Let $ f$ be an integrable real-valued function on the closed interval $ [0, 1]$ such that $$\int_{0}^{1} x^{m}f(x) dx=\begin{cases} 0 \;\; \text{for}\; m=0,1,\ldots,n-1;\\ 1\;\; \text{for}\; m=n. \end{cases} $$ Show that $|f(x)|\geq2^{n}(n+1)$ on a set of positive measure.

Bisector of one angle of triangle $ABC$ is equal to the bisector of its external angle at the same vertex (see figure). Find the difference between the other two angles of the triangle. [img]https://cdn.artofproblemsolving.com/attachments/c/3/d2efeb65544c45a15acccab8db05c8314eb5f2.png[/img]

Given the positive integer $n$, find the integer $f(n)$ so that $f(n)$ is the next positive integer that is always a number whose all digits are divisible by $n$.

Let $ABCD$ be a cyclic quadrilateral and let $K,L,M,N,S,T$ the midpoints of $AB, BC, CD, AD, AC, BD$ respectively. Prove that the circumcenters of $KLS, LMT, MNS, NKT$ form a cyclic quadrilateral which is similar to $ABCD$.

The polynomial $R(x)$ is the remainder upon dividing $x^{2007}$ by $x^2-5x+6$. $R(0)$ can be expressed as $ab(a^c-b^c)$. Find $a+c-b$.

What is the largest number $N$ for which there exist $N$ consecutive positive integers such that the sum of the digits in the $k$-th integer is divisible by $k$ for $1 \le k \le N$ ? (S Tokarev)

Find all real numbers $x,y$ greater than $1$, satisfying the condition that the numbers $\sqrt{x-1}+\sqrt{y-1}$ and $\sqrt{x+1}+\sqrt{y+1}$ are nonconsecutive integers.

A sphere with ray $R$ is cut by two parallel planes. such that the center of the sphere is outside the region determined by these planes. Let $S_{1}$ and $S_{2}$ be the areas of the intersections, and $d$ the distance between these planes. Find the area of the intersection of the sphere with the plane parallel with these two planes, with equal distance from them.

Consider the expanded form of $\left(x+\frac{1}{2\sqrt[4]{x}}\right)^n$, put all items in number (from high power to low power). If the coefficients of the first three items are arithmetic sequence, then the number of items with an integral power is________.

Let $k>1$ be a real number. Calculate: (a) $L=\lim_{n\to \infty} \int_0^1\left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n\text{d} x.$ (b) $\lim_{n\to \infty} n\left\lbrack L- \int_0^1\left( \frac{k}{\sqrt[n]{x}+k-1}\right)^n\text{d} x\right\rbrack.$

[b]a)[/b] Show that the number $ \sqrt{9-\sqrt{77}}\cdot\sqrt {2}\cdot\left(\sqrt{11}-\sqrt{7}\right)\cdot\left( 9+\sqrt{77}\right) $ is natural. [b]b)[/b] Consider two real numbers $ x,y $ such that $ xy=6 $ and $ x,y>2. $ Show that $ x+y<5. $

Let $\Gamma$ be a circle and $P$ a point outside of $\Gamma$ . A tangent from $P$ to the circle intersects it in $A$. Another line through $P$ intersects $\Gamma$ at the points $B$ and $C$. The bisector of $\angle APB$ intersects $AB$ at $D$ and $AC$ at $E$. Prove that the triangle $ADE$ is isosceles.

Let $n$ be a natural number. Prove that the number of all non-isosceles triangles with lengths of their sides equal to natural numbers and a perimeter $2n$ is $[\frac{n^2-6n+12}{12}]$.

Find all pairs $(p,q)$ of prime numbers such that $$ p(p^2 - p - 1) = q(2q + 3) .$$

Let $M$ be the set of natural numbers $k$ for which there exists a natural number $n$ such that $$3^n \equiv k\pmod n.$$ Prove that $M$ has infinitely many elements.

How many valence electrons are in the pyrophosphate ion, $\ce{P2O7}^{4-}?$ $ \textbf{(A) } 48\qquad\textbf{(B) } 52\qquad\textbf{(C) } 54\qquad\textbf{(D) } 56\qquad $

A sequence $ (x_n)$ is given by $ x_1\equal{}2$ and $ nx_n\equal{}2(2n\minus{}1)x_{n\minus{}1}$ for $ n>1$. Prove that $ x_n$ is an integer for every $ n \in \mathbb{N}$.

Let $x,y,z$ be real numbers such that, $x \geq y \geq z >0$. Prove that $$\frac{x^2-y^2}{z}+\frac{z^2-y^2}{x}+\frac{x^2-z^2}{y} \geq 3x-4y+z$$

Julia bakes a cake in the shape of a unit square. Each minute, Julia makes two cuts through the cake as follows: [list] [*] she picks a [b]square[/b] piece $\mathcal{S}$ of the cake with no cuts through its interior; then [*] she slices the entire cake along the two lines parallel to the sides of the cake passing through the center of $\mathcal{S}$. [/list] She does not move any pieces of cake during this process. After eight minutes, she has a grid of $9^2 = 81$ pieces of cake. (The pieces can be various sizes.) Compute the number of distinct grids that she could have ended up with. Two grids are the same if they have the same set of cuts; in particular, two grids that differ by a rotation or reflection are distinct. [i]Proposed by Sean Li[/i]

Find all integers $x,y$ such that $x^2(y-1)+y^2(x-1) = 1$.

Let \( P_1(x) \) and \( P_2(x) \) be monic quadratic trinomials, and let \( A_1 \) and \( A_2 \) be the vertices of the parabolas \( y = P_1(x) \) and \( y = P_2(x) \), respectively. Let \( m(g(x)) \) denote the minimum value of the function \( g(x) \). It is known that the differences \( m(P_1(P_2(x))) - m(P_1(x)) \) and \( m(P_2(P_1(x))) - m(P_2(x)) \) are equal positive numbers. Find the angle between the line \( A_1A_2 \) and the $x$-axis. \\

Find the maximum value of $n$, such that there exist $n$ pairwise distinct positive numbers $x_1,x_2,\cdots,x_n$, satisfy $$x_1^2+x_2^2+\cdots+x_n^2=2017$$

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

Given regular $45$-gon. Can you mark its corners with the digits $\{0,1,...,9\}$ in such a way, that for every pair of digits there would be a side with both ends marked with those digits?

Given a natural number $n\ge 3$, determine all strictly increasing sequences $a_1<a_2<\cdots<a_n$ such that $\text{gcd}(a_1,a_2)=1$ and for any pair of natural numbers $(k,m)$ satisfy $n\ge m\ge 3$, $m\ge k$, $$\frac{a_1+a_2+\cdots +a_m}{a_k}$$ is a positive integer.