Suppose $f(x)$ is a polynomial with integer coefficients satisfying the condition $0 \le f(c) \le 1997$ for each $c \in \{0, 1, ..., 1998\}$. Is is true that $f(0) = f(1) = ... = f(1998)$? (variation of [url=https://artofproblemsolving.com/community/c6h49788p315649]1997 IMO Shortlist p12[/url])

An international firm has 250 employees, each of whom speaks several languages. For each pair of employees, $(A,B)$, there is a language spoken by $A$ and not $B$, and there is another language spoken by $B$ but not $A$. At least how many languages must be spoken at the firm?

\[A=3\sum_{m=1}^{n^2}(\frac12-\{\sqrt{m}\})\] where $n$ is an positive integer. Find the largest $k$ such that $n^k$ divides $[A]$.

In the puzzle below, dots are placed in some of the empty squares (one dot per square) so that each number gives the combined number of dots in its row and column. How many dots must be placed to complete the puzzle? [asy] size(5cm,0); draw((0,0)--(5,0)); draw((0,1)--(5,1)); draw((0,2)--(5,2)); draw((0,3)--(5,3)); draw((0,4)--(5,4)); draw((0,5)--(5,5)); draw((0,0)--(0,5)); draw((1,0)--(1,5)); draw((2,0)--(2,5)); draw((3,0)--(3,5)); draw((4,0)--(4,5)); draw((5,0)--(5,5)); label("$1$",(2.5,4.5)); label("$2$",(4.5,4.5)); label("$3$",(2.5,1.5)); label("$4$",(2.5,3.5)); label("$5$",(4.5,3.5)); label("$6$",(0.5,2.5)); label("$7$",(3.5,0.5)); [/asy]

Circles $\mathcal{P}$ and $\mathcal{Q}$ have radii $1$ and $4$, respectively, and are externally tangent at point $A$. Point $B$ is on $\mathcal{P}$ and point $C$ is on $\mathcal{Q}$ so that line $BC$ is a common external tangent of the two circles. A line $\ell$ through $A$ intersects $\mathcal{P}$ again at $D$ and intersects $\mathcal{Q}$ again at $E$. Points $B$ and $C$ lie on the same side of $\ell$, and the areas of $\triangle DBA$ and $\triangle ACE$ are equal. This common area is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [asy] import cse5; pathpen=black; pointpen=black; size(6cm); pair E = IP(L((-.2476,1.9689),(0.8,1.6),-3,5.5),CR((4,4),4)), D = (-.2476,1.9689); filldraw(D--(0.8,1.6)--(0,0)--cycle,gray(0.7)); filldraw(E--(0.8,1.6)--(4,0)--cycle,gray(0.7)); D(CR((0,1),1)); D(CR((4,4),4,150,390)); D(L(MP("D",D(D),N),MP("A",D((0.8,1.6)),NE),1,5.5)); D((-1.2,0)--MP("B",D((0,0)),S)--MP("C",D((4,0)),S)--(8,0)); D(MP("E",E,N)); [/asy]

Let $ABCD$ be a trapezoid (quadrilateral with one pair of parallel sides) such that $AB < CD$. Suppose that $AC$ and $BD$ meet at $E$ and $AD$ and $BC$ meet at $F$. Construct the parallelograms $AEDK$ and $BECL$. Prove that $EF$ passes through the midpoint of the segment $KL$.

Two circles $\omega_1$ and $\omega_2$ intersecting at points $X{}$ and $Y{}$ are inside the circle $\Omega$ and touch it at points $A{}$ and $B{}$, respectively; the segments $AB$ and $XY$ intersect. The line $AB$ intersects the circles $\omega_1$ and $\omega_2$ again at points $C{}$ and $D{}$, respectively. The circle inscribed in the curved triangle $CDX$ touches the side $CD$ at the point $Z{}$. Prove that $XZ$ is a bisector of $\angle AXB{}$.

$N$ teams participated in a national basketball championship in which every two teams played exactly one game. Of the $N$ teams, $251$ are from California. It turned out that a Californian team Alcatraz is the unique Californian champion (Alcatraz has won more games against Californian teams than any other team from California). However, Alcatraz ended up being the unique loser of the tournament because it lost more games than any other team in the nation! What is the smallest possible value for $N$?

Find the ordered triple of natural numbers $(x,y,z)$ such that $x \le y \le z$ and $x^x+y^y+z^z = 3382.$ [i]Proposed by Evan Fang

In trapezoid $ABCD$: $BC <AD, AB = CD, K$ is midpoint of $AD, M$ is midpoint of $CD, CH$ is height. Prove that lines $AM, CK$ and $BH$ intersect at one point.

Two given circles $k_1$ and $k_2$ intersect at points $P$ and $Q$. Construct a segment $AB$ through $P$ with the endpoints at $k_1$ and $k_2$ for which $AP \cdot PB$ is maximal.

Below is a $4\times4$ grid. We wish to fill in the grid such that each row, each column, and each $2\times2$ square outlined by the double lines contains the digits 1 through 4. The first row has already been filled in. Find, with proof, the number of ways we can complete the rest of the grid. \[ \begin{tabular}{||c|c||c|c||}\hline\hline 1 & 2 & 3 & 4\\ \hline &&&\\ \hline\hline &&&\\ \hline &&&\\ \hline\hline \end{tabular} \]

Points $A$, $B$, $C$, $D$ and $E$ are placed on the circle. In how many ways can the resulting five arcs be designated by the letters $a$, $b$, $c$, $d$ and $e$, if it is forbidden to designate an arc with the same letter as one of its ends? (For example, an arc with ends $A$ and $B$ cannot be designated by the letter $a$ or $b$.)

Circles $\omega_1$ and $\omega_2$ intersect at two points $P$ and $Q$, and their common tangent line closer to $P$ intersects $\omega_1$ and $\omega_2$ at points $A$ and $B$, respectively. The line parallel to line $AB$ that passes through $P$ intersects $\omega_1$ and $\omega_2$ for the second time at points $X$ and $Y$, respectively. Suppose $PX = 10, PY = 14,$ and $PQ = 5$. Then the area of trapezoid $XABY$ is $m\sqrt{n}$ where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. Find $m + n$.

Let $n$ be a positive integer, and let $A$ be a subset of $\{ 1,\cdots ,n\}$. An $A$-partition of $n$ into $k$ parts is a representation of n as a sum $n = a_1 + \cdots + a_k$, where the parts $a_1 , \cdots , a_k $ belong to $A$ and are not necessarily distinct. The number of different parts in such a partition is the number of (distinct) elements in the set $\{ a_1 , a_2 , \cdots , a_k \} $. We say that an $A$-partition of $n$ into $k$ parts is optimal if there is no $A$-partition of $n$ into $r$ parts with $r<k$. Prove that any optimal $A$-partition of $n$ contains at most $\sqrt[3]{6n}$ different parts.

In Wonderland, the government of each country consists of exactly $a$ men and $b$ women, where $a$ and $b$ are fixed natural numbers and $b > 1$. For improving of relationships between countries, all possible working groups consisting of exactly one government member from each country, at least $n$ among whom are women, are formed (where $n$ is a fixed non-negative integer). The same person may belong to many working groups. Find all possibilities how many countries can be in Wonderland, given that the number of all working groups is prime.

Let triangle $ABC$ be a triangle right at $B$. The inscribed circle is tangent to sides $BC,CA,AB$ at points $D,E,F$, respectively. Let $CF$ intersect the circle at the point $P$. If $\angle APB=90^{\circ}$, find the value of $\dfrac{CP+CD}{PF}$. [i](tatari/nightmare)[/i]

On the board the number 1 is written. If on the board there is n, write $ n^2 $ or $ (n+1)^2 $ or $ (n+2)^2 $. Can we arrive at a number, devisible by 2015?

Find all $3$-digit positive integers that are $34$ times the sum of their digits.

Put $S = 2^1 + 3^5 + 4^9 + 5^{13} + ... + 505^{2013} + 506^{2017}$. The last digit of $S$ is (A): $1$ (B): $3$ (C): $5$ (D): $7$ (E): None of the above.

There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a [i]committee[/i] to a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most how many commitees are there in this community? $ \textbf{(A)}\ 120 \qquad\textbf{(B)}\ 380 \qquad\textbf{(C)}\ 570 \qquad\textbf{(D)}\ 1020 \qquad\textbf{(E)}\ 1140 $

Show that in a non-equilateral triangle, the following statements are equivalent: $(a)$ The angles of the triangle are in arithmetic progression. $(b)$ The common tangent to the Nine-point circle and the Incircle is parallel to the Euler Line.

For $n \in \mathbb{N}$, consider non-negative valued functions $f$ on $\{1,2, \cdots , n\}$ satisfying $f(i) \geqslant f(j)$ for $i>j$ and $\sum_{i=1}^{n} (i+ f(i))=2023.$ Choose $n$ such that $\sum_{i=1}^{n} f(i)$ is at least. How many such functions exist in that case?

The function ƒ. which is defined for all real numbers satisfies: $$f(x+y)+f(x-y)=2f(x)+2f(y)$$ Prove that $f(0) = 0$, $f(-x) = f(x)$, $f(2x) = 4 f (x)$, $$f(x + y + z) = f(x + y) + f(y + z) + f(z + x) -f(x) - f(y) -f(z).$$

Let $ a_0$, $ a_1$, $ a_2$, $ \ldots$ be a sequence of positive integers such that the greatest common divisor of any two consecutive terms is greater than the preceding term; in symbols, $ \gcd (a_i, a_{i \plus{} 1}) > a_{i \minus{} 1}$. Prove that $ a_n\ge 2^n$ for all $ n\ge 0$. [i]Proposed by Morteza Saghafian, Iran[/i]