Call a set $A$ of integers [i]non-isolated[/i], if for every $a\in A$ at least one of the numbers $a-1$ and $a+1$ also belongs to $A$. Prove that the number of five-element non-isolated subsets of $\{1, 2,\ldots ,n\}$ is $(n-4)^2$.

Find all finite sets $M \subset R, |M| \ge 2$, satisfying the following condition: [i]for all $a, b \in M, a \ne b$, the number $a^3 - \frac{4}{9}b$ also belongs to $M$. [/i] (I. Voronovich)

Let $N$ be a 3-digit number with three distinct non-zero digits. We say that $N$ is [i]mediocre[/i] if it has the property that when all six 3-digit permutations of $N$ are written down, the average is $N$. For example, $N = 481$ is mediocre, since it is the average of $\{418, 481, 148, 184, 814, 841\}$. Determine the largest mediocre number.

Show that if the differential equation $$M(x,y)\, dx +N(x,y) \, dy =0$$ is both homogeneous and exact, then the solution $y=y(x)$ satisfies that $xM(x,y)+yN(x,y)$ is constant.

Each of the numbers in the set \(A = \{1,2, \cdots, 2017\}\) is colored either red or white. Prove that for \(n \geq 18\), there exists a coloring of the numbers in \(A\) such that any of its n-term arithmetic sequences contains both colors.

In square $ABCD$, $\overline{AC}$ and $\overline{BD}$ meet at point $E$. Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$. If $\overline{AF}$ meets $\overline{ED}$ at point $G$, and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$.

How many positive integers can fill the blank in the sentence below? "One positive integer is $\underline{~~~~~}$ more than twice another, and the sum of the two numbers is 28." $\textbf{(A)} ~6\qquad\textbf{(B)} ~7\qquad\textbf{(C)} ~8\qquad\textbf{(D)} ~9\qquad\textbf{(E)} ~10\qquad$

Let $M$ be the midpoint of the side $BC$ of triangle $ABC$. The bisector of the exterior angle of point $A$ intersects the side $BC$ in $D$. Let the circumcircle of triangle $ADM$ intersect the lines $AB$ and $AC$ in $E$ and $F$ respectively. If the midpoint of $EF$ is $N$, prove that $MN\parallel AD$.

Let $a$, $b$ be two distinct real numbers and let $c$ be a positive real numbers such that $a^4 - 2019a = b^4 - 2019b = c$. Prove that $- \sqrt{c} < ab < 0$.

There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$? (Boris Frenkin)

A box whose shape is a parallelepiped can be completely filled with cubes of side $1.$ If we put in it the maximum possible number of cubes, each of volume $2$, with the sides parallel to those of the box, then exactly $40$ percent of the volume of the box is occupied. Determine the possible dimensions of the box.

Side lengths $a,b,c$ of a triangle satisfy $\frac{a^3+b^3+c^3}{a+b+c}= c^2$. Find the measure of the angle opposite to side $c$.

Two circles $C_1$ and $C_2$ touch each other externally in a point $P$. At point $C_1$ there is a point $Q$ such that the tangent line in $Q$ at $C_1$ intersects the circle $C_2$ at points $A$ and $B$. The line $QP$ still intersects $C_2$ at point $C$. Prove that triangle $ABC$ is isosceles.

Find all triples of primes $(p,q,r)$ such that $p^q=2021+r^3$

If $x \ge 0$, then $\sqrt{x \sqrt{x \sqrt{x}}} = $ $ \textbf{(A)}\ x\sqrt{x}\qquad\textbf{(B)}\ x\sqrt[4]{x}\qquad\textbf{(C)}\ \sqrt[8]{x}\qquad\textbf{(D)}\ \sqrt[8]{x^{3}}\qquad\textbf{(E)}\ \sqrt[8]{x^{7}} $

Let $a, b, c$ be the side lengths of a triangle. Prove that $2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)$

How many integers $100 \le x \le 999$ have the property that, among the six digits in $\lfloor 280 + \frac{x}{100} \rfloor$ and $x$, exactly two are identical?

Find the sum of all integer bases $b>9$ for which $17_b$ is a divisor of $97_b.$

Points $ A_{1}$, $ B_{1}$, $ C_{1}$ are chosen on the sides $ BC$, $ CA$, $ AB$ of a triangle $ ABC$ respectively. The circumcircles of triangles $ AB_{1}C_{1}$, $ BC_{1}A_{1}$, $ CA_{1}B_{1}$ intersect the circumcircle of triangle $ ABC$ again at points $ A_{2}$, $ B_{2}$, $ C_{2}$ respectively ($ A_{2}\neq A, B_{2}\neq B, C_{2}\neq C$). Points $ A_{3}$, $ B_{3}$, $ C_{3}$ are symmetric to $ A_{1}$, $ B_{1}$, $ C_{1}$ with respect to the midpoints of the sides $ BC$, $ CA$, $ AB$ respectively. Prove that the triangles $ A_{2}B_{2}C_{2}$ and $ A_{3}B_{3}C_{3}$ are similar.

Let $ABC$ be an acute triangle. Find the locus of the points $M$, in the interior of $\bigtriangleup ABC$, such that $AB-FG= \frac{MF.AG+MG.BF}{CM}$, where $F$ and $G$ are the feet of the perpendiculars from $M$ to the lines $BC$ and $AC$, respectively.

Let $n\geq 3$ be integer. Given a regular $n$-polygon $P$ with side length 4 on the plane $z=0$ in the $xyz$-space.Llet $G$ be a circumcenter of $P$. When the center of the sphere $B$ with radius 1 travels round along the sides of $P$, denote by $K_n$ the solid swept by $B$. Answer the following questions. (1) Take two adjacent vertices $P_1,\ P_2$ of $P$. Let $Q$ be the intersection point between the perpendicular dawn from $G$ to $P_1P_2$, prove that $GQ>1$. (2) (i) Express the area of cross section $S(t)$ in terms of $t,\ n$ when $K_n$ is cut by the plane $z=t\ (-1\leq t\leq 1)$. (ii) Express the volume $V(n)$ of $K_n$ in terms of $n$. (3) Denote by $l$ the line which passes through $G$ and perpendicular to the plane $z=0$. Express the volume $W(n)$ of the solid by generated by a rotation of $K_n$ around $l$ in terms of $n$. (4) Find $\lim_{n\to\infty} \frac{V(n)}{W(n)} .$

An ant climbs either two inches or three inches each day. In how many ways can the ant climb twelve inches, if the order of its climbing sequence matters? $\textbf{(A) }8\qquad\textbf{(B) }9\qquad\textbf{(C) }10\qquad\textbf{(D) }12\qquad\textbf{(E) }14$

Triangle $\vartriangle ABC$ is isosceles with $AB = AC$ and $\angle ABC = 2\angle BAC$. Compute $\frac{AB}{BC}$ .

Find all non decreasing functions or non increasing function $f \colon \mathbb{R} \to \mathbb{R}$ such that for all $x,y \in \mathbb{R}$ $$ f(x+f(y))=f(x)+f(y) \text{ or } f(f(f(x)))+y$$ .

There are $n$ points in the page such that no three of them are collinear.Prove that number of triangles that vertices of them are chosen from these $n$ points and area of them is 1,is not greater than $\frac23(n^2-n)$.