Found problems: 27
2021 AMC 12/AHSME Fall, 2
What is the area of the shaded figure shown below?
[asy]
size(200);
defaultpen(linewidth(0.4)+fontsize(12));
pen s = linewidth(0.8)+fontsize(8);
pair O,X,Y;
O = origin;
X = (6,0);
Y = (0,5);
fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2));
for (int i=1; i<7; ++i)
{
draw((i,0)--(i,5), gray+dashed);
label("${"+string(i)+"}$", (i,0), 2*S);
if (i<6)
{
draw((0,i)--(6,i), gray+dashed);
label("${"+string(i)+"}$", (0,i), 2*W);
}
}
label("$0$", O, 2*SW);
draw(O--X+(0.15,0), EndArrow);
draw(O--Y+(0,0.15), EndArrow);
draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5);
[/asy]
1991 AIME Problems, 13
A drawer contains a mixture of red socks and blue socks, at most 1991 in all. It so happens that, when two socks are selected randomly without replacement, there is a probability of exactly $1/2$ that both are red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data?
2001 Stanford Mathematics Tournament, 12
A binary string is a string consisting of only 0’s and 1’s (for instance, 001010, 101, etc.). What is the probability that a randomly chosen binary string of length 10 has 2 consecutive 0’s? Express your answer as a fraction.
1993 AMC 12/AHSME, 24
A box contains $3$ shiny pennies and $4$ dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is $\frac{a}{b}$ that it will take more than four draws until the third shiny penny appears and $\frac{a}{b}$ is in lowest terms, then $a+b=$
$ \textbf{(A)}\ 11 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 35 \qquad\textbf{(D)}\ 58 \qquad\textbf{(E)}\ 66 $
2002 AMC 12/AHSME, 7
How many three-digit numbers have at least one $2$ and at least one $3$?
$\textbf{(A) }52\qquad\textbf{(B) }54\qquad\textbf{(C) }56\qquad\textbf{(D) }58\qquad\textbf{(E) }60$
2014 Math Prize For Girls Problems, 18
For how many integers $k$ such that $0 \le k \le 2014$ is it true that the binomial coefficient $\binom{2014}{k}$ is a multiple of 4?
2013 AMC 10, 13
How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit?
$\textbf{(A) }52\qquad
\textbf{(B) }60\qquad
\textbf{(C) }66\qquad
\textbf{(D) }68\qquad
\textbf{(E) }70\qquad$
2004 Purple Comet Problems, 9
How many positive integers less that $200$ are relatively prime to either $15$ or $24$?
2001 AMC 10, 23
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
$ \displaystyle \textbf{(A)} \ \frac {3}{10} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {1}{2} \qquad \textbf{(D)} \ \frac {3}{5} \qquad \textbf{(E)} \ \frac {7}{10}$
2013 Harvard-MIT Mathematics Tournament, 4
Spencer is making burritos, each of which consists of one wrap and one filling. He has enough filling for up to four beef burritos and three chicken burritos. However, he only has five wraps for the burritos; in how many orders can he make exactly five burritos?
1983 AIME Problems, 7
Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen - all choices of three being equally likely - and are sent off to slay a troublesome dragon. Let $P$ be the probability that at least two of the three had been sitting next to each other. If $P$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
2010 CHMMC Fall, 1
Susan plays a game in which she rolls two fair standard six-sided dice with sides labeled one
through six. She wins if the number on one of the dice is three times the number on the other
die. If Susan plays this game three times, compute the probability that she wins at least once.
2010 AMC 8, 25
Everyday at school, Jo climbs a flight of $6$ stairs. Joe can take the stairs $1,2$, or $3$ at a time. For example, Jo could climb $3$, then $1$, then $2$. In how many ways can Jo climb the stairs?
$ \textbf{(A)}\ 13 \qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 20\qquad\textbf{(D)}\ 22\qquad\textbf{(E)}\ 24 $
1999 All-Russian Olympiad, 5
An equilateral triangle of side $n$ is divided into equilateral triangles of side $1$. Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle.
2001 AMC 12/AHSME, 11
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
$ \displaystyle \textbf{(A)} \ \frac {3}{10} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {1}{2} \qquad \textbf{(D)} \ \frac {3}{5} \qquad \textbf{(E)} \ \frac {7}{10}$
2014 AMC 12/AHSME, 13
A fancy bed and breakfast inn has $5$ rooms, each with a distinctive color-coded decor. One day $5$ friends arrive to spend the night. There are no other guests that night. The friends can room in any combination they wish, but with no more than $2$ friends per room. In how many ways can the innkeeper assign the guests to the rooms?
$\textbf{(A) }2100\qquad
\textbf{(B) }2220\qquad
\textbf{(C) }3000\qquad
\textbf{(D) }3120\qquad
\textbf{(E) }3125\qquad$
2008 AMC 10, 20
The faces of a cubical die are marked with the numbers $ 1$, $ 2$, $ 2$, $ 3$, $ 3$, and $ 4$. The faces of a second cubical die are marked with the numbers $ 1$, $ 3$, $ 4$, $ 5$, $ 6$, and $ 8$. Both dice are thrown. What is the probability that the sum of the two top numbers will be $ 5$, $ 7$, or $ 9$ ?
$ \textbf{(A)}\ \frac {5}{18} \qquad \textbf{(B)}\ \frac {7}{18} \qquad \textbf{(C)}\ \frac {11}{18} \qquad \textbf{(D)}\ \frac {3}{4} \qquad \textbf{(E)}\ \frac {8}{9}$
2011 AIME Problems, 12
Nine delegates, three each from three different countries, randomly select chairs at a round table that seats nine people. Let the probability that each delegate sits next to at least one delegate from another country be $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2021 AMC 10 Fall, 2
What is the area of the shaded figure shown below?
[asy]
size(200);
defaultpen(linewidth(0.4)+fontsize(12));
pen s = linewidth(0.8)+fontsize(8);
pair O,X,Y;
O = origin;
X = (6,0);
Y = (0,5);
fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2));
for (int i=1; i<7; ++i)
{
draw((i,0)--(i,5), gray+dashed);
label("${"+string(i)+"}$", (i,0), 2*S);
if (i<6)
{
draw((0,i)--(6,i), gray+dashed);
label("${"+string(i)+"}$", (0,i), 2*W);
}
}
label("$0$", O, 2*SW);
draw(O--X+(0.15,0), EndArrow);
draw(O--Y+(0,0.15), EndArrow);
draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5);
[/asy]
1998 Harvard-MIT Mathematics Tournament, 5
How many positive integers less than $1998$ are relatively prime to $1547$? (Two integers are relatively prime if they have no common factors besides 1.)
2013 AMC 8, 8
A fair coin is tossed 3 times. What is the probability of at least two consecutive heads?
$\textbf{(A)}\ \frac18 \qquad \textbf{(B)}\ \frac14 \qquad \textbf{(C)}\ \frac38 \qquad \textbf{(D)}\ \frac12 \qquad \textbf{(E)}\ \frac34$
2001 AMC 12/AHSME, 16
A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe?
$ \textbf{(A)} \ 8! \qquad \textbf{(B)} \ 2^8 \cdot 8! \qquad \textbf{(C)} \ (8!)^2 \qquad \textbf{(D)} \ \frac {16!}{2^8} \qquad \textbf{(E)} \ 16!$
2008 Vietnam National Olympiad, 5
What is the total number of natural numbes divisible by 9 the number of digits of which does not exceed 2008 and at least two of the digits are 9s?
2009 Indonesia MO, 1
In a drawer, there are at most $ 2009$ balls, some of them are white, the rest are blue, which are randomly distributed. If two balls were taken at the same time, then the probability that the balls are both blue or both white is $ \frac12$. Determine the maximum amount of white balls in the drawer, such that the probability statement is true?
2002 AMC 10, 20
How many three-digit numbers have at least one $2$ and at least one $3$?
$\textbf{(A) }52\qquad\textbf{(B) }54\qquad\textbf{(C) }56\qquad\textbf{(D) }58\qquad\textbf{(E) }60$