Practice (EndingDigits,TheDivideByNineMethod,MODBasic)

2730

Mary plans to ask Joe to water the flowers during her summer vacation. Joe has a $10\%$ chance of forgetting this chore. If the flowers have an $85\%$ survival rate when watered but only a $20\%$ survival rate when not watered, what is the probability that the flowers will die upon Mary's return?

2731

The germination rates of two different seeds are measured at $90\%$ and $80\%$, respectively. Find the probability that

both will germinate
at least one will germinate
exactly one will germinate

2732

A bug crawls from $A$ along a grid. It never goes backward, it crawls towards all the other possible directions with equal probability. For example:

At $A$, it may crawl to either $B$ or $D$ with a 50-50 chance
At $E$ (coming from $D$), it may crawl to $B$, $F$, or $H$ with a $\frac{1}{3}$ chance each
At $C$ (coming from $B$), it will crawl to $F$ for sure

The questions are, from $A$:

What is the probability of it landing at $E$ in 2 steps?
What is the probability of it landing at $F$ in 3 steps?
What is the probability of it landing at $G$ in 4 steps?

2733

The probability that Alice can solve a given problem is $1/2$. Beth has $1/3$ chance to solve the same problem. Carol's chance to solve it is $1/4$. If all them work on this problem independently, what is the probability that one and only one of them solves it?

2734

Let $a, b, c, m, n, p, k$ be positive real numbers that satisfy $a+m = b+n = c+p=k$. Show that $an+bp+cm < k^2$.

2735

Joe shoots the same target four times. If there is an $\frac{80}{81}$ chance he can hit the target at least once, what is his probability of hit the target in a single shoot?

2736

Fifteen guards and five prisoners stand in a single row. To ensure security, every prisoner must be escorted by two guards, one on each side. The five extra guards can stand anywhere in the row. How many different arrangements can be made?

2738

There are $n$ circles on the plane. Every pair of two circles intersect at two points. No three circles pass the same point. How many regions do these circles divide the whole plane into?

2739

Find the least non-negative residue of $70! \pmod{5183}$.

2740

Compute $50^{250} \pmod{83}$ .

2741

What is the last digit of $7^{222}$?

2742

A four digit number is divisible by all the even numbers strictly between $10$ and $20$. This four digit number plus the sum of its own digits equals a perfect square. Find this four digit number.

2743

For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. For how many positive integers $n$ is it true that both $P(n) = \sqrt{n}$ and $P(n+48) = \sqrt{n+48}$?

2744

$\textbf{Cover the Board}$

Joe cuts off the top left corner and the bottom right corner of an $8\times 8$ board, and then tries to cover the remaining board using thirty-one $1\times 2$ smaller pieces. Is it possible? Note: a smaller piece can be rotated, but cannot be further broken up.

2745

$\textbf{Cover the Board (II)}$

Joe cuts off a $2\times 2$ corner from an $8\times 8$ board, and then tries to cover the remaining part using $15$ L-shaped grids made of $4$ grids as shown. Is it possible?

2746

Show that if an $m\times n$ grid can be completely covered by some L-shaped smaller grids consist of 4 unit grids without overlapping, then the value of $mn$ must be a multiple of 8.

2747

Show that among any $6$ people in the world, there must exist $3$ people who either know each other or do not know each other.

2748

There are $6$ points in the $3$-D space. No three points are on the same line and no four points are one the same plane. Hence totally $15$ segments can be created among these points. Show that if each of these $15$ segments is colored either black or white, there must exist a triangle whose sides are of same color.

2749

Seventeen people correspond by mail with one another - each one with all the rest. In their letters only three different topics are discussed. Each pair of correspondents deals with only one of these topics. Prove that there are at least three people who write to each other about the same topic.

2750

Let positive integers $a_1, a_2, \cdots, a_{31}, b_1, b_2, \cdots, b_{31}$ satisfy the following conditions: - $a_1 < a_2 < a_3 < \cdots < a_{31}$ - $b_1 < b_2 < b_3 < \cdots < b_{31}$ - $a_1 + a_2+a_3+\cdots + a_{31} = b_1 + b_2 + b_3 + \cdots + b_{31}=2015$ Find the maximum value of $S=\mid a_1 - b_1 \mid + \mid a_2 - b_2 \mid + \cdots + \mid a_{31}-b_{31}\mid$.

2751

In chess, a knight can move between the two opposite corner squares of a $2 \times 3$ block. The $2\times 3$ block can be either horizontal or vertical, and can be either direction of where the knight stands. In a $4\times 4$ grid, how many squares can a knight visit, including its starting square, in a series of moves without stopping at the same square twice?

2752

Let $f(x)= \sqrt{ax^2+bx}$. For how many real values of $a$ is there at least one positive value of $b$ for which the domain of $f$ and the range of $f$ are the same set?

2753

If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$

2754

How many perfect squares are divisors of the product $1! \cdot 2! \cdot 3! \cdot \cdots \cdot 9!$?

2755

Objects $A$ and $B$ move simultaneously in the coordinate plane via a sequence of steps, each of length one. Object $A$ starts at $(0,0)$ and each of its steps is either right or up, both equally likely. Object $B$ starts at $(5,7)$ and each of its steps is either to the left or down, both equally likely. Which of the following is closest to the probability that the objects meet?

Practice (EndingDigits,TheDivideByNineMethod,MODBasic)

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