Found problems: 3632
2018 USAMO, 2
Find all functions $f:(0,\infty) \rightarrow (0,\infty)$ such that \[f\left(x+\frac{1}{y}\right)+f\left(y+\frac{1}{z}\right) + f\left(z+\frac{1}{x}\right) = 1\] for all $x,y,z >0$ with $xyz =1$.
1988 AMC 12/AHSME, 28
An unfair coin has probability $p$ of coming up heads on a single toss. Let $w$ be the probability that, in $5$ independent toss of this coin, heads come up exactly $3$ times. If $w = 144 / 625$, then
$ \textbf{(A)}\ p\text{ must be }2/5$
$ \textbf{(B)}\ p\text{ must be }3/5$
$ \textbf{(C)}\ p\text{ must be greater than }3/5$
$ \textbf{(D)}\ p\text{ is not uniquely determined}$
$ \textbf{(E)}\ \text{there is no value of }p\text{ for which }w = 144/625$
2014 AMC 8, 5
Margie's car can go $32$ miles on a gallon of gas, and gas currently costs $\$4$ per gallon. How many miles can Margie drive on $\$20$ worth of gas?
$\textbf{(A) }64\qquad\textbf{(B) }128\qquad\textbf{(C) }160\qquad\textbf{(D) }320\qquad \textbf{(E) }640$
1959 AMC 12/AHSME, 3
If the diagonals of a quadrilateral are perpendicular to each other, the figure would always be included under the general classification:
$ \textbf{(A)}\ \text{rhombus} \qquad\textbf{(B)}\ \text{rectangles} \qquad\textbf{(C)}\ \text{square} \qquad\textbf{(D)}\ \text{isosceles trapezoid}\qquad\textbf{(E)}\ \text{none of these} $
2011 AMC 12/AHSME, 22
Let $R$ be a square region and $n \ge 4$ an integer. A point $X$ in the interior of $R$ is called [i]n-ray partitional[/i] if there are $n$ rays emanating from $X$ that divide $R$ into $n$ triangles of equal area. How many points are 100-ray partitional but not 60-ray partitional?
$\textbf{(A)}\ 1500 \qquad
\textbf{(B)}\ 1560 \qquad
\textbf{(C)}\ 2320 \qquad
\textbf{(D)}\ 2480 \qquad
\textbf{(E)}\ 2500$
2012 ELMO Shortlist, 8
Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair $(x,y)$ denote the complex number $x+y\omega$ for $\omega=e^{2\pi i/3}$. We define an $\omega$-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an $\omega$-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a [i]tasteful tiling[/i] is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order).
a) Prove that if an $\omega$-chessboard polygon can be tiled by lozenges, then it can be done so tastefully.
b) Prove that such a tasteful tiling is unique.
[i]Victor Wang.[/i]
1977 AMC 12/AHSME, 30
[asy]
for (int i=0; i<9; ++i) {
draw(dir(10+40*i)--dir(50+40*i));
}
draw(dir(50) -- dir(90));
label("$a$", dir(50) -- dir(90), N);
draw(dir(10) -- dir(90));
label("$b$", dir(10) -- dir(90), SW);
draw(dir(-70) -- dir(90));
label("$d$", dir(-70) -- dir(90), E);
//Credit to MSTang for the diagram[/asy]
If $a,b,$ and $d$ are the lengths of a side, a shortest diagonal and a longest diagonal, respectively, of a regular nonagon (see adjoining figure), then
$\textbf{(A) }d=a+b\qquad\textbf{(B) }d^2=a^2+b^2\qquad\textbf{(C) }d^2=a^2+ab+b^2\qquad$
$\textbf{(D) }b=\frac{a+d}{2}\qquad \textbf{(E) }b^2=ad$
1988 AMC 12/AHSME, 20
In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY$ / $YZ$, in this rectangle is
[asy]
size(180);
defaultpen(linewidth(0.7)+fontsize(10));
pair A=(0,1), B=(0,-1), C=(2,-1), D=(2,1), E=(1,-1), F=(1,1), G=(.8,.6);
pair X=(4,sqrt(5)), Y=(4,-sqrt(5)), Z=(4+2/sqrt(5),-sqrt(5)), W=(4+2/sqrt(5),sqrt(5)), T=(4,0), U=(4+2/sqrt(5),-4/sqrt(5)), V=(4+2/sqrt(5),1/sqrt(5));
draw(A--B--C--D--A^^B--F^^E--D^^A--G^^rightanglemark(A,G,F));
draw(X--Y--Z--W--X^^T--V--X^^Y--U);
label("A", A, NW);
label("B", B, SW);
label("C", C, SE);
label("D", D, NE);
label("E", E, S);
label("F", F, N);
label("G", G, E);
label("X", X, NW);
label("Y", Y, SW);
label("Z", Z, SE);
label("W", W, NE);
[/asy]
$ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 1+2\sqrt{3}\qquad\textbf{(C)}\ 2\sqrt{5}\qquad\textbf{(D)}\ \frac{8+4\sqrt{3}}{3}\qquad\textbf{(E)}\ 5 $
1998 USAMO, 6
Let $n \geq 5$ be an integer. Find the largest integer $k$ (as a function of $n$) such that there exists a convex $n$-gon $A_{1}A_{2}\dots A_{n}$ for which exactly $k$ of the quadrilaterals $A_{i}A_{i+1}A_{i+2}A_{i+3}$ have an inscribed circle. (Here $A_{n+j} = A_{j}$.)
1969 AMC 12/AHSME, 27
A particle moves so that its speed for the second and subsequent miles varies inversely as the integral number of miles already traveled. For each subsequent mile the speed is constant. If the second mile is traversed in $2$ hours, then the time, in hours, needed to traverse the $n$th mile is:
$\textbf{(A) }\dfrac2{n-1}\qquad
\textbf{(B) }\dfrac{n-1}2\qquad
\textbf{(C) }\dfrac2n\qquad
\textbf{(D) }2n\qquad
\textbf{(E) }2(n-1)$
2014 USAMTS Problems, 2:
Find all triples $(x, y, z)$ such that $x, y, z, x - y, y - z, x - z$ are all prime positive integers.
1959 AMC 12/AHSME, 16
The expression $\frac{x^2-3x+2}{x^2-5x+6}\div \frac{x^2-5x+4}{x^2-7x+12}$, when simplified is:
$ \textbf{(A)}\ \frac{(x-1)(x-6)}{(x-3)(x-4)} \qquad\textbf{(B)}\ \frac{x+3}{x-3}\qquad\textbf{(C)}\ \frac{x+1}{x-1}\qquad\textbf{(D)}\ 1\qquad\textbf{(E)}\ 2$
2006 AMC 10, 13
A player pays $ \$ 5$ to play a game. A die is rolled. If the number on the die is odd, the game is lost. If the number on the die is even, the die is rolled again. In this case the player wins if the second number matches the first and loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the player should pay.)
$ \textbf{(A) } \$ 12 \qquad \textbf{(B) } \$ 30 \qquad \textbf{(C) } \$ 50\qquad \textbf{(D) } \$ 60 \qquad \textbf{(E) } \$ 100$
2010 USAMO, 3
The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.
2006 AIME Problems, 2
Let set $\mathcal{A}$ be a 90-element subset of $\{1,2,3,\ldots,100\},$ and let $S$ be the sum of the elements of $\mathcal{A}$. Find the number of possible values of $S$.
2024 AMC 12/AHSME, 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 $$\dfrac{F_2}{F_1}+\dfrac{F_4}{F_2}+\dfrac{F_6}{F_3}+\cdots+\dfrac{F_{20}}{F_{10}}?$$
$\textbf{(A) }318 \qquad\textbf{(B) }319\qquad\textbf{(C) }320\qquad\textbf{(D) }321\qquad\textbf{(E) }322$
2025 AIME, 6
An isosceles trapezoid has an inscribed circle tangent to each of its four sides. The radius of the circle is $3$, and the area of the trapezoid is $72$. Let the parallel sides of the trapezoid have lengths $r$ and $s$, with $r \neq s$. Find $r^2+s^2$
1976 AMC 12/AHSME, 26
[asy]
size(150);
dotfactor=4;
draw(circle((0,0),4));
draw(circle((10,-6),3));
pair O,A,P,Q;
O = (0,0);
A = (10,-6);
P = (-.55, -4.12);
Q = (10.7, -2.86);
dot("$O$", O, NE);
dot("$O'$", A, SW);
dot("$P$", P, SW);
dot("$Q$", Q, NE);
draw((2*sqrt(2),2*sqrt(2))--(10 + 3*sqrt(2)/2, -6 + 3*sqrt(2)/2)--cycle);
draw((-1.68*sqrt(2),-2.302*sqrt(2))--(10 - 2.6*sqrt(2)/2, -6 - 3.4*sqrt(2)/2)--cycle);
draw(P--Q--cycle);
//Credit to happiface for the diagram[/asy]
In the adjoining figure, every point of circle $\mathit{O'}$ is exterior to circle $\mathit{O}$. Let $\mathit{P}$ and $\mathit{Q}$ be the points of intersection of an internal common tangent with the two external common tangents. Then the length of $PQ$ is
$\textbf{(A) }\text{the average of the lengths of the internal and external common tangents}\qquad$
$\textbf{(B) }\text{equal to the length of an external common tangent if and only if circles }\mathit{O}\text{ and }\mathit{O'}\text{ have equal radii}\qquad$
$\textbf{(C) }\text{always equal to the length of an external common tangent}\qquad$
$\textbf{(D) }\text{greater than the length of an external common tangent}\qquad$
$\textbf{(E) }\text{the geometric mean of the lengths of the internal and external common tangents}$
1982 USAMO, 4
Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every integer $n$.
2017 AMC 10, 24
For certain real numbers $a$, $b$, and $c$, the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of $g(x)$ is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is $f(1)$?
$\textbf{(A)}\ -9009 \qquad\textbf{(B)}\ -8008 \qquad\textbf{(C)}\ -7007 \qquad\textbf{(D)}\ -6006 \qquad\textbf{(E)}\ -5005$
2004 Germany Team Selection Test, 2
Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.
2015 AMC 10, 1
What is the value of $(2^0-1+5^2+0)^{-1}\times 5$?
$\textbf{(A) }-125\qquad\textbf{(B) }-120\qquad\textbf{(C) }\dfrac15\qquad\textbf{(D) }\dfrac5{24}\qquad\textbf{(E) }25$
2014 Purple Comet Problems, 27
Five men and fi ve women stand in a circle in random order. The probability that every man stands next to at least one woman is $\tfrac m n$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
2002 AIME Problems, 11
Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12.$ A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P,$ which is 7 units from $\overline{BG}$ and 5 units from $\overline{BC}.$ The beam continues to be reflected off the faces of the cube. The length of the light path from the time it leaves point $A$ until it next reaches a vertex of the cube is given by $m\sqrt{n},$ where $m$ and $n$ are integers and $n$ is not divisible by the square of any prime. Find $m+n.$
1993 AMC 8, 2
When the fraction $\dfrac{49}{84}$ is expressed in simplest form, then the sum of the numerator and the denominator will be
$\text{(A)}\ 11 \qquad \text{(B)}\ 17 \qquad \text{(C)}\ 19 \qquad \text{(D)}\ 33 \qquad \text{(E)}\ 133$