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

## Instructions

1. All rules of a regular AMC 10 apply.

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 $k$.

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

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

## 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 $|x|=2$, 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

Evaluate $\sum_{i,j,k=1}^{7} ijk \pmod{5}$, where $\sum_{i,j,k=1}^{7} ijk$ is the sum of all products $ijk$ when $1 \leq i,j,k \leq 7$.

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

## Problem 8

Given that $258741=27*7*37*37$, evaluate $(\overline{.143})(\overline{.258741})$

$\textbf{(A)}\ \frac{1}{999} \qquad\textbf{(B)}\ \frac{1}{99} \qquad\textbf{(C)}\ \frac{1}{27} \qquad\textbf{(D)}\ \frac{1}{9} \qquad\textbf{(E)}\ \frac{1}{3}$

## Problem 9

Find the number of solutions to $x^{2021}+x^{2020}+x^{2019} . . . +x+1=\frac{1}{1-x}$.

$\textbf{(A) } 0 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 2022 \qquad \textbf{(D) } 2023 \qquad \textbf{(E) } 2024$

## 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

A circle $O$ has points $B$, $C$, $D$, $E$, $F$, $G$ on the circumference, in that order. $\overline{CF}$, $\overline{EB}$, and $\overline{GD}$ meet at the point $A$. $\overline{BD}$ intersects $\overline{AC}$ at $H$. Given that $\Delta AHD$ is similar to $\Delta AFB$, $\overline{AH}=5$, $\overline{AB}=9$, $\overline{BC}=7$. Find $\overline{CD}$.

$\textbf{(A)}\ \frac{35}{6} \qquad\textbf{(B)}\ \frac{35}{5} \qquad\textbf{(C)}\ \frac{35}{4} \qquad\textbf{(D)}\ \frac{35}{3} \qquad\textbf{(E)}\ \frac{35}{2}$

## Problem 12

How many ways can the number $2\times3\times5\times7\times11\times13\times17\times19\times23$ be written as a sum of at least 2 consecutive integers?

$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 255 \qquad\textbf{(C)}\ 256 \qquad\textbf{(D)}\ 511 \qquad\textbf{(E)}\ 512$

## Problem 13

What is the maximum amount of acute angles in a $20$-gon?

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

## Problem 14

Calculate: $$5^3+6^3+7^3+11\times7^2+12\times6^2+13\times5^2.$$

$\textbf{(A)}\ 684 \qquad\textbf{(B)}\ 1200 \qquad\textbf{(C)}\ 1350 \qquad\textbf{(D)}\ 1620 \qquad\textbf{(E)}\ 1980$