How many digits are in the base-ten representation of $8^5 \cdot 5^{10} \cdot 15^5$? $\textbf{(A)}~14\qquad\textbf{(B)}~15\qquad\textbf{(C)}~16\qquad\textbf{(D)}~17\qquad\textbf{(E)}~18\qquad$

Let $ p_1, p_2, p_3$ and $ p_4$ be four different prime numbers satisying the equations $ 2p_1 \plus{} 3p_2 \plus{} 5p_3 \plus{} 7p_4 \equal{} 162$ $ 11p_1 \plus{} 7p_2 \plus{} 5p_3 \plus{} 4p_4 \equal{} 162$ Find all possible values of the product $ p_1p_2p_3p_4$

Consider the set of all permutations, $\mathcal{P}$, of $\{1,2,\ldots,2022\}$. For permutation $P\in \mathcal{P}$, let $P_1$ denote the first element in $P$. Let $\text{sgn}(P)$ denote the sign of the permutation. Compute the following number modulo 1000: $$\displaystyle\sum_{P\in\mathcal{P}}\dfrac{P_1\cdot\text{sgn}(P)^{P_1}}{2020!}.$$ (The [i]sign[/i] of a permutation $P$ is $(-1)^k$, where $k$ is the minimum number of two-element swaps needed to reach that permutation). [i]Proposed by Nairit Sarkar[/i]

Given three numbers $x, y, z$ denote the absolute values of the differences of each pair by $x_1,y_1, z_1$. From $x_1, y_1, z_1$ form in the same fashion the numbers $x_2, y_2, z_2$, etc. It is known that $x_n = x,y_n = y, z_n = z$ for some $n$. Find $y$ and $z$ if $x = 1$.

Let $\Delta ABC$ be an acute-angled triangle and let $H$ be its orthocentre. Let $G_1, G_2$ and $G_3$ be the centroids of the triangles $\Delta HBC , \Delta HCA$ and $\Delta HAB$ respectively. If the area of $\Delta G_1G_2G_3$ is $7$ units, what is the area of $\Delta ABC $?

Let $N_0$ denote the set of nonnegative integers and $Z$ the set of all integers. Let a function $f : N_0 \times Z \to Z$ satisfy the conditions (i) $f(0, 0) = 1$, $f(0, 1) = 1$ (ii) for all $k, k \ne 0, k \ne 1$, $f(0, k) = 0$ and (iii) for all $n \ge 1$ and $k, f(n, k) = f(n -1, k) + f(n- 1, k - 2n)$. Find the value of $$\sum_{k=0}^{2009 \choose 2} f(2008, k)$$

Suppose $a_1, a_2, a_3, \dots,$ is a sequence of real numbers such that $$a_n = \frac{a_{n-1}a_{n-2}}{3a_{n-2}-2a_{n-1}}$$ for all $n \ge 3$. If $a_1 = 1$ and $a_{10} = 10$, what is $a_{19}$? [i]Proposed by Howard Halim[/i]

What are the sign and units digit of the product of all the odd negative integers strictly greater than $-2015$? $\textbf{(A) } \text{It is a negative number ending with a 1.}$ $\textbf{(B) } \text{It is a positive number ending with a 1.}$ $\textbf{(C) } \text{It is a negative number ending with a 5.}$ $\textbf{(D) } \text{It is a positive number ending with a 5.}$ $\textbf{(E) } \text{It is a negative number ending with a 0.}$

Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$. [i]Proposed by Andy Xu[/i]

The number $2019$ has the following nice properties: (a) It is the sum of the fourth powers of fuve distinct positive integers. (b) It is the sum of six consecutive positive integers. In fact, $2019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4$ (1) $2019 = 334 + 335 + 336 + 337 + 338 + 339$ (2) Prove that $2019$ is the smallest number that satises [b]both [/b] (a) and (b). (You may assume that (1) and (2) are correct!)

Find the number of polynomials $P(x)$ of degree $6$ whose coefficients are in the set $\{1,2,\ldots,1999\}$ and which are divisible by $x^3+x^2+x+1$.

Given a set of positive integers $R$, we define the [i]friend set[/i] of $R$ to be all positive integers that are divisible by at least one number in $R$. The friend set of $R$ is denoted by $\mathcal{F}(S_1)=\mathcal{F}(S_2)$. Show that $S_1=S_2$.

A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0),$ $(2020, 0),$ $(2020, 2020),$ and $(0, 2020)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{2}$. (A point $(x, y)$ is a lattice point if $x$ and $y$ are both integers.) What is $d$ to the nearest tenth$?$ $\textbf{(A) } 0.3 \qquad \textbf{(B) } 0.4 \qquad \textbf{(C) } 0.5 \qquad \textbf{(D) } 0.6 \qquad \textbf{(E) } 0.7$

For distinct real numbers $a_1,a_2,...,a_n$, we calculate the $\frac{n(n-1)}{2}$ sums $a_i +a_j$ with $1 \le i < j \le n$, and sort them in ascending order. Find all integers $n \ge 3$ for which there exist $a_1,a_2,...,a_n$, for which this sequence of $\frac{n(n-1)}{2}$ sums form an arithmetic progression (i.e. the dierence between consecutive terms is constant).

Let $p$ be a prime number and $n, k$ and $q$ natural numbers, where $q\le \frac{n -1}{p-1}$ should be. Let $M$ be the set of all integers $m$ from $0$ to $n$, for which $m-k$ is divisible by $p$. Show that $$\sum_{m \in M} (-1) ^m {n \choose m}$$ is divisible by $p^q$.

Let $x_0,x_1,\dots, x_{n-1}$ be real numbers such that $0<|x_0|<|x_1|<\dots<|x_{n-1}|$. We will write the sum of the elements of each one of the $2^n$ subsets of $\{x_0,x_1,\dots,x_{n-1}\}$ in a paper. Prove that the $2^n$ written numbers are consecutive elements of a arithmetic progression if and only if the ratios $$|\frac{x_i}{x_j}|, 0\leq j<i\leq n-1$$ are equal(s) to the ratio(s) obtained with the numbers $2^0,2^1,\dots,2^{n-1}$. Note: The sum of the elements of the empty set is $0$.

Find all functions $f(x)$ from nonnegative reals to nonnegative reals such that $f(f(x))=x^4$ and $f(x)\leq Cx^2$ for some constant $C$.

Circles $c_1$ and $c_2$ with centres $O_1$and $O_2$, respectively, intersect at points $A$ and $B$ so that the centre of each circle lies outside the other circle. Line $O_1A$ intersects circle $c_2$ again at point $P_2$ and line $O_2A$ intersects circle $c_1$ again at point $P_1$. Prove that the points $O_1,O_2, P_1, P_2$ and $B$ are concyclic

There are two bowls on a table, one white and one black. In the white bowl there $2019$ balls. Players $A$ and $B$ play a game where they make every other move ($A$ begins). One move consists is $\bullet$ to move one or your balls from one bowl to the other, or $\bullet$ to remove a ball from the white bowl, with the condition that the resulting position (that is, the number of bullets in the two bowls) have not occurred before. The player who has no valid move to make loses. Can any of the players be sure to win? If so, which one?

Let $a$ and $b$ be positive integers. Suppose that there are infinitely many pairs of positive integers $(m,n)$ for which $m^2+an+b$ and $n^2+am+b$ are both perfect squares. Prove that $a$ divides $2b$. [i]Holden Mui[/i]

[b]p1.[/b] $ x,y,z$ are required to be non-negative whole numbers, find all solutions to the pair of equations $$x+y+z=40$$ $$2x + 4y + 17z = 301.$$ [b]p2.[/b] Let $P$ be a point lying between the sides of an acute angle whose vertex is $O$. Let $A,B$ be the intersections of a line passing through $P$ with the sides of the angle. Prove that the triangle $AOB$ has minimum area when $P$ bisects the line segment $AB$. [b]p3.[/b] Find all values of $x$ for which $|3x-2|+|3x+1|=3$. [b]p4.[/b] Prove that $x^2+y^2+z^2$ cannot be factored in the form $$(ax + by + cz) (dx + ey + fz),$$ $a, b, c, d, e, f$ real. [b]p5.[/b] Let $f(x)$ be a continuous function for all real values of $x$ such that $f(a)\le f(b)$ whenever $a\le b$. Prove that, for every real number $r$, the equation $$x + f(x) = r$$ has exactly one solution. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. What is $OP$? $\textbf{(A)} ~\frac{23}{8}\qquad\textbf{(B)} ~\frac{29}{10}\qquad\textbf{(C)} ~\frac{35}{12}\qquad\textbf{(D)} ~\frac{73}{25}\qquad\textbf{(E)} ~3$

Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$; b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$. Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if \[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\] [i]Evan O'Dorney.[/i]

Suppose that $p_1<p_2<\dots <p_{15}$ are prime numbers in arithmetic progression, with common difference $d$. Prove that $d$ is divisible by $2,3,5,7,11$ and $13$.

In equilateral $\triangle ABC$ let points $D$ and $E$ trisect $\overline{BC}$. Then $\sin \left( \angle DAE \right)$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$, where $a$ and $c$ are relatively prime positive integers, and $b$ is an integer that is not divisible by the square of any prime. Find $a+b+c$.