Kara rolls a six-sided die, and if on that first roll she rolls an $n$, she rolls the die $n-1$ more times. She then computes that the product of all her rolls, including the first, is $8$. How many distinct sequences of rolls could Kara have rolled? [i]Proposed by Andrew Wu[/i]

The first term of a sequence is $2014$. Each succeeding term is the sum of the cubes of the digits of the previous term. What is the $2014$ th term of the sequence?

For each positive integer $n$, define \[g(n) = \gcd\left\{0! n!, 1! (n-1)!, 2 (n-2)!, \ldots, k!(n-k)!, \ldots, n! 0!\right\}.\] Find the sum of all $n \leq 25$ for which $g(n) = g(n+1)$.

Consider the operation $*$ defined by the following table: \[\begin{tabular}{c|cccc} * & 1 & 2 & 3 & 4 \\ \hline 1 & 1 & 2 & 3 & 4 \\ 2 & 2 & 4 & 1 & 3 \\ 3 & 3 & 1 & 4 & 2 \\ 4 & 4 & 3 & 2 & 1 \end{tabular}\] For example, $3*2=1$. Then $(2*4)*(1*3)=$ $\text{(A)}\ 1 \qquad \text{(B)}\ 2 \qquad \text{(C)}\ 3 \qquad \text{(D)}\ 4 \qquad \text{(E)}\ 5$

Can you find three polynomials $P,Q,R$ of three variables $x,y,z$, providing the condition: a)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-2x+1)^3 = 1$ b)$P(x-y+z)^3 + Q(y-z-1)^3 +R(z-x+1)^3 = 1$ for all $x,y,z$?

There are $1000$ inhabitants in a settlement. Every evening every inhabitant tells all his friends all the news he had heard the previous day. Every news becomes finally known to every inhabitant. Prove that it is possible to choose $90$ of inhabitants so, that if you tell them a news simultaneously, it will be known to everybody in $10$ days.

In a $2^n \times 2^n$ square with $n$ positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)

The increasing sequence $1,3,4,9,10,12,13\cdots$ consists of all those positive integers which are powers of 3 or sums of distinct powers of 3. Find the $100^{\text{th}}$ term of this sequence.

Let $ABC$ be an arbitrary triangle. A regular $n$-gon is constructed outward on the three sides of $\triangle ABC$. Find all $n$ such that the triangle formed by the three centres of the $n$-gons is equilateral.

Let $n$ be a positive integer. A panel of dimenisions $2n\times2n$ is divided in $4n^2$ squares with dimensions $1\times1$. What is the highest possible number of diagonals that can be drawn in $1\times1$ squares, such that each two diagonals have no common points.

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

Let the midline of $\triangle ABC$ parallel to $BC$ intersect the circumcircle $\Gamma$ of $\triangle ABC$ at $P$, $Q$, and the tangent of $\Gamma$ at $A$ intersects $BC$ at $T$. Show that $\measuredangle BTQ = \measuredangle PTA$.

Consider a natural number $n \ge 3$. A convex polygon with $n$ sides is entirely placed inside a square with side length 1. Prove that we can always find three vertices of this polygon, the triangle formed by which has area not greater than $\frac{8}{n^2}$.

Let $x, y, z > 0$ be real numbers such that $xyz + xy + yz + zx = 4$. Prove that $x + y + z \ge 3$.

[i]20 problems for 20 minutes.[/i] [b]p1.[/b] Evaluate $=\frac{1}{2 \cdot 3 \cdot 4}+\frac{1}{3 \cdot 4 \cdot 5}$. [b]p2.[/b] A regular hexagon and a regular $n$-sided polygon have the same perimeter. If the ratio of the side length of the hexagon to the side length of the $n$-sided polygon is $2 : 1$, what is $n$? [b]p3.[/b] How many nonzero digits are there in the decimal representation of $2 \cdot 10\cdot 500 \cdot 2500$? [b]p4.[/b] When the numerator of a certain fraction is increased by $2012$, the value of the fraction increases by $2$. What is the denominator of the fraction? [b]p5.[/b] Sam did the computation $1 - 10 \cdot a + 22$, where $a$ is some real number, except he messed up his order of operations and computed the multiplication last; that is, he found the value of $(1 - 10) \cdot (a + 22)$ instead. Luckily, he still ended up with the right answer. What is $a$? [b]p6.[/b] Let $n! = n \cdot(n-1) \cdot\cdot\cdot 2 \cdot 1$. For how many integers $n$ between $1$ and $100$ inclusive is $n!$ divisible by $36$? [b]p7.[/b] Simplify the expression $\sqrt{\frac{3 \cdot 27^3}{27 \cdot 3^3}}$ [b]p8.[/b] Four points $A,B,C,D$ lie on a line in that order such that $\frac{AB}{CB}=\frac{AD}{CD}$ . Let $M$ be the midpoint of segment $AC$. If $AB = 6$, $BC = 2$, compute $MB \cdot MD$. [b]p9.[/b] Allan has a deck with $8$ cards, numbered $1$, $1$, $2$, $2$, $3$, $3$, $4$, $4$. He pulls out cards without replacement, until he pulls out an even numbered card, and then he stops. What is the probability that he pulls out exactly $2$ cards? [b]p10.[/b] Starting from the sequence $(3, 4, 5, 6, 7, 8, ... )$, one applies the following operation repeatedly. In each operation, we change the sequence $$(a_1, a_2, a_3, ... , a_{a_1-1}, a_{a_1} , a_{a_1+1},...)$$ to the sequence $$(a_2, a_3, ... , a_{a_1} , a_1, a_{a_1+1}, ...) .$$ (In other words, for a sequence starting with$ x$, we shift each of the next $x-1$ term to the left by one, and put x immediately to the right of these numbers, and keep the rest of the terms unchanged. For example, after one operation, the sequence is $(4, 5, 3, 6, 7, 8, ... )$, and after two operations, the sequence becomes $(5, 3, 6, 4, 7, 8,... )$. How many operations will it take to obtain a sequence of the form $(7, ... )$ (that is, a sequence starting with $7$)? [b]p11.[/b] How many ways are there to place $4$ balls into a $4\times 6$ grid such that no column or row has more than one ball in it? (Rotations and reflections are considered distinct.) [b]p12.[/b] Point $P$ lies inside triangle $ABC$ such that $\angle PBC = 30^o$ and $\angle PAC = 20^o$. If $\angle APB$ is a right angle, find the measure of $\angle BCA$ in degrees. [b]p13.[/b] What is the largest prime factor of $9^3 - 4^3$? [b]p14.[/b] Joey writes down the numbers $1$ through $10$ and crosses one number out. He then adds the remaining numbers. What is the probability that the sum is less than or equal to $47$? [b]p15.[/b] In the coordinate plane, a lattice point is a point whose coordinates are integers. There is a pile of grass at every lattice point in the coordinate plane. A certain cow can only eat piles of grass that are at most $3$ units away from the origin. How many piles of grass can she eat? [b]p16.[/b] A book has 1000 pages numbered $1$, $2$, $...$ , $1000$. The pages are numbered so that pages $1$ and $2$ are back to back on a single sheet, pages $3$ and $4$ are back to back on the next sheet, and so on, with pages $999$ and $1000$ being back to back on the last sheet. How many pairs of pages that are back to back (on a single sheet) share no digits in the same position? (For example, pages $9$ and $10$, and pages $89$ and $90$.) [b]p17.[/b] Find a pair of integers $(a, b)$ for which $\frac{10^a}{a!}=\frac{10^b}{b!}$ and $a < b$. [b]p18.[/b] Find all ordered pairs $(x, y)$ of real numbers satisfying $$\begin{cases} -x^2 + 3y^2 - 5x + 7y + 4 = 0 \\ 2x^2 - 2y^2 - x + y + 21 = 0 \end{cases}$$ [b]p19.[/b] There are six blank fish drawn in a line on a piece of paper. Lucy wants to color them either red or blue, but will not color two adjacent fish red. In how many ways can Lucy color the fish? [b]p20.[/b] There are sixteen $100$-gram balls and sixteen $99$-gram balls on a table (the balls are visibly indistinguishable). You are given a balance scale with two sides that reports which side is heavier or that the two sides have equal weights. A weighing is defined as reading the result of the balance scale: For example, if you place three balls on each side, look at the result, then add two more balls to each side, and look at the result again, then two weighings have been performed. You wish to pick out two different sets of balls (from the $32$ balls) with equal numbers of balls in them but different total weights. What is the minimal number of weighings needed to ensure this? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

$a,b,c,d$ are positive reals with sum $1$. Show that $\frac{a^2}{a+b}+\frac{b^2}{b+c}+\frac{c^2}{c+d}+\frac{d^2}{d+a} \ge \frac{1}{2}$ with equality iff $a=b=c=d=\frac{1}{4}$.

The first three terms of an arithmetic sequence are $ 2x\minus{}3$, $ 5x\minus{}11$, and $ 3x\plus{}1$ respectively. The $ n$th term of the sequence is $ 2009$. What is $ n$? $ \textbf{(A)}\ 255 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 1506 \qquad \textbf{(E)}\ 8037$

Find the greatest possible value of $pq + r$, where p, q, and r are (not necessarily distinct) prime numbers satisfying $pq + qr + rp = 2016$.

Let $ ABCD$ be a (not self-intersecting) quadrilateral satisfying $ \measuredangle DAB \equal{} \measuredangle BCD\neq 90^{\circ}$. Let $ X$ and $ Y$ be the orthogonal projections of the point $ D$ on the lines $ AB$ and $ BC$, and let $ Z$ and $ W$ be the orthogonal projections of the point $ B$ on the lines $ CD$ and $ DA$. Establish the following facts: [b]a)[/b] The quadrilateral $ XYZW$ is an isosceles trapezoid such that $ XY\parallel ZW$. [b]b)[/b] Let $ M$ be the midpoint of the segment $ AC$. Then, the lines $ XZ$ and $ YW$ pass through the point $ M$. [b]c)[/b] Let $ N$ be the midpoint of the segment $ BD$, and let $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$, $ W^{\prime}$ be the midpoints of the segments $ AB$, $ BC$, $ CD$, $ DA$. Then, the point $ M$ lies on the circumcircles of the triangles $ W^{\prime}X^{\prime}N$ and $ Y^{\prime}Z^{\prime}N$. [hide="Notice"][i]Notice.[/i] This problem has been discussed at http://www.mathlinks.ro/Forum/viewtopic.php?t=172417 .[/hide]

We say that a positive integer is [i]decomposed [/i] if it is prime and also If a line is drawn separating it into two numbers, those two numbers are never composite. For example 1997 is [i]decomposed [/i] since it is prime, it is divided into: $1$, $997$; $19$, $97$; $199$, $7$ and none of those numbers are compound. How many [i]decomposed [/i] numbers are there between $2000$ and $3000$?

Given that $a_1, a_2, \ldots,a_{2020}$ are integers, find the maximal number of subsequences $a_i,a_{i+1}, ..., a_j$ ($0<i\leq j<2021$) with with sum $2021$

I have 6 friends and during a vacation I met them during several dinners. I found that I dined with all the 6 exactly on 1 day; with every 5 of them on 2 days; with every 4 of them on 3 days; with every 3 of them on 4 days; with every 2 of them on 5 days. Further every friend was present at 7 dinners and every friend was absent at 7 dinners. How many dinners did I have alone?

If 46 squares are colored red in a $9\times 9$ board, show that there is a $2\times 2$ block on the board in which at least 3 of the squares are colored red.

Find the lengths of the principal axes of the ellipsoid \[\sum_{i\le j}x_i x_j=1\]

A block $Z$ is formed by gluing one face of a solid cube with side length 6 onto one of the circular faces of a right circular cylinder with radius $10$ and height $3$ so that the centers of the square and circle coincide. If $V$ is the smallest convex region that contains Z, calculate $\lfloor\operatorname{vol}V\rfloor$ (the greatest integer less than or equal to the volume of $V$).