AMC10 2027 - Self-practice Set 6

Six beads, three red, two white, and one blue, are arranged randomly in a line. What is the probability that no neighboring beads have the same color?

Four cubes with edge lengths $1,$ $2,$ $3,$ and $4$ are stacked as shown. What is the length of the portion of $\overline{XY}$ contained in the cube with edge length $3?$

Positive integers $p,q$ satisfy $\frac59\lt \frac pq\lt \frac47$, with $q$ as small as possible. Find $q-p$.

Mary chooses an even four-digit positive integer $n$ and lists all its positive divisors in increasing order: $1,2,\ldots,n/2,n$. She has just written $323$ as a divisor. What is the smallest possible value of the next divisor?

Three distinct vertices of a cube are chosen uniformly at random. What is the probability that the plane through them contains a point in the interior of the cube?

A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. The areas of the three triangles are $3,$ $7,$ and $7,$ as shown. What is the area of the shaded quadrilateral?

Real numbers $a,b,c$ satisfy $a+b+c=2$, $a^2+b^2+c^2=12$. Find the difference between the maximum and minimum possible values of $a$.

How many positive integers $n\lt 100$ have the property that $n!$ is divisible by the sum of the first $n$ positive integers?

A number in $[0,1]$ is chosen using a fair coin. If the first toss is heads, toss again and choose $0$ for heads or $1$ for tails. If the first toss is tails, choose a real number uniformly from $[0,1]$.

Two numbers $x,y$ are chosen independently by this procedure. What is the probability that $|x-y|\gt \dfrac12$?

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 $m+n?$

How many rectangular boxes with integer side lengths $1\le a\le b\le c$ have volume equal to their total surface area?

Let $x,y$ be two-digit integers such that $y$ is obtained by reversing the digits of $x$. Suppose $x^2-y^2=m^2$ for a positive integer $m$. What is $x+y+m$?

Eight people sit around a circular table and independently flip fair coins. Those with heads stand and those with tails remain seated. What is the probability that no two adjacent people stand?

In $\triangle ABC$ with a right angle at $C,$ point $D$ lies in the interior of $\overline{AB}$ and point $E$ lies in the interior of $\overline{BC}$ so that $AC=CD,$ $DE=EB,$ and the ratio $AC:DE=4:3.$ What is the ratio $AD:DB?$

Positive integers $a\ge b\ge c$ satisfy $a^2-b^2-c^2+ab=2011$ and $a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$. Find $a$.

Let $n$ be the least integer satisfying
$$n\gt 1000,\qquad \gcd(63,n+120)=21,\qquad \gcd(n+63,120)=60.$$
What is the sum of the digits of $n$?

Rachel and Robert start together at the start line of a circular track. Rachel runs counterclockwise with a $90$-second lap, and Robert clockwise with an $80$-second lap. At a uniformly chosen time between $10$ and $11$ minutes after the start, a photograph captures one-quarter of the track centered on the start line. What is the probability that both runners appear?

Trapezoid $ ABCD $ has parallel sides $ \overline{AB} $ of length $ 33 $ and $ \overline {CD} $ of length $ 21 .$ The other two sides are of lengths $ 10 $ and $ 14 .$ The angles $ A $ and $ B $ are acute. What is the length of the shorter diagonal of $ ABCD ?$

Positive real numbers $a,b$ satisfy $\sqrt{\log_{10}a}+\sqrt{\log_{10}b}+\log_{10}\sqrt a+\log_{10}\sqrt b=100$. All four summands are positive integers. Find $ab$.

For a positive integer $n$, the number $110n^3$ has exactly $110$ positive divisors. How many positive divisors does $81n^4$ have?