suppose that polynomial $p(x)=x^{2010}\pm x^{2009}\pm...\pm x\pm 1$ does not have a real root. what is the maximum number of coefficients to be $-1$?(14 points)

Let $ ABC$ be a triangle, $ B_1$ the midpoint of side $ AB$ and $ C_1$ the midpoint of side $ AC$. Let $ P$ be the point of intersection ($ \neq A$) of the circumcircles of triangles $ ABC_1$ and $ AB_1C$. Let $ Q$ be the point of intersection ($ \neq A$) of the line $ AP$ and the circumcircle of triangle $ AB_1C_1$. Prove that $ \frac{AP}{AQ} \equal{} \frac{3}{2}$.

Fifteen elephants stand in a row. Their weights are expressed by integer numbers of kilograms. The sum of the weight of each elephant (except the one on the extreme right) and the doubled weight of its right neighbour is exactly $15$ tonnes. Determine the weight of each elephant. (F.L. Nazarov)

Big Welk writes the letters of the alphabet in order, and starts again at $A$ each time he gets to $Z.$ What is the $4^3$-rd letter that he writes down?

Let $a$, $b$, $c$ be the three roots of the equation $x^3-(k+1)x^2+kx+12=0$, where $k$ is a real number. If $(a-2)^3+(b-2)^3+(c-2)^3=-18$, find the value of $k$.

Which answer choice correctly fills the blank in the statement below? "The probability of flipping heads on a fair coin is the equal to the probability of rolling a $\underline{~~~~~~~~~~~}$ on a fair dice." $\textbf{(A) }\text{prime number}\qquad\textbf{(B) }\text{number divisible by 3}\qquad\textbf{(C) }\text{number with four factors}\qquad\textbf{(D) }2~\text{or}~3\qquad\textbf{(E) }4$

Let $p$ prime and $m$ a positive integer. Determine all pairs $( p,m)$ satisfying the equation: $ p(p+m)+p=(m+1)^3$

[b]p1.[/b] Steven has just learned about polynomials and he is struggling with the following problem: expand $(1-2x)^7$ as $a_0 +a_1x+...+a_7x^7$ . Help Steven solve this problem by telling him what $a_1 +a_2 +...+a_7$ is. [b]p2.[/b] Each element of the set ${2, 3, 4, ..., 100}$ is colored. A number has the same color as any divisor of it. What is the maximum number of colors? [b]p3.[/b] Fuchsia is selecting $24$ balls out of $3$ boxes. One box contains blue balls, one red balls and one yellow balls. They each have a hundred balls. It is required that she takes at least one ball from each box and that the numbers of balls selected from each box are distinct. In how many ways can she select the $24$ balls? [b]p4.[/b] Find the perfect square that can be written in the form $\overline{abcd} - \overline{dcba}$ where $a, b, c, d$ are non zero digits and $b < c$. $\overline{abcd}$ is the number in base $10$ with digits $a, b, c, d$ written in this order. [b]p5.[/b] Steven has $100$ boxes labeled from $ 1$ to $100$. Every box contains at most $10$ balls. The number of balls in boxes labeled with consecutive numbers differ by $ 1$. The boxes labeled $1,4,7,10,...,100$ have a total of $301$ balls. What is the maximum number of balls Steven can have? [b]p6.[/b] In acute $\vartriangle ABC$, $AB=4$. Let $D$ be the point on $BC$ such that $\angle BAD = \angle CAD$. Let $AD$ intersect the circumcircle of $\vartriangle ABC$ at $X$. Let $\Gamma$ be the circle through $D$ and $X$ that is tangent to $AB$ at $P$. If $AP = 6$, compute $AC$. [b]p7.[/b] Consider a $15\times 15$ square decomposed into unit squares. Consider a coloring of the vertices of the unit squares into two colors, red and blue such that there are $133$ red vertices. Out of these $133$, two vertices are vertices of the big square and $32$ of them are located on the sides of the big square. The sides of the unit squares are colored into three colors. If both endpoints of a side are colored red then the side is colored red. If both endpoints of a side are colored blue then the side is colored blue. Otherwise the side is colored green. If we have $196$ green sides, how many blue sides do we have? [b]p8.[/b] Carl has $10$ piles of rocks, each pile with a different number of rocks. He notices that he can redistribute the rocks in any pile to the other $9$ piles to make the other $9$ piles have the same number of rocks. What is the minimum number of rocks in the biggest pile? [b]p9.[/b] Suppose that Tony picks a random integer between $1$ and $6$ inclusive such that the probability that he picks a number is directly proportional to the the number itself. Danny picks a number between $1$ and $7$ inclusive using the same rule as Tony. What is the probability that Tony’s number is greater than Danny’s number? [b]p10.[/b] Mike wrote on the board the numbers $1, 2, ..., n$. At every step, he chooses two of these numbers, deletes them and replaces them with the least prime factor of their sum. He does this until he is left with the number $101$ on the board. What is the minimum value of $n$ for which this is possible? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

An arbitrary prime $ p$ is given. If an integer sequence $ (n_1 , n_2 , \cdots , n_k )$ satisfying the conditions - For all $ i\equal{} 1, 2, \cdots , k$, $ n_i \geq \frac{p\plus{}1}{2}$ - For all $ i\equal{} 1, 2, \cdots , k$, $ p^{n_i} \minus{} 1$ is divisible by $ n_{i\plus{}1}$, and $ \frac{p^{n_i} \minus{} 1}{n_{i\plus{}1}}$ is coprime to $ n_{i\plus{}1}$. Let $ n_{k\plus{}1} \equal{} n_1$. exists not for $ k\equal{}1$, but exists for some $ k \geq 2$, then call the prime a good prime. Prove that a prime is good iff it is not $ 2$.

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be a function which satisfies the following: [list][*] $f(m)=m$, for all $m\in\mathbb{Z}$;[*] $f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}$, for all $a, b, c, d\in\mathbb{Z}$ such that $|ad-bc|=1$, $c>0$ and $d>0$;[*] $f$ is monotonically increasing.[/list] (a) Prove that the function $f$ is unique. (b) Find $f(\frac{\sqrt{5}-1}{2})$.

Let $ABCD$ be a cyclic quadrilateral and let $E$ and $F$ be the points on the sides $AB$ and $CD$ respectively such that $AE : EB = CF : FD$. Point $P$ on the segment EF satsfies $EP : PF = AB : CD$. Prove that the ratio of the areas of $\vartriangle APD$ and $\vartriangle BPC$ does not depend on the choice of $E$ and $F$.

There are $2^{10}=1024$ possible 10-letter strings in which each letter is either an A or a B. Find the number of such strings that do not have more than 3 adjacent letters that are identical.

Each diagonal of the base and each lateral edge of a $9$-gonal pyramid is colored either green or red. Show that there must exist a triangle with the vertices at vertices of the pyramid having all three sides of the same color.

A $\text{0.3 kg}$ apple falls from rest through a height of $\text{40 cm}$ onto a flat surface. Upon impact, the apple comes to rest in $\text{0.1 s}$, and $\text{4 cm}^2$ of the apple comes into contact with the surface during the impact. What is the average pressure exerted on the apple during the impact? Ignore air resistance. (A) $\text{67,000 Pa}$ (B) $\text{21,000 Pa}$ (C) $\text{6,700 Pa}$ (D) $\text{210 Pa}$ (E) $\text{67 Pa}$

In a convex pentagon, let the perpendicular line from a vertex to the opposite side be called an altitude. Prove that if four of the altitudes are concurrent at a point then the fifth altitude also passes through this point.

Prove that there is no natural number n such that the sum of all the digits of the number m, where $m = n(2n-1)$ is equal to $2000$.

solve $x^2+31=y^3$ in integers

Consider $2009$ cards, each having one gold side and one black side, lying on parallel on a long table. Initially all cards show their gold sides. Two player, standing by the same long side of the table, play a game with alternating moves. Each move consists of choosing a block of $50$ consecutive cards, the leftmost of which is showing gold, and turning them all over, so those which showed gold now show black and vice versa. The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player? [i]Proposed by Michael Albert, Richard Guy, New Zealand[/i]

Find all the functions $ f: \mathbb{N}\rightarrow \mathbb{N}$ such that \[ 3f(f(f(n))) \plus{} 2f(f(n)) \plus{} f(n) \equal{} 6n, \quad \forall n\in \mathbb{N}.\]

A $1\times 2019$ board is filled with numbers $1, 2, \dots, 2019$ in an increasing order. In each step, three consecutive tiles are selected, then one of the following operations is performed: $\text{(i)}$ the number in the middle is increased by $2$ and its neighbors are decreased by $1$, or $\text{(ii)}$ the number in the middle is decreased by $2$ and its neighbors are increased by $1$. After several such operations, the board again contains all the numbers $1, 2,\dots, 2019$. Prove that each number is in its original position.

Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations $ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$ $ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$ Find all possible values of the product $ p_1p_2p_3p_4$

It is known that for some integers $a_{2021},a_{2020},...,a_1,a_0$ the expression $$a_{2021}n^{2021}+a_{2020}n^{2020}+...+a_1n+a_0$$ is divisible by $2021$ for any arbitrary integer $n$. Is it required that each of the numbers $a_{2021},a_{2020},...,a_1,a_0$ also divisible by $2021$?

Prove that $[\sqrt{n}+\sqrt{n+1}]=[\sqrt{4n+1}]$ for all $n \in N$.

Prove that \[ \lim_{n \to \infty} n \left( \frac{\pi}{4} - n \int_0^1 \frac{x^n}{1+x^{2n}} \, dx \right) = \int_0^1 f(x) \, dx , \] where $f(x) = \frac{\arctan x}{x}$ if $x \in \left( 0,1 \right]$ and $f(0)=1$. [i]Dorin Andrica, Mihai Piticari[/i]

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC,AC,AB$ at $A_{1},B_{1},C_{1}$ . let $AI,BI,CI$ meets $BC,AC,AB$ at $A_{2},B_{2},C_{2}$. let $A'$ is a point on $AI$ such that $A_{1}A'\perp B_{2}C_{2}$ .$B',C'$ respectively. prove that two triangle $A'B'C',A_{1}B_{1}C_{1}$ are equal.