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?

Let $ABC$ be a triangle with $AB=20, AC=21,$ and $\angle BAC = 90^{\circ}.$ Suppose $\Gamma_1$ is the unique circle centered at $B$ and passing through $A,$ and $\Gamma_2$ is the unique circle centered at $C$ and passing through $A.$ Points $E$ and $F$ are selected on $\Gamma_1$ and $\Gamma_2,$ respectively, such that $E, A, F$ are collinear in that order. The tangent to $\Gamma_1$ at $E$ and the tangent to $\Gamma_2$ at $F$ intersect at $P$. Given that $PA \bot BC$, compute the area of $PBC$.

Find the number of positive integers for which the product of digits and the sum of digits are the same and equal to $8$.

In natural numbers $m,n$ Solve : $n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4$

A football is covered by some polygonal pieces of leather which are sewed up by three different colors threads. It features as follows: i) any edge of a polygonal piece of leather is sewed up with an equal-length edge of another polygonal piece of leather by a certain color thread; ii) each node on the ball is vertex to exactly three polygons, and the three threads joint at the node are of different colors. Show that we can assign to each node on the ball a complex number (not equal to $1$), such that the product of the numbers assigned to the vertices of any polygonal face is equal to $1$.

Angle $A$ of triangle $ABC$ is $33^o$. A straight line passing through $A$ perpendicular to $AC$ intersects straight line $BC$ at point $D$ so that $CD = 2AB$. What is angle $C$ of triangle $ABC$? (Please list all options.)

A function $g: \mathbb{Z} \to \mathbb{Z}$ is called adjective if $g(m)+g(n)>max(m^2,n^2)$ for any pair of integers $m$ and $n$. Let $f$ be an adjective function such that the value of $f(1)+f(2)+\dots+f(30)$ is minimized. Find the smallest possible value of $f(25)$.

We say that a prism is [i]binary[/i] if there exists a labelling of the vertices of the prism with integers from the set $\{-1,1\}$ such that the product of the numbers assigned to the vertices of each face (base or lateral face) is equal to $-1$. a) Prove that any [i]binary[/i] prism has the number of total vertices divisible by 8; b) Prove that any prism with 2000 vertices is [i]binary[/i].

A regular tetrahedral massless frame whose side length is physically variable (with the constraint of the tetrahedron being regular) is dipped in a soap solution of surface tension $T$, taken outside and allowed to settle after a little wiggle.\\ The soap film is formed such that there is no volume in space that is enclosed by any of the surfaces soap film and all the soap film surfaces are planar. You may assume the configuration of the soap film without proof.\\ Now 4 point charges of charge $q$ are fixed at the vertices of the tetrahedron.\\ The system now sets into motion with the shape and nature of soap film being unaltered at all times.\\ [list] [*] Find the side length of the tetrahedron for which the system attains mechanical equilibrium. [/*] [*] Find the differential equation(s) governing the side length with respect to time.[/*] [*] If the amplitude of oscillations are very small, find the time period of oscillations.[/*] [/list]

A closed pentagonal line is inscribed in a sphere of the diameter $1$, and has all edges of length $\ell$. Prove that $\ell \le \sin \frac{2\pi}{5}$ .

Find all non-negative integers $a, b, c$ such that the roots of equations: $\begin{cases}x^2 - 2ax + b = 0 \\ x^2- 2bx + c = 0 \\ x^2 - 2cx + a = 0 \end{cases}$ are non-negative integers.

Let $a$ and $b$ be integers and $p$ a prime. For each positive integer k, define$ A_k = \{n \in Z^+ |p^k$ divides $a^n - b^n\}$. Show that if $A_1$ is nonempty then $A_k$ is nonempty for all positive integers $k$

Let $\mathbb{Q}$ denote the set of rational numbers. A nonempty subset $S$ of $\mathbb{Q}$ has the following properties: (a) $0$ is not in $S$; (b) for each $s_1,s_2$ in $S$, the rational number $s_1/s_2$ is in $S$; (c) there exists a nonzero number $q\in \mathbb{Q} \backslash S$ that has the property that every nonzero number in $\mathbb{Q} \backslash S$ is of the form $qs$ for some $s$ in $S$. Prove that if $x$ belongs to $S$, then there exists elements $y,z$ in $S$ such that $x=y+z$.

Consider the matrices $$A = \left(\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}\right) \\ \mbox{ and } \\ B = \left(\begin{matrix} 1 & 0 \\ 2 & 1 \end{matrix}\right).$$ Let $k\geq 1$ an integer. Prove that for any nonzero $i_1,i_2,\dots,i_{k-1},j_1,j_2,\dots,j_k$ and any integers $i_0,i_k$ it holds that $$A^{i_0}B^{j_1}A^{i_1}B^{j_2}\cdots A^{i_{k-1}}B^{i_k}A^{i_k} \not = I.$$

A team of four students goes to LMT, and each student brings a lunch. However, on the bus, the students’ lunches get mixed up, and during lunch time, each student chooses a random lunch to eat (no two students may eat the same lunch). What is the probability that each student chooses his or her own lunch correctly?

Find all positive integer bases $b \ge 9$ so that the number \[ \frac{{\overbrace{11 \cdots 1}^{n-1 \ 1's}0\overbrace{77 \cdots 7}^{n-1\ 7's}8\overbrace{11 \cdots 1}^{n \ 1's}}_b}{3} \] is a perfect cube in base 10 for all sufficiently large positive integers $n$. [i]Proposed by Yang Liu[/i]

Find the sum of $$\frac{\sigma(n) \cdot d(n)}{ \phi (n)}$$ over all positive $n$ that divide $ 60$. Note: The function $d(i)$ outputs the number of divisors of $i$, $\sigma (i)$ outputs the sum of the factors of $i$, and $\phi (i)$ outputs the number of positive integers less than or equal to $i$ that are relatively prime to $i$.

An $n\times n$ board is coloured in $n$ colours such that the main diagonal (from top-left to bottom-right) is coloured in the first colour; the two adjacent diagonals are coloured in the second colour; the two next diagonals (one from above and one from below) are coloured in the third colour, etc; the two corners (top-right and bottom-left) are coloured in the $n$-th colour. It happens that it is possible to place on the board $n$ rooks, no two attacking each other and such that no two rooks stand on cells of the same colour. Prove that $n=0\pmod{4}$ or $n=1\pmod{4}$.

Points $M,N,P,Q$ are taken on the sides $AB,BC,CD,DA$ respectively of a square $ABCD$ such that $AM=BN=CP=DQ=\frac1nAB$. Find the ratio of the area of the square determined by the lines $MN,NP,PQ,QM$ to the ratio of $ABCD$.

The traffic on a certain east-west highway moves at a constant speed of 60 miles per hour in both directions. An eastbound driver passes 20 west-bound vehicles in a five-minute interval. Assume vehicles in the westbound lane are equally spaced. Which of the following is closest to the number of westbound vehicles present in a 100-mile section of highway? $\text{(A)} \ 100 \qquad \text{(B)} \ 120 \qquad \text{(C)} \ 200 \qquad \text{(D)} \ 240 \qquad \text{(E)} \ 400$

What is the least number of weighings needed to determine the sum of weights of $13$ watermelons such that exactly two watermelons should be weighed in each weigh? $ \textbf{a)}\ 7 \qquad\textbf{b)}\ 8 \qquad\textbf{c)}\ 9 \qquad\textbf{d)}\ 10 \qquad\textbf{e)}\ 11 $

Prove that in a topological space X , if all discrete subspaces have compact closure , then X is compact.

Find all triples $(a,b, c)$ of real numbers for which there exists a non-zero function $f: R \to R$, such that $$af(xy + f(z)) + bf(yz + f(x)) + cf(zx + f(y)) = 0$$ for all real $x, y, z$. (E. Barabanov)

Let $ X,Y,Z$ be the midpoints of the small arcs $ BC,CA,AB$ respectively (arcs of the circumcircle of $ ABC$). $ M$ is an arbitrary point on $ BC$, and the parallels through $ M$ to the internal bisectors of $ \angle B,\angle C$ cut the external bisectors of $ \angle C,\angle B$ in $ N,P$ respectively. Show that $ XM,YN,ZP$ concur.

Four congruent rectangles are placed as shown. The area of the outer square is $ 4$ times that of the inner square. What is the ratio of the length of the longer side of each rectangle to the length of its shorter side? [asy]unitsize(6mm); defaultpen(linewidth(.8pt)); path p=(1,1)--(-2,1)--(-2,2)--(1,2); draw(p); draw(rotate(90)*p); draw(rotate(180)*p); draw(rotate(270)*p);[/asy]$ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ \sqrt {10} \qquad \textbf{(C)}\ 2 \plus{} \sqrt2 \qquad \textbf{(D)}\ 2\sqrt3 \qquad \textbf{(E)}\ 4$