Question bank AMC10 2018

What is the value of $$
\left(\left((2+1)^{-1}+1\right)^{-1}+1\right)^{-1}+1?
$$

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?

A unit of blood expires after $$
10! = 10 \cdot 9 \cdot 8 \cdots 1
$$ seconds. Yasin donates a unit of blood at noon on January $1.$ On what day does his unit of blood expire?

How many ways can a student schedule $3$ mathematics courses — algebra, geometry, and number theory — in a $6$-period day if no two mathematics courses can be taken in consecutive periods?

(What courses the student takes during the other $3$ periods is of no concern here.)

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 was 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 $d?$

Sangho uploaded a video to a website where viewers can vote that they like or dislike a video. Each video begins with a score of $0,$ and the score increases by $1$ for each like vote and decreases by $1$ for each dislike vote.

At one point Sangho saw that his video had a score of $90,$ and that $65\%$ of the votes cast on his video were like votes. How many votes had been cast on Sangho’s video at that point?

For how many (not necessarily positive) integer values of $n$ is the following value an integer? $$
4000 \cdot \left(\dfrac{2}{5}\right)^n
$$

Joe has a collection of $23$ coins, consisting of $5$-cent coins, $10$-cent coins, and $25$-cent coins. He has $3$ more $10$-cent coins than $5$-cent coins, and the total value of his collection is $320$ cents. How many more $25$-cent coins does Joe have than $5$-cent coins?

All of the triangles in the diagram below are similar to isosceles triangle $ABC,$ in which $AB=AC.$ Each of the $7$ smallest triangles has area $1,$ and $\triangle ABC$ has area $40.$ What is the area of trapezoid $DBCE?$

Suppose that real number $x$ satisfies $$
\sqrt{49-x^2}-\sqrt{25-x^2}=3.
$$ What is the value of $$
\sqrt{49-x^2}+\sqrt{25-x^2}?
$$

When $7$ fair standard $6$-sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as $$
\dfrac{n}{6^{7}},
$$ where $n$ is a positive integer. What is $n?$

How many ordered pairs of real numbers $(x,y)$ satisfy the following system of equations? $$
\begin{cases} ~x+3y&=3 \\ ~\big||x|-|y|\big|&=1 \end{cases}
$$

A paper triangle with sides of lengths $3,$ $4,$ and $5$ inches, as shown, is folded so that point $A$ falls on point $B.$ What is the length in inches of the crease?

What is the greatest integer less than or equal to $$
\dfrac{3^{100}+2^{100}}{3^{96}+2^{96}}?
$$

Two circles of radius $5$ are externally tangent to each other and are internally tangent to a circle of radius $13$ at points $A$ and $B,$ as shown in the diagram. The distance $AB$ can be written in the form $\frac{m}{n},$ where $m$ and $n$ are relatively prime positive integers. What is $m+n?$

Right triangle $ABC$ has leg lengths $AB=20$ and $BC=21.$ Including $\overline{AB}$ and $\overline{BC},$ how many line segments with integer length can be drawn from vertex $B$ to a point on hypotenuse $\overline{AC}?$

Let $S$ be a set of $6$ integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a < b,$ then $b$ is not a multiple of $a.$ What is the least possible value of an element in $S?$

How many nonnegative integers can be written in the form $$
a_7\cdot3^7+a_6\cdot3^6+a_5\cdot3^5
$$$$
+a_4\cdot3^4+a_3\cdot3^3+a_2\cdot3^2
$$$$
+a_1\cdot3^1+a_0\cdot3^0,
$$ where $a_i\in \{-1,0,1\}$ for $0\le i \le 7?$

A number $m$ is randomly selected from the set $$
\{11,13,15,17,19\},
$$ and a number $n$ is randomly selected from $$
\{1999,2000,2001,\ldots,2018\}.
$$ What is the probability that $m^n$ has a units digit of $1?$

A scanning code consists of a $7 \times 7$ grid of squares, with some of its squares colored black and the rest colored white. There must be at least one square of each color in this grid of $49$ squares.

A scanning code is called *symmetric* if its look does not change when the entire square is rotated by a multiple of $90^{\circ}$ counterclockwise around its center, nor when it is reflected across a line joining opposite corners or a line joining midpoints of opposite sides.

What is the total number of possible symmetric scanning codes?