To [i]dissect [/i] a polygon means to divide it into several regions by cutting along finitely many line segments. For example, the diagram below shows a dissection of a hexagon into two triangles and two quadrilaterals: [img]https://cdn.artofproblemsolving.com/attachments/0/a/378e477bcbcec26fc90412c3eada855ae52b45.png[/img] An [i]integer-ratio[/i] right triangle is a right triangle whose side lengths are in an integer ratio. For example, a triangle with sides $3,4,5$ is an[i] integer-ratio[/i] right triangle, and so is a triangle with sides $\frac52 \sqrt3 ,6\sqrt3, \frac{13}{2} \sqrt3$. On the other hand, the right triangle with sides$ \sqrt2 ,\sqrt5, \sqrt7$ is not an [i]integer-ratio[/i] right triangle. Determine, with proof, all integers $n$ for which it is possible to completely [i]dissect [/i] a regular $n$-sided polygon into [i]integer-ratio[/i] right triangles.

In the isosceles trapezoid with the area of $28$, a circle of radius $2$ is inscribed. Find the length of the side of the trapezoid.

(a). Given any natural number \(N\), prove that there exists a strictly increasing sequence of \(N\) positive integers in harmonic progression. (b). Prove that there cannot exist a strictly increasing infinite sequence of positive integers which is in harmonic progression.

There are a number of cards on a table. A number is written on each card. The "pick and replace" operation involves the following: two random cards are taken from the table and replaced by one new card. If the numbers $a$ and $b$ appear on the two packed cards, the number $a + b + ab$ is set on the new card. If we start with ten cards with the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ and $10$ respectively, what value(s) can the number have that "grab and replace" nine times is on the only card still on the table? Prove your answer

Let $M$ be a matrix with $r$ rows and $c$ columns. Each entry of $M$ is a nonnegative integer. Let $a$ be the average of all $rc$ entries of $M$. If $r > {(10 a + 10)}^c$, prove that $M$ has two identical rows.

Let $a, b, c$ be sides of a triangle whose perimeter does not exceed $2 \cdot \pi.$, Prove that $\sin a, \sin b, \sin c$ are sides of a triangle.

Assume that it is possible to color more than half of the surfaces of a given polyhedron so that no two colored surfaces have a common edge. (a) Describe one polyhedron with the above property. (b) Prove that one cannot inscribe a sphere touching all the surfaces of a polyhedron with the above property.

If $0<a,b<1$ and $p,q\geq 0 ,\ p+q=1$ are real numbers , then prove that: \[a^pb^q+(1-a)^p(1-b)^q\le 1\]

Twenty-one rectangles of size $3\times 1$ are placed on an $8\times 8$ chessboard, leaving only one free unit square. What position can the free square lie at?

Evaluate $\lim_{x \to +\infty}\sqrt{x^2+x+1}\frac{x-ln(e^x+x)}{x}$.

In a country with $n$ towns, all the direct flights are of double destinations (back and forth). There are $r>2014$ rootes between different pairs of towns, that include no more than one intermediate stop (direction of each root matters). Find the minimum possible value of $n$ and the minimum possible $r$ for that value of $n$.

In the coordinate plane the points with both coordinates being rational numbers are called rational points. For any positive integer $n$, is there a way to use $n$ colours to colour all rational points, every point is coloured one colour, such that any line segment with both endpoints being rational points contains the rational points of every colour?

Call a finite set of positive integers [i]independent [/i] if its elements are pairwise coprime, and [i]nice [/i] if the arithmetic mean of the elements of every non-empty subset of it is an integer. a) Prove that for any positive integer $n$ there is an $n$-element set of positive integers which is both independent and nice. b) Is there an infinite set of positive integers whose every independent subset is nice and which has an $n$-element independent subset for every positive integer $n$?

Let $f:\mathbb{R} \to \mathbb{R}$ be a monotonic function and $a,b,c,d$ be real numbers with $a$ and $c$ nonzero. Prove that if the equalities [center]$\int\limits_x^{x+\sqrt{3}} f(t) \mathrm{d}t=ax+b$ and $\int\limits_x^{x+\sqrt{2}} f(t) \mathrm{d}t=cx+d$[/center] hold for every real number $x,$ then $f$ is a polynomial function of degree one.

Two different numbers are randomly selected from the set ${ - 2, -1, 0, 3, 4, 5}$ and multiplied together. What is the probability that the product is $0$? $\textbf{(A) }\dfrac{1}{6}\qquad\textbf{(B) }\dfrac{1}{5}\qquad\textbf{(C) }\dfrac{1}{4}\qquad\textbf{(D) }\dfrac{1}{3}\qquad \textbf{(E) }\dfrac{1}{2}$

the positive divisors $d_1,d_2,\cdots,d_k$ of a positive integer $n$ are ordered \[1=d_1<d_2<\cdots<d_k=n\] Suppose $d_7^2+d_{15}^2=d_{16}^2$. Find all possible values of $d_{17}$.

Each side of a triangle $ABC$ is divided into three equal parts, and the middle segment in each of the sides is painted green. In the exterior of $\triangle ABC$ three equilateral triangles are constructed, in such a way that the three green segments are sides of these triangles. Denote by $A',B',C'$ the vertices of these new equilateral triangles that don’t belong to the edges of $\triangle ABC$, respectively. Let $A'',B'',C''$ be the points symmetric to $A',B',C'$ with respect to $BC,CA,AB$. (a) Prove that $\triangle A'B'C'$ and $\triangle A''B''C''$ are equilateral. (b) Prove that $ABC,A'B'C'$, and $A''B''C''$ have a common centroid.

Find all nonnegative integers $m$ such that \[a_m=(2^{2m+1})^2+1 \] is divisible by at most two different primes.

Let $D$ be a point inside an acute triangle $ABC$, such that $\angle BAD = \angle DBC$ and $\angle DAC = \angle BCD$. Let $P$ be a point on the circumcircle of the triangle $ADB$. Suppose $P$ are itself outside the triangle $ABC$. A line through $P$ intersects the ray $BA$ in $X$ and ray $CA$ in $Y$, so that $\angle XPB = \angle PDB$. Show that $BY$ and $CX$ intersect on $AD$.

Let $n$ be an integer and $p$ a prime greater than 2. Show that: $$(p-1)^nn!|(p^n-1)(p^n-p)(p^n-p^2)\cdots(p^n-p^{n-1}).$$

Let $A, B, C, D$ be four different points on a line $\ell$, such that $AB = BC = CD$. In one of the semiplanes determined by the line $\ell$, the points $P$ and $Q$ are chosen in such a way that the triangle $CPQ$ is equilateral with its vertices named clockwise. Let $M$ and $N$ be two points on the plane such that the triangles $MAP$ and $NQD$ are equilateral (the vertices are also named clockwise). Find the measure of the angle $\angle MBN$.

A clock has an hour, minute, and second hand, all of length $1$. Let $T$ be the triangle formed by the ends of these hands. A time of day is chosen uniformly at random. What is the expected value of the area of $T$? [i]Proposed by Dylan Toh[/i]

Let $a,b,c,d, e$ be real numbers such that they simultaneously satisfy the following equations $$a+b+c+d+e=8$$ $$a^2+b^2+c^2+d^2+e^2=16$$ Determine the smallest and largest value that $a$ can take.

A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$, in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

We will say that a set of real numbers $A = (a_1,... , a_{17})$ is stronger than the set of real numbers $B = (b_1, . . . , b_{17})$, and write $A >B$ if among all inequalities $a_i > b_j$ the number of true inequalities is at least $3$ times greater than the number of false. Prove that there is no chain of sets $A_1, A_2, . . . , A_N$ such that $A_1>A_2> \cdots A_N>A_1$. Remark: For 11.4, the constant $3$ is changed to $2$ and $N=3$ and $17$ is changed to $m$ and $n$ in the definition (the number of elements don't have to be equal).