Take the complete 2024 AMC 12B as a timed practice test. You'll have 75 minutes to solve all 25 questions under realistic AMC conditions. Submit your test to see your official-style score and review detailed solutions.
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
Balls numbered 1,2,3,… are deposited in 5 bins, labeled A, B, C, D, and E, using the following procedure. Ball 1 is deposited in bin A, and balls 2 and 3 are deposited in bin B. The next 3 balls are deposited in bin C, the next 4 in bin D, and so on, cycling back to bin A after balls are deposited in bin E. (For example, balls numbered 22,23,…,28 are deposited in bin B at step 7 of this process.) In which bin is ball 2024 deposited?
In the following expression, Melanie changed some of the plus signs to minus signs: 1+3+5+7+⋯+97+99 When the new expression was evaluated, it was negative. What is the least number of plus signs that Melanie could have changed to minus signs?
The national debt of the United States is on track to reach 5⋅1013 dollars by 2033. How many digits does this number of dollars have when written as a numeral in base 5? (The approximation of log105 as 0.7 is sufficient for this problem.)
In the figure below WXYZ is a rectangle with WX=4 and WZ=8. Point M lies on XY, point A lies on YZ, and ∠WMA is a right angle. The areas of △WXM and △WAZ are equal. What is the area of △WMA?
A dartboard is the region B in the coordinate plane consisting of points (x,y) such that ∣x∣+∣y∣≤8. A target T is the region where (x2+y2−25)2≤49. A dart is thrown and lands at a random point in B. The probability that the dart lands in T can be expressed as nm⋅π, where m and n are relatively prime positive integers. What is m+n?
A list of 9 real numbers consists of 1,2.2,3.2,5.2,6.2, and 7, as well as x, y, and z, with x≤y≤z. The range of the list is 7, and the mean and the median are both positive integers. How many ordered triples (x,y,z) are possible?
Suppose z is a complex number with positive imaginary part, with real part greater than 1, and with ∣z∣=2. In the complex plane, the four points 0, z, z2, and z3 are the vertices of a quadrilateral with area 15. What is the imaginary part of z?
A group of 16 people will be partitioned into 4 indistinguishable 4-person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as 3rM, where r and M are positive integers and M is not divisible by 3. What is r?
Integers a and b are randomly chosen without replacement from the set of integers with absolute value not exceeding 10. What is the probability that the polynomial x3+ax2+bx+6 has 3 distinct integer roots?
Equilateral △ABC with side length 14 is rotated about its center by angle θ, where 0<θ<60∘, to form △DEF. See the figure. The area of hexagon ADBECF is 913. What is tanθ?
Suppose A, B, and C are points in the plane with AB=40 and AC=42, and let x be the length of the line segment from A to the midpoint of BC. Define a function f by letting f(x) be the area of △ABC. Then the domain of f is an open interval (p,q), and the maximum value r of f(x) occurs at x=s. What is p+q+r+s?
The measures of the smallest angles of three different right triangles sum to 90∘. All three triangles have side lengths that are primitive Pythagorean triples. Two of them are 3−4−5 and 5−12−13. What is the perimeter of the third triangle?
A right pyramid has regular octagon ABCDEFGH with side length 1 as its base and apex V. Segments AV and DV are perpendicular. What is the square of the height of the pyramid?
What is the number of ordered triples (a,b,c) of positive integers, with a≤b≤c≤9, such that there exists a non-degenerate triangle △ABC with an integer inradius for which a, b, and c are the lengths of the altitudes from A to BC, B to AC, and C to AB, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.)
Pablo will decorate each of 6 identical white balls with either a striped or a dotted pattern, using either red or blue paint. He will decide on the color and pattern for each ball by flipping a fair coin for each of the 12 decisions he must make. After the paint dries, he will place the 6 balls in an urn. Frida will randomly select one ball from the urn and note its color and pattern. The events "the ball Frida selects is red" and "the ball Frida selects is striped" may or may not be independent, depending on the outcome of Pablo's coin flips. The probability that these two events are independent can be written as nm, where m and n are relatively prime positive integers. What is m? (Recall that two events A and B are independent if P(A and B)=P(A)⋅P(B).)
