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2024 AMC 12A | 1 | What is the value of $9901 \cdot 101 - 99 \cdot 10101$? | 2 |
2024 AMC 12A | 2 | A model used to estimate the time it will take to hike to the top of the mountain on a trail is of the form $T = aL + bG$, where $a$ and $b$ are constants, $T$ is the time in minutes, $L$ is the length of the trail in miles, and $G$ is the altitude gain in feet. The model estimates that it will take $69$ minutes to hike to the top if a trail is $1.5$ miles long and ascends $800$ feet, as well as if a trail is $1.2$ miles long and ascends $1100$ feet. How many minutes does the model estimates it will take to hike to the top if the trail is $4.2$ miles long and ascends $4000$ feet? | 246 |
2024 AMC 12A | 3 | The number $2024$ is written as the sum of not necessarily distinct two-digit numbers. What is the least number of two-digit numbers needed to write this sum? | 21 |
2024 AMC 12A | 4 | What is the least value of $n$ such that $n!$ is a multiple of $2024$? | 23 |
2024 AMC 12A | 5 | A data set containing $20$ numbers, some of which are $6$, has mean $45$. When all the 6s are removed, the data set has mean $66$. How many 6s were in the original data set? | 7 |
2024 AMC 12A | 6 | The product of three integers is $60$. What is the least possible positive sum of the three integers? | 3 |
2024 AMC 12A | 7 | In $\triangle ABC$, $\angle ABC = 90^\circ$ and $BA = BC = \sqrt{2}$. Points $P_1, P_2, \ldots, P_{2024}$ lie on hypotenuse $\overline{AC}$ so that $AP_1 = P_1P_2 = P_2P_3 = \cdots = P_{2023}P_{2024} = P_{2024}C$. What is the length of the vector sum $\overrightarrow{BP_1} + \overrightarrow{BP_2} + \overrightarrow{BP_3} + \cdots + \overrightarrow{BP_{2024}}$? | 2024 |
2024 AMC 12A | 8 | How many angles $\theta$ with $0 \leq \theta \leq 2\pi$ satisfy $\log(\sin(3\theta)) + \log(\cos(2\theta)) = 0$? | 0 |
2024 AMC 12A | 9 | Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$? | 8 |
2024 AMC 12A | 10 | Let $\alpha$ be the radian measure of the smallest angle in a $3-4-5$ right triangle. Let $\beta$ be the radian measure of the smallest angle in a $7-24-25$ right triangle. In terms of $\alpha$, what is $\beta$? | \frac{\pi}{2} - 2\alpha |
2024 AMC 12A | 11 | There are exactly $K$ positive integers $b$ with $5 \leq b \leq 2024$ such that the base-$b$ integer $2024_b$ is divisible by $16$ (where $16$ is in base ten). What is the sum of the digits of $K$? | 20 |
2024 AMC 12A | 12 | The first three terms of a geometric sequence are the integers $a$, $720$, and $b$, where $a < 720 < b$. What is the sum of the digits of the least possible value of $b$? | 21 |
2024 AMC 12A | 13 | The graph of $y = e^{x+1} + e^{-x} - 2$ has an axis of symmetry. What is the reflection of the point $(-1, \frac{1}{2})$ over this axis? | (0, \frac{1}{2}) |
2024 AMC 12A | 15 | The roots of $x^3 + 2x^2 - x + 3$ are $p$, $q$, and $r$. What is the value of $(p^2 + 4)(q^2 + 4)(r^2 + 4)$? | 125 |
2024 AMC 12A | 16 | A set of $12$ tokens — $3$ red, $2$ white, $1$ blue, and $6$ black — is to be distributed at random to $3$ game players, $4$ tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? | 389 |
2024 AMC 12A | 17 | Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$? | 276 |
2024 AMC 12A | 19 | Cyclic quadrilateral $ABCD$ has lengths $BC = CD = 3$ and $DA = 5$ with $\angle CDA = 120^\circ$. What is the length of the shorter diagonal of $ABCD$? | \frac{39}{7} |
2024 AMC 12A | 20 | Points $P$ and $Q$ are chosen uniformly and independently at random on sides $\overline{AB}$ and $\overline{AC}$, respectively, of equilateral triangle $\triangle ABC$. Which of the following intervals contains the probability that the area of $\triangle APQ$ is less than half the area of $\triangle ABC$? | [\frac{3}{4}, \frac{7}{8}] |
2024 AMC 12A | 21 | Suppose that $a_1 = 2$ and the sequence $(a_n)$ satisfies the recurrence relation $\frac{a_n - 1}{n - 1} = \frac{a_{n-1} + 1}{(n - 1) + 1}$ for all $n \geq 2$. What is the greatest integer less than or equal to $\sum_{n=1}^{100} a_n^2$? | 338551 |
2024 AMC 12A | 23 | What is the value of $\tan^2 \frac{\pi}{16}\cdot\tan^2 \frac{3\pi}{16} + \tan^2 \frac{\pi}{16}\cdot\tan^2 \frac{5\pi}{16} + \tan^2 \frac{3\pi}{16}\cdot\tan^2 \frac{7\pi}{16} + \tan^2 \frac{5\pi}{16}\cdot\tan^2 \frac{7\pi}{16}$? | 68 |
2024 AMC 12A | 24 | A disphenoid is a tetrahedron whose triangular faces are congruent to one another. What is the least total surface area of a disphenoid whose faces are scalene triangles with integer side lengths? | 15\sqrt{7} |
2024 AMC 12A | 25 | A graph is symmetric about a line if the graph remains unchanged after reflection in that line. For how many quadruples of integers $(a, b, c, d)$, where $|a|, |b|, |c|, |d| \leq 5$ and $c$ and $d$ are not both $0$, is the graph of $y = \frac{ax + b}{cx + d}$ symmetric about the line $y = x$? | 1292 |
2024 AMC 12B | 1 | 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? | 2022 |
2024 AMC 12B | 2 | What is $10! - 7! \cdot 6!$? | 0 |
2024 AMC 12B | 3 | For how many integer values of $x$ is $|2x| \leq 7\pi$? | 21 |
2024 AMC 12B | 4 | Balls numbered $1, 2, 3, \ldots$ 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, $22, 23, \ldots, 28$ are deposited in bin $B$ at step 7 of this process.) In which bin is ball $2024$ deposited? | D |
2024 AMC 12B | 5 | 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? | 15 |
2024 AMC 12B | 6 | 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.) | 20 |
2024 AMC 12B | 8 | What value of $x$ satisfies $\frac{\log_2 x \cdot \log_3 x}{\log_2 x + \log_3 x} = 2$? | 36 |
2024 AMC 12B | 9 | 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 $\frac{m}{n} \cdot \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$? | 71 |
2024 AMC 12B | 10 | 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? | 3 |
2024 AMC 12B | 11 | Let $x_n = \sin^2 (n^5)$. What is the mean of $x_1, x_2, x_3, \cdots, x_{90}$? | \frac{91}{180} |
2024 AMC 12B | 12 | 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 $z$? | \frac{3}{2} |
2024 AMC 12B | 13 | There are real numbers $x, y, h$ and $k$ that satisfy the system of equations $x^2 + y^2 - 6x - 8y = h$ and $x^2 + y^2 - 10x + 4y = k$. What is the minimum possible value of $h + k$? | -34 |
2024 AMC 12B | 14 | How many different remainders can result when the $100$th power of an integer is divided by $125$? | 5 |
2024 AMC 12B | 15 | 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 $\triangle ABC$? | \log_2 \frac{7}{\sqrt{3}} |
2024 AMC 12B | 16 | 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 $r$? | 5 |
2024 AMC 12B | 17 | 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 $x^3 + ax^2 + bx + 6$ has $3$ distinct integer roots? | \frac{1}{105} |
2024 AMC 12B | 18 | 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 $\frac{F_2}{F_1} + \frac{F_4}{F_2} + \frac{F_6}{F_3} + \cdots + \frac{F_{20}}{F_{10}}$? | 320 |
2024 AMC 12B | 20 | 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 $\overline{BC}$. Define a function $f$ by letting $f(x)$ be the area of $\triangle ABC$. Then the domain of $f$ is an open interval $(p, q)$, and the maximum value $r$ of $f(x)$ occurs at $x = 8$. What is $p + q + r + s$? | 911 |
2024 AMC 12B | 21 | 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? | 154 |
2024 AMC 12B | 22 | Let $\triangle ABC$ 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? | 15 |
2024 AMC 12B | 23 | A right pyramid has regular octagon $ABCDEFGH$ with side length $1$ as its base and apex $V$. Segments $\overline{AV}$ and $\overline{DV}$ are perpendicular. What is the square of the height of the pyramid? | \frac{1 + \sqrt{2}}{2} |
2024 AMC 12B | 24 | 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 $\triangle ABC$ with an integer inradius for which $a, b$, and $c$ are the lengths of the altitudes from $A$ to $\overline{BC}$, $B$ to $\overline{AC}$, and $C$ to $\overline{AB}$, respectively? (Recall that the inradius of a triangle is the radius of the largest possible circle that can be inscribed in the triangle.) | 3 |
2024 AMC 12B | 25 | 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 $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m$? (Recall that two events $A$ and $B$ are independent if $P(A \text{ and } B) = P(A) \cdot P(B)$.) | 247 |
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All problems copyrighted by the Mathematical Association of America's American Mathematics Competitions
Source:
- https://artofproblemsolving.com/wiki/index.php/2024_AMC_12A_Problems
- https://artofproblemsolving.com/wiki/index.php/2024_AMC_12B_Problems
Removed problems with figures:
- 12A: problem 14,18,22
- 12B: problem 7, 19
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