# Difference between revisions of "Lcz's Mock AMC 10A Problems"

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Find the number of solutions to <math>x^{2021}+x^{2020}+x^{2019} . . . +x+1=\frac{1}{1-x}</math>. | Find the number of solutions to <math>x^{2021}+x^{2020}+x^{2019} . . . +x+1=\frac{1}{1-x}</math>. | ||

− | <math>\textbf{(A) } | + | <math>\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2022 \qquad \textbf{(D) } 2023 \qquad \textbf{(E) } 2024</math> |

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

## Revision as of 12:38, 1 July 2020

## Contents

- 1 Instructions
- 2 Sample Problems lol
- 3 Problem 1
- 4 Problem 2
- 5 Problem 3
- 6 Problem 4
- 7 Problem 5
- 8 Problem 6
- 9 Problem 7
- 10 Problem 8
- 11 Problem 9
- 12 Problem 10
- 13 Problem 11
- 14 Problem 12
- 15 Problem 13
- 16 Problem 14
- 17 Problem 15
- 18 Problem 16
- 19 Problem 17
- 20 Problem 18
- 21 Problem 19
- 22 Problem 20
- 23 Problem 21
- 24 Problem 22
- 25 Problem 23
- 26 Problem 24
- 27 Problem 25

## Instructions

1. All rules of a regular AMC 10 apply.

2. Please submit your answers in a DM to me (Lcz).

3. Don't cheat.

Here's the problems!

## Sample Problems lol

Given that , can be expressed as , where the are an increasing sequence of positive integers. Find .

NOTE THAT THESE PROBLEMS ARE DEFINETELY NOT ORDERED BY DIFFICULTY YET LMAO

## Problem 1

Find the value of .

## Problem 2

If , and , find the sum of all possible values of .

## Problem 3

What is ?

## Problem 4

Find the sum of all ordered pairs of positive integer and such that

(1)

(2)

(3)

## Problem 5

Find if .

## Problem 6

Given that is prime, find the number of factors of .

## Problem 7

Evaluate , where is the sum of all products when .

## Problem 8

Given that , evaluate

## Problem 9

Find the number of solutions to .

## Problem 10

Jack and Jill play a (bad) game on a number line which contains the integers. Jack starts at , and Jill starts at . Every turn, the judge flip a standard six sided die. If the number rolled is a square number, Jack moves to the right units. Otherwise, Jill moves to the left units. Find the probability for which Jack and Jill pass each other for the first time in moves.

## Problem 11

There is a circle O, and there are points B, C, D, E, F, G on the circumference, in that order. CG EB and FD meet at the point A. BD intersects AC at H. Given that triangle AHD is similar to triangle AFB, AH=5, AB=9, BC=7. Find CD.