The radii of the circumscribed circle and the inscribed circle of a regular $n$-gon, $n\ge 3$ are denoted by $R_n$ and $r_n$, respectively. Prove that \[\frac{r_n}{R_n}\ge\left(\frac{r_{n+1}}{R_{n+1}}\right)^2.\]

What is the last digit of $2021^{2021}$? [i]Proposed by Yifan Kang[/i]

Let $\{a_n\}_{n=1}^{\infty}$ be a sequence of positive integers such that $a_1=1$. For each $n \geq 1$, $a_{n+1}$ is the smallest positive integer, distinct from $a_1,a_2,...,a_n$, such that $\gcd(a_{n+1}a_n+1,a_i)=1$ for each $i=1,2,...,n$. Prove that every positive integer appears in $\{a_n\}_{n=1}^{\infty}$.

For a natural number $n$, denote by $s(n)$ the sum of all positive divisors of n. Prove that for every $n > 1$ the product $s(n - 1)s(n)s(n + 1)$ is even.

Through the centre of a cube (tetrahedron, icosahedron) draw a straight line in such a way that the sum of the squares of its distances from the vertices is a) minimal, b) maximal.

Let $u_1, u_2, \dots, u_{2019}$ be real numbers satisfying \[u_{1}+u_{2}+\cdots+u_{2019}=0 \quad \text { and } \quad u_{1}^{2}+u_{2}^{2}+\cdots+u_{2019}^{2}=1.\] Let $a=\min \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$ and $b=\max \left(u_{1}, u_{2}, \ldots, u_{2019}\right)$. Prove that \[ a b \leqslant-\frac{1}{2019}. \]

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

To each vertex of a given triangulation of the two-dimensional sphere, we assign a convex subset of the plane. Assume that the three convex sets corresponding to the three vertices of any two-dimensional face of the triangulation have at least one point in common. Show that there exist four vertices such that the corresponding convex sets have at least one point in common.

Evaluate $\int_0^1 \{x(1-x)\}^{\frac 32}dx.$ [i]2010 Hirosaki University School of Medicine entrance exam[/i]

Fix a positive integer $n.{}$ Consider an $n{}$-point set $S{}$ in the plane. An [i]eligible[/i] set is a non-empty set of the form $S\cap D,{}$ where $D$ is a closed disk in the plane. In terms of $n,$ determine the smallest possible number of eligible subsets $S{}$ may contain. [i]Proposed by Cristi Săvescu[/i]

Prove that in triangle $ABC$, radical center of its excircles lies on line $GI$, which $G$ is Centroid of triangle $ABC$, and $I$ is the incenter.

$n \in \mathbb {N^+}$ Prove that the following equation can be expressed as a polynomial about $n$. $$\left[2\sqrt {1}\right]+\left[2\sqrt {2}\right]+\left[2\sqrt {3}\right]+ . . . +\left[2\sqrt {n^2}\right]$$

Find the smallest integer $b$ with the following property: For each way of colouring exactly $b$ squares of an $8 \times 8$ chessboard green, one can place $7$ bishops on $7$ green squares so that no two bishops attack each other.

The decimal digits of a natural number $A$ form an increasing sequence (from left to right). Find the sum of the digits of $9A$.

A circle is inscribed in an trapezoid. Show that the diagonals of the trapezoid intersect at a point on the diameter of the circle perpendicular to the two parallel sides.

Let $n\ge3$ positive integers $a_1,\ldots,a_n$ be written on a circle so that each of them divides the sum of its two neighbors. Let us denote $$S_n=\frac{a_n+a_2}{a_1}+\frac{a_1+a_3}{a_2}+\ldots+\frac{a_{n-2}+a_n}{a_{n-1}}+\ldots+\frac{a_{n-1}+a_1}{a_n}.$$Determine the minimum and maximum values of $S_n$.

Find all integes $a,b,c,d$ that form an arithmetic progression satisfying $d-c+1$ is prime number and $a+b^2+c^3=d^2b$

Let $\mathbb R$ be the set of real numbers. We denote by $\mathcal F$ the set of all functions $f\colon\mathbb R\to\mathbb R$ such that $$f(x + f(y)) = f(x) + f(y)$$ for every $x,y\in\mathbb R$ Find all rational numbers $q$ such that for every function $f\in\mathcal F$, there exists some $z\in\mathbb R$ satisfying $f(z)=qz$.

[b]10.[/b] Given a triangle $ABC$, construct outwards over the sides $AB, BC, CA$ similiar isosceles triangles $ABC_{1}, BCA_{1}$ and $CAB_{1}$. Prove that the straight lines $AA_{1}. BB_{1}$ and $CC_{1}$ are concurrent. Is this statemente true in elliptic and hyperbolic geometry, too? [b](G. 19)[/b]

Let $\triangle ABC$ be an acute triangle with $AB =AC$ and let $D$ be a point on the side $BC$. The circle with centre $D$ passing through $C$ intersects $\odot(ABD)$ at points $P$ and $Q$, where $Q$ is the point closer to $B$. The line $BQ$ intersects $AD$ and $AC$ at points $X$ and $Y$ respectively. Prove that quadrilateral $PDXY$ is cyclic.

The natural number $A$ has three digits added to its right. The resulting number turned out to be equal to the sum of all natural numbers from $1$ to $A$. Find $A$.

A sequence of squares is made of identical square tiles. The edge of each square is one tile length longer than the edge of the previous square. The first three squares are shown. How many more tiles does the seventh square require than the sixth? [asy] path p=origin--(1,0)--(1,1)--(0,1)--cycle; draw(p); draw(shift(3,0)*p); draw(shift(3,1)*p); draw(shift(4,0)*p); draw(shift(4,1)*p); draw(shift(7,0)*p); draw(shift(7,1)*p); draw(shift(7,2)*p); draw(shift(8,0)*p); draw(shift(8,1)*p); draw(shift(8,2)*p); draw(shift(9,0)*p); draw(shift(9,1)*p); draw(shift(9,2)*p); [/asy] $ \text{(A)}\ 11\qquad\text{(B)}\ 12\qquad\text{(C)}\ 13\qquad\text{(D)}\ 14\qquad\text{(E)}\ 15 $

$N$ unfair coins (with heads and tails on the sides) are thrown, with the $k^{th}$ coin has got a chance of $\frac{1}{2k + 1}$ to land on tails.How high is the probability that an odd number of coins will show tails?

The integer sequence $(s_i)$ "having pattern 2016'" is defined as follows: $\circ$ The first member $s_1$ is 2. $\circ$ The second member $s_2$ is the least positive integer exceeding $s_1$ and having digit 0 in its decimal notation. $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 1 in its decimal notation. $\circ$ The third member $s_3$ is the least positive integer exceeding $s_2$ and having digit 6 in its decimal notation. The following members are defined in the same way. The required digits change periodically: $2 \rightarrow 0 \rightarrow 1 \rightarrow 6 \rightarrow 2 \rightarrow 0 \rightarrow \ldots$. The first members of this sequence are the following: $2; 10; 11; 16; 20; 30; 31; 36; 42; 50$.\\ Does this sequence contain a) 2001, b) 2006?

Let $w$ and $z$ be complex numbers such that $|w| = 1$ and $|z| = 10$. Let $\theta = \arg\left(\tfrac{w-z}{z}\right)$. The maximum possible value of $\tan^2 \theta$ can be written as $\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. (Note that $\arg(w)$, for $w \neq 0$, denotes the measure of the angle that the ray from $0$ to $w$ makes with the positive real axis in the complex plane.