In a $100\times25$ rectangle table, fill in a positive real number in each blank. Let the number in the $i$th line, the $j$th column be $x_{i,j}(i=1,2,\cdots,100,j=1,2,\cdots,25)$ (shown in Fig.1 ). Then, we rearrange the numbers in each column: $x'_{1,j}\geq x'_{2,j}\geq\cdots\geq x'_{100,j}(j=1,2,\cdots,25)$ (shown in Fig.2 ). Find the minumum value of $k$, satisfying: As long as $\sum_{j=1}^{25}x_{i,j}\leq1$ for numbers in Fig.1 ($i=1,2,\cdots,100$), then $\sum_{j=1}^{25}x'_{i,j}\leq1$ for $i\geq k$ in Fig.2. $$\textbf{Fig.1}\\ \begin{tabular}{|c|c|c|c|} \hline $x_{1,1}$&$x_{1,2}$&$\cdots$&$x_{1,25}$\\ \hline $x_{2,1}$&$x_{2,2}$&$\cdots$&$x_{2,25}$\\ \hline $\cdots$&$\cdots$&$\cdots$&$\cdots$\\ \hline $x_{100,1}$&$x_{100,2}$&$\cdots$&$x_{100,25}$\\ \hline \end{tabular} \qquad\textbf{Fig.2}\\ \begin{tabular}{|c|c|c|c|} \hline $x'_{1,1}$&$x'_{1,2}$&$\cdots$&$x'_{1,25}$\\ \hline $x'_{2,1}$&$x'_{2,2}$&$\cdots$&$x'_{2,25}$\\ \hline $\cdots$&$\cdots$&$\cdots$&$\cdots$\\ \hline $x'_{100,1}$&$x'_{100,2}$&$\cdots$&$x'_{100,25}$\\ \hline \end{tabular}$$

How many spanning trees does the following graph (with $6$ vertices and $9$ edges) have? (A spanning tree is a subset of edges that spans all of the vertices of the original graph, but does not contain any cycles.) [img]https://cdn.artofproblemsolving.com/attachments/0/4/0e53e0fbb141b66a7b1c08696be2c5dfe68067.png[/img]

Let $n$ be a positive integer, and let $A_{n}$ be the the set of all positive integers $a\le n$ such that $n|a^{n}+1$. a) Find all $n$ such that $A_{n}\neq \emptyset$ b) Find all $n$ such that $|{A_{n}}|$ is even and non-zero. c) Is there $n$ such that $|{A_{n}}| = 130$?

Compute the number of ordered quintuples of nonnegative integers $(a_1,a_2,a_3,a_4,a_5)$ such that $0\leq a_1,a_2,a_3,a_4,a_5\leq 7$ and $5$ divides $2^{a_1}+2^{a_2}+2^{a_3}+2^{a_4}+2^{a_5}$.

George has two coins, one of which is fair and the other of which always comes up heads. Jacob takes one of them at random and flips it twice. Given that it came up heads both times, what is the probability that it is the coin that always comes up heads?

Let $ n$ be a positive integer. Given an integer coefficient polynomial $ f(x)$, define its [i]signature modulo $ n$[/i] to be the (ordered) sequence $ f(1), \ldots , f(n)$ modulo $ n$. Of the $ n^n$ such $ n$-term sequences of integers modulo $ n$, how many are the signature of some polynomial $ f(x)$ if a) $ n$ is a positive integer not divisible by the square of a prime. b) $ n$ is a positive integer not divisible by the cube of a prime.

A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?

Determine all the triples $(a, b, c)$ of positive real numbers such that the system \[ax + by -cz = 0,\]\[a \sqrt{1-x^2}+b \sqrt{1-y^2}-c \sqrt{1-z^2}=0,\] is compatible in the set of real numbers, and then find all its real solutions.

If $a,b,c$ are the sides of a triangle whose perimeter is equal to 2 then prove that: a) $abc+\frac{28}{27}\geq ab+bc+ac$; b) $abc+1<ab+bc+ac$ See also [url]http://www.mathlinks.ro/Forum/viewtopic.php?t=47939&view=next[/url] (problem 1) :)

[u]Polyhedron Hopping[/u] [b]p1.[/b] Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started? [b]p2.[/b] In terms of $k$, for $k > 0$ how likely is he to be back where he started after $2k$ minutes? [b]p3.[/b] While Travis is having fun on cubes, Sherry is hopping in the same manner on an octahedron. An octahedron has six vertices and eight regular triangular faces. After ve minutes, how likely is Sherry to be one edge away from where she started? [b]p4.[/b] In terms of $k$, for $k > 0$, how likely is it that after $k$ minutes Sherry is at the vertex opposite the vertex where she started?

Define an ordered triple $ (A, B, C)$ of sets to be minimally intersecting if $ |A \cap B| \equal{} |B \cap C| \equal{} |C \cap A| \equal{} 1$ and $ A \cap B \cap C \equal{} \emptyset$. For example, $ (\{1,2\},\{2,3\},\{1,3,4\})$ is a minimally intersecting triple. Let $ N$ be the number of minimally intersecting ordered triples of sets for which each set is a subset of $ \{1,2,3,4,5,6,7\}$. Find the remainder when $ N$ is divided by $ 1000$. [b]Note[/b]: $ |S|$ represents the number of elements in the set $ S$.

For an arbitrary prime number $p$, show that there exists infinitely many multiples of $p$ that can be expressed as the form $$\frac{x^2+y+1}{x+y^2+1}$$ Where $x, y$ are some positive integers.

Decide if exists a convex hexagon with all sides longer than $1$ and all nine of its diagonals are less than $2$ in length.

Prove that, if a pentagon (five-sided polygon) inscribed in a circle has equal angles, then its sides are equal.

We will say that a natural number is [i]acceptable[/i] if it has at most $9$ distinct prime divisors. There is a stack of $100!=1\times2\times\cdots\times100$ stones. A [i]legal move[/i] consists in removing $k$ stones from the stack, where $k$ is an acceptable number. Two players, Lucas and Nicolas, take turns making legal moves; Lucas starts the game. The one who removes the last stone wins. Determine which of the players has a winning strategy and describe this strategy.

A sequence of positive integers is given such that the sum of any $6$ consecutive terms does not exceed $11$. Prove that for any positive integer $a$ in the sequence one can find consecutive terms with sum $a$