AMC10 2027 - Self-practice Set 3

The figure below shows a square and four equilateral triangles, with each triangle having a side lying on a side of the square, such that each triangle has side length $2$ and the third vertices of the triangles meet at the center of the square. The region inside the square but outside the triangles is shaded. What is the area of the shaded region?

How many real ordered pairs solve $x+3y=3$ and $\bigl||x|-|y|\bigr|=1$?

Let $n$ be a five-digit positive integer. Let $q,r$ be the quotient and remainder when $n$ is divided by $100$. How many such $n$ have $q+r$ divisible by $11$?

Two points are chosen independently and uniformly on a circle of radius $r\gt 0$. What is the probability that their distance is at most $r$?

In the figure below, $N$ congruent semicircles are drawn along a diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the small semicircles. The ratio $A : B$ is $1 : 18.$ What is $N?$

Find the sum of real numbers $m$ for which $x^2-2|x|+2=m$ has exactly three distinct real solutions.

Which expression below is guaranteed not to be prime for every prime $p$?

Bins are numbered $1,2,3,\ldots$. A red ball and a green ball are tossed independently into them, each ball landing in bin $k$ with probability $2^{-k}$. What is the probability that the red ball lands in a higher-numbered bin?

A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle?

Distinct real numbers $x,y$ satisfy $y+4=(x-2)^2$ and $x+4=(y-2)^2$. Find $x^2+y^2$.

Let $a_1,a_2,\ldots,a_{2018}$ be increasing positive integers satisfying
$$a_1+a_2+\cdots+a_{2018}=2018^{2018}.$$
What is the remainder when $a_1^3+a_2^3+\cdots+a_{2018}^3$ is divided by $6$?

Seven fair standard six-sided dice are rolled. The probability that their sum is $10$ is $\dfrac n{6^7}$, where $n$ is a positive integer. What is $n$?

A regular hexagon has side length $6.$ Congruent arcs with radius $3$ are drawn with the center at each of the vertices, creating circular sectors as shown. The region inside the hexagon but outside the sectors is shaded as shown. What is the area of the shaded region?

How many integers $x$ satisfy $x^4-51x^2+50\lt 0$?

Bob computes $10!$ in decimal and writes the result in base $12$. This representation has exactly $k$ trailing zeros. What is the decimal value of $k$?

The number $21!=51\,090\,942\,171\,709\,440\,000$ has over $60\,000$ positive integer divisors. One is chosen uniformly at random. What is the probability that it is odd?

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 $DBCE?$

Distinct digits $a,b,c$ are given. Find the largest possible sum of the distinct real roots of $(x-a)(x-b)+(x-b)(x-c)=0$.

How many of the first $2018$ terms of the sequence $101,1001,10001,100001,\ldots$ are divisible by $101$?

A dartboard is a regular octagon divided into regions as shown. A dart is equally likely to land anywhere on the board. What is the probability that it lands in the central square?

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