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Difference between revisions of "2019 AMC 10C Problems"

(Added a page on my mock amc 10)
 
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Given that <math>a+m=2017</math>, <math>m+c=2018</math>, and <math>a+c=2019</math>, find the value of <math>a+m+c+10</math>.
 
Given that <math>a+m=2017</math>, <math>m+c=2018</math>, and <math>a+c=2019</math>, find the value of <math>a+m+c+10</math>.
 +
  
 
<math>\mathrm{(A) \ } 3027\qquad \mathrm{(B) \ } 3037\qquad \mathrm{(C) \ } 4037\qquad \mathrm{(D) \ } 6054\qquad \mathrm{(E) \ } 6064</math>
 
<math>\mathrm{(A) \ } 3027\qquad \mathrm{(B) \ } 3037\qquad \mathrm{(C) \ } 4037\qquad \mathrm{(D) \ } 6054\qquad \mathrm{(E) \ } 6064</math>
 +
 +
[[2019 AMC 10C Problems/Problem 1|Solution]]
 +
  
 
==Problem 2==
 
==Problem 2==
  
 
Three distinct vertices of a regular pentagon are chosen at random. What is the probability that they form an obtuse triangle?
 
Three distinct vertices of a regular pentagon are chosen at random. What is the probability that they form an obtuse triangle?
 +
 +
 
<math>\mathrm{(A) \ } \frac{1}{10}\qquad \mathrm{(B) \ } \frac{1}{5}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{4}{5}\qquad \mathrm{(E) \ } \frac{9}{10}</math>
 
<math>\mathrm{(A) \ } \frac{1}{10}\qquad \mathrm{(B) \ } \frac{1}{5}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{4}{5}\qquad \mathrm{(E) \ } \frac{9}{10}</math>
 +
 +
[[2019 AMC 10C Problems/Problem 2|Solution]]
 +
  
 
==Problem 3==
 
==Problem 3==
  
 
The numbers from 1 to 10 are written on 10 slips of paper, with 1 number on each slip of paper, and put in a bag. Maria will randomly draw 2 slips from the bag without replacement. What is the probability that she will obtain one prime number and one composite number?
 
The numbers from 1 to 10 are written on 10 slips of paper, with 1 number on each slip of paper, and put in a bag. Maria will randomly draw 2 slips from the bag without replacement. What is the probability that she will obtain one prime number and one composite number?
 +
 +
 
<math>\mathrm{(A) \ } \frac{2}{15}\qquad \mathrm{(B) \ }  \frac{2}{5}\qquad \mathrm{(C) \ } \frac{4}{9}\qquad \mathrm{(D) \ } \frac{5}{9}\qquad \mathrm{(E) \ } \frac{8}{9}</math>
 
<math>\mathrm{(A) \ } \frac{2}{15}\qquad \mathrm{(B) \ }  \frac{2}{5}\qquad \mathrm{(C) \ } \frac{4}{9}\qquad \mathrm{(D) \ } \frac{5}{9}\qquad \mathrm{(E) \ } \frac{8}{9}</math>
 +
 +
[[2019 AMC 10C Problems/Problem 3|Solution]]
 +
  
 
==Problem 4==
 
==Problem 4==
Line 20: Line 34:
 
In the figure below, the part of the 1st square that is not contained in the 2nd square will be shaded, the part of the 3rd square that is not contained in the 4th square will be shaded, the part of the 5th square that is not contained in the 6th square will be shaded, and so on to infinity. What is the ratio of the shaded area to the non-shaded area?
 
In the figure below, the part of the 1st square that is not contained in the 2nd square will be shaded, the part of the 3rd square that is not contained in the 4th square will be shaded, the part of the 5th square that is not contained in the 6th square will be shaded, and so on to infinity. What is the ratio of the shaded area to the non-shaded area?
  
−
[asy]
 
−
/* Geogebra to Asymptote conversion, documentation at artofproblemsolving.com/Wiki go to User:Azjps/geogebra */
 
−
import graph; size(5cm);
 
−
real labelscalefactor = 0.5; /* changes label-to-point distance */
 
−
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); /* default pen style */
 
−
pen dotstyle = black; /* point style */
 
−
real xmin = -7.92, xmax = 11.8, ymin = -4.55, ymax = 7.67;  /* image dimensions */
 
  
 +
<math>\mathrm{(A) \ } \frac{1}{3}\qquad \mathrm{(B) \ } \frac{1}{2}\qquad \mathrm{(C) \ } \frac{3}{2}\qquad \mathrm{(D) \ } 2\qquad \mathrm{(E) \ } 3</math>
  
−
draw((0,0)--(4,0)--(4,4)--(0,4)--cycle, linewidth(2));
+
[[2019 AMC 10C Problems/Problem 4|Solution]]
−
draw((0,2)--(2,0)--(4,2)--(2,4)--cycle, linewidth(2));
 
−
draw((1,3)--(1,1)--(3,1)--(3,3)--cycle, linewidth(2));
 
−
draw((2,3)--(3,2)--(2,1)--(1,2)--cycle, linewidth(2));
 
−
draw((2.5,2.5)--(1.5,2.5)--(1.5,1.5)--(2.5,1.5)--cycle, linewidth(2));
 
−
/* draw figures */
 
−
draw((0,0)--(4,0), linewidth(2));
 
−
draw((4,0)--(4,4), linewidth(2));
 
−
draw((4,4)--(0,4), linewidth(2));
 
−
draw((0,4)--(0,0), linewidth(2));
 
−
draw((0,2)--(2,0), linewidth(2));
 
−
draw((2,0)--(4,2), linewidth(2));
 
−
draw((4,2)--(2,4), linewidth(2));
 
−
draw((2,4)--(0,2), linewidth(2));
 
−
draw((1,3)--(1,1), linewidth(2));
 
−
draw((1,1)--(3,1), linewidth(2));
 
−
draw((3,1)--(3,3), linewidth(2));
 
−
draw((3,3)--(1,3), linewidth(2));
 
−
draw((2,3)--(3,2), linewidth(2));
 
−
draw((3,2)--(2,1), linewidth(2));
 
−
draw((2,1)--(1,2), linewidth(2));
 
−
draw((1,2)--(2,3), linewidth(2));
 
−
draw((2.5,2.5)--(1.5,2.5), linewidth(2));
 
−
draw((1.5,2.5)--(1.5,1.5), linewidth(2));
 
−
draw((1.5,1.5)--(2.5,1.5), linewidth(2));
 
−
draw((2.5,1.5)--(2.5,2.5), linewidth(2));
 
−
clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle);
 
−
/* end of picture */
 
−
[/asy]
 
  
−
<math>\mathrm{(A) \ } \frac{1}{3}\qquad \mathrm{(B) \ } \frac{1}{2}\qquad \mathrm{(C) \ } \frac{3}{2}\qquad \mathrm{(D) \ } 2\qquad \mathrm{(E) \ } 3</math>
 
  
 
==Problem 5==
 
==Problem 5==
  
 
The numeral representation of the product <math>99...9 \times 999...9</math>, where the former number has <math>n</math> 9’s and the latter number has <math>n+1</math> 9’s, where <math>n</math> is some positive integer, has a sum of digits of 630. Find <math>n</math>.
 
The numeral representation of the product <math>99...9 \times 999...9</math>, where the former number has <math>n</math> 9’s and the latter number has <math>n+1</math> 9’s, where <math>n</math> is some positive integer, has a sum of digits of 630. Find <math>n</math>.
 +
 +
 
<math>\mathrm{(A) \ } 68\qquad \mathrm{(B) \ } 69\qquad \mathrm{(C) \ } 70\qquad \mathrm{(D) \ } 71\qquad \mathrm{(E) \ } 72</math>
 
<math>\mathrm{(A) \ } 68\qquad \mathrm{(B) \ } 69\qquad \mathrm{(C) \ } 70\qquad \mathrm{(D) \ } 71\qquad \mathrm{(E) \ } 72</math>
 +
 +
[[2019 AMC 10C Problems/Problem 5|Solution]]
 +
  
 
==Problem 6==
 
==Problem 6==
  
−
A function <math>f(x)</math>is defined for all positive integers n>2 as <math>f(n)=f(n-1) \times f(n-2)</math>. Given that <math>f(1)=2</math> and <math>f(2)=3</math>, the value of <math>f(12)</math>can be expressed in the form of <math>2^{p}3^{q}</math>,where <math>p</math> and <math>q</math> are positive integers. Find <math>|p-q|</math>.
+
A function <math>f(x)</math>is defined for all positive integers <math>n>2</math> as <math>f(n)=f(n-1) \times f(n-2)</math>. Given that <math>f(1)=2</math> and <math>f(2)=3</math>, the value of <math>f(12)</math>can be expressed in the form of <math>2^{p}3^{q}</math>,where <math>p</math> and <math>q</math> are positive integers. Find <math>|p-q|</math>.
 +
 
 +
 
 
<math>\mathrm{(A) \ } 21\qquad \mathrm{(B) \ } 34\qquad \mathrm{(C) \ } 55\qquad \mathrm{(D) \ } 89\qquad \mathrm{(E) \ } 144</math>
 
<math>\mathrm{(A) \ } 21\qquad \mathrm{(B) \ } 34\qquad \mathrm{(C) \ } 55\qquad \mathrm{(D) \ } 89\qquad \mathrm{(E) \ } 144</math>
 +
 +
[[2019 AMC 10C Problems/Problem 6|Solution]]
 +
  
 
==Problem 7==
 
==Problem 7==
  
 
Given that <math>n^4 = 22,667,121</math> for a positive integer <math>n</math>, find the sum of the digits of <math>n</math>.
 
Given that <math>n^4 = 22,667,121</math> for a positive integer <math>n</math>, find the sum of the digits of <math>n</math>.
 +
  
 
<math>\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 18</math>
 
<math>\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 18</math>
 +
 +
[[2019 AMC 10C Problems/Problem 7|Solution]]
 +
  
 
==Problem 8==
 
==Problem 8==
  
 
When rolling 12 standard 6-sided dice, the probability that the sum of the numbers rolled on the 12 dice is 69 can be expressed in <math>\frac{N}{6^{12}}</math>. Find the sum of the digits of <math>N</math>.
 
When rolling 12 standard 6-sided dice, the probability that the sum of the numbers rolled on the 12 dice is 69 can be expressed in <math>\frac{N}{6^{12}}</math>. Find the sum of the digits of <math>N</math>.
 +
  
 
<math>\mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 10\qquad \mathrm{(C) \ } 13\qquad \mathrm{(D) \ } 14\qquad \mathrm{(E) \ } 15</math>
 
<math>\mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 10\qquad \mathrm{(C) \ } 13\qquad \mathrm{(D) \ } 14\qquad \mathrm{(E) \ } 15</math>
 +
 +
[[2019 AMC 10C Problems/Problem 8|Solution]]
 +
  
 
==Problem 9==
 
==Problem 9==
  
 
A gear with radius 34.5 rolls around another the circumference of a larger gear with radius 103.5. How many revolutions around the larger gear would the smaller gear have completed by the time it makes 3 complete rotations?
 
A gear with radius 34.5 rolls around another the circumference of a larger gear with radius 103.5. How many revolutions around the larger gear would the smaller gear have completed by the time it makes 3 complete rotations?
 +
  
 
<math>\mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \frac{3}{4}\qquad \mathrm{(D) \ } 1\qquad \mathrm{(E) \ } \frac{6}{5}</math>
 
<math>\mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \frac{3}{4}\qquad \mathrm{(D) \ } 1\qquad \mathrm{(E) \ } \frac{6}{5}</math>
 +
 +
[[2019 AMC 10C Problems/Problem 9|Solution]]
 +
  
 
==Problem 10==
 
==Problem 10==
  
 
Al and Bob play a game of Bottle Flip. Each of them takes turns flipping a water bottle and the first person to not land the bottle is deemed the loser. Al has a <math>\frac{4}{5}</math> chance of landing the bottle every time he flips it, and Bob has a <math>\frac{3}{4}</math> chance of landing the bottle every time he flips it. Given that Al goes first, what is the probability that he wins the game?
 
Al and Bob play a game of Bottle Flip. Each of them takes turns flipping a water bottle and the first person to not land the bottle is deemed the loser. Al has a <math>\frac{4}{5}</math> chance of landing the bottle every time he flips it, and Bob has a <math>\frac{3}{4}</math> chance of landing the bottle every time he flips it. Given that Al goes first, what is the probability that he wins the game?
 +
  
 
<math>\mathrm{(A) \ } \frac{1}{3}\qquad \mathrm{(B) \ } \frac{3}{8}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{5}{8} \qquad \mathrm{(E) \ } \frac{2}{3}</math>
 
<math>\mathrm{(A) \ } \frac{1}{3}\qquad \mathrm{(B) \ } \frac{3}{8}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{5}{8} \qquad \mathrm{(E) \ } \frac{2}{3}</math>
 +
 +
[[2019 AMC 10C Problems/Problem 10|Solution]]
 +
  
 
==Problem 11==
 
==Problem 11==
  
 
How many base 3 positive integers <math>n</math> with nonzero digits have the property that the sum of the digits of <math>n</math> is 10?
 
How many base 3 positive integers <math>n</math> with nonzero digits have the property that the sum of the digits of <math>n</math> is 10?
 +
 +
 
<math>\mathrm{(A) \ } 74\qquad \mathrm{(B) \ } 79\qquad \mathrm{(C) \ } 88\qquad \mathrm{(D) \ } 89\qquad \mathrm{(E) \ } 97</math>
 
<math>\mathrm{(A) \ } 74\qquad \mathrm{(B) \ } 79\qquad \mathrm{(C) \ } 88\qquad \mathrm{(D) \ } 89\qquad \mathrm{(E) \ } 97</math>
 +
 +
[[2019 AMC 10C Problems/Problem 11|Solution]]
 +
  
 
==Problem 12==
 
==Problem 12==
  
 
In music, a whole octave consists of 12 semitones: C, C#, D, D#, E, F, F#, G, G#, A, A#, and B. Consist a certain four-note chord to be pretty if all four notes are from the same octave and no two notes are either <math>1</math> semitone or <math>11</math> semitones apart. How many four-note chords are not pretty?
 
In music, a whole octave consists of 12 semitones: C, C#, D, D#, E, F, F#, G, G#, A, A#, and B. Consist a certain four-note chord to be pretty if all four notes are from the same octave and no two notes are either <math>1</math> semitone or <math>11</math> semitones apart. How many four-note chords are not pretty?
 +
  
 
<math>\textbf{(A)}\ 117 \qquad\textbf{(B)}\ 369 \qquad\textbf{(C)}\ 390 \qquad\textbf{(D)}\ 425 \qquad\textbf{(E)}\ 440</math>
 
<math>\textbf{(A)}\ 117 \qquad\textbf{(B)}\ 369 \qquad\textbf{(C)}\ 390 \qquad\textbf{(D)}\ 425 \qquad\textbf{(E)}\ 440</math>
 +
 +
[[2019 AMC 10C Problems/Problem 12|Solution]]
 +
  
 
==Problem 13==
 
==Problem 13==
  
−
How many positive integers <math>N</math> where 1≤ N ≤20cannot be the number of 0s at the end of the decimal representation of <math>x!</math>, where <math>x</math> is a positive integer?
+
How many positive integers <math>N</math> where <math>1 \leq N \leq 20</math> cannot be the number of 0s at the end of the decimal representation of <math>x!</math>, where <math>x</math> is a positive integer?
 +
 
 +
 
 
<math>\mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6</math>
 
<math>\mathrm{(A) \ } 2\qquad \mathrm{(B) \ } 3\qquad \mathrm{(C) \ } 4\qquad \mathrm{(D) \ } 5\qquad \mathrm{(E) \ } 6</math>
 +
 +
[[2019 AMC 10C Problems/Problem 13|Solution]]
 +
  
 
==Problem 14==
 
==Problem 14==
  
 
Let <math>M</math> denote the largest integer power of 16 that allows <math>100^{2019}-16^{M}</math> to be a positive integer and let <math>N</math> denote the smallest integer power of 100 that allows <math>16^{2019}-100^{N}</math> to be a negative integer. In which of the following ranges does <math>M+N</math> lie?
 
Let <math>M</math> denote the largest integer power of 16 that allows <math>100^{2019}-16^{M}</math> to be a positive integer and let <math>N</math> denote the smallest integer power of 100 that allows <math>16^{2019}-100^{N}</math> to be a negative integer. In which of the following ranges does <math>M+N</math> lie?
 +
  
 
<math>\mathrm{(A) \ } [2000,3000]\qquad \mathrm{(B) \ } [3000,4000]\qquad \mathrm{(C) \ } [4000,5000]\qquad \mathrm{(D) \ } [5000,6000]\qquad \mathrm{(E) \ } [6000,7000]</math>
 
<math>\mathrm{(A) \ } [2000,3000]\qquad \mathrm{(B) \ } [3000,4000]\qquad \mathrm{(C) \ } [4000,5000]\qquad \mathrm{(D) \ } [5000,6000]\qquad \mathrm{(E) \ } [6000,7000]</math>
 +
 +
[[2019 AMC 10C Problems/Problem 14|Solution]]
 +
  
 
==Problem 15==
 
==Problem 15==
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Let <math>N</math> be <math>2019^{2019^{2019^{2019...}}}</math>. (2019 <math>2019</math>'s)
 
Let <math>N</math> be <math>2019^{2019^{2019^{2019...}}}</math>. (2019 <math>2019</math>'s)
 
Find the remainder when <math>N</math> is divided by <math>7</math>.
 
Find the remainder when <math>N</math> is divided by <math>7</math>.
 +
 +
 
<math>\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 6</math>
 
<math>\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 6</math>
 +
 +
[[2019 AMC 10C Problems/Problem 15|Solution]]
 +
  
 
==Problem 16==
 
==Problem 16==
  
 
Given that :
 
Given that :
−
1.The sum of the roots of <math>ax^3+bx^2+cx+d=0</math>, where <math>a, b, c, </math> and <math>d</math> are integers no more than 1000 and no less than -1000, is <math>7</math>.
+
1.The sum of the roots of <math>ax^3+bx^2+cx+d=0</math>, where <math>a, b, c, </math> and <math>d</math> are integers no more than 1000 and no less than <math>-1000</math>, is <math>7</math>.
−
2.The product of the roots in <math>ax^4+bx^3+cx+abcd=0</math>, where <math>a, b, c,</math> and <math>d</math> are integers no more than 1000 and no less than -1000, is <math>420</math>.
+
2.The product of the roots in <math>ax^4+bx^3+cx+abcd=0</math>, where <math>a, b, c,</math> and <math>d</math> are integers no more than 1000 and no less than <math>-1000</math>, is <math>420</math>.
 
Find the number of ordered pairs <math>(a, b, c, d)</math> that satisfy these conditions.
 
Find the number of ordered pairs <math>(a, b, c, d)</math> that satisfy these conditions.
 +
  
 
<math>\mathrm{(A) \ } 27\qquad \mathrm{(B) \ } 54\qquad \mathrm{(C) \ } 108\qquad \mathrm{(D) \ } 162\qquad \mathrm{(E) \ } 216</math>
 
<math>\mathrm{(A) \ } 27\qquad \mathrm{(B) \ } 54\qquad \mathrm{(C) \ } 108\qquad \mathrm{(D) \ } 162\qquad \mathrm{(E) \ } 216</math>
 +
 +
[[2019 AMC 10C Problems/Problem 16|Solution]]
 +
  
 
==Problem 17==
 
==Problem 17==
  
 
Define a subset of the first ten positive integers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} to be spicy if the sum of the squares of the elements in the subset is divisible by 5. How many spicy subsets are there?
 
Define a subset of the first ten positive integers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} to be spicy if the sum of the squares of the elements in the subset is divisible by 5. How many spicy subsets are there?
 +
  
 
<math>\mathrm{(A) \ } 35\qquad \mathrm{(B) \ } 70\qquad \mathrm{(C) \ } 140\qquad \mathrm{(D) \ } 210\qquad \mathrm{(E) \ } 280</math>
 
<math>\mathrm{(A) \ } 35\qquad \mathrm{(B) \ } 70\qquad \mathrm{(C) \ } 140\qquad \mathrm{(D) \ } 210\qquad \mathrm{(E) \ } 280</math>
 +
 +
[[2019 AMC 10C Problems/Problem 17|Solution]]
 +
  
 
==Problem 18==
 
==Problem 18==
  
 
A tripod has three legs of length 3 feet, 4 feet, and 4 feet. It is set up so that the angle between any two legs is <math>90^{\circ}</math>. The height from the top of the tripod to the ground, in feet, can be expressed in the form <math>\dfrac{a\sqrt{b}}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, <math>a</math> and <math>c</math> are relatively prime, and <math>b</math> is not divisible by the square of a prime. What is <math>a+b+c</math>?
 
A tripod has three legs of length 3 feet, 4 feet, and 4 feet. It is set up so that the angle between any two legs is <math>90^{\circ}</math>. The height from the top of the tripod to the ground, in feet, can be expressed in the form <math>\dfrac{a\sqrt{b}}{c}</math>, where <math>a</math>, <math>b</math>, and <math>c</math> are positive integers, <math>a</math> and <math>c</math> are relatively prime, and <math>b</math> is not divisible by the square of a prime. What is <math>a+b+c</math>?
 +
  
 
<math>\textbf{(A)}\ 53 \qquad\textbf{(B)}\ 57 \qquad\textbf{(C)}\ 69 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 89</math>
 
<math>\textbf{(A)}\ 53 \qquad\textbf{(B)}\ 57 \qquad\textbf{(C)}\ 69 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 89</math>
 +
 +
[[2019 AMC 10C Problems/Problem 18|Solution]]
 +
  
 
==Problem 19==
 
==Problem 19==
  
 
Triangle <math>ABC</math> has <math>AB=10</math>, <math>BC=14</math>, and <math>AC=22</math>. The line <math>AN</math>, where <math>N</math> is a point on <math>BC</math>, divides the triangle into two halves with equal perimeters. Let <math>N</math> be the length of <math>AN</math>. Find the greatest integer less or equal to <math>N</math>.
 
Triangle <math>ABC</math> has <math>AB=10</math>, <math>BC=14</math>, and <math>AC=22</math>. The line <math>AN</math>, where <math>N</math> is a point on <math>BC</math>, divides the triangle into two halves with equal perimeters. Let <math>N</math> be the length of <math>AN</math>. Find the greatest integer less or equal to <math>N</math>.
 +
 +
 
<math>\mathrm{(A) \ } 18\qquad \mathrm{(B) \ } 19\qquad \mathrm{(C) \ } 20\qquad \mathrm{(D) \ } 21\qquad \mathrm{(E) \ } 22</math>
 
<math>\mathrm{(A) \ } 18\qquad \mathrm{(B) \ } 19\qquad \mathrm{(C) \ } 20\qquad \mathrm{(D) \ } 21\qquad \mathrm{(E) \ } 22</math>
 +
 +
[[2019 AMC 10C Problems/Problem 19|Solution]]
 +
  
 
==Problem 20==
 
==Problem 20==
  
 
How many five digit positive integers have the property that every two adjacent digits are consecutive? For example, the integer 32345 has this property, while the integer 98623 does not.
 
How many five digit positive integers have the property that every two adjacent digits are consecutive? For example, the integer 32345 has this property, while the integer 98623 does not.
 +
 +
 
<math>\mathrm{(A) \ } 116\qquad \mathrm{(B) \ } 120\qquad \mathrm{(C) \ } 125\qquad \mathrm{(D) \ } 127\qquad \mathrm{(E) \ } 144</math>
 
<math>\mathrm{(A) \ } 116\qquad \mathrm{(B) \ } 120\qquad \mathrm{(C) \ } 125\qquad \mathrm{(D) \ } 127\qquad \mathrm{(E) \ } 144</math>
 +
 +
[[2019 AMC 10C Problems/Problem 20|Solution]]
 +
  
 
==Problem 21==
 
==Problem 21==
Line 158: Line 208:
 
Jack eats candy while abiding to the following rule:
 
Jack eats candy while abiding to the following rule:
 
On Day <math>n</math>, he eats exactly <math>n^3</math> pieces of candy if <math>n</math> is odd and exactly <math>n^2</math> pieces of candy if <math>n</math> is even. Let <math>N</math> be the number of candies Jack has eaten after Day 2019. Find the last two digits of <math>N</math>.
 
On Day <math>n</math>, he eats exactly <math>n^3</math> pieces of candy if <math>n</math> is odd and exactly <math>n^2</math> pieces of candy if <math>n</math> is even. Let <math>N</math> be the number of candies Jack has eaten after Day 2019. Find the last two digits of <math>N</math>.
 +
  
 
<math>\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 90</math>
 
<math>\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 90</math>
 +
 +
[[2019 AMC 10C Problems/Problem 21|Solution]]
 +
  
 
==Problem 22==
 
==Problem 22==
  
−
100 chicks are sitting in a circle. The first chick in the circle says the number 1, then the chick 2 seats away from the first chick says the number 2, then the chick 3 seats away from the chicken that said the number 2 says the number 3, and so on. The process will always go clockwise. Some of the chicks in the circle will say more than one number while others might not even say a number at all. The process stops when the 1001th number is said. How many numbers would the chick that said 1001 have said by that point (including 1001)?
+
<math>100</math> chicks are sitting in a circle. The first chick in the circle says the number <math>1</math>, then the chick <math>2</math> seats away from the first chick says the number <math>2</math>, then the chick <math>3</math> seats away from the chicken that said the number <math>2</math> says the number <math>3</math>, and so on. The process will always go clockwise. Some of the chicks in the circle will say more than one number while others might not even say a number at all. The process stops when the <math>1001</math>th number is said. How many numbers would the chick that said <math>1001</math> have said by that point (including <math>1001</math>)?
 +
 
 +
 
 
<math>\mathrm{(A) \ } 20\qquad \mathrm{(B) \ } 21\qquad \mathrm{(C) \ } 30\qquad \mathrm{(D) \ } 31\qquad \mathrm{(E) \ } 41</math>
 
<math>\mathrm{(A) \ } 20\qquad \mathrm{(B) \ } 21\qquad \mathrm{(C) \ } 30\qquad \mathrm{(D) \ } 31\qquad \mathrm{(E) \ } 41</math>
 +
 +
[[2019 AMC 10C Problems/Problem 22|Solution]]
 +
  
 
==Problem 23==
 
==Problem 23==
  
 
Bernado has an infinite amount of red, blue, orange, pink, yellow, purple, and black blocks. He puts them in the 2 by 2019 grid such that adjacent blocks are of different colors. What is the hundreds digit of the number of ways he can put the blocks in?
 
Bernado has an infinite amount of red, blue, orange, pink, yellow, purple, and black blocks. He puts them in the 2 by 2019 grid such that adjacent blocks are of different colors. What is the hundreds digit of the number of ways he can put the blocks in?
 +
  
 
<math>\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4</math>
 
<math>\textbf{(A)}\ 0\qquad\textbf{(B)}\ 1\qquad\textbf{(C)}\ 2\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ 4</math>
 +
 +
[[2019 AMC 10C Problems/Problem 23|Solution]]
 +
  
 
==Problem 24==
 
==Problem 24==
  
 
Let <math>S</math> be the set of points <math>(x,y,z)</math> in the 3-dimensional coordinate plane such that <math>x</math>, <math>y</math>, and <math>z</math> are integers, <math>0 \le x \le 2</math>, <math>0 \le y \le 2</math>, and <math>0 \le z \le 1</math>. How many tetrahedrons of positive volume can be formed by choosing four points in <math>S</math> as vertices of the tetrahedron?
 
Let <math>S</math> be the set of points <math>(x,y,z)</math> in the 3-dimensional coordinate plane such that <math>x</math>, <math>y</math>, and <math>z</math> are integers, <math>0 \le x \le 2</math>, <math>0 \le y \le 2</math>, and <math>0 \le z \le 1</math>. How many tetrahedrons of positive volume can be formed by choosing four points in <math>S</math> as vertices of the tetrahedron?
 +
 +
 
<math>\mathrm{(A) \ } 2436\qquad \mathrm{(B) \ } 2472\qquad \mathrm{(C) \ } 2580\qquad \mathrm{(D) \ } 2664\qquad \mathrm{(E) \ } 3060</math>
 
<math>\mathrm{(A) \ } 2436\qquad \mathrm{(B) \ } 2472\qquad \mathrm{(C) \ } 2580\qquad \mathrm{(D) \ } 2664\qquad \mathrm{(E) \ } 3060</math>
 +
 +
[[2019 AMC 10C Problems/Problem 24|Solution]]
 +
  
 
==Problem 25==
 
==Problem 25==
  
−
Let <math>N</math> be the least positive integer <math>x</math> such that <math>\lfloor \frac{x^{8}}{x-1} \rfloor</math> is a multiple of 10000. Find the sum of the digits of <math>N</math>. (Note: <math>\lfloor x \rfloor</math> denotes the greatest integer less than or equal to <math>x</math>. )
+
Let <math>N</math> be the least positive integer <math>x</math> such that <math>\left\lfloor \frac{x^{8}}{x-1}\right\rfloor</math> is a multiple of 10000. Find the sum of the digits of <math>N</math>. (Note: <math>\lfloor x \rfloor</math> denotes the greatest integer less than or equal to <math>x</math>. )
 +
 
  
 
<math>\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 24 \qquad\textbf{(E)}\ 36</math>
 
<math>\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 24 \qquad\textbf{(E)}\ 36</math>
 +
 +
[[2019 AMC 10C Problems/Problem 25|Solution]]
 +
 +
 +
 +
 +
 +
 +
[[AMC 8 2019(Mock) Problems]]
 +
[[AMC 12C 2020 Problems]]
 +
 +
[[AIME 2020(MOCK) Problems]]
 +
 +
[[F = MA 2020 (Mock) Problems]]
 +
 +
[[AMC 10 2021 (Mock) Problems]]

Latest revision as of 11:14, 29 November 2021

Here are the problems from the 2019 AMC 10C, a mock contest created by the AoPS user fidgetboss_4000.

Problem 1

Given that $a+m=2017$, $m+c=2018$, and $a+c=2019$, find the value of $a+m+c+10$.


$\mathrm{(A) \ } 3027\qquad \mathrm{(B) \ } 3037\qquad \mathrm{(C) \ } 4037\qquad \mathrm{(D) \ } 6054\qquad \mathrm{(E) \ } 6064$

Solution


Problem 2

Three distinct vertices of a regular pentagon are chosen at random. What is the probability that they form an obtuse triangle?


$\mathrm{(A) \ } \frac{1}{10}\qquad \mathrm{(B) \ } \frac{1}{5}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{4}{5}\qquad \mathrm{(E) \ } \frac{9}{10}$

Solution


Problem 3

The numbers from 1 to 10 are written on 10 slips of paper, with 1 number on each slip of paper, and put in a bag. Maria will randomly draw 2 slips from the bag without replacement. What is the probability that she will obtain one prime number and one composite number?


$\mathrm{(A) \ } \frac{2}{15}\qquad \mathrm{(B) \ } \frac{2}{5}\qquad \mathrm{(C) \ } \frac{4}{9}\qquad \mathrm{(D) \ } \frac{5}{9}\qquad \mathrm{(E) \ } \frac{8}{9}$

Solution


Problem 4

In the figure below, the part of the 1st square that is not contained in the 2nd square will be shaded, the part of the 3rd square that is not contained in the 4th square will be shaded, the part of the 5th square that is not contained in the 6th square will be shaded, and so on to infinity. What is the ratio of the shaded area to the non-shaded area?


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

Solution


Problem 5

The numeral representation of the product $99...9 \times 999...9$, where the former number has $n$ 9’s and the latter number has $n+1$ 9’s, where $n$ is some positive integer, has a sum of digits of 630. Find $n$.


$\mathrm{(A) \ } 68\qquad \mathrm{(B) \ } 69\qquad \mathrm{(C) \ } 70\qquad \mathrm{(D) \ } 71\qquad \mathrm{(E) \ } 72$

Solution


Problem 6

A function $f(x)$is defined for all positive integers $n>2$ as $f(n)=f(n-1) \times f(n-2)$. Given that $f(1)=2$ and $f(2)=3$, the value of $f(12)$can be expressed in the form of $2^{p}3^{q}$,where $p$ and $q$ are positive integers. Find $|p-q|$.


$\mathrm{(A) \ } 21\qquad \mathrm{(B) \ } 34\qquad \mathrm{(C) \ } 55\qquad \mathrm{(D) \ } 89\qquad \mathrm{(E) \ } 144$

Solution


Problem 7

Given that $n^4 = 22,667,121$ for a positive integer $n$, find the sum of the digits of $n$.


$\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 18$

Solution


Problem 8

When rolling 12 standard 6-sided dice, the probability that the sum of the numbers rolled on the 12 dice is 69 can be expressed in $\frac{N}{6^{12}}$. Find the sum of the digits of $N$.


$\mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 10\qquad \mathrm{(C) \ } 13\qquad \mathrm{(D) \ } 14\qquad \mathrm{(E) \ } 15$

Solution


Problem 9

A gear with radius 34.5 rolls around another the circumference of a larger gear with radius 103.5. How many revolutions around the larger gear would the smaller gear have completed by the time it makes 3 complete rotations?


$\mathrm{(A) \ } \frac{1}{2}\qquad \mathrm{(B) \ } \frac{2}{3}\qquad \mathrm{(C) \ } \frac{3}{4}\qquad \mathrm{(D) \ } 1\qquad \mathrm{(E) \ } \frac{6}{5}$

Solution


Problem 10

Al and Bob play a game of Bottle Flip. Each of them takes turns flipping a water bottle and the first person to not land the bottle is deemed the loser. Al has a $\frac{4}{5}$ chance of landing the bottle every time he flips it, and Bob has a $\frac{3}{4}$ chance of landing the bottle every time he flips it. Given that Al goes first, what is the probability that he wins the game?


$\mathrm{(A) \ } \frac{1}{3}\qquad \mathrm{(B) \ } \frac{3}{8}\qquad \mathrm{(C) \ } \frac{1}{2}\qquad \mathrm{(D) \ } \frac{5}{8} \qquad \mathrm{(E) \ } \frac{2}{3}$

Solution


Problem 11

How many base 3 positive integers $n$ with nonzero digits have the property that the sum of the digits of $n$ is 10?


$\mathrm{(A) \ } 74\qquad \mathrm{(B) \ } 79\qquad \mathrm{(C) \ } 88\qquad \mathrm{(D) \ } 89\qquad \mathrm{(E) \ } 97$

Solution


Problem 12

In music, a whole octave consists of 12 semitones: C, C#, D, D#, E, F, F#, G, G#, A, A#, and B. Consist a certain four-note chord to be pretty if all four notes are from the same octave and no two notes are either $1$ semitone or $11$ semitones apart. How many four-note chords are not pretty?


$\textbf{(A)}\ 117 \qquad\textbf{(B)}\ 369 \qquad\textbf{(C)}\ 390 \qquad\textbf{(D)}\ 425 \qquad\textbf{(E)}\ 440$

Solution


Problem 13

How many positive integers $N$ where $1 \leq N \leq 20$ cannot be the number of 0s at the end of the decimal representation of $x!$, where $x$ is a positive integer?


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

Solution


Problem 14

Let $M$ denote the largest integer power of 16 that allows $100^{2019}-16^{M}$ to be a positive integer and let $N$ denote the smallest integer power of 100 that allows $16^{2019}-100^{N}$ to be a negative integer. In which of the following ranges does $M+N$ lie?


$\mathrm{(A) \ } [2000,3000]\qquad \mathrm{(B) \ } [3000,4000]\qquad \mathrm{(C) \ } [4000,5000]\qquad \mathrm{(D) \ } [5000,6000]\qquad \mathrm{(E) \ } [6000,7000]$

Solution


Problem 15

Let $N$ be $2019^{2019^{2019^{2019...}}}$. (2019 $2019$'s) Find the remainder when $N$ is divided by $7$.


$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } 2\qquad \mathrm{(C) \ } 3\qquad \mathrm{(D) \ } 4\qquad \mathrm{(E) \ } 6$

Solution


Problem 16

Given that : 1.The sum of the roots of $ax^3+bx^2+cx+d=0$, where $a, b, c,$ and $d$ are integers no more than 1000 and no less than $-1000$, is $7$. 2.The product of the roots in $ax^4+bx^3+cx+abcd=0$, where $a, b, c,$ and $d$ are integers no more than 1000 and no less than $-1000$, is $420$. Find the number of ordered pairs $(a, b, c, d)$ that satisfy these conditions.


$\mathrm{(A) \ } 27\qquad \mathrm{(B) \ } 54\qquad \mathrm{(C) \ } 108\qquad \mathrm{(D) \ } 162\qquad \mathrm{(E) \ } 216$

Solution


Problem 17

Define a subset of the first ten positive integers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} to be spicy if the sum of the squares of the elements in the subset is divisible by 5. How many spicy subsets are there?


$\mathrm{(A) \ } 35\qquad \mathrm{(B) \ } 70\qquad \mathrm{(C) \ } 140\qquad \mathrm{(D) \ } 210\qquad \mathrm{(E) \ } 280$

Solution


Problem 18

A tripod has three legs of length 3 feet, 4 feet, and 4 feet. It is set up so that the angle between any two legs is $90^{\circ}$. The height from the top of the tripod to the ground, in feet, can be expressed in the form $\dfrac{a\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers, $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of a prime. What is $a+b+c$?


$\textbf{(A)}\ 53 \qquad\textbf{(B)}\ 57 \qquad\textbf{(C)}\ 69 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 89$

Solution


Problem 19

Triangle $ABC$ has $AB=10$, $BC=14$, and $AC=22$. The line $AN$, where $N$ is a point on $BC$, divides the triangle into two halves with equal perimeters. Let $N$ be the length of $AN$. Find the greatest integer less or equal to $N$.


$\mathrm{(A) \ } 18\qquad \mathrm{(B) \ } 19\qquad \mathrm{(C) \ } 20\qquad \mathrm{(D) \ } 21\qquad \mathrm{(E) \ } 22$

Solution


Problem 20

How many five digit positive integers have the property that every two adjacent digits are consecutive? For example, the integer 32345 has this property, while the integer 98623 does not.


$\mathrm{(A) \ } 116\qquad \mathrm{(B) \ } 120\qquad \mathrm{(C) \ } 125\qquad \mathrm{(D) \ } 127\qquad \mathrm{(E) \ } 144$

Solution


Problem 21

Jack eats candy while abiding to the following rule: On Day $n$, he eats exactly $n^3$ pieces of candy if $n$ is odd and exactly $n^2$ pieces of candy if $n$ is even. Let $N$ be the number of candies Jack has eaten after Day 2019. Find the last two digits of $N$.


$\textbf{(A)}\ 16 \qquad\textbf{(B)}\ 30 \qquad\textbf{(C)}\ 40 \qquad\textbf{(D)}\ 80 \qquad\textbf{(E)}\ 90$

Solution


Problem 22

$100$ chicks are sitting in a circle. The first chick in the circle says the number $1$, then the chick $2$ seats away from the first chick says the number $2$, then the chick $3$ seats away from the chicken that said the number $2$ says the number $3$, and so on. The process will always go clockwise. Some of the chicks in the circle will say more than one number while others might not even say a number at all. The process stops when the $1001$th number is said. How many numbers would the chick that said $1001$ have said by that point (including $1001$)?


$\mathrm{(A) \ } 20\qquad \mathrm{(B) \ } 21\qquad \mathrm{(C) \ } 30\qquad \mathrm{(D) \ } 31\qquad \mathrm{(E) \ } 41$

Solution


Problem 23

Bernado has an infinite amount of red, blue, orange, pink, yellow, purple, and black blocks. He puts them in the 2 by 2019 grid such that adjacent blocks are of different colors. What is the hundreds digit of the number of ways he can put the blocks in?


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

Solution


Problem 24

Let $S$ be the set of points $(x,y,z)$ in the 3-dimensional coordinate plane such that $x$, $y$, and $z$ are integers, $0 \le x \le 2$, $0 \le y \le 2$, and $0 \le z \le 1$. How many tetrahedrons of positive volume can be formed by choosing four points in $S$ as vertices of the tetrahedron?


$\mathrm{(A) \ } 2436\qquad \mathrm{(B) \ } 2472\qquad \mathrm{(C) \ } 2580\qquad \mathrm{(D) \ } 2664\qquad \mathrm{(E) \ } 3060$

Solution


Problem 25

Let $N$ be the least positive integer $x$ such that $\left\lfloor \frac{x^{8}}{x-1}\right\rfloor$ is a multiple of 10000. Find the sum of the digits of $N$. (Note: $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. )


$\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 15 \qquad\textbf{(D)}\ 24 \qquad\textbf{(E)}\ 36$

Solution




AMC 8 2019(Mock) Problems AMC 12C 2020 Problems

AIME 2020(MOCK) Problems

F = MA 2020 (Mock) Problems

AMC 10 2021 (Mock) Problems

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