AMC10 2027 - Self-practice Set 5

Let $M=\dfrac{3^{100}+2^{100}}{3^{96}+2^{96}}$. What is the greatest integer not exceeding $M$?

Bernardo uniformly selects three distinct digits from $\{1,2,\ldots,9\}$ and places them in descending order to form a three-digit number. Silvia independently does the same with three distinct digits from $\{1,2,\ldots,8\}$. What is the probability that Bernardo's number is larger?

A cylindrical pipe has diameter $30\text{ cm}$ and height $80\text{ cm}$. A gray stripe of horizontal width $3\text{ cm}$ winds exactly twice around the pipe as shown. What is its area in $\text{cm}^2$?

Mr.Brown drives for a positive integer number of hours at an average 55 km/h. His odometer changes from $\overline{abc}$ to $\overline{cba}$, where $a\ge1$ and $a+b+c\le7$. Find $a^2+b^2+c^2$.

For a positive integer $n$, let $S(n)$ be the sum of its decimal digits; for example, $S(6920)=17$. Suppose $S(n)=1274$. Which of the following could equal $S(n+1)$?

Al, Bill, and Cal are randomly assigned three distinct integers from $1$ through $10$, with all ordered assignments equally likely. What is the probability that Al's number is a multiple of Bill's and Bill's number is a multiple of Cal's?

A sphere is tangent to both bases and the lateral surface of a right circular conical frustum as shown. The frustum's volume is twice the sphere's volume. What is the ratio of the lower base radius to the upper base radius?

Distinct nonzero real numbers $x,y$ satisfy $x+\frac2x=y+\frac2y$. Find $xy$.

Let $S=1!\cdot2!\cdot3!\cdot4!\cdot5!\cdot6!\cdot7!\cdot8!\cdot9!\cdot10!$. How many positive divisors of $S$ are perfect squares?

The numbers $1,2,\ldots,9$ are randomly placed into the nine squares of a $3\times3$ grid, each used once. What is the probability that every row sum and every column sum is odd?

In rectangle $ABCD,$ $DC = 2 \cdot CB$ and points $E$ and $F$ lie on $\overline{AB}$ so that $\overline{ED}$ and $\overline{FD}$ trisect $\angle ADC$ as shown. What is the ratio of the area of $\triangle DEF$ to the area of rectangle $ABCD?$

An arithmetic progression has first three terms $2x-3$, $5x-11$, $3x+1$, and its $n$ th term is 2009. Find $n$.

What is the largest possible sum of the digits in the base-$7$ representation of a positive integer less than $2019$?

Two dice have faces labeled $1$ through $6$. All twelve labels are removed, mixed in a bag, and randomly reattached to the twelve faces, one label per face. The two fair dice are then rolled independently. What is the probability that the two top labels sum to $7$?

Six rectangular plates are arranged on a circular table of radius $4$ as shown. Each plate is $1$ unit wide and $x$ units long. What is $x$?

A rectangle has positive integer side lengths in cm, area $A$ in square cm, and perimeter $P$ in cm. Which value cannot equal $A+P$?

How many positive integers $n$ make $n^2-19n+99$ a perfect square?

Real numbers $x,y$ are chosen independently and uniformly from $[0,1]$. Which value is closest to the probability that $x,y,1$ are the side lengths of an obtuse triangle?

In $\triangle ABC,$ $AB=6,$ $AC=8,$ $BC=10,$ and $D$ is the midpoint of $\overline{BC}.$ What is the sum of the radii of the circles inscribed in $\triangle ADB$ and $\triangle ADC?$

How many ordered pairs of positive integers $(m,n)$ satisfy $\frac4m+\frac2n=1$?