2018 Memorial Day Mock AMC 10 Discussion Forum

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0 replies

jlacosta

Apr 2, 2025

0 replies

i General Discussion

QIDb602 2

N Jun 23, 2018 by QIDb602

This is a thread for general discussion of the 2018 Memorial Day Mock AMC 10.

Please discuss individual problems in the corresponding threads.

Official solutions are in the attachments right after the statistics.

Statistics:

2 replies

QIDb602

Jun 18, 2018

QIDb602

Jun 23, 2018

22. Two Quadratics

QIDb602 6

N Jan 28, 2021 by fukano_2

The quadratic equations $y=ax^2+20x+32$ and $y=x^2+bx+16$ share the same vertex in the $xy$ -coordinate plane. If $a$ and $b$ are integers, what is $a+b$ ?

$\textbf{(A)}\ 9\ \ \ \ \ \ \textbf{(B)}\ 10\ \ \ \ \ \ \textbf{(C)}\ 12\ \ \ \ \ \ \textbf{(D)}\ 14\ \ \ \ \ \ \textbf{(E)}\ 21$

6 replies

QIDb602

Jun 18, 2018

fukano_2

Jan 28, 2021

12. Three Circles

QIDb602 1

N Jan 28, 2021 by fukano_2

\item In triangle $ABC$ , $AB=BC=1$ and $m\angle ABC=90^{\circ}$ . Non-overlapping congruent circles $D$ , $E$ , and $F$ are placed inside triangle $ABC$ so that each circle is internally tangent to triangle $ABC$ at two different points. If circle $E$ is externally tangent to circles $D$ and $F$ , what is the radius of circle $D$ ?

IMAGE

$\textbf{(A)}\ \displaystyle\frac{2-\sqrt{2}}{4}\ \ \ \ \ \ \textbf{(B)}\ \displaystyle\frac{5-\sqrt{2}}{23}\ \ \ \ \ \ \textbf{(C)}\ 3-2\sqrt{2}\ \ \ \ \ \ \textbf{(D)}\ \displaystyle\frac{4-\sqrt{2}}{14}\ \ \ \ \ \ \textbf{(E)}\ \displaystyle\frac{3-\sqrt{2}}{7}$

1 reply

QIDb602

Jun 18, 2018

fukano_2

Jan 28, 2021

9. Multiplication Game

QIDb602 1

N Dec 30, 2020 by GammaZero

Ashley and Ben are playing a game. At the beginning of the game, $n=1$ . Starting with Ashley, the players then take turns multiplying $n$ by an integer between 2 and 9, inclusive. The first player to get a product of at least 100 wins. If neither player makes any mistakes, how many different integers can Ashley multiply $n$ by on her first turn in order to win?

$\textbf{(A)}\ 1\ \ \ \ \ \ \textbf{(B)}\ 2\ \ \ \ \ \ \textbf{(C)}\ 3\ \ \ \ \ \ \textbf{(D)}\ 4\ \ \ \ \ \ \textbf{(E)}\ 5$

1 reply

QIDb602

Jun 18, 2018

GammaZero

Dec 30, 2020

7. Solve for a and b

QIDb602 1

N Dec 30, 2020 by GammaZero

$a$ and $b$ are positive integers in the equation $a^2+4b=b^2+a$ , and $a>1$ . What is the only possible value of $a+b$ ?

$\textbf{(A)}\ 7\ \ \ \ \ \ \textbf{(B)}\ 8\ \ \ \ \ \ \textbf{(C)}\ 10\ \ \ \ \ \ \textbf{(D)}\ 12\ \ \ \ \ \ \textbf{(E)}\ 24$

1 reply

QIDb602

Jun 18, 2018

GammaZero

Dec 30, 2020

4. Picture Frame

QIDb602 1

N Dec 30, 2020 by GammaZero

A picture frame is made by cutting out a $7\times10\times1$ rectangular prism from the center of a $9\times12\times1$ rectangular prism. What is the surface area of the picture frame?

$\textbf{(A)}\ 128\ \ \ \ \ \ \textbf{(B)}\ 136\ \ \ \ \ \ \textbf{(C)}\ 144\ \ \ \ \ \ \textbf{(D)}\ 152\ \ \ \ \ \ \textbf{(E)}\ 160$

1 reply

QIDb602

Jun 18, 2018

GammaZero

Dec 30, 2020

15. Area of a Kite

QIDb602 2

N Dec 26, 2020 by star32

In the diagram shown below, square $ABCD$ has a side length of 10. Point $E$ is the midpoint of $\overline{BC}$ and point $F$ is the midpoint of $\overline{CD}$ . If $m\angle EJD=m\angle EKA=m\angle FGB=m\angle FHA=75^{\circ}$ , the area of the shaded quadrilateral is equal to $\frac{r\sqrt{t}}{s}$ where $r$ , $s$ , and $t$ are positive integers, $r$ and $s$ are relatively prime, and $t$ is not evenly divisible by the square of any integer greater than 1. What is $r+s+t$ ?

IMAGE

$\textbf{(A)}\ 31\ \ \ \ \ \ \textbf{(B)}\ 34\ \ \ \ \ \ \textbf{(C)}\ 37\ \ \ \ \ \ \textbf{(D)}\ 43\ \ \ \ \ \ \textbf{(E)}\ 49$

2 replies

QIDb602

Jun 18, 2018

star32

Dec 26, 2020

6. Factoring Factorials

QIDb602 1

N Sep 21, 2020 by Ninjasolver0201

$n$ is a positive integer. The decimal representation of $n!$ ends in $r$ zeros. The decimal representation of $n!+(n-1)!$ ends in $s$ zeros. Which of the following values of $n$ causes $s$ to be greater than $r$ ?

$\textbf{(A)}\ 2014\ \ \ \ \ \ \textbf{(B)}\ 2015\ \ \ \ \ \ \textbf{(C)}\ 2016\ \ \ \ \ \ \textbf{(D)}\ 2017\ \ \ \ \ \ \textbf{(E)}\ 2018$

1 reply

QIDb602

Jun 18, 2018

Ninjasolver0201

Sep 21, 2020

20. Two Variables

QIDb602 2

N Aug 23, 2018 by TaBellaria

$q$ and $r$ are positive integers in the following equation. There is only one possible value of $q+r$ . Let $S(n)$ represent the sum of the digits of positive integer $n$ . What is $S(q+r)$ ?

$\displaystyle\frac{q}{r}+\frac{r}{q}=\frac{qr}{144}$
$\textbf{(A)}\ 5\ \ \ \ \ \ \textbf{(B)}\ 6\ \ \ \ \ \ \textbf{(C)}\ 8\ \ \ \ \ \ \textbf{(D)}\ 9\ \ \ \ \ \ \textbf{(E)}\ 11$

2 replies

QIDb602

Jun 18, 2018

TaBellaria

Aug 23, 2018

11. Remainders

QIDb602 5

N Jun 21, 2018 by dragonsbaseball26

How many numbers from the set $\{8^2, 18^2, 28^2, \dots, 2018^2\}$ have a remainder of 64 when divided by 100?

$\textbf{(A)}\ 40\ \ \ \ \ \ \textbf{(B)}\ 41\ \ \ \ \ \ \textbf{(C)}\ 403\ \ \ \ \ \ \textbf{(D)}\ 404\ \ \ \ \ \ \textbf{(E)}\ 405$

5 replies

QIDb602

Jun 18, 2018

dragonsbaseball26

Jun 21, 2018

18. Powers and Roots

QIDb602 1

N Jun 21, 2018 by Miziziriz

How many integer values of $n$ between 1 and 100, inclusive, cause the value of the following expression to be an integer?

$\displaystyle\frac{\sqrt[2]{n^{3n}}}{\sqrt[3]{n^{2n}}}$
$\textbf{(A)}\ 18\ \ \ \ \ \ \textbf{(B)}\ 19\ \ \ \ \ \ \textbf{(C)}\ 20\ \ \ \ \ \ \textbf{(D)}\ 21\ \ \ \ \ \ \textbf{(E)}\ 22$

1 reply

QIDb602

Jun 18, 2018

Miziziriz

Jun 21, 2018