AMC 10A 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 of 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 were 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 value of $4000 \cdot \left(\tfrac{2}{5}\right)^n$ an integer?

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 $DBCE$?

The figure shows isosceles $\triangle ABC$ ($A$ at top, $B$ lower-left, $C$ lower-right). Points $D$ on $\overline{AB}$ and $E$ on $\overline{AC}$ with $\overline{DE} \parallel \overline{BC}$ split the figure into $\triangle ADE$ (top) and trapezoid $DBCE$ (bottom). The interior of $\triangle ABC$ is tiled by triangles similar to $\triangle ABC$: a zig-zag row of $7$ smallest triangles appears just below $\overline{DE}$ inside the trapezoid, and a mirror zig-zag of $7$ smallest triangles (plus one tiny apex triangle) appears inside $\triangle ADE$.

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 $\frac{n}{6^{7}},$ where $n$ is a positive integer. What is $n$?

How many ordered pairs of real numbers $(x, y)$ satisfy the following system of equations? $x + 3y = 3$ $\big||x| - |y|\big| = 1$

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

The triangle has $A$ at the lower-left (the right angle is at $C$, lower-right), $B$ at the upper-right, $C$ at the lower-right, with $BC = 3$, $AC = 4$, and $AB = 5$.

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

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. The distance $AB$ can be written in the form $\tfrac{m}{n}$, 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 $a_7 \cdot 3^7 + a_6 \cdot 3^6 + a_5 \cdot 3^5 + a_4 \cdot 3^4 + a_3 \cdot 3^3 + a_2 \cdot 3^2 + a_1 \cdot 3^1 + a_0 \cdot 3^0,$ where $a_i \in \{-1, 0, 1\}$ for $0 \leq i \leq 7$?

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 $\textit{symmetric}$ 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 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 $C_n - B_n = A_n^2$?