A sit of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one? $ \text{(A)}\ 10 \qquad \text{(B)}\ 11 \qquad \text{(C)}\ 18 \qquad \text{(D)}\ 19 \qquad \text{(E)}\ 20$

Let $p$ be an odd prime number. For every integer $a,$ define the number $S_a = \sum^{p-1}_{j=1} \frac{a^j}{j}.$ Let $m,n \in \mathbb{Z},$ such that $S_3 + S_4 - 3S_2 = \frac{m}{n}.$ Prove that $p$ divides $m.$ [i]Proposed by Romeo Meštrović, Montenegro[/i]

Writing down the first $4$ rows of the Pascal triangle in the usual way and then adding up the numbers in vertical columns, we obtain $7$ numbers as shown above. If we repeat this procedure with the first $1024$ rows of the Pascal triangle, how many of the $2047$ numbers thus obtained will be odd? [img]https://cdn.artofproblemsolving.com/attachments/8/a/4dc4a815d8b002c9f36a6da7ad6e1c11a848e9.png[/img]

Given a positive integer $n > 1$. Find the smallest integer of the form $\frac{n^a-n^b}{n^c-n^d}$ for all positive integers $a,b,c,d$.

Prove that the volume $V$ and the lateral area $S$ of a right circular cone satisfy the inequality \[\left( \frac{6V}{\pi}\right)^2 \leq \left( \frac{2S}{\pi \sqrt 3}\right)^3\] When does equality occur?

Three cevians divided the triangle into six triangles, the areas of which are marked in the figure. 1) Prove that $S_1 \cdot S_2 \cdot S_3 =Q_1 \cdot Q_2 \cdot Q_3$. 2) Determine whether it is true that if $S_1 = S_2 = S_3$, then $Q_1 = Q_2 = Q_3$. [img]https://cdn.artofproblemsolving.com/attachments/c/d/3e847223b24f783551373e612283e10e477e62.png[/img]

The integer 287 is exactly divisible by $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 6$

Let $ABC$ be a nonisosceles triangle. Point $O$ is its circumcenter, and point $K$ is the center of the circumcircle $w$ of triangle $BCO$. The altitude of $ABC$ from $A$ meets $w$ at a point $P$. The line $PK$ intersects the circumcircle of $ABC$ at points $E$ and $F$. Prove that one of the segments $EP$ and $FP$ is equal to the segment $PA$.

Determine $ x,y,z>0$ for which $ x^3y\plus{}3<\equal{}4z, y^3z\plus{}3<\equal{}4x,z^3x\plus{}3<\equal{}4y.$

Find all real numbers of the form $\sqrt[p]{2021+\sqrt[q]{a}}$ that can be expressed as a linear combination of roots of unity with rational coefficients, where $p$ and $q$ are (possible the same) prime numbers, and $a>1$ is an integer, which is not a $q$-th power.

For a triangle $\triangle ABC$ with an obtuse angle $\angle A$ , let $E , F$ be feet of altitudes from $B , C$ on sides $AC , AB$ respectively. The tangents from $B , C$ to circumcircle of triangle $\triangle ABC$ intersect line $EF$ at points $K , L$ respectively and we know that $\angle CLB=135$. Point $R$ lies on segment $BK$ in such a way that $KR=KL$ and let $S$ be a point on line $BK$ such that $K$ is between $B , S$ and $\angle BLS=135$. Prove that the circle with diameter $RS$ is tangent to circumcircle of triangle $\triangle ABC$. [i]Proposed by Mehran Talaei[/i]

For any odd prime $p$ and any integer $n,$ let $d_p (n) \in \{ 0,1, \dots, p-1 \}$ denote the remainder when $n$ is divided by $p.$ We say that $(a_0, a_1, a_2, \dots)$ is a [i]p-sequence[/i], if $a_0$ is a positive integer coprime to $p,$ and $a_{n+1} =a_n + d_p (a_n)$ for $n \geqslant 0.$ (a) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_n >b_n$ for infinitely many $n,$ and $b_n > a_n$ for infinitely many $n?$ (b) Do there exist infinitely many primes $p$ for which there exist $p$-sequences $(a_0, a_1, a_2, \dots)$ and $(b_0, b_1, b_2, \dots)$ such that $a_0 <b_0,$ but $a_n >b_n$ for all $n \geqslant 1?$ [I]United Kingdom[/i]

Let $ n \geq 3$ be a positive integer and let $ m \geq 2^{n\minus{}1}\plus{}1$. Prove that for each family of nonzero distinct subsets $ (A_j)_{j \in \overline{1, m}}$ of $ \{1, 2, ..., n\}$ there exist $ i$, $ j$, $ k$ such that $ A_i \cup A_j \equal{} A_k$.

2 Points $ P\left(a,\ \frac{1}{a}\right),\ Q\left(2a,\ \frac{1}{2a}\right)\ (a > 0)$ are on the curve $ C: y \equal{}\frac{1}{x}$. Let $ l,\ m$ be the tangent lines at $ P,\ Q$ respectively. Find the area of the figure surrounded by $ l,\ m$ and $ C$.

The $1 \times 1$ cells located around the perimeter of a $4 \times 4$ square are filled with the numbers $1, 2, \ldots, 12$ so that the sums along each of the four sides are equal. In the upper left corner cell is the number $1$, in the upper right - the number $5$, and in the lower right - the number $11$. [img]https://i.ibb.co/PM0ry1D/Kyiv-City-MO-2021-Round-1-10-2.png[/img] Under these conditions, what number can be located in the last corner cell? [i]Proposed by Mariia Rozhkova[/i]

Prove that the points symmetric to the vertices of a triangle with respect to the opposite side are collinear if and only if the distance from the orthocenter to the circumcenter is twice the circumradius.

Let $D_n$ be the set of divisors of $n$. Find all natural $n$ such that it is possible to split $D_n$ into two disjoint sets $A$ and $G$, both containing at least three elements each, such that the elements in $A$ form an arithmetic progression while the elements in $G$ form a geometric progression.

A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac13$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.

Two circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$. Line $\ell$ is tangent to $\omega_1$ at $P$ and to $\omega_2$ at $Q$ so that $A$ is closer to $\ell$ than $B$. Let $X$ and $Y$ be points on major arcs $\overarc{PA}$ (on $\omega_1$) and $AQ$ (on $\omega_2$), respectively, such that $AX/PX = AY/QY = c$. Extend segments $PA$ and $QA$ through $A$ to $R$ and $S$, respectively, such that $AR = AS = c\cdot PQ$. Given that the circumcenter of triangle $ARS$ lies on line $XY$, prove that $\angle XPA = \angle AQY$.

Prove that among any $n$ positive integers one can always find some (at least one) whose sum is divisible by $n$.

Let $ n$ be a positive integer. Determine the number of triangles (non congruent) with integral side lengths and the longest side length is $ n$.

The positive integers \( a \) and \( b \) are coprime and such that there exist positive integers \( m_2 \) and \( m_5 \) for which \( am_2 + b \) is a perfect square of a positive integer, and \( am_5 + b \) is a perfect fifth power of a positive integer. Does there always exist a positive integer \( n \) for which \( an + b \) is a perfect \( k \)-th power of a positive integer, if: a) \( k = 7 \); b) \( k = 10 \)?

A straight line tangent to a circle circumscribed about an isosceles triangle $ABC$ ($AB = AC$) at point $B$ intersects straight line $AC$ at point $P$, $E$ is the midpoint of $AB$ (fig.). What is the projection of $DE$ onto $AB$ if $PA = a$? [img]https://cdn.artofproblemsolving.com/attachments/e/3/59c67e8f5eb3d399656d86613bc699c8baf1c1.png[/img]

Let $n$ be a positive integer. (1) For a positive integer $k$ such that $1\leq k\leq n$, Show that : \[\int_{\frac{k-1}{2n}\pi}^{\frac{k}{2n}\pi} \sin 2nt\cos t\ dt=(-1)^{k+1}\frac{2n}{4n^2-1}(\cos \frac{k}{2n}\pi +\cos \frac{k-1}{2n}\pi).\] (2) Find the area $S_n$ of the part expressed by a parameterized curve $C_n: x=\sin t,\ y=\sin 2nt\ (0\leq t\leq \pi).$ If necessary, you may use ${\sum_{k=1}^{n-1} \cos \frac{k}{2n}\pi =\frac 12(\frac{1}{\tan \frac{\pi}{4n}}-1})\ (n\geq 2).$ (3) Find $\lim_{n\to\infty} S_n.$

Problems $15, 16$, and $17$ all refer to the following: In the very center of the Irenic Sea lie the beautiful Nisos Isles. In $1998$ the number of people on these islands is only 200, but the population triples every $25$ years. Queen Irene has decreed that there must be at least $1.5$ square miles for every person living in the Isles. The total area of the Nisos Isles is $24,900$ square miles. 15. Estimate the population of Nisos in the year $2050$. $ \text{(A)}\ 600\qquad\text{(B)}\ 800\qquad\text{(C)}\ 1000\qquad\text{(D)}\ 2000\qquad\text{(E)}\ 3000 $