Practice (11)

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How many positive integers less than $1998$ are relatively prime to $1547$?

Find the smallest positive integer $n$ so that $107n$ has the same last two digits as $n$.

Determine all pairs $(a, b)$ of real numbers such that $10, a, b, ab$ is an arithmetic progression.

Let $$f(r) = \displaystyle\sum_{j=2}^{2008}\frac{1}{j^r} = \frac{1}{2^r}+\frac{1}{3^r}+\cdots+\frac{1}{2016^r}$$ Find $$\sum_{k=2}^{\infty}f(k)$$

Let $P(x)$ be a polynomial with degree 2008 and leading coeffi\u000ecient 1 such that $P(0) = 2007, P(1) = 2006, P(2) = 2005, \cdots, P(2007) = 0$. Determine the value of $P(2008)$. You may use factorials in your answer.

Evaluate the infinite sum $\displaystyle\sum_{n=1}^{\infty}\frac{n}{n^4+4}$.

Solve the equation $$\sqrt{x+\sqrt{4x+\sqrt{16x+\sqrt{\cdots+\sqrt{4^{2008}x+3}}}}}-\sqrt{x}=1$$

Three of the roots of $x^4 + ax^2 + bx + c = 0$ are $2$, $−3$, and $5$. Find the value of $a + b + c$.

What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?

Twenty-seven players are randomly split into three teams of nine. Given that Zack is on a different team from Mihir and Mihir is on a different team from Andrew, what is the probability that Zack and Andrew are on the same team?

A square in the $xy$-plane has area $A$, and three of its vertices have $x$-coordinates $2$, $0$, and $18$ in some order. Find the sum of all possible values of $A$.

Find the number of eight-digit positive integers that are multiples of $9$ and have all distinct digits.

Compute the smallest positive integer n for which $$\sqrt{100+\sqrt{n}}+\sqrt{100-\sqrt{n}}$$

is an integer.

Call a polygon normal if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?

Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical by expressing it as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible distinct values of ab can be expressed in the form $q\cdot 15!$ for some rational number $q$. Find $q$.

Equilateral triangle $ABC$ has circumcircle $\omega$. Points $D$ and $E$ are chosen on minor arcs $AB$ and $AC$ of $\omega$ respectively such that $BC = DE$. Given that triangle $ABE$ has area $3$ and triangle $ACD$ has area $4$, find the area of triangle $ABC$.

$20$ players are playing in a Super Smash Bros. Melee tournament. They are ranked $1$ - $20$, and player $n$ will always beat player $m$ if $n < m$. Out of all possible tournaments where each player plays $18$ distinct other players exactly once, one is chosen uniformly at random. Find the expected number of pairs of players that win the same number of games.

Real numbers x, y, and z are chosen from the interval $[−1, 1]$ independently and uniformly at random. What is the probability that $$|x| + |y| + |z| + |x + y + z| = |x + y| + |y + z| + |z + x|$$

How many distinct permutations of the letters of the word REDDER are there that do not contain a palindromic substring of length at least two? (A substring is a contiguous block of letters that is part of the string. A string is palindromic if it is the same when read backwards.)

Your math friend Steven rolls five fair icosahedral dice (each of which is labelled $1$, $2$, $\cdots$, $20$ on its sides). He conceals the results but tells you that at least half of the rolls are $20$. Suspicious, you examine the first two dice and find that they show $20$ and $19$ in that order. Assuming that Steven is truthful, what is the probability that all three remaining concealed dice show $20$?

Reimu and Sanae play a game using $4$ fair coins. Initially both sides of each coin are white. Starting with Reimu, they take turns to color one of the white sides either red or green. After all sides are colored, the 4 coins are tossed. If there are more red sides showing up, then Reimu wins, and if there are more green sides showing up, then Sanae wins. However, if there is an equal number of red sides and green sides, then neither of them wins. Given that both of them play optimally to maximize the probability of winning, what is the probability that Reimu wins? 

Yannick is playing a game with $100$ rounds, starting with $1$ coin. During each round, there is a $n\%$ chance that he gains an extra coin, where $n$ is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?

Contessa is taking a random lattice walk in the plane, starting at $(1, 1)$. (In a random lattice walk, one moves up, down, left, or right $1$ unit with equal probability at each step.) If she lands on a point of the form $(6m, 6n)$ for $m$, $n\in\mathbb{Z}$, she ascends to heaven, but if she lands on a point of the form $(6m+ 3, 6n+ 3)$ for $m,\ n\in\mathbb{Z}$, she descends to hell. What is the probability that she ascends to heaven? 

A point $P$ lies at the center of square $ABCD$. A sequence of points $\{P_n\}$ is determined by $P_0 = P$, and given point $P_i$ , point $P_{i+1}$ is obtained by reflecting $P_i$ over one of the four lines $AB$, $BC$, $CD$, $DA$, chosen uniformly at random and independently for each $i$. What is the probability that $P_8 = P$? 

In an election for the Peer Pressure High School student council president, there are $2019$ voters and two candidates Alice and Celia (who are voters themselves). At the beginning, Alice and Celia both vote for themselves, and Alice’s boyfriend Bob votes for Alice as well. Then one by one, each of the remaining $2016$ voters votes for a candidate randomly, with probabilities proportional to the current number of the respective candidate’s votes. For example, the first undecided voter David has a $2/3$ probability of voting for Alice and a $1/3$ probability of voting for Celia. What is the probability that Alice wins the election (by having more votes than Celia)?