Found problems: 3632
2017 AMC 10, 3
Real numbers $x$, $y$, and $z$ satisfy the inequalities
$$0<x<1,\qquad-1<y<0,\qquad\text{and}\qquad1<z<2.$$
Which of the following numbers is nessecarily positive?
$\textbf{(A) } y+x^2 \qquad \textbf{(B) } y+xz \qquad \textbf{(C) }y+y^2 \qquad \textbf{(D) }y+2y^2 \qquad\\
\textbf{(E) } y+z$
1976 AMC 12/AHSME, 8
A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. What is the probability that the distance from the point to the origin is at most two units?
$\textbf{(A) }\frac{13}{81}\qquad\textbf{(B) }\frac{15}{81}\qquad\textbf{(C) }\frac{13}{64}\qquad\textbf{(D) }\frac{\pi}{16}\qquad \textbf{(E) }\text{the square of a rational number}$
2012 AMC 10, 18
The closed curve in the figure is made up of $9$ congruent circular arcs each of length $\frac{2\pi}{3}$, where each of the centers of the corresponding circles is among the vertices of a regular hexagon of side $2$. What is the area enclosed by the curve?
[asy]
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label("$\circ$",(0.865,0.5));
label("$\circ$",(-0.865,0.5));
label("$\circ$",(0.865,-0.5));
label("$\circ$",(-0.865,-0.5));
label("$\circ$",(0,-1));
dot((0,1.5));
dot((-0.4325,0.75));
dot((0.4325,0.75));
dot((-0.4325,-0.75));
dot((0.4325,-0.75));
dot((-0.865,0));
dot((0.865,0));
dot((-1.2975,-0.75));
dot((1.2975,-0.75));
draw(Arc((0,1),0.5,210,-30));
draw(Arc((0.865,0.5),0.5,150,270));
draw(Arc((0.865,-0.5),0.5,90,-150));
draw(Arc((0.865,-0.5),0.5,90,-150));
draw(Arc((0,-1),0.5,30,150));
draw(Arc((-0.865,-0.5),0.5,330,90));
draw(Arc((-0.865,0.5),0.5,-90,30));
[/asy]
$ \textbf{(A)}\ 2\pi+6\qquad\textbf{(B)}\ 2\pi+4\sqrt3 \qquad\textbf{(C)}\ 3\pi+4 \qquad\textbf{(D)}\ 2\pi+3\sqrt3+2 \qquad\textbf{(E)}\ \pi+6\sqrt3 $
1961 AMC 12/AHSME, 20
The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants:
${{ \textbf{(A)}\ \text{I and II} \qquad\textbf{(B)}\ \text{II and III} \qquad\textbf{(C)}\ \text{I and III} \qquad\textbf{(D)}\ \text{III and IV} }\qquad\textbf{(E)}\ \text{I and IV} } $
1969 AMC 12/AHSME, 31
Let $OABC$ be a unit square in the $xy$-plane with $O(0,0),A(1,0),B(1,1)$ and $C(0,1)$. Let $u=x^2-y^2$ and $v=2xy$ be a transformation of the $xy$-plane into the $uv$-plane. The transform (or image) of the square is:
[asy]
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draw((-2.5,0)--(2.5,0),EndArrow(size=7));
draw((0,-3)--(0,3),EndArrow(size=7));
label("$O$",(0,0),SW);
label("$u$",(2.5,0),E);
label("$v$",(0,3),N);
draw((0,2)--(1,0)--(0,-2)--(-1,0)--cycle);
label("$(0,2)$",(0,2),NE);
label("$(1,0)$",(1,0),SE);
label("$(0,-2)$",(0,-2),SE);
label("$(-1,0)$",(-1,0),SW);
label("$\textbf{(A)}$",(-2,1.5));
[/asy]
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(8));
draw((-2.5,0)--(2.5,0),EndArrow(size=7));
draw((0,-3)--(0,3),EndArrow(size=7));
label("$O$",(0,0),SW);
label("$u$",(2.5,0),E);
label("$v$",(0,3),N);
draw((0,2)..(1,0)..(0,-2)^^(0,-2)..(-1,0)..(0,2));
label("$(0,2)$",(0,2),NE);
label("$(1,0)$",(1,0),SE);
label("$(0,-2)$",(0,-2),SE);
label("$(-1,0)$",(-1,0),SW);
label("$\textbf{(B)}$",(-2,1.5));
[/asy]
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(8));
draw((-2.5,0)--(2.5,0),EndArrow(size=7));
draw((0,-3)--(0,3),EndArrow(size=7));
label("$O$",(0,0),SW);
label("$u$",(2.5,0),E);
label("$v$",(0,3),N);
draw((0,2)--(1,0)--(-1,0)--cycle);
label("$(0,2)$",(0,2),NE);
label("$(1,0)$",(1,0),S);
label("$(-1,0)$",(-1,0),S);
label("$\textbf{(C)}$",(-2,1.5));
[/asy]
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(8));
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draw((0,-3)--(0,3),EndArrow(size=7));
label("$O$",(0,0),SW);
label("$u$",(2.5,0),E);
label("$v$",(0,3),N);
draw((0,2)..(1/2,3/2)..(1,0)--(-1,0)..(-1/2,3/2)..(0,2));
label("$(0,2)$",(0,2),NE);
label("$(1,0)$",(1,0),S);
label("$(-1,0)$",(-1,0),S);
label("$\textbf{(D)}$",(-2,1.5));
[/asy]
[asy]
size(150);
defaultpen(linewidth(0.8)+fontsize(8));
draw((-2.5,0)--(2.5,0),EndArrow(size=7));
draw((0,-3)--(0,3),EndArrow(size=7));
label("$O$",(0,0),SW);
label("$u$",(2.5,0),E);
label("$v$",(0,3),N);
draw((0,1)--(1,0)--(0,-1)--(-1,0)--cycle);
label("$(0,1)$",(0,1),NE);
label("$(1,0)$",(1,0),SE);
label("$(0,-1)$",(0,-1),SE);
label("$(-1,0)$",(-1,0),SW);
label("$\textbf{(E)}$",(-2,1.5));
[/asy]
1979 AMC 12/AHSME, 18
To the nearest thousandth, $\log_{10}2$ is $.301$ and $\log_{10}3$ is $.477$. Which of the following is the best approximation of $\log_5 10$?
$\textbf{(A) }\frac{8}{7}\qquad\textbf{(B) }\frac{9}{7}\qquad\textbf{(C) }\frac{10}{7}\qquad\textbf{(D) }\frac{11}{7}\qquad\textbf{(E) }\frac{12}{7}$
2007 AMC 10, 3
An aquarium has a rectangular base that measures 100 cm by 40 cm and has a height of 50 cm. It is filled with water to a height of 40 cm. A brick with a rectangular base that measures 40 cm by 20 cm and a height of 10 cm is placed in the aquarium. By how many centimeters does the water rise?
$ \textbf{(A)}\ 0.5 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 1.5 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 2.5$
2016 AIME Problems, 4
An $a\times b\times c$ rectangular box is built from $a\cdot b \cdot c$ unit cubes. Each unit cube is colored red, green, or yellow. Each of the $a$ layers of size $1\times b \times c$ parallel to the $(b\times c)$-faces of the box contains exactly $9$ red cubes, exactly 12 green cubes, and some yellow cubes. Each of the $b$ layers of size $a\times 1 \times c$ parallel to the $(a\times c)$-faces of the box contains exactly 20 green cubes, exactly 25 yellow cubes, and some red cubes. Find the smallest possible volume of the box.
1971 AMC 12/AHSME, 34
An ordinary clock in a factory is running slow so that the minute hand passes the hour hand at the usual dial position($12$ o'clock, etc.) but only every $69$ minutes. At time and one-half for overtime, the extra pay to which a $\textdollar 4.00$ per hour worker should be entitled after working a normal $8$ hour day by that slow running clock, is
$\textbf{(A) }\textdollar 2.30\qquad\textbf{(B) }\textdollar 2.60\qquad\textbf{(C) }\textdollar 2.80\qquad\textbf{(D) }\textdollar 3.00\qquad \textbf{(E) }\textdollar 3.30$
2012 AMC 12/AHSME, 15
A $3\times3$ square is partitioned into $9$ unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is the rotated $90^\circ$ clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. What is the probability that the grid is now entirely black?
$ \textbf{(A)}\ \dfrac{49}{512}
\qquad\textbf{(B)}\ \dfrac{7}{64}
\qquad\textbf{(C)}\ \dfrac{121}{1024}
\qquad\textbf{(D)}\ \dfrac{81}{512}
\qquad\textbf{(E)}\ \dfrac{9}{32}
$
2008 AMC 10, 5
Which of the following is equal to the product
\[ \frac {8}{4}\cdot\frac {12}{8}\cdot\frac {16}{12}\cdots\frac {4n \plus{} 4}{4n}\cdots\frac {2008}{2004}?
\]$ \textbf{(A)}\ 251 \qquad \textbf{(B)}\ 502 \qquad \textbf{(C)}\ 1004 \qquad \textbf{(D)}\ 2008 \qquad \textbf{(E)}\ 4016$
2020 CHMMC Winter (2020-21), 1
A right triangle $ABC$ is inscribed in the circular base of a cone. If two of the side lengths of $ABC$ are $3$ and $4$, and the distance from the vertex of the cone to any point on the circumference of the base is $3$, then the minimum possible volume of the cone can be written as $\frac{m\pi\sqrt{n}}{p}$, where $m$, $n$, and $p$ are positive integers, $m$ and $p$ are relatively prime, and $n$ is squarefree. Find $m + n + p$.
2014 AMC 12/AHSME, 2
Orvin went to the store with just enough money to buy $30$ balloons. When he arrived, he discovered that the store had a special sale on balloons: buy $1$ balloon at the regular price and get a second at $\frac{1}{3}$ off the regular price. What is the greatest number of balloons Orvin could buy?
$ \textbf {(A) } 33 \qquad \textbf {(B) } 34 \qquad \textbf {(C) } 36 \qquad \textbf {(D) } 38 \qquad \textbf {(E) } 39$
2017 AMC 12/AHSME, 15
Let $f(x)=\sin x+2\cos x+3\tan x$, using radian measure for the variable $x$. In what interval does the smallest positive value of $x$ for which $f(x)=0$ lie?
$\textbf{(A) } (0,1) \qquad \textbf{(B) } (1,2) \qquad \textbf{(C) } (2,3) \qquad \textbf{(D) } (3,4) \qquad \textbf{(E) } (4,5)$
2000 AMC 8, 17
The operation $\otimes$ is defined for all nonzero numbers by $a\otimes b = \dfrac{a^2}{b}$. Determine $[(1\otimes 2)\otimes 3] - [1\otimes (2\otimes 3)]$.
$\text{(A)}\ -\dfrac{2}{3} \qquad \text{(B)}\ -\dfrac{1}{4} \qquad \text{(C)}\ 0 \qquad \text{(D)}\ \dfrac{1}{4} \qquad \text{(E)}\ \dfrac{2}{3}$
1984 USAMO, 4
A difficult mathematical competition consisted of a Part I and a Part II with a combined total of $28$ problems. Each contestant solved $7$ problems altogether. For each pair of problems, there were exactly two contestants who solved both of them. Prove that there was a contestant who, in Part I, solved either no problems or at least four problems.
2008 AMC 12/AHSME, 21
Two circles of radius 1 are to be constructed as follows. The center of circle $ A$ is chosen uniformly and at random from the line segment joining $ (0,0)$ and $ (2,0)$. The center of circle $ B$ is chosen uniformly and at random, and independently of the first choice, from the line segment joining $ (0,1)$ to $ (2,1)$. What is the probability that circles $ A$ and $ B$ intersect?
$ \textbf{(A)} \; \frac{2\plus{}\sqrt{2}}{4} \qquad \textbf{(B)} \; \frac{3\sqrt{3}\plus{}2}{8} \qquad \textbf{(C)} \; \frac{2 \sqrt{2} \minus{} 1}{2} \qquad \textbf{(D)} \; \frac{2\plus{}\sqrt{3}}{4} \qquad \textbf{(E)} \; \frac{4 \sqrt{3} \minus{} 3}{4}$
2016 USAMO, 6
Integers $n$ and $k$ are given, with $n\ge k\ge2$. You play the following game against an evil wizard.
The wizard has $2n$ cards; for each $i=1,\ldots,n$, there are two cards labeled $i$. Initially, the wizard places all cards face down in a row, in unknown order.
You may repeatedly make moves of the following form: you point to any $k$ of the cards. The wizard then turns those cards face up. If any two of the cards match, the game is over and you win. Otherwise, you must look away, while the wizard arbitrarily permutes the $k$ chosen cards and then turns them back face-down. Then, it is your turn again.
We say this game is [i]winnable[/i] if there exist some positive integer $m$ and some strategy that is guaranteed to win in at most $m$ moves, no matter how the wizard responds.
For which values of $n$ and $k$ is the game winnable?
2012 Olympic Revenge, 3
Let $G$ be a finite graph. Prove that one can partition $G$ into two graphs $A \cup B=G$ such that if we erase all edges conecting a vertex from $A$ to a vertex from $B$, each vertex of the new graph has even degree.
2010 AMC 10, 22
Eight points are chosen on a circle, and chords are drawn connecting every pair of points. No three chords intersect in a single point inside the circle. How many triangles with all three vertices in the interior of the circle are created?
$ \textbf{(A)}\ 28 \qquad
\textbf{(B)}\ 56 \qquad
\textbf{(C)}\ 70 \qquad
\textbf{(D)}\ 84 \qquad
\textbf{(E)}\ 140$
1960 AMC 12/AHSME, 8
The number $2.5252525...$ can be written as a fraction. When reduced to lowest terms the sum of the numerator and denominator of this fraction is:
$ \textbf{(A) }7\qquad\textbf{(B)} 29\qquad\textbf{(C) }141\qquad\textbf{(D) }349\qquad\textbf{(E) }\text{none of these} $
2002 AMC 12/AHSME, -1
This test and the matching AMC 10P were developed for the use of a group of Taiwan schools, in early January of 2002. When Taiwan had taken the contests, the AMC released the questions here as a set of practice questions for the 2002 AMC 10 and AMC 12 contests.
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 $r?$
$\textbf{(A) }5 \qquad\textbf{(B) }6\qquad\textbf{(C) }7\qquad\textbf{(D) }8\qquad\textbf{(E) }9$
1991 AMC 12/AHSME, 8
Liquid X does not mix with water. Unless obstructed, it spreads out on the surface of water to form a circular film $0.1$ cm thick. A rectangular box measuring $6$ cm by $3$ cm by $12$ cm is filled with liquid X. Its contents are poured onto a large body of water. What will be the radius, in centimeters, of the resulting circular film?
$ \textbf{(A)}\ \frac{\sqrt{216}}{\pi}\qquad\textbf{(B)}\ \sqrt{\frac{216}{\pi}}\qquad\textbf{(C)}\ \sqrt{\frac{2160}{\pi}}\qquad\textbf{(D)}\ \frac{216}{\pi}\qquad\textbf{(E)}\ \frac{2160}{\pi} $
2006 AIME Problems, 5
The number \[ \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006} \] can be written as $a\sqrt{2}+b\sqrt{3}+c\sqrt{5},$ where $a, b,$ and $c$ are positive integers. Find $a\cdot b\cdot c$.