Found problems: 3632
2020 AMC 12/AHSME, 10
In unit square $ABCD,$ the inscribed circle $\omega$ intersects $\overline{CD}$ at $M,$ and $\overline{AM}$ intersects $\omega$ at a point $P$ different from $M.$ What is $AP?$
$\textbf{(A) } \frac{\sqrt5}{12} \qquad \textbf{(B) } \frac{\sqrt5}{10} \qquad \textbf{(C) } \frac{\sqrt5}{9} \qquad \textbf{(D) } \frac{\sqrt5}{8} \qquad \textbf{(E) } \frac{2\sqrt5}{15}$
2008 AMC 10, 11
While Steve and LeRoy are fishing $ 1$ mile from shore, their boat springs a leak, and water comes in at a constant rate of $ 10$ gallons per minute. The boat will sink if it takes in more than $ 30$ gallons of water. Steve starts rowing toward the shore at a constant rate of $ 4$ miles per hour while LeRoy bails water out of the boat. What is the slowest rate, in gallons per minute, at which LeRoy can bail if they are to reach the shore without sinking?
$ \textbf{(A)}\ 2 \qquad
\textbf{(B)}\ 4 \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 8 \qquad
\textbf{(E)}\ 10$
2013 AMC 12/AHSME, 3
When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$?
$\textbf{(A) }146\qquad \textbf{(B) } 147\qquad\textbf{(C) } 148\qquad\textbf{(D) }149\qquad\textbf{(E) }150$
2014 AMC 10, 18
A list of $11$ positive integers has a mean of $10$, a median of $9$, and a unique mode of $8$. What is the largest possible value of an integer in the list?
${ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 31\qquad\textbf{(D)}}\ 33\qquad\textbf{(E)}\ 35 $
2022 AMC 10, 3
How many three-digit positive integers have an odd number of even digits?
$\textbf{(A) }150\qquad\textbf{(B) }250\qquad\textbf{(C) }350\qquad\textbf{(D) }450\qquad\textbf{(E) }550$
1974 AMC 12/AHSME, 6
For positive real numbers $x$ and $y$ define $x*y=\frac{x\cdot y}{x+y}$; then
$ \textbf{(A)}\ \text{"*" is commutative but not associative} \\ \qquad\textbf{(B)}\ \text{"*" is associative but not commutative} \\ \qquad\textbf{(C)}\ \text{"*" is neither commutative nor associative} \\ \qquad\textbf{(D)}\ \text{"*" is commutative and associative} \\ \qquad\textbf{(E)}\ \text{none of these} $
1987 AMC 12/AHSME, 8
In the figure the sum of the distances $AD$ and $BD$ is
[asy]
draw((0,0)--(13,0)--(13,4)--(10,4));
draw((12.5,0)--(12.5,.5)--(13,.5));
draw((13,3.5)--(12.5,3.5)--(12.5,4));
label("A", (0,0), S);
label("B", (13,0), SE);
label("C", (13,4), NE);
label("D", (10,4), N);
label("13", (6.5,0), S);
label("4", (13,2), E);
label("3", (11.5,4), N);
[/asy]
$ \textbf{(A)}\ \text{between 10 and 11} \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ \text{between 15 and 16} \qquad\textbf{(D)}\ \text{between 16 and 17} \qquad\textbf{(E)}\ 17 $
2024 AMC 10, 3
For how many integer values of $x$ is $|2x|\leq 7\pi?$
$\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$
2023 AMC 12/AHSME, 6
When the roots of the polynomial \[P(x)=\prod_{i=1}^{10}(x-i)^{i}\] are removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of those intervals is $P(x)$ positive?
$\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$
2022 AIME Problems, 10
Three spheres with radii $11$, $13$, and $19$ are mutually externally tangent. A plane intersects the spheres in three congruent circles centered at $A$, $B$, and $C$, respectively, and the centers of the spheres all lie on the same side of this plane. Suppose that $AB^2 = 560$. Find $AC^2$.
1978 AMC 12/AHSME, 13
If $a,b,c,$ and $d$ are non-zero numbers such that $c$ and $d$ are the solutions of $x^2+ax+b=0$ and $a$ and $b$ are the solutions of $x^2+cx+d=0$, then $a+b+c+d$ equals
$\textbf{(A) }0\qquad\textbf{(B) }-2\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad \textbf{(E) }(-1+\sqrt{5})/2$
2008 AMC 12/AHSME, 15
On each side of a unit square, an equilateral triangle of side length 1 is constructed. On each new side of each equilateral triangle, another equilateral triangle of side length 1 is constructed. The interiors of the square and the 12 triangles have no points in common. Let $ R$ be the region formed by the union of the square and all the triangles, and $ S$ be the smallest convex polygon that contains $ R$. What is the area of the region that is inside $ S$ but outside $ R$?
$ \textbf{(A)} \; \frac{1}{4} \qquad \textbf{(B)} \; \frac{\sqrt{2}}{4} \qquad \textbf{(C)} \; 1 \qquad \textbf{(D)} \; \sqrt{3} \qquad \textbf{(E)} \; 2 \sqrt{3}$
1993 AMC 12/AHSME, 27
The sides of $\triangle ABC$ have lengths $6, 8$ and $10$. A circle with center $P$ and radius $1$ rolls around the inside of $\triangle ABC$, always remaining tangent to at least one side of the triangle. When $P$ first returns to its original position, through what distance has $P$ traveled?
[asy]
draw((0,0)--(8,0)--(8,6)--(0,0));
draw(Circle((4.5,1),1));
draw((4.5,2.5)..(5.55,2.05)..(6,1), EndArrow);
dot((0,0));
dot((8,0));
dot((8,6));
dot((4.5,1));
label("A", (0,0), SW);
label("B", (8,0), SE);
label("C", (8,6), NE);
label("8", (4,0), S);
label("6", (8,3), E);
label("10", (4,3), NW);
label("P", (4.5,1), NW);
[/asy]
$ \textbf{(A)}\ 10 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 14 \qquad\textbf{(D)}\ 15 \qquad\textbf{(E)}\ 17 $
2024 AMC 12/AHSME, 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.)
$
\textbf{(A) }18 \qquad
\textbf{(B) }20 \qquad
\textbf{(C) }22 \qquad
\textbf{(D) }24 \qquad
\textbf{(E) }26 \qquad
$
1966 AMC 12/AHSME, 39
In base $R_1$ the expanded fraction $F_1$ becomes $0.373737...$, and the expanded fraction $F_2$ becomes $0.737373...$. In base $R_2$ fraction $F_1$, when expanded, becomes $0.252525...$, while fraction $F_2$ becomes $0.525252...$. The sum of $R_1$ and $R_2$, each written in base ten is:
$\text{(A)}\ 24 \qquad
\text{(B)}\ 22\qquad
\text{(C)}\ 21\qquad
\text{(D)}\ 20\qquad
\text{(E)}\ 19$
1995 AIME Problems, 6
Let $n=2^{31}3^{19}.$ How many positive integer divisors of $n^2$ are less than $n$ but do not divide $n$?
2017 USAJMO, 6
Let $P_1$, $P_2$, $\dots$, $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$, other than $(1,0)$. Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$, $R_2$, $\dots$, $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$. Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$, and so on, until we have labeled all of the blue points $B_1, \dots, B_n$. Show that the number of counterclockwise arcs of the form $R_i \to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \dots, R_n$ of the red points.
1978 AMC 12/AHSME, 30
In a tennis tournament, $n$ women and $2n$ men play, and each player plays exactly one match with every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is $7/5$, then $n$ equals
$\textbf{(A) }2\qquad\textbf{(B) }4\qquad\textbf{(C) }6\qquad\textbf{(D) }7\qquad \textbf{(E) }\text{none of these}$
2009 AMC 10, 20
Andrea and Lauren are $ 20$ kilometers apart. They bike toward one another with Andrea traveling three times as fast as Lauren, and the distance between them decreasing at a rate of $ 1$ kilometer per minute. After $ 5$ minutes, Andrea stops biking because of a flat tire and waits for Lauren. After how many minutes from the time they started to bike does Lauren reach Andrea?
$ \textbf{(A)}\ 20 \qquad
\textbf{(B)}\ 30 \qquad
\textbf{(C)}\ 55 \qquad
\textbf{(D)}\ 65 \qquad
\textbf{(E)}\ 80$
1991 AMC 12/AHSME, 21
If $f\left(\frac{x}{x - 1}\right) = \frac{1}{x}$ for all $x \ne 0,1$ and $0 < \theta < \frac{\pi}{2}$, then $f(\sec^{2}\theta) =$
$ \textbf{(A) }\sin^{2}\theta\qquad\textbf{(B) }\cos^{2}\theta\qquad\textbf{(C) }\tan^{2}\theta\qquad\textbf{(D) }\cot^{2}\theta\qquad\textbf{(E) }\csc^{2}\theta $
2006 AMC 12/AHSME, 14
Elmo makes $ N$ sandwiches for a fundraiser. For each sandwich he uses $ B$ globs of peanut butter at 4 cents per glob and $ J$ blobs of jam at 5 cents per glob. The cost of the peanut butter and jam to make all the sandwiches is $ \$$2.53. Assume that $ B, J,$ and $ N$ are all positive integers with $ N > 1$. What is the cost of the jam Elmo uses to make the sandwiches?
$ \textbf{(A) } \$1.05 \qquad \textbf{(B) } \$1.25 \qquad \textbf{(C) } \$1.45 \qquad \textbf{(D) } \$1.65 \qquad \textbf{(E) } \$1.85$
2007 AMC 12/AHSME, 10
A triangle with side lengths in the ratio $ 3: 4: 5$ is inscribed in a circle of radius $ 3.$ What is the area of the triangle?
$ \textbf{(A)}\ 8.64 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 5\pi \qquad \textbf{(D)}\ 17.28 \qquad \textbf{(E)}\ 18$
2002 AMC 8, 22
Six cubes, each an inch on an edge, are fastened together, as shown. Find the total surface area in square inches. Include the top, bottom and sides.
[asy]/* AMC8 2002 #22 Problem */
draw((0,0)--(0,1)--(1,1)--(1,0)--cycle);
draw((0,1)--(0.5,1.5)--(1.5,1.5)--(1,1));
draw((1,0)--(1.5,0.5)--(1.5,1.5));
draw((0.5,1.5)--(1,2)--(1.5,2));
draw((1.5,1.5)--(1.5,3.5)--(2,4)--(3,4)--(2.5,3.5)--(2.5,0.5)--(1.5,.5));
draw((1.5,3.5)--(2.5,3.5));
draw((1.5,1.5)--(3.5,1.5)--(3.5,2.5)--(1.5,2.5));
draw((3,4)--(3,3)--(2.5,2.5));
draw((3,3)--(4,3)--(4,2)--(3.5,1.5));
draw((4,3)--(3.5,2.5));
draw((2.5,.5)--(3,1)--(3,1.5));[/asy]
$ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 26\qquad\textbf{(D)}\ 30\qquad\textbf{(E)}\ 36$
2016 AMC 10, 14
How many squares whose sides are parallel to the axes and whose vertices have coordinates that are integers lie entirely within the region bounded by the line $y=\pi x$, the line $y=-0.1$ and the line $x=5.1?$
$\textbf{(A)}\ 30 \qquad
\textbf{(B)}\ 41 \qquad
\textbf{(C)}\ 45 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 57$
1997 AMC 8, 13
Three bags of jelly beans contain 26, 28, and 30 beans. The ratios of yellow beans to all beans in each of these bags are $50\%$, $25\%$, and $20\%$, respectively. All three bags of candy are dumped into one bowl. Which of the following is closest to the ratio of yellow jelly beans to all beans in the bowl?
$\textbf{(A)}\ 31\% \qquad \textbf{(B)}\ 32\% \qquad \textbf{(C)}\ 33\% \qquad \textbf{(D)}\ 35\% \qquad \textbf{(E)}\ 95\%$