Found problems: 3632
2010 AMC 10, 8
Tony works $ 2$ hours a day and is paid $ \$0.50$ per hour for each full year of his age. During a six month period Tony worked $ 50$ days and earned $ \$630$. How old was Tony at the end of the six month period?
$ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 11 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 13 \qquad \textbf{(E)}\ 14$
2017 AMC 12/AHSME, 12
There are $10$ horses, named Horse 1, Horse 2, $\ldots$, Horse 10. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time 0 all the horses are together at the starting point on the track. The horses start running in the same direction, and they keep running around the circular track at their constant speeds. The least time $S > 0$, in minutes, at which all $10$ horses will again simultaneously be at the starting point is $S = 2520$. Let $T>0$ be the least time, in minutes, such that at least $5$ of the horses are again at the starting point. What is the sum of the digits of $T$?
$\textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 5\qquad\textbf{(E)}\ 6$
1968 AMC 12/AHSME, 9
The sum of the real values of $x$ satisfying the equality $|x+2|=2|x-2|$ is:
$\textbf{(A)}\ \dfrac{1}{3} \qquad
\textbf{(B)}\ \dfrac{2}{3} \qquad
\textbf{(C)}\ 6 \qquad
\textbf{(D)}\ 6\dfrac{1}{3} \qquad
\textbf{(E)}\ 6\dfrac{2}{3} $
1979 USAMO, 1
Determine all non-negative integral solutions $ (n_{1},n_{2},\dots , n_{14}) $ if any, apart from permutations, of the Diophantine Equation \[n_{1}^{4}+n_{2}^{4}+\cdots+n_{14}^{4}=1,599.\]
1992 AMC 12/AHSME, 15
Let $i = \sqrt{-1}$. Define a sequence of complex numbers by $z_{1} = 0, z_{n+1} = z_{n}^{2}+i$ for $n \ge 1$. In the complex plane, how far from the origin is $z_{111}$?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ \sqrt{2}\qquad\textbf{(C)}\ \sqrt{3}\qquad\textbf{(D)}\ \sqrt{110}\qquad\textbf{(E)}\ \sqrt{2^{55}} $
2010 Danube Mathematical Olympiad, 1
Determine all integer numbers $n\ge 3$ such that the regular $n$-gon can be decomposed into isosceles triangles by non-intersecting diagonals.
2013 AIME Problems, 6
Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold any five of her textbooks. If Melinda packs her textbooks into these boxes in random order, the probability that all three mathematics textbooks end up in the same box can be written as $\frac{m}{n}$, where $m$ and $n$ Are relatively prime positive integers. Find $m+n$.
2019 AIME Problems, 4
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
1998 AMC 8, 7
$ 100\times 19.98\times 1.998\times 1000\equal{} $
$ \text{(A)}\ (1.998)^{2}\qquad\text{(B)}\ (19.98)^{2}\qquad\text{(C)}\ (199.8)^{2}\qquad\text{(D)}\ (1998)^{2}\qquad\text{(E)}\ (19980)^{2} $
2001 AMC 10, 23
A box contains exactly five chips, three red and two white. Chips are randomly removed one at a time without replacement until all the red chips are drawn or all the white chips are drawn. What is the probability that the last chip drawn is white?
$ \displaystyle \textbf{(A)} \ \frac {3}{10} \qquad \textbf{(B)} \ \frac {2}{5} \qquad \textbf{(C)} \ \frac {1}{2} \qquad \textbf{(D)} \ \frac {3}{5} \qquad \textbf{(E)} \ \frac {7}{10}$
2015 AIME Problems, 11
Triangle $ABC$ has positive integer side lengths with $AB=AC$. Let $I$ be the intersection of the bisectors of $\angle B$ and $\angle C$. Suppose $BI=8$. Find the smallest possible perimeter of $\triangle ABC$.
2018 AMC 12/AHSME, 11
A paper triangle with sides of lengths 3, 4, and 5 inches, as shown, is folded so that point $A$ falls on point $B$. What is the length in inches of the crease?
[asy]
draw((0,0)--(4,0)--(4,3)--(0,0));
label("$A$", (0,0), SW);
label("$B$", (4,3), NE);
label("$C$", (4,0), SE);
label("$4$", (2,0), S);
label("$3$", (4,1.5), E);
label("$5$", (2,1.5), NW);
fill(origin--(0,0)--(4,3)--(4,0)--cycle, gray(0.9));
[/asy]
$\textbf{(A) } 1+\frac12 \sqrt2 \qquad \textbf{(B) } \sqrt3 \qquad \textbf{(C) } \frac74 \qquad \textbf{(D) } \frac{15}{8} \qquad \textbf{(E) } 2 $
2012 Balkan MO, 4
Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold:
(i) $f(n!)=f(n)!$ for every positive integer $n$,
(ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.
1968 AMC 12/AHSME, 13
If $m$ and $n$ are the roots of $x^2+mx+n=0$, $m\ne0$, $n\ne0$, then the sum of the roots is:
$\textbf{(A)}\ -\dfrac{1}{2} \qquad
\textbf{(B)}\ -1 \qquad
\textbf{(C)}\ \dfrac{1}{2} \qquad
\textbf{(D)}\ 1 \qquad
\textbf{(E)}\ \text{Undetermined} $
2021 AMC 12/AHSME Spring, 18
Let $z$ be a complex number satisfying $12\lvert z\rvert^2 = 2 \lvert z+2 \rvert ^2+\lvert z^2+1\rvert ^2+31.$ What is the value of $z+\frac{6}{z}?$
$\textbf{(A) }-2\qquad\textbf{(B) }-1\qquad\textbf{(C) }\frac{1}{2}\qquad\textbf{(D) }1\qquad\textbf{(E) }4$
2019 AMC 10, 20
As shown in the figure, line segment $\overline{AD}$ is trisected by points $B$ and $C$ so that $AB=BC=CD=2.$ Three semicircles of radius $1,$ $\overarc{AEB},\overarc{BFC},$ and $\overarc{CGD},$ have their diameters on $\overline{AD},$ and are tangent to line $EG$ at $E,F,$ and $G,$ respectively. A circle of radius $2$ has its center on $F. $ The area of the region inside the circle but outside the three semicircles, shaded in the figure, can be expressed in the form
\[\frac{a}{b}\cdot\pi-\sqrt{c}+d,\]
where $a,b,c,$ and $d$ are positive integers and $a$ and $b$ are relatively prime. What is $a+b+c+d$?
[asy]
size(6cm);
filldraw(circle((0,0),2), gray(0.7));
filldraw(arc((0,-1),1,0,180) -- cycle, gray(1.0));
filldraw(arc((-2,-1),1,0,180) -- cycle, gray(1.0));
filldraw(arc((2,-1),1,0,180) -- cycle, gray(1.0));
dot((-3,-1));
label("$A$",(-3,-1),S);
dot((-2,0));
label("$E$",(-2,0),NW);
dot((-1,-1));
label("$B$",(-1,-1),S);
dot((0,0));
label("$F$",(0,0),N);
dot((1,-1));
label("$C$",(1,-1), S);
dot((2,0));
label("$G$", (2,0),NE);
dot((3,-1));
label("$D$", (3,-1), S);
[/asy]
$\textbf{(A) } 13 \qquad\textbf{(B) } 14 \qquad\textbf{(C) } 15 \qquad\textbf{(D) } 16\qquad\textbf{(E) } 17$
2020 AMC 12/AHSME, 21
How many positive integers $n$ satisfy$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
$\textbf{(A) } 2 \qquad\textbf{(B) } 4 \qquad\textbf{(C) } 6 \qquad\textbf{(D) } 30 \qquad\textbf{(E) } 32$
2015 AMC 8, 18
An arithmetic sequence is a sequence in which each term after the first is obtained by adding a constant to the previous term. For example, $2,5,8,11,14$ is an arithmetic sequence with five terms, in which the first term is $2$ and the constant added is $3$. Each row and each column in this $5\times5$ array is an arithmetic sequence with five terms. What is the value of $X$?
$\textbf{(A) }21\qquad\textbf{(B) }31\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad \textbf{(E) }42$
[asy]
size(3.85cm);
label("$X$",(2.5,2.1),N);
for (int i=0; i<=5; ++i)
draw((i,0)--(i,5), linewidth(.5));
for (int j=0; j<=5; ++j)
draw((0,j)--(5,j), linewidth(.5));
void draw_num(pair ll_corner, int num)
{
label(string(num), ll_corner + (0.5, 0.5), p = fontsize(19pt));
}
draw_num((0,0), 17);
draw_num((4, 0), 81);
draw_num((0, 4), 1);
draw_num((4,4), 25);
void foo(int x, int y, string n)
{
label(n, (x+0.5,y+0.5), p = fontsize(19pt));
}
foo(2, 4, " ");
foo(3, 4, " ");
foo(0, 3, " ");
foo(2, 3, " ");
foo(1, 2, " ");
foo(3, 2, " ");
foo(1, 1, " ");
foo(2, 1, " ");
foo(3, 1, " ");
foo(4, 1, " ");
foo(2, 0, " ");
foo(3, 0, " ");
foo(0, 1, " ");
foo(0, 2, " ");
foo(1, 0, " ");
foo(1, 3, " ");
foo(1, 4, " ");
foo(3, 3, " ");
foo(4, 2, " ");
foo(4, 3, " ");
[/asy]
1960 AMC 12/AHSME, 5
The number of distinct points common to the graphs of $x^2+y^2=9$ and $y^2=9$ is:
$ \textbf{(A) }\text{infinitely many} \qquad\textbf{(B) } \text{four}\qquad\textbf{(C) }\text{two}\qquad\textbf{(D) }\text{one}\qquad\textbf{(E) }\text{none} $
1978 AMC 12/AHSME, 14
If an integer $n > 8$ is a solution of the equation $x^2 - ax+b=0$ and the representation of $a$ in the base-$n$ number system is $18$, then the base-$n$ representation of $b$ is
$\textbf{(A)}\ 18 \qquad
\textbf{(B)}\ 20 \qquad
\textbf{(C)}\ 80 \qquad
\textbf{(D)}\ 81 \qquad
\textbf{(E)}\ 280$
1987 AMC 12/AHSME, 22
A ball was floating in a lake when the lake froze. The ball was removed (without breaking the ice), leaving a hole $24$ cm across as the top and $8$ cm deep. What was the radius of the ball (in centimeters)?
$ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 13 \qquad\textbf{(D)}\ 8\sqrt{3} \qquad\textbf{(E)}\ 6\sqrt{6} $
1993 Balkan MO, 1
Let $a,b,c,d,e,f$ be six real numbers with sum 10, such that \[ (a-1)^2+(b-1)^2+(c-1)^2+(d-1)^2+(e-1)^2+(f-1)^2 = 6. \] Find the maximum possible value of $f$.
[i]Cyprus[/i]
2009 AMC 10, 19
Circle $ A$ has radius $ 100$. Circle $ B$ has an integer radius $ r<100$ and remains internally tangent to circle $ A$ as it rolls once around the circumference of circle $ A$. The two circles have the same points of tangency at the beginning and end of circle $ B$'s trip. How many possible values can $ r$ have?
$ \textbf{(A)}\ 4 \qquad
\textbf{(B)}\ 8 \qquad
\textbf{(C)}\ 9 \qquad
\textbf{(D)}\ 50 \qquad
\textbf{(E)}\ 90$
2010 AMC 12/AHSME, 22
Let $ ABCD$ be a cyclic quadrilateral. The side lengths of $ ABCD$ are distinct integers less than $ 15$ such that $ BC\cdot CD\equal{}AB\cdot DA$. What is the largest possible value of $ BD$?
$ \textbf{(A)}\ \sqrt{\frac{325}{2}} \qquad \textbf{(B)}\ \sqrt{185} \qquad \textbf{(C)}\ \sqrt{\frac{389}{2}} \qquad \textbf{(D)}\ \sqrt{\frac{425}{2}} \qquad \textbf{(E)}\ \sqrt{\frac{533}{2}}$
1977 AMC 12/AHSME, 9
[asy]
size(120);
path c = Circle((0, 0), 1);
pair A = dir(20), B = dir(130), C = dir(240), D = dir(330);
draw(c);
pair F = 3(A-B) + B;
pair G = 3(D-C) + C;
pair E = intersectionpoints(B--F, C--G)[0];
draw(B--E--C--A);
label("$A$", A, NE);
label("$B$", B, NW);
label("$C$", C, SW);
label("$D$", D, SE);
label("$E$", E, E);
//Credit to MSTang for the diagram[/asy]
In the adjoining figure $\measuredangle E=40^\circ$ and arc $AB$, arc $BC$, and arc $CD$ all have equal length. Find the measure of $\measuredangle ACD$.
$\textbf{(A) }10^\circ\qquad\textbf{(B) }15^\circ\qquad\textbf{(C) }20^\circ\qquad\textbf{(D) }\left(\frac{45}{2}\right)^\circ\qquad \textbf{(E) }30^\circ$