AMC 12B 2024

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?

What is \(10 ! - 7 ! \cdot 6 !\)?

For how many integer values of \(x\) is \(\| 2 x \| \leq 7 \pi ?\)

Balls numbered \(1 , 2 , 3 , \cdots\) 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 \(B\). The next three 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, \(22 , 23 , \cdots , 28\) 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: \( 1 + 3 + 5 + 7 + \cdots.c + 97 + 99 \) 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 \(5 \cdot 10^13\) dollars by \(2033\). How many digits does this number of dollars have when written as a numeral in base \(5\)? (The approximation of \(\log_10 5\) as \(0.7\) is sufficient for this problem.)

In the figure below \(W X Y Z\) is a rectangle with \(W X = 4\) and \(W Z = 8\). Point \(M\) lies \(\overline{X Y}\), point \(A\) lies on \(\overline{Y Z}\), and \(\angle W M A\) is a right angle. The areas of \(\triangle W X M\) and \(\triangle W A Z\) are equal. What is the area of \(\triangle W M A\)?

What value of \(x\) satisfies \( \dfrac{\log_2 x \cdot \log_3 x}{\log_2 x + \log_3 x} = 2 ? \)

A dartboard is the region \(B\) in the coordinate plane consisting of points \(( x , y )\) such that \(\| x \| + \| y \| \leq 8\) . A target \(T\) is the region where \(( x^2 + y^2 - 25 )^2 \leq 49 .\) A dart is thrown and lands at a random point in \(B\). The probability that the dart lands in \(T\) can be expressed as \(\dfrac{m}{n} \cdot \pi ,\) where \(m\) and \(n\) are relatively prime positive integers. What is \(m + n ?\)

In a long line of people, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?

A list of 9 real numbers consists of \(1\), \(2.2\), \(3.2\), \(5.2\), \(6.2\), and \(7\), as well as \(x , y , z\) with \(x \leq y \leq z\). The range of the list is \(7\), and the mean and median are both positive integers. How many ordered triples \(( x , y , z )\) are possible?

Let \(x_n = \sin^2 ( n^\circ )\). What is the mean of \(x_1 , x_2 , x_3 , \cdots.c , x_90\)?

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 \(0\), \(z\), \(z^2\), and \(z^3\) are the vertices of a quadrilateral with area \(15\). What is the imaginary part of \(z\)?

There are real numbers \(x , y , h\) and \(k\) that satisfy the system of equations \( x^2 + y^2 - 6 x - 8 y = h \) \( x^2 + y^2 - 10 x + 4 y = k \) What is the minimum possible value of \(h + k\)?

How many different remainders can result when the \(100\)th power of an integer is divided by \(125\)?

A triangle in the coordinate plane has vertices \(A ( \log_21 , \log_22 )\), \(B ( \log_23 , \log_24 )\), and \(C ( \log_27 , \log_28 )\). What is the area of \(\triangle A B C\)?

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 \(3^r M\), 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 \(x^3 + a x^2 + b x + 6\) has \(3\) distinct integer roots?

The Fibonacci numbers are defined by \(F_1 = 1 ,\)\(F_2 = 1 ,\) and \(F_n = F_{n - 1} + F_{n - 2}\) for \(n \geq 3 .\) What is \( \dfrac{F_2}{F_1} + \dfrac{F_4}{F_2} + \dfrac{F_6}{F_3} + \cdots.c + \dfrac{F_20}{F_10} ? \)

Equilateral \(\triangle A B C\) with side length \(14\) is rotated about its center by angle \(\theta\), where \(0 < \theta < 60^\circ\), to form \(\triangle D E F\). See the figure. The area of hexagon \(A D B E C F\) is \(91 \sqrt{3}\). What is \(\tan \theta\)?

What is \(10! - 7! \cdot 6!\)?

Suppose \(A\), \(B\), and \(C\) are points in the plane with \(A B = 40\) and \(A C = 42\), and let \(x\) be the length of the line segment from \(A\) to the midpoint of \(\overline{B C}\). Define a function \(f\) by letting \(f ( x )\) be the area of \(\triangle A B C\). 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 \(90^\circ\). 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?

Let \(\triangle A B C\) be a triangle with integer side lengths and the property that \(\angle B = 2 \angle A\). What is the least possible perimeter of such a triangle?

A right pyramid has regular octagon \(A B C D E F G H\) with side length \(1\) as its base and apex \(V .\) Segments \(\overline{A V}\) and \(\overline{D V}\) 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 \(a \leq b \leq c \leq 9\), such that there exists a (non-degenerate) triangle \(\triangle A B C\) with an integer inradius for which \(a\), \(b\), and \(c\) are the lengths of the altitudes from \(A\) to \(\overline{B C}\), \(B\) to \(\overline{A C}\), and \(C\) to \(\overline{A B}\), 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 \(\dfrac{m}{n} ,\) where \(m\) and \(n\) are relatively prime positive integers. What is \(m ?\) (Recall that two events \(A\) and \(B\) are independent if \(P ( A \text{ and } B ) = P ( A ) \cdot P ( B ) .\))

For how many integer values of \(x\) is \(|2x| \leq 7 \pi\)?

Balls numbered \(1, 2, 3, ...\) are placed in bins \(A\), \(B\), \(C\), \(D\), and \(E\) so that the first ball is placed in \(A\), the next two are placed in \(B\), the next three are placed in \(C\), the next four are placed in \(D\), the next five are placed in \(E\), and then the next six go in \(A\), etc. For example, \(22, 23, ..., 28\) are placed in \(B\). Which bin contains ball 2024?

In the following expression, Melanie changed some of the plus signs to minus signs: \( 1 + 3 + 5 + 7 + \cdots + 97 + 99 \) 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 \(5 \cdot 10^{13}\) dollars by 2033. How many digits does this number of dollars have when written as a numeral in base 5? (The approximation of \(\log_{10} 5\) as \(0.7\) is sufficient for this problem.)

In the figure below \(W X Y Z\) is a rectangle with \(W X = 4\) and \(W Z = 8\). Point \(M\) lies on \(\overline{X Y}\), point \(A\) lies on \(\overline{Y Z}\), and \(\angle W M A\) is a right angle. The areas of \(\triangle W X M\) and \(\triangle W A Z\) are equal. What is the area of \(\triangle W M A\)?

What value of \(x\) satisfies \( \dfrac{\log_2 x \cdot \log_3 x}{\log_2 x + \log_3 x} = 2 ? \)

A dartboard is the region B in the coordinate plane consisting of points \((x, y)\) such that \(|x| + |y| \leq 8\). A target T is the region where \((x^2 + y^2 - 25)^2 \leq 49\). A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as \(\dfrac{m}{n} \pi\), where \(m\) and \(n\) are relatively prime positive integers. What is \(m + n\)?

A list of 9 real numbers consists of \(1\), \(2.2\), \(3.2\), \(5.2\), \(6.2\), \(7\), as well as \(x, y, z\) with \(x \leq y \leq z\). The range of the list is 7, and the mean and median are both positive integers. How many ordered triples \((x, y, z)\) are possible?

Let \(x_n = \sin^2 (n^{\circ})\). What is the mean of \(x_1, x_2, x_3, ..., x_{90}\)?

Let \(z\) be a complex number with real part greater than 1 and \(|z| = 2\). In the complex plane, the four points \(0\), \(z\), \(z^2\), and \(z^3\) are the vertices of a quadrilateral with area 15. What is the imaginary part of \(z\)?

There are real numbers \(x\), \(y\), \(h\) and \(k\) that satisfy the system of equations \( x^2 + y^2 - 6x - 8y = h \) \( x^2 + y^2 - 10x + 4y = k \) What is the minimum possible value of \(h + k\)?

How many different remainders can result when the 100th power of an integer is divided by 125?

A triangle in the coordinate plane has vertices \(A(\log_2 1, \log_2 2)\), \(B(\log_2 3, \log_2 4)\), and \(C(\log_2 7, \log_2 8)\). What is the area of \(\triangle A B C\)?

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 \(3^r M\), 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 \(x^3 + a x^2 + b x + 6\) has 3 distinct integer roots?

The Fibonacci numbers are defined by \(F_1 = 1\), \(F_2 = 1\), and \(F_n = F_{n-1} + F_{n-2}\) for \(n \geq 3\). What is \( \dfrac{F_2}{F_1} + \dfrac{F_4}{F_2} + \dfrac{F_6}{F_3} + \cdots + \dfrac{F_{20}}{F_{10}} ? \)

Equilateral triangle \(A B C\) with side length 14 is rotated about its center by angle \(\theta\) to form \(\triangle D E F\), where \(0 < \theta < 60^{\circ}\). The area of hexagon \(A D B E C F\) is \(91 \sqrt{3}\). What is \(\tan \theta\)?

Suppose \(A\), \(B\), and \(C\) are points in the plane with \(A B = 40\) and \(A C = 42\), and let \(x\) be the length of the line segment from \(A\) to the midpoint of \(\overline{B C}\). Define a function \(f\) by letting \(f(x)\) be the area of \(\triangle A B C\). 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 \(90^{\circ}\). 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?

Let \(\triangle A B C\) be a triangle with integer side lengths and the property that \(\angle B = 2 \angle A\). What is the least possible perimeter of such a triangle?

A right pyramid has regular octagon \(A B C D E F G H\) with side length 1 as its base and apex \(V\). Segments \(\overline{A V}\) and \(\overline{D V}\) 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 \(a \leq b \leq c \leq 9\), such that there exists a (non-degenerate) triangle \(\triangle A B C\) with an integer inradius for which \(a\), \(b\), and \(c\) are the lengths of the altitudes from \(A\) to \(\overline{B C}\), \(B\) to \(\overline{A C}\), and \(C\) to \(\overline{A B}\), 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 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 \(\dfrac{m}{n}\), where \(m\) and \(n\) are relatively prime positive integers. What is \(m\)? (Recall that two events \(A\) and \(B\) are independent if \(P(A \text{ and } B) = P(A) \cdot P(B)\).)