The odd positive integers $1,3,5,7,\cdots,$ are arranged into in five columns continuing with the pattern shown on the right. Counting from the left, the column in which $ 1985$ appears in is the [asy] int i,j; for(i=0; i<4; i=i+1) { label(string(16*i+1), (2*1,-2*i)); label(string(16*i+3), (2*2,-2*i)); label(string(16*i+5), (2*3,-2*i)); label(string(16*i+7), (2*4,-2*i)); } for(i=0; i<3; i=i+1) { for(j=0; j<4; j=j+1) { label(string(16*i+15-2*j), (2*j,-2*i-1)); }} dot((0,-7)^^(0,-9)^^(2*4,-8)^^(2*4,-10)); for(i=-10; i<-6; i=i+1) { for(j=1; j<4; j=j+1) { dot((2*j,i)); }} [/asy] $ \textbf{(A)} \text{ first} \qquad \textbf{(B)} \text{ second} \qquad \textbf{(C)} \text{ third} \qquad \textbf{(D)} \text{ fourth} \qquad \textbf{(E)} \text{ fifth}$

$E$ is the midpoint of the side $AB$ of the square $ABCD$, and $F, G$ are any points on the sides $BC$, $CD$ such that $EF$ is parallel to $AG$. Show that $FG$ touches the inscribed circle of the square.

There are $m$ identical rectangular chocolate bars and $n$ people. Each chocolate bar may be cut into two (possibly unequal) pieces at most once. For which $m$ and $n$ is it possible to split the chocolate evenly among all the people? [i]Selected from the Kvant Magazine (D. Bugaenko and N. Konstantinov)[/i]

A shape is created by joining seven unit cubes, as shown. What is the ratio of the volume in cubic units to the surface area in square units? [asy] import three; defaultpen(linewidth(0.8)); real r=0.5; currentprojection=orthographic(1,1/2,1/4); draw(unitcube, white, thick(), nolight); draw(shift(1,0,0)*unitcube, white, thick(), nolight); draw(shift(1,-1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,-1)*unitcube, white, thick(), nolight); draw(shift(2,0,0)*unitcube, white, thick(), nolight); draw(shift(1,1,0)*unitcube, white, thick(), nolight); draw(shift(1,0,1)*unitcube, white, thick(), nolight);[/asy] $\textbf{(A)} \:1 : 6 \qquad\textbf{ (B)}\: 7 : 36 \qquad\textbf{(C)}\: 1 : 5 \qquad\textbf{(D)}\: 7 : 30\qquad\textbf{ (E)}\: 6 : 25$

Let $x, y$ be positive integers such that $\frac{x^2}{y}+\frac{y^2}{x}$ is an integer. Prove that $y|x^2$.

The Fibonacci sequence is defined by $F_0=F_1=1$ and $F_{n+2}=F_{n+1}+F_n$ for $n\ge0$. Prove that the sum of $2000$ consecutive terms of the Fibonacci sequence is never a term of the sequence.

Let $a$ and $b$ be integers. Find all polynomial with integer coefficients sucht that $P(n)$ divides $P(an+b)$ for infinitely many positive integer $n$

What is the volume of the region in three-dimensional space defined by the inequalities $|x|+|y|+|z|\le1$ and $|x|+|y|+|z-1|\le1$? $\textbf{(A)}\ \frac{1}{6} \qquad \textbf{(B)}\ \frac{1}{3} \qquad \textbf{(C)}\ \frac{1}{2} \qquad \textbf{(D)}\ \frac{2}{3} \qquad \textbf{(E)}\ 1$

In cube $ABCD-A_1B_1C_1D_1$, the degree of dihedral angle $A-BD_1-A_1$ is________.

Let $ABC$ be an acute-angled triangle and let $P$ be the intersection of the tangents at $B$ and $C$ of the circumscribed circle of $\vartriangle ABC$. The line through $A$ perpendicular on $AB$ and cuts the line perpendicular on $AC$ through $C$ at $X$. The line through $A$ perpendicular on $AC$ cuts the line perpendicular on $AB$ through $B$ at $Y$. Show that $AP \perp XY$.

Solve in positive integers the following equation; $xy(x+y-10)-3x^2-2y^2+21x+16y=60$

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 $bc - ad = 93$?