# 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 $\overline{CD}$.

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

WE ARE DEFINETELY GOING TO THROW THIS PROBLEM AWAY

RIGHT

RIGHT

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

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

## Problem 15

Sally is going home from school. First, she visits the river to the east of her school, and must take that water to the farm. The school and the farm are on the same side of the river, which runs north to south, but the school is 5 miles from it and the farm is 2 miles from the river. The school and the farm are 5 miles apart. After visiting the farm, she must visit another river that runs east to west. The farm is on the same side of the river as her house, and the farm is 3 miles from the river and her home is 9 miles. She must go to the river to get water then go back home. Her home and the farm are 10 miles apart. What is the shortest distance Sally has to go?