# Difference between revisions of "G285 2021 MC10A"

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==Problem 9== | ==Problem 9== | ||

− | + | If a real number <math>k</math> is <math>happy</math> , <math>k^3+5k-3 \ge (k-1)^4</math>. If a real number <math>l</math> is <math>unhappy</math> , <math>l^3+5l^2 \ge 2064l</math>. If a number is neither <math>happy</math> or <math>unhappy</math>, it will be <math>neutral</math>. What is the probability that <math>3</math> randomly selected numbers from the interval <math>[1,100]</math> are <math>happy</math> , <math>unhappy</math>, and <math>neutral</math>, in any given order? | |

<math>\textbf{(A)}\ \frac{20,007}{5,000,000}\qquad\textbf{(B)}\ \frac{2}{9}\qquad\textbf{(C)}\ \frac{6,669}{1,000,000}\qquad\textbf{(D)}\ \frac{247}{35,937}\qquad\textbf{(E)}\ \frac{494}{11,979}</math> | <math>\textbf{(A)}\ \frac{20,007}{5,000,000}\qquad\textbf{(B)}\ \frac{2}{9}\qquad\textbf{(C)}\ \frac{6,669}{1,000,000}\qquad\textbf{(D)}\ \frac{247}{35,937}\qquad\textbf{(E)}\ \frac{494}{11,979}</math> | ||

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==Problem 10== | ==Problem 10== | ||

+ | Suppose the area of <math>\triangle ABC</math> is equal to the sum of its side lengths. Let point <math>D</math> be on the circumcircle of <math>\triangle ABC</math> such that <math>AD</math> is a diameter. If <math>E</math> is the center of the circumcircle, and <math>I</math> is the center of the incircle of <math>\triangle ABC</math>, and <math>CI=4</math>, find <math>EI</math>. | ||

+ | |||

+ | <math>\textbf{(A)}\ 0\qquad\textbf{(B)}\ \frac{1}{2}\qquad\textbf{(C)}\ \frac{3}{4}\qquad\textbf{(D)}\ \frac{5}{2}\qquad\textbf{(E)}\ 2</math> | ||

+ | |||

+ | [[G285 MC10A Problems/Problem 10|Solution]] | ||

+ | |||

+ | ==Problem 11== | ||

+ | If <math>abcd_{11}</math> is a palindrome in base <math>7</math>, and <math>dcba</math> expressed in base <math>10</math> does not begin with a nonzero digit, find the difference between the largest and smallest possible sum of <math>a+b+c+d</math>. | ||

+ | |||

+ | <math>\textbf{(A)}\ 8\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12</math> | ||

+ | |||

+ | [[G285 MC10A Problems/Problem 11|Solution]] |

## Revision as of 00:22, 15 May 2021

## Contents

## Problem 1

What is the smallest value of that minimizes ?

## Problem 2

Suppose the set denotes . Then, a subset of length is chosen. All even digits in the subset are then are put into group , and the odd digits are put in . Then, one number is selected at random from either or with equal chances. What is the probability that the number selected is a perfect square, given ?

## Problem 3

Let be a unit square. If points and are chosen on and respectively such that the area of . What is ?

## Problem 4

What is the smallest value of for which

## Problem 5

Let a recursive sequence be denoted by such that and . Suppose for . Let an infinite arithmetic sequence be such that . If is prime, for what value of will ?

## Problem 6

Find

## Problem 7

A regular tetrahedron has length . Suppose on the center of each surface, a hemisphere of diameter is constructed such that the hemisphere falls inside the volume of the figure. If the ratio between the radius of the largest sphere that can be inscribed inside the old tetrahedron and new tetrahedron , where is square free, and . Find .

## Problem 8

If can be expressed as , where is square free and , find if and .

## Problem 9

If a real number is , . If a real number is , . If a number is neither or , it will be . What is the probability that randomly selected numbers from the interval are , , and , in any given order?

## Problem 10

Suppose the area of is equal to the sum of its side lengths. Let point be on the circumcircle of such that is a diameter. If is the center of the circumcircle, and is the center of the incircle of , and , find .

## Problem 11

If is a palindrome in base , and expressed in base does not begin with a nonzero digit, find the difference between the largest and smallest possible sum of .