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Difference between revisions of "AMC 10 2021 (Mock) Problems"

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==Problem 1==
 
==Problem 1==
  
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Given that <math>A + B - C = 2020, B + C - A = 2021,</math> and <math>A + C - B = 2022,</math> what is the value of <math>A + B + C - 2020 - 2021 - 2022</math>?
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Given that your mom + B - C = 2020, B + C - A = 2021,<math> and </math>A + C - B = 2022,<math> what is the value of </math>A + B + C - 2020 - 2021 - 2022<math>?
  
  
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<math>\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2020\qquad \mathrm{(C) \ } 2021\qquad \mathrm{(D) \ } 2022\qquad \mathrm{(E) \ } 6063</math>
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</math>\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2020\qquad \mathrm{(C) \ } 2021\qquad \mathrm{(D) \ } 2022\qquad \mathrm{(E) \ } 6063$
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==Problem 2==
 
==Problem 2==
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Let a quadratic <math>Ax^2 + Bx + C</math>, where <math>A, B</math> and <math>C</math> are nonzero, have two distinct, real roots <math>a</math> and <math>b</math>. Given that <math>\frac{1}{a}</math> <math>+</math> <math>\frac{1}{b}</math> <math>=</math> <math>\frac{1}{6}</math>, which of the following conditions must be true?
 
Let a quadratic <math>Ax^2 + Bx + C</math>, where <math>A, B</math> and <math>C</math> are nonzero, have two distinct, real roots <math>a</math> and <math>b</math>. Given that <math>\frac{1}{a}</math> <math>+</math> <math>\frac{1}{b}</math> <math>=</math> <math>\frac{1}{6}</math>, which of the following conditions must be true?
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<math>\mathrm{(A) \ } B^2 - 4A = 0 \qquad \mathrm{(B) \ } B - 24A \ge 0 \qquad \mathrm{(C) \ } B^3 - A^2 \ge 24 \qquad \mathrm{(D) \ } A + B = 6 \qquad \mathrm{(E) \ } B + 24A - C = 24</math>
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==Problem 6==
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Two glass containers stand on a tabletop. The first one is a solution consisting of <math>80\%</math> water and <math>20\%</math> alcohol. The second solution consists of <math>35\%</math> water and <math>65\%</math> alcohol. A third container is placed on the same table and must be a mixture of the first and second solutions in a particular ratio. After the third container is filled as per these conditions, it consists of an equal amount of alcohol and water. What is the ratio of the amount of the first solution mixed in the third container and the amount of the second solution mixed in the third container?
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<math>\mathrm{(A) \ } 1 : 4\qquad \mathrm{(B) \ } 1 : 3 \qquad \mathrm{(C) \ } 1 : 2 \qquad \mathrm{(D) \ } 2 : 1 \qquad \mathrm{(E) \ } 3 : 1</math>
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==Problem 7==
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What is the value of <math>1 + 3 - 5 + 7 + 9 - 11 + … + 475 + 477 - 479</math>?
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<math>\mathrm{(A) \ } 16,340 \qquad \mathrm{(B) \ } 17,280 \qquad \mathrm{(C) \ } 18,220 \qquad \mathrm{(D) \ } 19,160 \qquad \mathrm{(E) \ } 20,100</math>
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==Problem 8==
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Let <math>\overline{AB}</math> be a segment in the real coordinate plane. How many points C exist on the same plane, such that <math>\triangle ABC</math> is isosceles?

Latest revision as of 18:42, 28 September 2023

Problem 1

Given that your mom + B - C = 2020, B + C - A = 2021,$and$A + C - B = 2022,$what is the value of$A + B + C - 2020 - 2021 - 2022$?$\mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 2020\qquad \mathrm{(C) \ } 2021\qquad \mathrm{(D) \ } 2022\qquad \mathrm{(E) \ } 6063$

Problem 2

A bag of marbles consists of $4$ red marbles and $3$ blue marbles. Each of these $7$ marbles are pulled out one at a time. What is the probability that the $5th$ marble pulled out is red?


$\mathrm{(A) \ } \frac{1}{5}\qquad \mathrm{(B) \ } \frac{3}{7}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{4}{7}\qquad \mathrm{(E) \ } \frac{4}{5}$


Problem 3

Meena has $11$ snakes, $6$ are purple and the rest are green. Some of the snakes are poisonous. She knows that $2$ of the poisonous snakes are green and the number of poisonous snakes which are purple is double the amount of poisonous snakes that are green. How many snakes are not poisonous?


$\mathrm{(A) \ } 3 \qquad \mathrm{(B) \ } 4 \qquad \mathrm{(C) \ } 5\qquad \mathrm{(D) \ }6\qquad \mathrm{(E) \ }7$


Problem 4

At a birthday celebration consisting of $K$ kids, each of the $K$ kids purchase $S$ soda cans except for the birthday kid, who purchases $2S$ soda cans. $S$ soda cans cost $Q$ quarters. How many dollars were spent total on the soda cans?

$\mathrm{(A) \ } \frac{(K - 1)Q}{6}\qquad \mathrm{(B) \ } \frac{(K - 1)Q}{4}\qquad \mathrm{(C) \ } \frac{KQ}{4}\qquad \mathrm{(D) \ } \frac{(K + 1)Q}{4}\qquad \mathrm{(E) \ } \frac{(K + 1)Q}{2}$


Problem 5

Let a quadratic $Ax^2 + Bx + C$, where $A, B$ and $C$ are nonzero, have two distinct, real roots $a$ and $b$. Given that $\frac{1}{a}$ $+$ $\frac{1}{b}$ $=$ $\frac{1}{6}$, which of the following conditions must be true?


$\mathrm{(A) \ } B^2 - 4A = 0 \qquad \mathrm{(B) \ } B - 24A \ge 0 \qquad \mathrm{(C) \ } B^3 - A^2 \ge 24 \qquad \mathrm{(D) \ } A + B = 6 \qquad \mathrm{(E) \ } B + 24A - C = 24$

Problem 6

Two glass containers stand on a tabletop. The first one is a solution consisting of $80\%$ water and $20\%$ alcohol. The second solution consists of $35\%$ water and $65\%$ alcohol. A third container is placed on the same table and must be a mixture of the first and second solutions in a particular ratio. After the third container is filled as per these conditions, it consists of an equal amount of alcohol and water. What is the ratio of the amount of the first solution mixed in the third container and the amount of the second solution mixed in the third container?

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


Problem 7

What is the value of $1 + 3 - 5 + 7 + 9 - 11 + … + 475 + 477 - 479$?


$\mathrm{(A) \ } 16,340 \qquad \mathrm{(B) \ } 17,280 \qquad \mathrm{(C) \ } 18,220 \qquad \mathrm{(D) \ } 19,160 \qquad \mathrm{(E) \ } 20,100$


Problem 8

Let $\overline{AB}$ be a segment in the real coordinate plane. How many points C exist on the same plane, such that $\triangle ABC$ is isosceles?

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