2018 AMC 10A

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?

Which of the following describes the set of values of $a$ for which the curves $x^2+y^2=a^2$ and $y=x^2-a$ in the real $xy$-plane intersect at exactly $3$ points?

Let $a,$ $b,$ $c,$ and $d$ be positive integers such that $$\gcd(a, b)=24,$$ $$\gcd(b, c)=36,$$ $$\gcd(c, d)=54,$$ and $$70 < \gcd(d, a) < 100.$$ Which of the following must be a divisor of $a?$

Farmer Pythagoras has a field in the shape of a right triangle. The right triangle’s legs have lengths $3$ and $4$ units. In the corner where those sides meet at a right angle, he leaves a small unplanted square $S$ so that from the air it looks like the right angle symbol. The rest of the field is planted. The shortest distance from $S$ to the hypotenuse is $2$ units. What fraction of the field is planted?

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120.$ Let $D$ be the midpoint of $\overline{AB},$ and let $E$ be the midpoint of $\overline{AC}.$ The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G,$ respectively. What is the area of quadrilateral $FDBG?$

For a positive integer $n$ and nonzero digits $a,$ $b,$ and $c,$ let $A_n$ be the $n$-digit integer each of whose digits is equal to $a$; let $B_n$ be the $n$-digit integer each of whose digits is equal to $b$; and let $C_n$ be the $2n$-digit (not $n$-digit) integer each of whose digits is equal to $c.$ What is the greatest possible value of $a + b + c$ for which there are at least two values of $n$ such that $$C_n - B_n = A_n^2?$$