If you increase a number $X$ by $20\%,$ you get $Y.$ By what percent must you decrease $Y$ to get $X?$

Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$? $\textbf{(A) } 7 \qquad\textbf{(B) } 9 \qquad\textbf{(C) } 10 \qquad\textbf{(D) } 11 \qquad\textbf{(E) } 12$

What is the smallest number of $3$-cell corners needed to be painted in a $6\times 6$ square so that it was impossible to paint more than one corner of it? (The painted corners should not overlap.)

Let F be the midpoint of side BC or triangle ABC. Construct isosceles right triangles ABD and ACE externally on sides AB and AC with the right angles at D and E respectively. Show that DEF is an isosceles right triangle.

Let be a natural number $ n\ge 2, $ a group $ G $ and two elements of it $ e_1,e_2 $ such that $ e_2e_1x=xe_2e_1, $ for any element $ x $ of $ G. $ Prove that $ \left( e_1xe_2 \right)^n =e_1x^ne_2, $ for any element $ x $ of $ G, $ if and only if $ e_2e_1=\left( e_2e_1\right)^n. $ [i]Ion Bursuc[/i]

The ring shown below is made out of 18 congruent regular hexagons. How many ways are there to tile the ring using tiles that consist of two hexagons, each congruent to any one of the 18 in the design, joined edge-to-edge? (The central hexagon, in black, is not to be covered with a tile and the ring cannot be rotated or reflected.)

Let $ABC$ be a triangle and $\ell_1,\ell_2$ be two parallel lines. Let $\ell_i$ intersects line $BC,CA,AB$ at $X_i,Y_i,Z_i$, respectively. Let $\Delta_i$ be the triangle formed by the line passed through $X_i$ and perpendicular to $BC$, the line passed through $Y_i$ and perpendicular to $CA$, and the line passed through $Z_i$ and perpendicular to $AB$. Prove that the circumcircles of $\Delta_1$ and $\Delta_2$ are tangent.

Sarah has five rings (numbered 1 through 5), each with ten rungs labeled $1$ through $10$. Rung $i$ is adjacent to rung $i+1$ for $1 \le i \le 9$, and rung $10$ is adjacent to rung $1$. How many ways can Sarah paint some (possibly none) of the rungs red such that in each ring, the red rungs form a contiguous block, and the total number of red rungs across the five rings is divisible by $11$? (For example, Sarah can paint rungs $8, 9, 10, 1, 2$ on ring $1$, rungs $3, 4, 5$ on ring $2$, no rungs on rings $3$ and $4$, and rungs $1,2,3$ on ring $5$.)

Let $ABC$ be a right-angled triangle ($\angle B = 90^\circ$). The excircle inscribed into the angle $A$ touches the extensions of the sides $AB$, $AC$ at points $A_1, A_2$ respectively; points $C_1, C_2$ are defined similarly. Prove that the perpendiculars from $A, B, C$ to $C_1C_2, A_1C_1, A_1A_2$ respectively, concur.

Consider a $m\times n$ rectangular board consisting of $mn$ unit squares. Two of its unit squares are called [i]adjacent[/i] if they have a common edge, and a [i]path[/i] is a sequence of unit squares in which any two consecutive squares are adjacent. Two parths are called [i]non-intersecting[/i] if they don't share any common squares. Each unit square of the rectangular board can be colored black or white. We speak of a [i]coloring[/i] of the board if all its $mn$ unit squares are colored. Let $N$ be the number of colorings of the board such that there exists at least one black path from the left edge of the board to its right edge. Let $M$ be the number of colorings of the board for which there exist at least two non-intersecting black paths from the left edge of the board to its right edge. Prove that $N^{2}\geq M\cdot 2^{mn}$.

There are $100$ white points on a circle. Asya and Borya play the following game: they alternate, starting with Asya, coloring a white point in green or blue. Asya wants to obtain as much as possible pairs of adjacent points of distinct colors, while Borya wants these pairs to be as less as possible. What is the maximal number of such pairs Asya can guarantee to obtain, no matter how Borya plays.

Jack and Jill run $10$ kilometers. They start at the same point, run $5$ kilometers up a hill, and return to the starting point by the same route. Jack has a $10$ minute head start and runs at the rate of $15$ km/hr uphill and $20$ km/hr downhill. Jill runs $16$ km/hr uphill and $22$ km/hr downhill. How far from the top of the hill are they when they pass going in opposite directions? $ \textbf{(A)}\ \frac{5}{4}\ km\qquad\textbf{(B)}\ \frac{35}{27}\ km\qquad\textbf{(C)}\ \frac{27}{20}\ km\qquad\textbf{(D)}\ \frac{7}{3}\ km\qquad\textbf{(E)}\ \frac{28}{9}\ km $

A circle touches the extensions of sides $CA$ and $CB$ of triangle $ABC$, and also touches side $AB$ of this triangle at point $P$. Prove that the radius of the circle tangent to segments $AP$, $CP$ and the circumscribed circle of this triangle is equal to the radius of the inscribed circle in this triangle.

A tetrahedron is given. Determine whether it is possible to put some 10 consecutive positive integers at 4 vertices and at 6 midpoints of the edges so that the number at the midpoint of each edge is equal to the arithmetic mean of two numbers at the endpoints of this edge.

Let $a_n$ be the number of such numbers $N$: sum of all digits of $N$ is $n$, and each digit can only be $1,3,4$. Prove that $a_{2n}$ is a perfect square for all $n\in\mathbb{Z}_+$.

The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$

Prove that if all dihedral angles of a tetrahedron are acute, then all its faces are acute-angled triangles.

Let $S_1$, $S_2$, $\dots$, $S_{125}$ be 125 sets of 5 numbers each, comprising $625$ distinct numbers. Let $m_i$ be the median of $S_i$. Let $M$ be the median of $m_1$, $m_2$, $\dots$, $m_{125}$. What is the greatest possible number of the 625 numbers that are less than $M$?

Let $ABC$ be an acute triangle with $AT, AS$ respectively are the internal, external angle bisector of $ABC$ and $T, S \in BC$. On the circle with diameter $TS$, take an arbitrary point $P$ that lies inside the triangle ABC. Denote $D, E, F, I$ as the incenter of triangle $PBC, PCA, PAB, ABC$. Prove that four lines $AD, BE, CF$ and $IP$ are concurrent.

On the legs $AC$ and $BC$ of an isosceles right-angled triangle with a right angle $C$, points $D$ and $E$ are taken, respectively, so that $CD = CE$. Perpendiculars on line $AE$ from points $C$ and $D$ intersect segment $AB$ at points $P$ and $Q$, respectively. Prove that $BP = PQ$.

Prove that if $ n $ is a natural number and the angle $ \alpha $ is not a multiple of $ \frac{180^{\circ}}{2^n} $, then $$\frac{1}{\sin 2\alpha} + \frac{1}{\sin 4\alpha} + \frac{1}{\sin 8\alpha} + ... + = ctg \alpha - ctg 2^n \alpha.$$

Of all the fractions $\frac{x}{y}$ that satisfy $$\frac{41}{2010}<\frac{x}{y}<\frac{1}{49}$$ find the one with the smallest denominator.

Let $A$ and $B$ be points with $AB=12$. A point $P$ in the plane of $A$ and $B$ is $\textit{special}$ if there exist points $X, Y$ such that [list] [*]$P$ lies on segment $XY$, [*]$PX : PY = 4 : 7$, and [*]the circumcircles of $AXY$ and $BXY$ are both tangent to line $AB$. [/list] A point $P$ that is not special is called $\textit{boring}$. Compute the smallest integer $n$ such that any two boring points have distance less than $\sqrt{n/10}$ from each other. [i]Proposed by Michael Ren[/i]

A [i]normal magic square[/i] of order $n$ is an arrangement of the integers from $1$ to $n^2$ in a square such that the $n$ numbers in each row, each column, and each of the two diagonals sum to a constant, called the [i]magic sum[/i] of the magic square. Compute the magic sum of a normal magic square of order $8$.

In triangle $ABC$, point $A_1$ lies on side $BC$ and point $B_1$ lies on side $AC$. Let $P$ and $Q$ be points on segments $AA_1$ and $BB_1$, respectively, such that $PQ$ is parallel to $AB$. Let $P_1$ be a point on line $PB_1$, such that $B_1$ lies strictly between $P$ and $P_1$, and $\angle PP_1C=\angle BAC$. Similarly, let $Q_1$ be the point on line $QA_1$, such that $A_1$ lies strictly between $Q$ and $Q_1$, and $\angle CQ_1Q=\angle CBA$. Prove that points $P,Q,P_1$, and $Q_1$ are concyclic. [i]Proposed by Anton Trygub, Ukraine[/i]