[et_pb_section fb_built="1" _builder_version="4.2.2" custom_padding="||42px|||"][et_pb_row _builder_version="4.2.2" module_alignment="center"][et_pb_column type="4_4" _builder_version="4.2.2"][et_pb_blurb _builder_version="4.2.2"][/et_pb_blurb][et_pb_blurb _builder_version="4.2.2"][/et_pb_blurb][et_pb_team_member name="American Mathematics contest 10 (AMC 10) - Algebra problems " _builder_version="4.2.2" header_level="h1" header_font="Acme||||||||" header_text_align="center" body_font="Acme||||||||" body_text_align="center" body_letter_spacing="1px" background_color="#032b35" background_layout="dark" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"] Try these AMC 10 Algebra Questions and check your knowledge![/et_pb_team_member][et_pb_image src="https://www.staging18.cheenta.com/wp-content/uploads/2020/01/AMC-8-Geometry-Questions-5.png" _builder_version="4.2.2"][/et_pb_image][et_pb_team_member name="AMC 10A, 2020, Problem 1" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What value of $x$ satisfies

[x- \frac{3}{4} = \frac{5}{12} - \frac{1}{3}?]$\textbf{(A)}\ -\frac{2}{3}\qquad\textbf{(B)}\ \frac{7}{36}\qquad\textbf{(C)}\ \frac{7}{12}\qquad\textbf{(D)}\ \frac{2}{3}\qquad\textbf{(E)}\ \frac{5}{6}$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 3" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Assuming $a\neq3$, $b\neq4$, and $c\neq5$, what is the value in simplest form of the following expression?[\frac{a-3}{5-c} \cdot \frac{b-4}{3-a} \cdot \frac{c-5}{4-b}]
$\textbf{(A) } -1 \qquad \textbf{(B) } 1 \qquad \textbf{(C) } \frac{abc}{60} \qquad \textbf{(D) } \frac{1}{abc} - \frac{1}{60} \qquad \textbf{(E) } \frac{1}{60} - \frac{1}{abc}$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 5" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the sum of all real numbers $x$ for which $|x^2-12x+34|=2?$

$\textbf{(A) } 12 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 18 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 25$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 14" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Real numbers $x$ and $y$ satisfy $x + y = 4$ and $x \cdot y = -2$. What is the value of[x + \frac{x^3}{y^2} + \frac{y^3}{x^2} + y?]$\textbf{(A)}\ 360\qquad\textbf{(B)}\ 400\qquad\textbf{(C)}\ 420\qquad\textbf{(D)}\ 440\qquad\textbf{(E)}\ 480$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 19" _builder_version="4.2.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the least possible value of[(x+1)(x+2)(x+3)(x+4)+2019]where $x$ is a real number?

$\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 4" _builder_version="4.4.0" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]A driver travels for $2$ hours at $60$ miles per hour, during which her car gets $30$ miles per gallon of gasoline. She is paid $dollar 0.50$ per mile, and her only expense is gasoline at $dollar2.00$ per gallon. What is her net rate of pay, in dollars per hour, after this expense?

$\textbf{(A) }20 \qquad\textbf{(B) }22 \qquad\textbf{(C) }24 \qquad\textbf{(D) } 25\qquad\textbf{(E) } 26$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 6" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]How many $4$-digit positive integers (that is, integers between $1000$ and $9999$, inclusive) having only even digits are divisible by $5?$

$\textbf{(A) } 80 \qquad \textbf{(B) } 100 \qquad \textbf{(C) } 125 \qquad \textbf{(D) } 200 \qquad \textbf{(E) } 500$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 7" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum?

$\textbf{(A) }2 \qquad\textbf{(B) } 5\qquad\textbf{(C) } 10\qquad\textbf{(D) } 25\qquad\textbf{(E) } 50$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 8" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the value of

$1+2+3-4+5+6+7-8+\cdots+197+198+199-200?$

$\textbf{(A) } 9,800 \qquad \textbf{(B) } 9,900 \qquad \textbf{(C) } 10,000 \qquad \textbf{(D) } 10,100 \qquad \textbf{(E) } 10,200$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 17" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Define[P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).]How many integers $n$ are there such that $P(n)\leq 0$?

$\textbf{(A) } 4900 \qquad \textbf{(B) } 4950\qquad \textbf{(C) } 5000\qquad \textbf{(D) } 5050 \qquad \textbf{(E) } 5100$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 18" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Define[P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).]How many integers $n$ are there such that $P(n)\leq 0$?

$\textbf{(A) } 4900 \qquad \textbf{(B) } 4950\qquad \textbf{(C) } 5000\qquad \textbf{(D) } 5050 \qquad \textbf{(E) } 5100$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 21" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]There exists a unique strictly increasing sequence of nonnegative integers $a_1 < a_2 < … < a_k$ such that[\frac{2^{289}+1}{2^{17}+1} = 2^{a_1} + 2^{a_2} + … + 2^{a_k}.]What is $k?$

$\textbf{(A) } 117 \qquad \textbf{(B) } 136 \qquad \textbf{(C) } 137 \qquad \textbf{(D) } 273 \qquad \textbf{(E) } 306$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 22" _builder_version="4.4.0" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]For how many positive integers $n \le 1000$ is[\left\lfloor \frac{998}{n} \right\rfloor+\left\lfloor \frac{999}{n} \right\rfloor+\left\lfloor \frac{1000}{n}\right \rfloor]not divisible by $3$? (Recall that $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.)

$\textbf{(A) } 22 \qquad\textbf{(B) } 23 \qquad\textbf{(C) } 24 \qquad\textbf{(D) } 25 \qquad\textbf{(E) } 26$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2020, Problem 24" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Let $n$ be the least positive integer greater than $1000$ for which[\gcd(63, n+120) =21\quad \text{and} \quad \gcd(n+63, 120)=60.]What is the sum of the digits of $n$?

$\textbf{(A) } 12 \qquad\textbf{(B) } 15 \qquad\textbf{(C) } 18 \qquad\textbf{(D) } 21\qquad\textbf{(E) } 24$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2020, Problem 1" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the value of[1-(-2)-3-(-4)-5-(-6)?]
$\textbf{(A)}\ -20 \qquad\textbf{(B)}\ -3 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 5 \qquad\textbf{(E)}\ 21$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2020, Problem 3" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$

$\textbf{(A)}\ 4:3 \qquad\textbf{(B)}\ 3:2 \qquad\textbf{(C)}\ 8:3 \qquad\textbf{(D)}\ 4:1 \qquad\textbf{(E)}\ 16:3$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2020, Problem 6" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Driving along a highway, Megan noticed that her odometer showed $15951$ (miles). This number is a palindrome-it reads the same forward and backward. Then $2$ hours later, the odometer displayed the next higher palindrome. What was her average speed, in miles per hour, during this $2$-hour period?

$\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 55 \qquad\textbf{(C)}\ 60 \qquad\textbf{(D)}\ 65 \qquad\textbf{(E)}\ 70$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2020, Problem 7" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]How many positive even multiples of $3$ less than $2020$ are perfect squares?

$\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 12$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2020, Problem 9" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]How many ordered pairs of integers $(x,y)$ satisfy the equation[x^{2020} + y^2 = 2y?]
$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{infinitely many}$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2020, Problem 12" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]The decimal representation of[\frac{1}{20^{20}}]consists of a string of zeros after the decimal point, followed by a $9$ and then several more digits. How many zeros are in that initial string of zeros after the decimal point?

$\textbf{(A)}\ 23 \qquad\textbf{(B)}\ 24 \qquad\textbf{(C)}\ 25 \qquad\textbf{(D)}\ 26 \qquad\textbf{(E)}\ 27$

[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2020, Problem 15" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Steve wrote the digits $1$, $2$, $3$, $4$, and $5$ in order repeatedly from left to right, forming a list of $10,000$ digits, beginning $123451234512\ldots.$ He then erased every third digit from his list (that is, the $3$rd, $6$th, $9$th, $\ldots$ digits from the left), then erased every fourth digit from the resulting list (that is, the $4$th, $8$th, $12$th, $\ldots$ digits from the left in what remained), and then erased every fifth digit from what remained at that point. What is the sum of the three digits that were then in the positions $2019, 2020, 2021$?

$\textbf{(A)} \text{ 7} \qquad \textbf{(B)} \text{ 9} \qquad \textbf{(C)} \text{ 10} \qquad \textbf{(D)} \text{ 11} \qquad \textbf{(E)} \text{ 12}$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2020, Problem 16" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Bela and Jenn play the following game on the closed interval $[0, n]$ of the real number line, where $n$ is a fixed integer greater than $4$. They take turns playing, with Bela going first. At his first turn, Bela chooses any real number in the interval $[0, n]$. Thereafter, the player whose turn it is chooses a real number that is more than one unit away from all numbers previously chosen by either player. A player unable to choose such a number loses. Using optimal strategy, which player will win the game?

$\textbf{(A)} \text{ Bela will always win.} \qquad \textbf{(B)} \text{ Jenn will always win.} \qquad \textbf{(C)} \text{Bela will win if and only if }n \text{ is odd.}$ $\textbf{(D)} \text{Jenn will win if and only if }n \text{ is odd.} \qquad \textbf{(E)} \text { Jenn will win if and only if } n>8.$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2020, Problem 22" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$?

$\textbf{(A) } 100 \qquad\textbf{(B) } 101 \qquad\textbf{(C) } 200 \qquad\textbf{(D) } 201 \qquad\textbf{(E) } 202$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2020, Problem 24" _builder_version="4.4.0" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]How many positive integers $n$ satisfy[\frac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?](Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)

$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 1" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the value of[2^{\left(0^{\left(1^9\right)}\right)}+\left(\left(2^0\right)^1\right)^9?]$\textbf{(A) } 0 \qquad\textbf{(B) } 1 \qquad\textbf{(C) } 2 \qquad\textbf{(D) } 3 \qquad\textbf{(E) } 4$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 2" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the hundreds digit of $(20!-15!)?$

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }5$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 3" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Ana and Bonita are born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$

$\textbf{(A) } 3 \qquad\textbf{(B) } 5 \qquad\textbf{(C) } 9 \qquad\textbf{(D) } 12 \qquad\textbf{(E) } 15$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 5" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the greatest number of consecutive integers whose sum is $45?$

$\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 9" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?

$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 15" builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and[a_n=\frac{a{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}]for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q ?$

$\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 18" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]For some positive integer $k$, the repeating base-$k$ representation of the (base-ten) fraction $\frac{7}{51}$ is $0.\overline{23}_k = 0.232323..._k$. What is $k$?

$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16 \qquad\textbf{(E) } 17$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 19" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the least possible value of[(x+1)(x+2)(x+3)(x+4)+2019]where $x$ is a real number?

$\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 24" _builder_version="4.4.0" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" hover_enabled="0" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Let $p$, $q$, and $r$ be the distinct roots of the polynomial $x^{3} - 22x^{2} + 80x - 67$. It is given that there exist real numbers $A$, $B$, and $C$ such that[\frac{1}{s^{3} - 22s^{2} + 80s - 67} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r}]for all $s\not\in{p,q,r}$. What is $\frac1A+\frac1B+\frac1C$?

$\textbf{(A) }243\qquad\textbf{(B) }244\qquad\textbf{(C) }245\qquad\textbf{(D) }246\qquad\textbf{(E) } 247$

[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 25" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]For how many integers $n$ between $1$ and $50$, inclusive, is[\frac{(n^2-1)!}{(n!)^{n}}]an integer? (Recall that $0!=1$.)

$\textbf{(A) } 31 \qquad \textbf{(B) } 32 \qquad \textbf{(C) } 33 \qquad \textbf{(D) } 34 \qquad \textbf{(E) } 35$

[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2019, Problem 2" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement?

$\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2019, Problem 3" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]In a high school with $500$ students, $40\%$ of the seniors play a musical instrument, while $30\%$ of the non-seniors do not play a musical instrument. In all, $46.8\%$ of the students do not play a musical instrument. How many non-seniors play a musical instrument?

$\textbf{(A) } 66 \qquad\textbf{(B) } 154 \qquad\textbf{(C) } 186 \qquad\textbf{(D) } 220 \qquad\textbf{(E) } 266$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2019, Problem 6" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]There is a real $n$ such that $(n+1)! + (n+2)! = n! \cdot 440$. What is the sum of the digits of $n$?

$\textbf{(A) }3\qquad\textbf{(B) }8\qquad\textbf{(C) }10\qquad\textbf{(D) }11\qquad\textbf{(E) }12$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2019, Problem 11" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $9:1$, and the ratio of blue to green marbles in Jar $2$ is $8:1$. There are $95$ green marbles in all. How many more blue marbles are in Jar $1$ than in Jar $2$?

$\textbf{(A) } 5 \qquad\textbf{(B) } 10 \qquad\textbf{(C) } 25 \qquad\textbf{(D) } 45 \qquad\textbf{(E) } 50$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2019, Problem 12" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the greatest possible sum of the digits in the base-seven representation of a positive integer less than $2019$?

$\textbf{(A) } 11 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 22 \qquad\textbf{(D) } 23 \qquad\textbf{(E) } 27$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2019, Problem 15" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Right triangles $T_1$ and $T_2$ have areas 1 and 2, respectively. A side of $T_1$ is congruent to a side of $T_2$, and a different side of $T_1$ is congruent to a different side of $T_2$. What is the square of the product of the other (third) sides of $T_1$ and $T_2$?

$\textbf{(A) } \frac{28}{3} \qquad\textbf{(B) }10\qquad\textbf{(C) } \frac{32}{3} \qquad\textbf{(D) } \frac{34}{3} \qquad\textbf{(E) }12$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2019, Problem 18" _builder_version="4.4.0" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Henry decides one morning to do a workout, and he walks $\frac{3}{4}$ of the way from his home to his gym. The gym is $2$ kilometers away from Henry's home. At that point, he changes his mind and walks $\frac{3}{4}$ of the way from where he is back toward home. When he reaches that point, he changes his mind again and walks $\frac{3}{4}$ of the distance from there back toward the gym. If Henry keeps changing his mind when he has walked $\frac{3}{4}$ of the distance toward either the gym or home from the point where he last changed his mind, he will get very close to walking back and forth between a point $A$ kilometers from home and a point $B$ kilometers from home. What is $|A-B|$?

$\textbf{(A) } \frac{2}{3} \qquad \textbf{(B) } 1 \qquad \textbf{(C) } 1\frac{1}{5} \qquad \textbf{(D) } 1\frac{1}{4} \qquad \textbf{(E) } 1\frac{1}{2}$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2019, Problem 19" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Let $S$ be the set of all positive integer divisors of $100,000.$ How many numbers are the product of two distinct elements of $S?$

$\textbf{(A) }98\qquad\textbf{(B) }100\qquad\textbf{(C) }117\qquad\textbf{(D) }119\qquad\textbf{(E) }121$[/et_pb_team_member][et_pb_team_member name="AMC 10B, 2019, Problem 24" builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Define a sequence recursively by $x_0=5$ and[x{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}]for all nonnegative integers $n.$ Let $m$ be the least positive integer such that[x_m\leq 4+\frac{1}{2^{20}}.]In which of the following intervals does $m$ lie?

$\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty)$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2018, Problem 1" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the value of[\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?]$\textbf{(A) } \frac58 \qquad \textbf{(B) }\frac{11}7 \qquad \textbf{(C) } \frac85 \qquad \textbf{(D) } \frac{18}{11} \qquad \textbf{(E) } \frac{15}8$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2018, Problem 2" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Liliane has $50\%$ more soda than Jacqueline, and Alice has $25\%$ more soda than Jacqueline. What is the relationship between the amounts of soda that Liliane and Alice have?

$\textbf{(A)}$ Liliane has $20\%$ more soda than Alice.

$\textbf{(B)}$ Liliane has $25\%$ more soda than Alice.

$\textbf{(C)}$ Liliane has $45\%$ more soda than Alice.

$\textbf{(D)}$ Liliane has $75\%$ more soda than Alice.

$\textbf{(E)}$ Liliane has $100\%$ more soda than Alice.[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2018, Problem 3" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]A unit of blood expires after $10!=10\cdot 9 \cdot 8 \cdots 1$ seconds. Yasin donates a unit of blood at noon of January 1. On what day does his unit of blood expire?

$\textbf{(A) }\text{January 2}\qquad\textbf{(B) }\text{January 12}\qquad\textbf{(C) }\text{January 22}\qquad\textbf{(D) }\text{February 11}\qquad\textbf{(E) }\text{February 12}$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2018, Problem 5" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]Alice, Bob, and Charlie were on a hike and were wondering how far away the nearest town was. When Alice said, "We are at least 6 miles away," Bob replied, "We are at most 5 miles away." Charlie then remarked, "Actually the nearest town is at most 4 miles away." It turned out that none of the three statements were true. Let $d$ be the distance in miles to the nearest town. Which of the following intervals is the set of all possible values of $d$?

$\textbf{(A) } (0,4) \qquad \textbf{(B) } (4,5) \qquad \textbf{(C) } (4,6) \qquad \textbf{(D) } (5,6) \qquad \textbf{(E) } (5,\infty)$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 9" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?

$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 9" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?

$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 9" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?

$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$[/et_pb_team_member][et_pb_team_member name="AMC 10A, 2019, Problem 9" _builder_version="4.3.2" custom_margin="20px||20px||false|false" custom_padding="10px|20px|20px|20px|false|false" border_radii="on|5px|5px|5px|5px" border_radii_image="on|5px|5px|5px|5px" box_shadow_style="preset1"]What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers?

$\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$[/et_pb_team_member][/et_pb_column][/et_pb_row][/et_pb_section][et_pb_section fb_built="1" _builder_version="4.2.2"][et_pb_row _builder_version="4.2.2"][et_pb_column type="4_4" _builder_version="4.2.2"][/et_pb_column][/et_pb_row][/et_pb_section]

### Cheenta. Passion for Mathematics

Advanced Mathematical Science. Taught by olympians, researchers and true masters of the subject.