Consider a random string $s$ of $10^{2020}$ base-ten digits (there can be leading zeroes). We say a substring $s' $ (which has no leading zeroes) is self-locating if $s' $ appears in $s$ at index $s' $ where the string is indexed at $ 1$. For example the substring $11$ in the string “$122352242411$” is selflocating since the $11$th digit is $ 1$ and the $12$th digit is $ 1$. Let the expected number of self-locating substrings in s be $G$. Compute $\lfloor G \rfloor$.

Let $A$ be a subset of $\{2,3, \ldots, 28 \}$ such that if $a \in A$, then the residue obtained when we divide $a^2$ by $29$ also belongs to $A$. Find the minimum possible value of $|A|$.

Let $n$ be a positive integer. Determine the smallest positive integer $k$ with the following property: it is possible to mark $k$ cells on a $2n \times 2n$ board so that there exists a unique partition of the board into $1 \times 2$ and $2 \times 1$ dominoes, none of which contain two marked cells.

Prove that $5^n$ has a block of $1976$ consecutive $0's$ in its decimal representation.

As shown in figure , a circle of radius $1$ passes through vertex $A$ of $\vartriangle ABC$ and is tangent to the side $BC$ at the point $D$ , intersect sides $AB$ and $AC$ at points $E$ and $F$ respectively . Also$ EF$ bisects $\angle AFD$, and $\angle ADC = 80^o$ , Is there a triangle that satisfies the condition, so that $\frac{AB+BC+CA}{AD^2}$ is an irrational number, and the irrational number is the root of a quadratic equation with integral coefficients? If it does not exist, please prove it; if it exists, find the quadratic equation that satisfies the condition. [img]https://cdn.artofproblemsolving.com/attachments/b/9/9e3b955b6d6df35832dd0c0a2d1d2a1e1cce94.png[/img]

Let $a_1,a_2,\ldots a_n,k$, and $M$ be positive integers such that $$\frac{1}{a_1}+\frac{1}{a_2}+\cdots+\frac{1}{a_n}=k\quad\text{and}\quad a_1a_2\cdots a_n=M.$$ If $M>1$, prove that the polynomial $$P(x)=M(x+1)^k-(x+a_1)(x+a_2)\cdots (x+a_n)$$ has no positive roots.

Given an integer $ n\ge 2$, find the maximal constant $ \lambda (n)$ having the following property: if a sequence of real numbers $ a_{0},a_{1},a_{2},\cdots,a_{n}$ satisfies $ 0 \equal{} a_{0}\le a_{1}\le a_{2}\le \cdots\le a_{n},$ and $ a_{i}\ge\frac {1}{2}(a_{i \plus{} 1} \plus{} a_{i \minus{} 1}),i \equal{} 1,2,\cdots,n \minus{} 1,$ then $ (\sum_{i \equal{} 1}^n{ia_{i}})^2\ge \lambda (n)\sum_{i \equal{} 1}^n{a_{i}^2}.$

A unit square is divided into polygons, so that all sides of a polygon are parallel to sides of the given square. If the total length of the segments inside the square (without the square) is $2n$ (where $n$ is a positive real number), prove that there exists a polygon whose area is greater than $\frac{1}{(n+1)^2}$.

Let's construct a family $\{K_n\}$ of geometric figures following the pattern shown in pictures: [center][img]https://cdn.artofproblemsolving.com/attachments/4/1/76d6cf2b7ec3bd69de7bf33e2a382885f744a0.png[/img][/center] where each hexagon (like the starting one) is constructed by cutting the two corners tops of a square, in such a way that the two figures removed are identical isosceles triangles, and the three resulting upper sides have the same length. Continuing like this, a pattern is produced with which we can build the figures $K_n$, for integer $n \ge 0$ . Then, we denote by $P_n$ and $A_n$ the perimeter and area of the figure $K_n$, respectively. If the side of square to build $K_0$ measures $x$: (a) Calculate $P_0$ and $A_0$ (in terms of the length $x$). (b) Find an explicit formula for $P_n$, and for $A_n$, in terms of $x$ and of $n$. Simplify your answers. (c) If $P_{2022} = A_{2022}$, find the measure of the six sides of the figure $K_0$, in its simplest form.

For the real pairs $(x,y)$ satisfying the equation $x^2 + y^2 + 2x - 6y = 6$, which of the following cannot be equal to $(x-1)^2 + (y-2)^2$? $ \textbf{(A)}\ 2 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 16 \qquad\textbf{(D)}\ 23 \qquad\textbf{(E)}\ 30 $

Let $ABC$ be a triangle and $M$ be the midpoint of side $BC$. The perpendicular bisector of $BC$ intersects the circumcircle of $\triangle ABC$ at points $K$ and $L$ ($K$ and $A$ lie on the opposite sides of $BC$). A circle passing through $L$ and $M$ intersects $AK$ at points $P$ and $Q$ ($P$ lies on the line segment $AQ$). $LQ$ intersects the circumcircle of $\triangle KMQ$ again at $R$. Prove that $BPCR$ is a cyclic quadrilateral.

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

$6$ people stand in a circle with water guns. Each person randomly selects another person to shoot. What is the probability that no pair of people shoots at each other?

Let $ABC$ be a triangle with $AC > AB$. Let $P$ be the intersection point of the perpendicular bisector of $BC$ and the internal angle bisector of $\angle{A}$. Construct points $X$ on $AB$ (extended) and $Y$ on $AC$ such that $PX$ is perpendicular to $AB$ and $PY$ is perpendicular to $AC$. Let $Z$ be the intersection point of $XY$ and $BC$. Determine the value of $\frac{BZ}{ZC}$.

If the line $L$ in the $xy$-plane has half the slope and twice the y-intercept of the line $y = \frac{2}{3} x + 4$, then an equation for $L$ is: $ \textbf{(A)}\ y = \frac{1}{3} x + 8 \qquad \textbf{(B)}\ y = \frac{4}{3} x + 2 \qquad \textbf{(C)}\ y = \frac{1}{3} x + 4 \qquad\\ \textbf{(D)}\ y = \frac{4}{3} x + 4 \qquad \textbf{(E)}\ y = \frac{1}{3} x + 2 $

Determine all functions $f: \mathbb{Q} \rightarrow \mathbb{Z} $ satisfying \[ f \left( \frac{f(x)+a} {b}\right) = f \left( \frac{x+a}{b} \right) \] for all $x \in \mathbb{Q}$, $a \in \mathbb{Z}$, and $b \in \mathbb{Z}_{>0}$. (Here, $\mathbb{Z}_{>0}$ denotes the set of positive integers.)

A domino is a $1 \times 2$ (or 2 $\times 1$) rectangular piece; namely, made up of two squares. There is an $8 \times 8$ board such that each domino can be cover exactly two of its squares. John places $n$ dominoes on the board, so that each one covers exactly two squares of the board and it is no longer possible to place a piece more without overlapping with any of those already placed. Determine the smallest value of $n$ for which the described situation is possible.

Let $p\ge 5$ be a prime number. Consider the function given by the formula $$f (x_1,..., x_p) = x_1 + 2x_2 +... + px_p.$$ Let $A_k$ denote the set of all these permutations $(a_1,..., a_p)$ of the set $\{1,..., p\}$, for integer number $f (a_1,..., a_p) - k$ is divisible by $p$ and $a_i \ne i$ for all $i \in \{1,..., p\}$. Prove that the sets $A_1$ and $A_4$ have the same number of elements.

Characterize all increasing sequences $(s_n)$ of positive reals for which there exists a set $A\subset \mathbb{R}$ with positive measure such that $\lambda(A\cap I)<\frac{s_n}{n}$ holds for every interval $I$ with length $1/n$, where $\lambda$ denotes the Lebesgue measure.

Let $H$ be the orthocenter of triangle $ABC$, and $D$ be the foot of $A$ onto $BC$. Given that $DB=3$, $DH=2$, and $DC=6$, calculate $HA$.

The diagonals $AC$ and $BD$ of a convex quadrilateral $ABCD$ intersect in point $M$. The angle bisector of $\angle ACD$ intersects the ray $\overrightarrow{BA}$ in point $K$. If $MA.MC+MA.CD=MB.MD$, prove that $\angle BKC=\angle CDB$.

The Fibonacci numbers are a sequence of numbers defined recursively as follows: $F_1=1$, $F_2=1$, and $F_n=F_{n-1}+F_{n-2}$. Using this definition, compute the sum $$\sum_{k=1}^{10}\frac{F_k}{F_{k+1}F_{k+2}}.$$

Let $ABCD$ be a trapezoid with $AB\parallel CD$. The bisectors of $\angle CDA$ and $\angle DAB$ meet at $E$, the bisectors of $\angle ABC$ and $\angle BCD$ meet at $F$, the bisectors of $\angle BCD$ and $\angle CDA$ meet at $G$, and the bisectors of $\angle DAB$ and $\angle ABC$ meet at $H$. Quadrilaterals $EABF$ and $EDCF$ have areas $24$ and $36$, respectively, and triangle $ABH$ has area $25$. Find the area of triangle $CDG$.

Diameter $ACE$ is divided at $C$ in the ratio $2:3$. The two semicircles, $ABC$ and $CDE$, divide the circular region into an upper (shaded) region and a lower region. The ratio of the area of the upper region to that of the lower region is [asy]pair A,B,C,D,EE; A = (0,0); B = (2,2); C = (4,0); D = (7,-3); EE = (10,0); fill(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)--arc((5,0),EE,A,CCW)--cycle,gray); draw(arc((2,0),A,C,CW)--arc((7,0),C,EE,CCW)); draw(circle((5,0),5)); dot(A); dot(B); dot(C); dot(D); dot(EE); label("$A$",A,W); label("$B$",B,N); label("$C$",C,E); label("$D$",D,N); label("$E$",EE,W); [/asy] $\textbf{(A)}\ 2:3 \qquad \textbf{(B)}\ 1:1 \qquad \textbf{(C)}\ 3:2 \qquad \textbf{(D)}\ 9:4 \qquad \textbf{(E)}\ 5:2$

The star shown is symmetric with respect to each of the six diagonals shown. All segments connecting the points $A_1, A_2, . . . , A_6$ with the centre of the star have the length $1$, and all the angles at $B_1, B_2, . . . , B_6$ indicated in the figure are right angles. Calculate the area of the star. [img]https://1.bp.blogspot.com/-Rso2aWGUq_k/XzcAm4BkAvI/AAAAAAAAMW0/277afcqTfCgZOHshf_6ce2XpinWWR4SZACLcBGAsYHQ/s0/2006%2BMohr%2Bp1.png[/img]