A cent, a stuiver ($5$ cent coin), a dubbeltje ($10$ cent coin), a kwartje ($25$ cent coin), a gulden ($100$ cent coin) and a rijksdaalder ($250$ cent coin) are divided among four children in such a way that each of them receives at least one of the six coins. How many such distributions are there?

Solid camphor is insoluble in water but is soluble in vegetable oil. The best explanation for this behavior is that camphor is a(n) ${ \textbf{(A)}\ \text{Ionic solid} \qquad\textbf{(B)}\ \text{Metallic solid} \qquad\textbf{(C)}\ \text{Molecular solid} \qquad\textbf{(D)}\ \text{Network solid} } $

For which natural number $n$ is it possible to draw $n$ line segments between vertices of a regular $2n$-gon so that every vertex is an endpoint for exactly one segment and these segments have pairwise different lengths?

There are $10$ teams, named $T_1$ through $T_{10},$ participating in a draft in which there are $20$ players available, named $P_1$ through $P_{20}.$ Suppose each team independent of the others has uniform random preference on the $20$ players. Team $T_1$ will draft their favorite player, and then each subsequent team $T_2, \ldots , T_{10}$ draft their favorite player among the ones not already drafted. Each team drafts exactly one player. Given that $P_1$ is among the $10$ favorite players for each team, the probability that $P_1$ is drafted can be written as $\tfrac{m}{n}$ where $m$ and $n$ are coprime positive integers. Find $m + n.$

Find the number of positive integers less than $1000$ that can be expressed as the difference of two integral powers of $2.$

Let $ \sigma_1 ,\sigma_2 $ be two permutations of order $ n $ such that $ \sigma_1 (k)=\sigma_2 (n-k+1) $ for $ k=\overline{1,n} . $ Prove that the number of inversions of $ \sigma_1 $ plus the number of inversions of $ \sigma_2 $ is $ \frac{n(n+1)}{2} . $

Calculate $ \int_0^{2\pi }\prod_{i=1}^{2002} cos^i (it) dt. $ [i]Dorin Andrica[/i]

Five children are divided into groups and in each group they take the hand forming a wheel to dance spinning. How many different wheels those children can form, if it is valid that there are groups of $1$ to $5$ children, and can there be any number of groups?

Given a triangle $ABC$, in which the angle $B$ is three times the angle $C$. On the side $AC$, point $D$ is chosen such that the angle $BDC$ is twice the angle $C$. Prove that $BD + BA = AC$.

For each positive integer $n$ we define $f (n)$ as the product of the sum of the digits of $n$ with $n$ itself. Examples: $f (19) = (1 + 9) \times 19 = 190$, $f (97) = (9 + 7) \times 97 = 1552$. Show that there is no number $n$ with $f (n) = 19091997$.

The rooms of a building are arranged in a $m\times n$ rectangular grid (as shown below for the $5\times 6$ case). Every room is connected by an open door to each adjacent room, but the only access to or from the building is by a door in the top right room. This door is locked with an elaborate system of $mn$ keys, one of which is located in every room of the building. A person is in the bottom left room and can move from there to any adjacent room. However, as soon as the person leaves a room, all the doors of that room are instantly and automatically locked. Find, with proof, all $m$ and $n$ for which it is possible for the person to collect all the keys and escape the building. [asy] unitsize(5mm); defaultpen(linewidth(.8pt)); fontsize(25pt); for(int i=0; i<=5; ++i) { for(int j=0; j<= 6; ++j) { draw((0,i)--(9,i)); draw((1.5*j,0)--(1.5*j,5)); }} dot((.75, .5)); label("$\ast$",(8.25,4.5)); dot((11, 3)); label("$\ast$",(11,1.75)); label("room with locked external door",(18,1.9)); label("starting position",(15.3,3)); [/asy]

In rectangle $ABCD$, $AB = 40$ and $AD = 30$. Let $C' $ be the reflection of $C$ over $BD$. Find the length of $AC'$.

Let $r_A,r_B,r_C$ rays from point $P$. Define circles $w_A,w_B,w_C$ with centers $X,Y,Z$ such that $w_a$ is tangent to $r_B,r_C , w_B$ is tangent to $r_A, r_C$ and $w_C$ is tangent to $r_A,r_B$. Suppose $P$ lies inside triangle $XYZ$, and let $s_A,s_B,s_C$ be the internal tangents to circles $w_B$ and $w_C$; $w_A$ and $w_C$; $w_A$ and $w_B$ that do not contain rays $r_A,r_B,r_C$ respectively. Prove that $s_A, s_B, s_C$ concur at a point $Q$, and also that $P$ and $Q$ are isotomic conjugates. [b]PS: The rays can be lines and the problem is still true.[/b]

Let $\, P(z) \,$ be a polynomial with complex coefficients which is of degree $\, 1992 \,$ and has distinct zeros. Prove that there exist complex numbers $\, a_1, a_2, \ldots, a_{1992} \,$ such that $\, P(z) \,$ divides the polynomial \[ \left( \cdots \left( (z-a_1)^2 - a_2 \right)^2 \cdots - a_{1991} \right)^2 - a_{1992}. \]

Two circles have a common internal tangent of length $17$ and a common external tangent of length $25$. Find the product of the radii of the two circles.

Inside a rectangle is inscribed a quadrilateral, which has a vertex on each side of the rectangle. Prove that the perimeter of the inscribed quadrilateral is not smaller than double the length of a diagonal of the rectangle. (V. V . Proizvolov , Moscow)

In the arrow-shaped polygon [see figure], the angles at vertices $A$, $C$, $D$, $E$ and $F$ are right angles, $BC = FG = 5$, $CD = FE = 20$, $DE = 10$, and $AB = AG$. The area of the polygon is closest to [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,10), C=(10,5), D=(30,5), E=(30,-5), F=(10,-5), G=(10,-10); draw(A--B--C--D--E--F--G--A); label("$A$", A, W); label("$B$", B, NE); label("$C$", C, S); label("$D$", D, NE); label("$E$", E, SE); label("$F$", F, N); label("$G$", G, SE); label("$5$", (11,7.5)); label("$5$", (11,-7.5)); label("$20$", (C+D)/2, N); label("$20$", (F+E)/2, S); label("$10$", (31,0)); [/asy] $ \textbf{(A)}\ 288\qquad\textbf{(B)}\ 291\qquad\textbf{(C)}\ 294\qquad\textbf{(D)}\ 297\qquad\textbf{(E)}\ 300 $

Let $ P_1(x), P_2(x), \ldots, P_n(x)$ be real polynomials, i.e. they have real coefficients. Show that there exist real polynomials $ A_r(x),B_r(x) \quad (r \equal{} 1, 2, 3)$ such that \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_1(x) \right)^2 \plus{} \left( B_1(x) \right)^2\] \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_2(x) \right)^2 \plus{} x \left( B_2(x) \right)^2\] \[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_3(x) \right)^2 \minus{} x \left( B_3(x) \right)^2\]

A [i]tetromino[/i] consists of four squares connected along their edges. There are five possible tetromino shapes, I, O, L, T, S, shown below, which can be rotated or flipped over. Three tetrominos are used to completely cover a $3\times 4$ rectangle. At least one of the titles is an S tile. What are the other two tiles? [img]https://i.imgur.com/9Nxq4y6.png[/img] $\textbf{(A) } \text{I and L} \qquad\textbf{(B) }\text{I and T} \qquad\textbf{(C) }\text{L and L}\qquad\textbf{(D) }\text{L and S} \qquad\textbf{(E) }\text{O and T}$\\

Let $ABC$ be an acute-angled triangle with $AB<AC$, and let $O,H$ be its circumcentre and orthocentre respectively. Points $Z,Y$ lie on segments $AB,AC$ respectively, such that \[\angle ZOB=\angle YOC = 90^{\circ}.\] The perpendicular line from $H$ to line $YZ$ meets lines $BO$ and $CO$ at $Q,R$ respectively. Let the tangents to the circumcircle of $\triangle AYZ$ at points $Y$ and $Z$ meet at point $T$. Prove that $Q, R, O, T$ are concyclic. [i]Proposed by Kazi Aryan Amin and K.V. Sudharshan[/i]

A forest grows up $p$ percent during a summer, but gets reduced by $x$ units between two summers. At the beginning of this summer, the size of the forest has been $a$ units. How large should $x$ be if we want the forest to increase $q$ times in $n$ years?

In the figure, there is a circle of radius $1$ such that the segment $AG$ is diameter and that line $AF$ is perpendicular to line $DC$. There are also two squares $ABDC$ and $DEGF$, where $B$ and $E$ are points on the circle, and the points $A$, $D$ and $E$ are collinear. What is the area of square $DEGF$? [img]https://cdn.artofproblemsolving.com/attachments/1/e/794da3bc38096ef5d5daaa01d9c0f8c41a6f84.png[/img]

In triangle $ABC$ it is known that $\angle ACB = 2\angle ABC$. Furthermore $P$ is an interior point of the triangle $ABC$ such that $AP = AC$ and $PB = PC$. Prove that $\angle BAC = 3 \angle BAP$.

Let $ n>1$ be a positive integer an $ a$ a real number. Determine all real solutions $ (x_1,x_2,\dots,x_n)$ to following system of equations: $ x_1\plus{}ax_2\equal{}0$ $ x_2\plus{}a^2x_3\equal{}0$ … $ x_k\plus{}a^kx_{k\plus{}1}\equal{}0$ … $ x_n\plus{}a^nx_1\equal{}0$

Petya and Vasya live in neighboring houses (see the plan in the figure). Vasya lives in the fourth entrance. It is known that Petya runs to Vasya by the shortest route (it is not necessary walking along the sides of the cells) and it does not matter from which side he runs around his house. Determine in which entrance he lives Petya . [img]https://cdn.artofproblemsolving.com/attachments/b/1/741120341a54527b179e95680aaf1c4b98ff84.png[/img]