Found problems: 93
2024 AMC 12/AHSME, 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?
$\textbf{(A) }20\qquad\textbf{(B) }21\qquad\textbf{(C) }22\qquad\textbf{(D) }23\qquad\textbf{(E) }24$
2024 AMC 10, 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?
$
\textbf{(A) }14 \qquad
\textbf{(B) }15 \qquad
\textbf{(C) }16 \qquad
\textbf{(D) }17 \qquad
\textbf{(E) }18 \qquad
$
2024 AMC 12/AHSME, 9
A dartboard is the region B in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8$. A target T is the region where $(x^2 + y^2 - 25)^2 \le 49$. A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as $\frac{m}{n} \pi$, where $m$ and $n$ are relatively prime positive integers. What is $m + n$?
$
\textbf{(A) }39 \qquad
\textbf{(B) }71 \qquad
\textbf{(C) }73 \qquad
\textbf{(D) }75 \qquad
\textbf{(E) }135 \qquad
$
2024 AMC 10, 11
In the figure below $WXYZ$ is a rectangle with $WX=4$ and $WZ=8$. Point $M$ lies $\overline{XY}$, point $A$ lies on $\overline{YZ}$, and $\angle WMA$ is a right angle. The areas of $\triangle WXM$ and $\triangle WAZ$ are equal. What is the area of $\triangle WMA$?
[asy]
pair X = (0, 0);
pair W = (0, 4);
pair Y = (8, 0);
pair Z = (8, 4);
label("$X$", X, dir(180));
label("$W$", W, dir(180));
label("$Y$", Y, dir(0));
label("$Z$", Z, dir(0));
draw(W--X--Y--Z--cycle);
dot(X);
dot(Y);
dot(W);
dot(Z);
pair M = (2, 0);
pair A = (8, 3);
label("$A$", A, dir(0));
dot(M);
dot(A);
draw(W--M--A--cycle);
markscalefactor = 0.05;
draw(rightanglemark(W, M, A));
label("$M$", M, dir(-90));
[/asy]
$
\textbf{(A) }13 \qquad
\textbf{(B) }14 \qquad
\textbf{(C) }15 \qquad
\textbf{(D) }16 \qquad
\textbf{(E) }17 \qquad
$
2024 AMC 10, 8
Let $N$ be the product of all the positive integer divisors of $42$. What is the units digit of $N$?
$
\textbf{(A) }0 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }4 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }8 \qquad
$
2024 AMC 10, 18
How many different remainders can result when the $100$th power of an integer is divided by $125$?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }25 \qquad
\textbf{(E) }125 \qquad
$
2024 AMC 12/AHSME, 13
The graph of $y=e^{x+1}+e^{-x}-2$ has an axis of symmetry. What is the reflection of the point $(-1,\tfrac{1}{2})$ over this axis?
$\textbf{(A) }\left(-1,-\frac{3}{2}\right)\qquad\textbf{(B) }(-1,0)\qquad\textbf{(C) }\left(-1,\tfrac{1}{2}\right)\qquad\textbf{(D) }\left(0,\frac{1}{2}\right)\qquad\textbf{(E) }\left(3,\frac{1}{2}\right)$
2024 AMC 12/AHSME, 15
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 $\triangle ABC$?
$
\textbf{(A) }\log_2\frac{\sqrt3}7\qquad
\textbf{(B) }\log_2\frac3{\sqrt7}\qquad
\textbf{(C) }\log_2\frac7{\sqrt3}\qquad
\textbf{(D) }\log_2\frac{11}{\sqrt7}\qquad
\textbf{(E) }\log_2\frac{11}{\sqrt3}\qquad
$
2024 AMC 12/AHSME, 9
Let $M$ be the greatest integer such that both $M + 1213$ and $M + 3773$ are perfect squares. What is the units digit of $M$?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }6 \qquad
\textbf{(E) }8 \qquad
$
2024 AMC 12/AHSME, 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 $x=s$. What is $p+q+r+s$?
$
\textbf{(A) }909\qquad
\textbf{(B) }910\qquad
\textbf{(C) }911\qquad
\textbf{(D) }912\qquad
\textbf{(E) }913\qquad
$
2024 AMC 10, 17
Two teams are in a best-two-out-of-three playoff: the teams will play at most $3$ games, and the winner of the playoff is the first team to win $2$ games. The first game is played on Team A's home field, and the remaining games are played on Team B's home field. Team A has a $\frac{2}{3}$ chance of winning at home, and its probability of winning when playing away from home is $p$. Outcomes of the games are independent. The probability that Team A wins the playoff is $\frac{1}{2}$. Then $p$ can be written in the form $\frac{1}{2}(m - \sqrt{n})$, where $m$ and $n$ are positive integers. What is $m + n$?
$\textbf{(A) } 10 \qquad \textbf{(B) } 11 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 13 \qquad \textbf{(E) } 14$
2024 AMC 12/AHSME, 14
How many different remainders can result when the $100$th power of an integer is divided by $125$?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }5 \qquad
\textbf{(D) }25 \qquad
\textbf{(E) }125 \qquad
$
2024 AMC 12/AHSME, 17
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$
\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad
$
2024 AMC 10, 6
A rectangle has integer side lengths and an area of $2024$. What is the least possible perimeter of the rectangle?
$
\textbf{(A) }160 \qquad
\textbf{(B) }180 \qquad
\textbf{(C) }222 \qquad
\textbf{(D) }228 \qquad
\textbf{(E) }390 \qquad
$
2024 AMC 10, 23
Integers $a$, $b$, and $c$ satisfy $ab + c = 100$, $bc + a = 87$, and $ca + b = 60$. What is $ab + bc + ca$?
$
\textbf{(A) }212 \qquad
\textbf{(B) }247 \qquad
\textbf{(C) }258 \qquad
\textbf{(D) }276 \qquad
\textbf{(E) }284 \qquad
$
2024 AMC 10, 23
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 $$\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?$$
$\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$
2024 AMC 10, 13
Two transformations are said to [i]commute[/i] if applying the first followed by the second gives the same result as applying the second followed by the first. Consider these four transformations of the coordinate plane:
- A translation $2$ units to the right
- A $90^\circ$- rotation counterclockwise about the origin.
- A reflection across the $x$-axis, and
- A dilation centered at the origin with scale factor $2$.
Of the $6$ pairs of distinct transformations from this list, how many commute?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) }5 \qquad
$
2024 AMC 12/AHSME, 24
What is the number of ordered triples $(a,b,c)$ of positive integers, with $a\le b\le c\le 9$, 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.)
$
\textbf{(A) }2\qquad
\textbf{(B) }3\qquad
\textbf{(C) }4\qquad
\textbf{(D) }5\qquad
\textbf{(E) }6\qquad
$
2024 AMC 12/AHSME, 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?
$\textbf{(A) }4\qquad\textbf{(B) }5\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad\textbf{(E) }8$
2024 AMC 10, 15
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?
$
\textbf{(A) }1 \qquad
\textbf{(B) }2 \qquad
\textbf{(C) }3 \qquad
\textbf{(D) }4 \qquad
\textbf{(E) infinitely many}\qquad
$
2024 AMC 12/AHSME, 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 deposted 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 despoited in bin $B$ at step 7 of this process.) In which bin is ball $2024$ deposited?
$\textbf{(A) }A\qquad\textbf{(B) }B\qquad\textbf{(C) }C\qquad\textbf{(D) }D\qquad\textbf{(E) }E$
2024 AMC 10, 10
Consider the following operation. Given a positive integer $n$, if $n$ is a multiple of $3$, then you replace $n$ by $\dfrac{n}3$. If $n$ is not a multiple of $3$, then you replace $n$ by $n + 10$. Then continue this process. For example, beginning with $n = 4$, this procedure gives $4 \to 14 \to 24 \to 8 \to 18 \to 6 \to 2 \to 12 \to \cdots$. Suppose you start with $n = 100$. What value results if you perform this operation exactly $100$ times?
$\textbf{(A) }10\qquad\textbf{(B) }20\qquad\textbf{(C) }30\qquad\textbf{(D) }40\qquad\textbf{(E) }50$
2024 AMC 10, 22
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 $r?$
$\textbf{(A) }5 \qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
2024 AMC 12/AHSME, 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?
$
\textbf{(A) }13 \qquad
\textbf{(B) }14 \qquad
\textbf{(C) }15 \qquad
\textbf{(D) }16 \qquad
\textbf{(E) }17 \qquad
$
2024 AMC 12/AHSME, 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?
$
\textbf{(A) }14 \qquad
\textbf{(B) }15 \qquad
\textbf{(C) }16 \qquad
\textbf{(D) }17 \qquad
\textbf{(E) }18 \qquad
$