We are given a supply of $a \times b$ tiles with $a$ and $b$ distinct positive integers. The tiles are to be used to tile a $28 \times 48$ rectangle. Find $a, b$ such that the tile has the smallest possible area and there is only one possible tiling. (If there are two distinct tilings, one of which is a reflection of the other, then we treat that as more than one possible tiling. Similarly for other symmetries.) Find $a, b$ such that the tile has the largest possible area and there is more than one possible tiling.

Let $p,q$ be prime numbers such that $n^{3pq}-n$ is a multiple of $3pq$ for [b]all[/b] positive integers $n$. Find the least possible value of $p+q$.

Let $ \sigma = (a_1, a_2, \cdots , a_n)$ be permutation of $ (1, 2 ,\cdots, n)$. A pair $ (a_i, a_j)$ is said to correspond to an [b]inversion[/b] of $\sigma$ if $ i<j$ but $ a_i>a_j$. How many permutations of $ (1,2,\cdots,n)$, $ n \ge 3$, have exactly [b]two[/b] inversions? For example, In the permutation $(2,4,5,3,1)$, there are 6 inversions corresponding to the pairs $ (2,1),(4,3),(4,1),(5,3),(5,1),(3,1)$.

Show that for a triangle we have \[ \max \{am_a,bm_b,cm_c\} \leq sR \] where $m_a$ denotes the length of median of side $BC$ and $s$ is half of the perimeter of the triangle.

What is the radius of a circle inscribed in a rhombus with diagonals of length $ 10$ and $ 24$? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 58/13 \qquad \textbf{(C)}\ 60/13 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$

Evaluate \[\int_0^1 x^{2005}e^{-x^2}dx\]

[b]p1.[/b] Solve the system of equations $$xy = 2x + 3y$$ $$yz = 2y + 3z$$ $$zx =2z+3x$$ [b]p2.[/b] For any integer $k$ greater than $1$ and any positive integer $n$ , prove that $n^k$ is the sum of $n$ consecutive odd integers. [b]p3.[/b] Determine all pairs of real numbers, $x_1$, $x_2$ with $|x_1|\le 1$ and $|x_2|\le 1$ which satisfy the inequality: $|x^2-1|\le |x-x_1||x-x_2|$ for all $x$ such that $|x| \ge 1$. [b]p4.[/b] Find the smallest positive integer having exactly $100$ different positive divisors. (The number $1$ counts as a divisor). [b]p5.[/b] $ABC$ is an equilateral triangle of side $3$ inches. $DB = AE = 1$ in. and $F$ is the point of intersection of segments $\overline{CD}$ and $\overline{BE}$ . Prove that $\overline{AF} \perp \overline{CD}$. [img]https://cdn.artofproblemsolving.com/attachments/f/a/568732d418f2b1aa8a4e8f53366df9fbc74bdb.png[/img] PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

If the solutions of the equation $x^2+px+q=0$ are the cubes of the solutions of the equation $x^2+mx+n=0$, then $\textbf{(A) }p=m^3+3mn\qquad\textbf{(B) }p=m^3-3mn\qquad$ $\textbf{(C) }p+q=m^3\qquad\textbf{(D) }\left(\frac{m}{n}\right)^2=\frac{p}{q}\qquad \textbf{(E) }\text{none of these}$

Let $ ABC$ be an acute-angled triangle. We consider the equilateral triangle $ A'UV$, where $ A' \in (BC)$, $ U\in (AC)$ and $ V\in(AB)$ such that $ UV \parallel BC$. We define the points $ B',C'$ in the same way. Prove that $ AA'$, $ BB'$ and $ CC'$ are concurrent.

Already the complete problem: Determine all pairs $(a,b)$ of integers different from $0$ for which it is possible to find a positive integer $x$ and an integer $y$ such that $x$ is relatively prime to $b$ and in the following list there is an infinity of integers: $\rightarrow\qquad\frac{a + xy}{b}$, $\frac{a + xy^2}{b^2}$, $\frac{a + xy^3}{b^3}$, $\ldots$, $\frac{a + xy^n}{b^n}$, $\ldots$ One idea? :arrow: [b][url=http://www.mathlinks.ro/Forum/viewtopic.php?t=61319]View all the problems from XIX Mexican Mathematical Olympiad[/url][/b]

Prove that if real numbers $a,b,c,d$ satisfy $a^2+b^2+(a+b)^2=c^2+d^2+(c+d)^2$, then they also satisfy $a^4+b^4+(a+b)^4=c^4+d^4+(c+d)^4$.

Circle $\omega$ is inscribed in quadrilateral $ABCD$ and is tangent to segments $BC, AD$ at $E,F$ , respectively.$DE$ intersects $\omega$ for the second time at $X$. if the circumcircle of triangle $DFX$ is tangent to lines $AB$ and $CD$ , prove that quadrilateral $AFXC$ is cyclic.

Suppose that the measurement of time during the day is converted to the metric system so that each day has $10$ metric hours, and each metric hour has $100$ metric minutes. Digital clocks would then be produced that would read $9{:}99$ just before midnight, $0{:}00$ at midnight, $1{:}25$ at the former $3{:}00$ $\textsc{am}$, and $7{:}50$ at the former $6{:}00$ $\textsc{pm}$. After the conversion, a person who wanted to wake up at the equivalent of the former $6{:}36$ $\textsc{am}$ would have to set his new digital alarm clock for $\text{A:BC}$, where $\text{A}$, $\text{B}$, and $\text{C}$ are digits. Find $100\text{A} + 10\text{B} + \text{C}$.

Let $a$, $n$ be positive integers such that $a^n$ is a perfect number. Prove that $$a^{\frac{n}{\mu}}> \frac{\mu}{2}$$ where $\mu$ denotes the number of distinct prime divisors of $a^n$

What is the largest possible number of vertices one can have in a graph that satisfies the following conditions: each vertex is connected to exactly $3$ other vertices, and there always exists a path of length less than or equal to $2$ between any two vertices?

[b]Problem Section #3 a) Circles $O_1$ and $O_2$ interest at two points $B$ and $C$, and $BC$ is the diameter of circle $O_1$. Construct a tangent line of circle $O_1$ at $C$ and intersecting circle $O_2$ at another point $A$. Join $AB$ to intersect circle $O_1$ at point $E$, then join $CE$ and extend it to intersect circle $O_2$ at point $F$. Assume $H$ is an arbitrary point on line segment $AF$. Join $HE$ and extend it to intersect circle $O_1$ at point $G$, and then join $BG$ and extend it to intersect the extend of $AC$ at point $D$. Prove: $\frac{AH}{HF}=\frac{AC}{CD}$.