2}(x_{n+1}+x_n)$ for $n=1,2,3,4$. Determine the value of $x_7$.

For how many integer value of $ m$ does the lines $ 13x\plus{}11y \equal{} 700$ and $ y \equal{} mx\minus{}1$ intersect in a point with integer valued coordinats? A. None B. 1 C. 2 D. 3 E. Infinitely many

Given positive integer $n$ and positive number $M$. For all arithmetic squence $a_1,a_2,\cdots,$ that $a_1^2+a_{n+1}^2\leq M$, find the maximum value of $S=a_{n+1}+a_{n+2}+\cdots,a_{2n+1}$.

[b]p1.[/b] (a) Show that for every set of three integers, we can find two of them whose average is also an integer. (b) Show that for every set of $5$ integers, there is a subset of three of them whose average is an integer. [b]p2.[/b] Let $f(x) = x^2 + ax + b$ and $g(x) = x^2 + cx + d$ be two different quadratic polynomials such that $f(7) + f(11) = g(7) + g(11)$. (a) Show that $f(9) = g(9)$. (b) Show that $x = 9$ is the only value of $x$ where $f(x) = g(x)$. [b]p3.[/b] Consider a rectangle $ABCD$ and points $E$ and $F$ on the sides $BC$ and $CD$, respectively, such that the areas of the triangles $ABE$, $ECF$, and $ADF$ are $11$, $3$, and $40$, respectively. Compute the area of rectangle $ABCD$. [img]https://cdn.artofproblemsolving.com/attachments/f/0/2b0bb188a4157894b85deb32d73ab0077cd0b7.png[/img] [b]p4.[/b] How many ways are there to put markers on a $8 \times 8$ checkerboard, with at most one marker per square, such that each of the $8$ rows and each of the $8$ columns contain an odd number of markers? [b]p5.[/b] A robot places a red hat or a blue hat on each person in a room. Each person can see the colors of the hats of everyone in the room except for his own. Each person tries to guess the color of his hat. No communication is allowed between people and each person guesses at the same time (so no timing information can be used, for example). The only information a person has is the color of each other person’s hat. Before receiving the hats, the people are allowed to get together and decide on their strategies. One way to think of this is that each of the $n$ people makes a list of all the possible combinations he could see (there are $2^{n-1}$ such combinations). Next to each combination, he writes what his guess will be for the color of his own hat. When the hats are placed, he looks for the combination on his list and makes the guess that is listed there. (a) Prove that if there are exactly two people in the room, then there is a strategy that guarantees that always at least one person gets the right answer for his hat color. (b) Prove that if you have a group of $2008$ people, then there is a strategy that guarantees that always at least $1004$ people will make a correct guess. (c) Prove that if there are $2009$ people, then there is no strategy that guarantees that always at least $1005$ people will make a correct guess. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Two equally oriented regular $2n$-gons $A_1A_2\ldots A_{2n}$ and $B_1B_2\ldots B_{2n}$ are given. The perpendicular bisectors $\ell_i$ of the segments $A_iB_i$ are drawn. Let the lines $\ell_i$ and $\ell_{i+1}$ intersect at the point $K_i$ (hereafter we reduce indices modulo $2n$). Denote by $m_i$ the line $K_iK_{i+n}$. Prove that $n{}$ lines $m_i$ intersect at one point and at that the angles between the lines $m_i$ and $m_{i+1}$ are equal. [i]Proposed by Chan Quang Hung (Vietnam)[/i]

Andrew has $10$ balls in a bag, each a different color. He randomly picks a ball from the bag $4$ times, with replacement. The expected number of distinct colors among the balls he picks is $\tfrac{p}{q}$, where $\gcd(p, q) = 1$ and $p, q > 0$. What is $p + q$?