Difference between revisions of "Lcz's Mock AMC 10A Problems"
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==Problem 7== | ==Problem 7== | ||
− | Given that <math>729=1011011001_2</math>, <math>2021^2</math> can be expressed as <math>2^a_1+2^a_2+2^a_3 . . . +2^a_k</math>, where the <math>a_i</math> are an increasing sequence of positive integers. Find <math>k</math>. | + | Given that <math>729=1011011001_2</math>, <math>2021^2</math> can be expressed as <math>2^{a_1}+2^{a_2}+2^{a_3} . . . +2^{a_k}</math>, where the <math>a_i</math> are an increasing sequence of positive integers. Find <math>k</math>. |
<math>\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 14</math> | <math>\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 14</math> |
Revision as of 16:05, 30 June 2020
Contents
[hide]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!
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
Given that , can be expressed as , where the are an increasing sequence of positive integers. Find .
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 13
How many -digit integers contain a substring of digits that is divisible by ? (For example, count in because it contains , but don't count in .)