There are $330$ seats in the first row of the auditorium. Some of these seats are occupied by $25$ viewers. Prove that among the pairwise distances between these viewers, there are two equal.

In the addition problem, each digit has been replaced by a letter. If different letters represent different digits then $C=$ [asy] unitsize(18); draw((-1,0)--(3,0)); draw((-3/4,1/2)--(-1/4,1/2)); draw((-1/2,1/4)--(-1/2,3/4)); label("$A$",(0.5,2.1),N); label("$B$",(1.5,2.1),N); label("$C$",(2.5,2.1),N); label("$A$",(1.5,1.1),N); label("$B$",(2.5,1.1),N); label("$A$",(2.5,0.1),N); label("$3$",(0.5,-.1),S); label("$0$",(1.5,-.1),S); label("$0$",(2.5,-.1),S); [/asy] $\text{(A)}\ 1 \qquad \text{(B)}\ 3 \qquad \text{(C)}\ 5 \qquad \text{(D)}\ 7 \qquad \text{(E)}\ 9$

Let $N$ be the set of positive integers. If $A,B,C \ne \emptyset$, $A \cap B = B \cap C = C \cap A = \emptyset$ and $A \cup B \cup C = N$, we say that $A,B,C$ are partitions of $N$. Prove that there are no partitions of $N, A,B,C$, that satisfy the following: (i) $\forall a \in A, b \in B$, we have $a + b + 1 \in C$ (ii) $\forall b \in B, c \in C$, we have $b + c + 1 \in A$ (iii) $\forall c \in C, a \in A$, we have $c + a + 1 \in B$

Consider the set of solutions of the equation $$x^2+y^3=z^2.$$ in positive integers. Is it finite or infinite? (Folklore)

Let $P$ be a point on the median $AM$ of a triangle $ABC$. Suppose that the tangents to the circumcircles of $ABP$ and $ACP$ at $B$ and $C$ respectively meet at $Q$. Show that $\angle PAB = \angle CAQ$.

As shown in the figure, $\odot O$ is the circumcircle of $\vartriangle ABC$, $\odot J$ is inscribed in $\odot O$ and is tangent to $AB$, $AC$ at points $D$ and E respectively, line segment $FG$ and $\odot O$ are tangent to point $A$, and $AF =AG=AD$, the circumscribed circle of $\vartriangle AFB$ intersects $\odot J$ at point $S$. Prove that the circumscribed circle of $\vartriangle ASG$ is tangent to $\odot J$. [img]https://cdn.artofproblemsolving.com/attachments/a/a/62d44e071ea9903ebdd68b43943ba1d93b4138.png[/img]

Let $m,n,a_1,a_2,\dots,a_n$ be positive integers and $r$ be a real number. Prove that the equation \[\lfloor a_1x\rfloor+\lfloor a_2x\rfloor+\cdots+\lfloor a_nx\rfloor=sx+r\] has exactly $ms$ solutions in $x$, where $s=a_1+a_2+\cdots+a_n+\frac1m$. [i]Linus Tang[/i]

In the figure, $ \angle EAB$ and $ \angle ABC$ are right angles. $ AB \equal{} 4, BC \equal{} 6, AE \equal{} 8$, and $ \overline{AC}$ and $ \overline{BE}$ intersect at $ D$. What is the difference between the areas of $ \triangle ADE$ and $ \triangle BDC$? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); pair A=(0,0), B=(4,0), C=(4,6), Ep=(0,8); pair D=extension(A,C,Ep,B); draw(A--C--B--A--Ep--B); label("$A$",A,SW); label("$B$",B,SE); label("$C$",C,N); label("$E$",Ep,N); label("$D$",D,2.5*N); label("$4$",midpoint(A--B),S); label("$6$",midpoint(B--C),E); label("$8$",(0,3),W);[/asy]$ \textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 9$

If $m=2+2\sqrt{44n^2+1}$ is an integer then show that it is also a perfect square. Here $n$ is a natural number.

Let $(a_1,a_2,a_3,...,a_{2011})$ be a permutation of the numbers $1,2,3,...,2011$. Show that there exist two numbers $j,k$ such that $1\leq{j}<k\leq2011$ and $|a_j-j|=|a_k-k|$

[b]9.[/b] Let $A_1, \dots , A_n$ and $B$ be ideals of an assoticative ring $R$ such that $B$ is contained in the set-union of the ideals $A_i$($i=1, \dots , n$) but not contained in the union of any $n-1$ of the ideals $A_i$. Show that, for some positive integer $k$, $B_k$ is contained in the intersection of the ideals $A_i$. [b](A. 19)[/b]

Let $f:\mathbb{R} \to \mathbb{R}$ a continuos function and $\alpha$ a real number such that $$\lim_{x\to\infty}f(x) = \lim_{x\to-\infty}f(x) = \alpha.$$ Prove that for any $r > 0,$ there exists $x,y \in \mathbb{R}$ such that $y-x = r$ and $f(x) = f(y).$

Circle $\Gamma$ has radius $10$, center $O$, and diameter $\overline{AB}$. Point $C$ lies on $\Gamma$ such that $AC = 12$. Let $P$ be the circumcenter of $\vartriangle AOC$. Line $AP$ intersects $\Gamma$ at $Q$, where $Q$ is different from $A$. Then the value of $\frac{AP}{AQ}$ can be expressed in the form $\frac{m}{n}$, where m and $n$ are relatively prime positive integers. Compute $m + n$.

If $n,p,q \in \mathbb{N}, p<q $ then $${{(p+q)n}\choose{n}} \sum \limits_{k=0}^n (-1)^k {{n}\choose{k}} {{(p+q-1)n}\choose{pn-k}}= {{(p+q)n}\choose{pn}} \sum \limits_{k=0}^{\left [\frac{n}{2} \right ]} (-1)^k {{pn}\choose{k}} {{(q-p)n}\choose{n-2k}} $$

Let $X$ be a non-empty and finite set, $A_1,...,A_k$ $k$ subsets of $X$, satisying: (1) $|A_i|\leq 3,i=1,2,...,k$ (2) Any element of $X$ is an element of at least $4$ sets among $A_1,....,A_k$. Show that one can select $[\frac{3k}{7}] $ sets from $A_1,...,A_k$ such that their union is $X$.

13. Let $\triangle T B D$ be a triangle with $T B=6, B D=8$, and $D T=7$. Let $I$ be the incenter of $\triangle T B D$, and let $T I$ intersect the circumcircle of $\triangle T B D$ at $M \neq T$. Let lines $T B$ and $M D$ intersect at $Y$, and let lines $T D$ and $M B$ intersect at $X$. Let the circumcircles of $\triangle Y B M$ and $\triangle X D M$ intersect at $Z \neq M$. If the area of $\triangle Y B Z$ is $x$ and the area of $\triangle X D Z$ is $y$, then the ratio $\frac{x}{y}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

Point $X$ is placed on segment $AB$ of a regular hexagon $ABCDEF$ such that the ratio of the area of $AXEF$ to the area of $XBCDE$ is $\frac12$. If $AB = 2018$, find $AX$.

Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived.

Let $ n > 1$ be an integer, and $ n$ can divide $ 2^{\phi(n)} \plus{} 3^{\phi(n)} \plus{} \cdots \plus{} n^{\phi(n)},$ let $ p_{1},p_{2},\cdots,p_{k}$ be all distinct prime divisors of $ n$. Show that $ \frac {1}{p_{1}} \plus{} \frac {1}{p_{2}} \plus{} \cdots \plus{} \frac {1}{p_{k}} \plus{} \frac {1}{p_{1}p_{2}\cdots p_{k}}$ is an integer. ( where $ \phi(n)$ is defined as the number of positive integers $ \leq n$ that are relatively prime to $ n$.)

Point $P$ lies inside of parallelogram $ABCD$. Perpendicular lines to $PA,PB,PC$ and $PD$ through $A,B,C$ and $D$ construct convex quadrilateral $XYZT$. Prove that $S_{XYZT}\geq 2S_{ABCD}$. [i]Proposed by Siamak Ahmadpour[/i]

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. [i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]

Let $x, y, z, t$ be real numbers such that $(x^2 + y^2 -1)(z^2 + t^2 - 1) > (xz + yt -1)^2$. Prove that $x^2 + y^2 > 1$.

Let $s(n)$ denote the number of $1$'s in the binary representation of $n$. Compute \[ \frac{1}{255}\sum_{0\leq n<16}2^n(-1)^{s(n)}. \]

Let $ f$ be a function for which $ f(x/3) \equal{} x^2 \plus{} x \plus{} 1$. Find the sum of all values of $ z$ for which $ f(3z) \equal{} 7$. $ \textbf{(A)}\ \minus{} 1/3 \qquad \textbf{(B)}\ \minus{} 1/9 \qquad \textbf{(C)}\ 0 \qquad \textbf{(D)}\ 5/9 \qquad \textbf{(E)}\ 5/3$

Prove that every $1000$-element subset $M$ of the set $\{0,1,...,2001\}$ contains either a power of two or two distinct numbers whose sum is a power of two.