Let $M=\{1,2,3,\ldots, 10000\}.$ Prove that there are $16$ subsets of $M$ such that for every $a \in M,$ there exist $8$ of those subsets that intersection of the sets is exactly $\{a\}.$

[b]Problem 3.[/b] Two points $M$ and $N$ are chosen inside a non-equilateral triangle $ABC$ such that $\angle BAM=\angle CAN$, $\angle ABM=\angle CBN$ and \[AM\cdot AN\cdot BC=BM\cdot BN\cdot CA=CM\cdot CN\cdot AB=k\] for some real $k$. Prove that: [b]a)[/b] We have $3k=AB\cdot BC\cdot CA$. [b]b)[/b] The midpoint of $MN$ is the medicenter of $\triangle ABC$. [i]Remark.[/i] The [b]medicenter[/b] of a triangle is the intersection point of the three medians: If $A_{1}$ is midpoint of $BC$, $B_{1}$ of $AC$ and $C_{1}$ of $AB$, then $AA_{1}\cap BB_{1}\cap CC_{1}= G$, and $G$ is called medicenter of triangle $ABC$. [i] Nikolai Nikolov[/i]

It is known that $a^{2005} + b^{2005}$ can be expressed as the polynomial of $a + b$ and $ab$. Find the coefficients' sum of this polynomial.

If $xy:yz:zx=6:8:12,$ and $x^3+y^3+z^3:xyz$ is $m:n$ where $m$ and $n$ are relatively prime positive integers, then find $m+n.$ [i]Proposed by Ada Tsui[/i]

Determine all polynomials $P$ with real coeffients such that $1 + P(x) = \frac12 (P(x -1) + P(x + 1))$ for all real $x$.

Let $p$ and $q$ be integers such that $q$ is nonzero. Prove that \[ \Bigl\lvert \frac{p}{q} - \sqrt{7} \Bigr\rvert \ge \frac{24 - 9\sqrt{7}}{q^2} \, . \]

Given a simple graph with $ 4038 $ vertices. Assume we arbitrarily choose $ 2019 $ vertices as a group (the other $ 2019 $ is another group, of course), there are always $ k $ edges that connect two groups. Find all possible value of $ k $.

Let $AA_1$, $CC_1$ be the altitudes of $\Delta ABC$, and $P$ be an arbitrary point of side $BC$. Point $Q$ on the line $AB$ is such that $QP = PC_1$, and point $R$ on the line $AC$ is such that $RP = CP$. Prove that $QA_1RA$ is a cyclic quadrilateral.

The [i]runcible[/i] positive integers are defined recursively as follows: [list] [*]$1$ and $2$ are runcible [*]If $a$ and $b$ are runcible (where $a$ and $b$ are not necessarily distinct) then $2a + 3b$ is runcible. [/list] Is $2024$ runcible?

Let $a, b, c, d$ be real numbers. Suppose that all the roots of $z^4+az^3+bz^2+cz+d=0$ are complex numbers lying on a circle in the complex plane centered at $0+0i$ and having radius $1$. The sum of the reciprocals of the roots is necessarily $ \textbf{(A)}\ a \qquad\textbf{(B)}\ b \qquad\textbf{(C)}\ c \qquad\textbf{(D)}\ -a \qquad\textbf{(E)}\ -b $

Let $k$,$n$ be positive integers, $k,n>1$, $k<n$ and a $n \times n$ grid of unit squares is given. Ana and Maya take turns in coloring the grid in the following way: in each turn, a unit square is colored black in such a way that no two black cells have a common side or vertex. Find the smallest positive integer $n$ , such that they can obtain a configuration in which each row and column contains exactly $k$ black cells. Draw one example.

Let $n$ and $k$ be two integers with $n>k\geqslant 1$. There are $2n+1$ students standing in a circle. Each student $S$ has $2k$ [i]neighbors[/i] - namely, the $k$ students closest to $S$ on the left, and the $k$ students closest to $S$ on the right. Suppose that $n+1$ of the students are girls, and the other $n$ are boys. Prove that there is a girl with at least $k$ girls among her neighbors. [i]Proposed by Gurgen Asatryan, Armenia[/i]

There are 1000 doors $D_1, D_2, . . . , D_{1000}$ and 1000 persons $P_1, P_2, . . . , P_{1000}$. Initially all the doors were closed. Person $P_1$ goes and opens all the doors. Then person $P_2$ closes door $D_2, D_4, . . . , D_{1000}$ and leaves the odd numbered doors open. Next $P_3$ changes the state of every third door, that is, $D_3, D_6, . . . , D_{999}$ . (For instance, $P_3$ closes the open door $D_3$ and opens the closed door D6, and so on). Similarly, $P_m$ changes the state of the the doors $D_m, D_{2m}, D_{3m}, . . . , D_{nm}, . . .$ while leaving the other doors untouched. Finally, $P_{1000}$ opens $D_{1000}$ if it was closed or closes it if it were open. At the end, how many doors will remain open?

How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?

An altitude $h$ of a triangle is increased by a length $m$. How much must be taken from the corresponding base $b$ so that the area of the new triangle is one-half that of the original triangle? ${{ \textbf{(A)}\ \frac{bm}{h+m}\qquad\textbf{(B)}\ \frac{bh}{2h+2m}\qquad\textbf{(C)}\ \frac{b(2m+h)}{m+h}\qquad\textbf{(D)}\ \frac{b(m+h)}{2m+h} }\qquad\textbf{(E)}\ \frac{b(2m+h)}{2(h+m)} } $

Suppose $n$ is an integer $\geq 2$. Determine the first digit after the decimal point in the decimal expansion of the number \[\sqrt[3]{n^{3}+2n^{2}+n}\]

Let $\vartriangle ABC$ be a triangle of which the side lengths are positive integers which are pairwise coprime. The tangent in $A$ to the circumcircle intersects line $BC$ in $D$. Prove that $BD$ is not an integer.

Prove that every natural number which is not an integer power of $2$ is the sum of two or more consecutive natural numbers.

Two disjoint circles $W_1(S_1,r_1)$ and $W_2(S_2,r_2)$ are given in the plane. Point $A$ is on circle $W_1$ and $AB,AC$ touch the circle $W_2$ at $B,C$ respectively. Find the loci of the incenter and orthocenter of triangle $ABC$.

Positive numbers $a, b, c, x, y, z$ satisfy that $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function $f(x,y,z) =\frac{x^2}{x+1}+\frac{y^2}{y+1}+\frac{z^2}{z+1}$.

$m,n,p$ are positive integers which do not simultaneously equal to zero. $3$D Cartesian space is divided into unit cubes by planes each perpendicular to one of $3$ axes and cutting corresponding axis at integer coordinates. Each unit cube is filled with an integer from $1$ to $60$. A filling of integers is called [i]Dien Bien[/i] if, for each rectangular box of size $\{2m+1,2n+1,2p+1\}$, the number in the unit cube which has common centre with the rectangular box is the average of the $8$ numbers of the $8$ unit cubes at the $8$ corners of that rectangular box. How many [i]Dien Bien[/i] fillings are there? Two fillings are the same if one filling can be transformed to the other filling via a translation. [hide]translation from [url=http://artofproblemsolving.com/community/c6h592875p3515526]here[/url][/hide]

Math teacher wrote in a table a polynomial $P(x)$ with integer coefficients and he said: "Today my daughter have a birthday.If in polynomial $P(x)$ we have $x=a$ where $a$ is the age of my daughter we have $P(a)=a$ and $P(0)=p$ where $p$ is a prime number such that $p>a$." How old is the daughter of math teacher?

Find all triplets of real numbers $(x, y, z)$ that satisfies the system of equations $x^5 = 2y^3 + y - 2$ $y^5 = 2z^3 + z - 2$ $z^5 = 2x^3 + x - 2$

consider $0<u<1$. find $\alpha > 0$ minimum such that there exists $\beta > 0$ satisfying $(1+x)^u +(1-x)^u \leq 2 - \frac{x^\alpha}{\beta} \forall 0<x<1$

Find all integer coefficient polynomial $P(x)$ such that for all positive integer $x$, we have $$\tau(P(x))\geq\tau(x)$$Where $\tau(n)$ denotes the number of divisors of $n$. Define $\tau(0)=\infty$. Note: you can use this conclusion. For all $\epsilon\geq0$, there exists a positive constant $C_\epsilon$ such that for all positive integer $n$, the $n$th smallest prime is at most $C_\epsilon n^{1+\epsilon}$. [i]Proposed by USJL[/i]