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

(→‎Problem 7)
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==Problem 7==
 
==Problem 7==
  
−
Find <math>\sum{i,j,k=1}^{7} ijk=1*1*1+1*1*2+1*1*3 . . . 1*1*7+1*2*1+1*2*2+1*2*3 . . . 2*1*1+2*1*2 . . .7*7*7 \pmod{5}</math>
+
Find <math>\sum_{i,j,k=1}^{7} ijk=1*1*1+1*1*2+1*1*3 . . . 1*1*7+1*2*1+1*2*2+1*2*3 . . . 2*1*1+2*1*2 . . .7*7*7 \pmod{5}</math>
  
 
<math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4</math>
 
<math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4</math>

Revision as of 19:24, 30 June 2020

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 $729=1011011001_2$, $2021^2$ can be expressed as $2^{a_1}+2^{a_2}+2^{a_3} . . . +2^{a_k}$, where the $a_i$ are an increasing sequence of positive integers. Find $k$.

$\textbf{(A)}\ 10 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 13 \qquad\textbf{(E)}\ 14$

Problem 1

Find the value of $2^{0+2+1}+2+0(2+(1))+20(21)$.

$\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 426 \qquad\textbf{(C)}\ 428 \qquad\textbf{(D)}\ 430 \qquad\textbf{(E)}\ 432$

Problem 2

If $|x-2|=0$, and $|y-3|=1$, find the sum of all possible values of $|xy|$.

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 16$

Problem 3

What is $1*2+2*3+3*4+4*5+5*6+6*7+7*8$?

$\textbf{(A)}\ 84 \qquad\textbf{(B)}\ 120 \qquad\textbf{(C)}\ 138 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 168$

Problem 4

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

(1) $|x-y| \geq 0$

(2) $x,y \leq 3$

(3) $xy \leq 8$

$\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 27 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 36 \qquad\textbf{(E)}\ 39$

Problem 5

Find $x$ if $x^3-3x^2+3x-1=x^3-2x^2+15x+35$.

$\textbf{(A)}\ -6 \qquad\textbf{(B)}\ -4 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 6$

Problem 6

Given that $5101$ is prime, find the number of factors of $104060401+20402+1$.

$\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8$

Problem 7

Find $\sum_{i,j,k=1}^{7} ijk=1*1*1+1*1*2+1*1*3 . . . 1*1*7+1*2*1+1*2*2+1*2*3 . . . 2*1*1+2*1*2 . . .7*7*7 \pmod{5}$

$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4$

Problem 8

Problem 9

Problem 10

Jack and Jill play a (bad) game on a number line which contains the integers. Jack starts at $-1$, and Jill starts at $18$. Every turn, the judge flip a standard six sided die. If the number rolled is a square number, Jack moves to the right $10$ units. Otherwise, Jill moves to the left $5$ units. Find the probability for which Jack and Jill pass each other for the first time in $3$ moves.

$\textbf{(A)}\ 4/27 \qquad\textbf{(B)}\ 2/9 \qquad\textbf{(C)}\ 1/3 \qquad\textbf{(D)}\ 4/9 \qquad\textbf{(E)}\ 2/3$

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

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