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

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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 <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>
==Problem 1==
Find the value of <math>2^{0+2+1}+2+0(2+(1))+20(21)</math>.
<math>\textbf{(A)}\ 8 \qquad\textbf{(B)}\ 426 \qquad\textbf{(C)}\ 428 \qquad\textbf{(D)}\ 430 \qquad\textbf{(E)}\ 432</math>
==Problem 2==
If <math>|x|=2</math>, and <math>|y-3|=1</math>, find the sum of all possible values of <math>|xy|</math>.
<math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 8 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 16</math>
==Problem 3==
What is <math>1*2+2*3+3*4+4*5+5*6+6*7+7*8</math>?
<math>\textbf{(A)}\ 84 \qquad\textbf{(B)}\ 120 \qquad\textbf{(C)}\ 138 \qquad\textbf{(D)}\ 150 \qquad\textbf{(E)}\ 168</math>
==Problem 4==
Find the sum of all ordered pairs of positive integer <math>x</math> and <math>y</math> such that
(1) <math>|x-y| \geq 0</math>
(2) <math>x,y \leq 3</math>
(3) <math>xy \leq 8</math>
<math>\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 27 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 36 \qquad\textbf{(E)}\ 39</math>
==Problem 5==
Find <math>x</math> if <math>x^3-3x^2+3x-1=x^3-2x^2+15x+35</math>.
<math>\textbf{(A)}\ -6 \qquad\textbf{(B)}\ -4 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 6</math>
==Problem 6==
Given that <math>5101</math> is prime, find the number of factors of <math>104060401+20402+1</math>.
<math>\textbf{(A)}\ 2 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 6 \qquad\textbf{(E)}\ 8</math>
==Problem 7==
Evaluate <math>\sum_{i,j,k=1}^{7} ijk \pmod{5}</math>, where <math>\sum_{i,j,k=1}^{7} ijk</math> is the sum of all products <math>ijk</math> when <math>1 \leq i,j,k \leq 7</math>.
<math>\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 4</math>
==Problem 8==
Given that <math>258741=27*7*37*37</math>, evaluate <math>(\overline{.143})(\overline{.258741})</math>
<math>\textbf{(A)}\ \frac{1}{999} \qquad\textbf{(B)}\ \frac{1}{99} \qquad\textbf{(C)}\ \frac{1}{27} \qquad\textbf{(D)}\ \frac{1}{9} \qquad\textbf{(E)}\ \frac{1}{3}</math>
==Problem 9==
Find the number of solutions to <math>x^{2021}+x^{2020}+x^{2019} . . . +x+1=\frac{1}{1-x}</math>.
<math>\textbf{(A) } 2020 \qquad \textbf{(B) } 2021 \qquad \textbf{(C) } 2022 \qquad \textbf{(D) } 2023 \qquad \textbf{(E) } 2024</math>
==Problem 10==
Jack and Jill play a (bad) game on a number line which contains the integers. Jack starts at <math>-1</math>, and Jill starts at <math>18</math>. Every turn, the judge flip a standard six sided die. If the number rolled is a square number, Jack moves to the right <math>10</math> units. Otherwise, Jill moves to the left <math>5</math> units. Find the probability for which Jack and Jill pass each other for the first time in <math>3</math> moves.
<math>\textbf{(A)}\ 4/27 \qquad\textbf{(B)}\ 2/9 \qquad\textbf{(C)}\ 1/3 \qquad\textbf{(D)}\ 4/9 \qquad\textbf{(E)}\ 2/3</math>
==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.
==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==

Latest revision as of 16:13, 16 July 2020

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