Found problems: 3632
2022 AMC 10, 17
One of the following numbers is not divisible by any prime number less than 10. Which is it?
(A) $2^{606} - 1 \ \ $ (B) $2^{606} + 1 \ \ $ (C) $2^{607} - 1 \ \ $ (D) $2^{607} + 1 \ \ $ (E) $2^{607} + 3^{607} \ \ $
2009 AMC 12/AHSME, 9
Suppose that $ f(x\plus{}3)\equal{}3x^2\plus{}7x\plus{}4$ and $ f(x)\equal{}ax^2\plus{}bx\plus{}c$. What is $ a\plus{}b\plus{}c$?
$ \textbf{(A)}\minus{}\!1 \qquad
\textbf{(B)}\ 0 \qquad
\textbf{(C)}\ 1 \qquad
\textbf{(D)}\ 2 \qquad
\textbf{(E)}\ 3$
2023 AIME, 5
Let $S$ be the set of all positive rational numbers $r$ such that when the two numbers $r$ and $55r$ are written as fractions in lowest terms, the sum of the numerator and denominator of one fraction is the same as the sum of the numerator and denominator of the other fraction. The sum of all the elements of $S$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$.
2009 AIME Problems, 1
Before starting to paint, Bill had $ 130$ ounces of blue paint, $ 164$ ounces of red paint, and $ 188$ ounces of white paint. Bill painted four equally sized stripes on a wall, making a blue stripe, a red stripe, a white stripe, and a pink stirpe. Pink is a mixture of red and white, not necessarily in equal amounts. When Bill finished, he had equal amounts of blue, red, and white paint left. Find the total number of ounces of paint Bill had left.
2016 AMC 12/AHSME, 12
All the numbers $1, 2, 3, 4, 5, 6, 7, 8, 9$ are written in a $3\times3$ array of squares, one number in each square, in such a way that if two numbers are consecutive then they occupy squares that share an edge. The numbers in the four corners add up to $18$. What is the number in the center?
$\textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 9$
2001 AIME Problems, 11
Club Truncator is in a soccer league with six other teams, each of which it plays once. In any of its 6 matches, the probabilities that Club Truncator will win, lose, or tie are each $\frac{1}{3}$. The probability that Club Truncator will finish the season with more wins than losses is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
2012 AMC 12/AHSME, 8
An [i]iterative average[/i] of the numbers $1$, $2$, $3$, $4$, and $5$ is computed in the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is the difference between the largest and smallest possible values that can be obtained using this procedure?
$ \textbf{(A)}\ \frac{31}{16}\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ \frac{17}{8}\qquad\textbf{(D)}\ 3\qquad\textbf{(E)}\ \frac{65}{16} $
1996 AMC 8, 13
In the fall of $1996$, a total of $800$ students participated in an annual school clean-up day. The organizers of the event expect that in each of the years $1997$, $1998$, and $1999$, participation will increase by $50 \%$ over the previous year. The number of participants the organizers will expect in the fall of $1999$ is
$\text{(A)}\ 1200 \qquad \text{(B)}\ 1500 \qquad \text{(C)}\ 2000 \qquad \text{(D)}\ 2400 \qquad \text{(E)}\ 2700$
1977 AMC 12/AHSME, 21
For how many values of the coefficient $a$ do the equations \begin{align*}x^2+ax+1=0 \\ x^2-x-a=0\end{align*} have a common real solution?
$\textbf{(A)}\ 0 \qquad
\textbf{(B)}\ 1 \qquad
\textbf{(C)}\ 2 \qquad
\textbf{(D)}\ 3 \qquad
\textbf{(E)}\ \text{infinitely many}$
2012 AMC 12/AHSME, 8
A dessert chef prepares the dessert for every day of a week starting with Sunday. The dessert each day is either cake, pie, ice cream, or pudding. The same dessert may not be served two days in a row. There must be cake on Friday because of a birthday. How many different dessert menus for the week are possible?
$ \textbf{(A)}\ 729\qquad\textbf{(B)}\ 972\qquad\textbf{(C)}\ 1024\qquad\textbf{(D)}\ 2187\qquad\textbf{(E)}\ 2304 $
2013 AMC 10, 22
Six spheres of radius $1$ are positioned so that their centers are at the vertices of a regular hexagon of side length $2$. The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
$ \textbf{(A)} \ \sqrt{2} \qquad \textbf{(B)} \ \frac{3}{2} \qquad \textbf{(C)} \ \frac{5}{3} \qquad \textbf{(D)} \ \sqrt{3} \qquad \textbf{(E)} \ 2$
1996 AMC 12/AHSME, 17
In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE = 6,$ and $AF = 2$. Which of the following is closest to the area of the rectangle $ABCD$?
[asy]
size(140);
pair A, B, C, D, E, F, X, Y;
real length = 12.5;
real width = 10;
A = origin;
B = (length, 0);
C = (length, width);
D = (0, width);
X = rotate(330, C)*B;
E = extension(C, X, A, B);
Y = rotate(30, C)*D;
F = extension(C, Y, A, D);
draw(E--C--F);
label("$2$", A--F, dir(180));
label("$6$", E--B, dir(270));
draw(A--B--C--D--cycle);
dot(A);dot(B);dot(C);dot(D);dot(E);dot(F);
label("$A$", A, dir(225));
label("$B$", B, dir(315));
label("$C$", C, dir(45));
label("$D$", D, dir(135));
label("$E$", E, dir(270));
label("$F$", F, dir(180));
[/asy]
$\textbf{(A)} \ 110 \qquad \textbf{(B)} \ 120 \qquad \textbf{(C)} \ 130 \qquad \textbf{(D)} \ 140 \qquad \textbf{(E)} \ 150$
2009 AIME Problems, 14
The sequence $ (a_n)$ satisfies $ a_0 \equal{} 0$ and $ \displaystyle a_{n \plus{} 1} \equal{} \frac85a_n \plus{} \frac65\sqrt {4^n \minus{} a_n^2}$ for $ n\ge0$. Find the greatest integer less than or equal to $ a_{10}$.
2010 USAJMO, 3
Let $AXYZB$ be a convex pentagon inscribed in a semicircle of diameter $AB$. Denote by $P$, $Q$, $R$, $S$ the feet of the perpendiculars from $Y$ onto lines $AX$, $BX$, $AZ$, $BZ$, respectively. Prove that the acute angle formed by lines $PQ$ and $RS$ is half the size of $\angle XOZ$, where $O$ is the midpoint of segment $AB$.
2014 AIME Problems, 9
Ten chairs are arranged in a circle. Find the number of subsets of this set of chairs that contain at least three adjacent chairs.
1960 AMC 12/AHSME, 15
Triangle I is equilateral with side $A$, perimeter $P$, area $K$, and circumradius $R$ (radius of the circumscribed circle). Triangle II is equilateral with side $a$, perimeter $p$, area $k$, and circumradius $r$. If $A$ is different from $a$, then:
$ \textbf{(A)}\ P:p = R:r \text{ } \text{only sometimes} \qquad\textbf{(B)}\ P:p = R:r \text{ } \text{always}\qquad$
$\textbf{(C)}\ P:p = K:k \text{ } \text{only sometimes} \qquad\textbf{(D)}\ R:r = K:k \text{ } \text{always}\qquad$
$\textbf{(E)}\ R:r = K:k \text{ } \text{only sometimes} $
1994 AMC 12/AHSME, 27
A bag of popping corn contains $\frac{2}{3}$ white kernels and $\frac{1}{3}$ yellow kernels. Only $\frac{1}{2}$ of the white kernels will pop, whereas $\frac{2}{3}$ of the yellow ones will pop. A kernel is selected at random from the bag, and pops when placed in the popper. What is the probability that the kernel selected was white?
$ \textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{5}{9} \qquad\textbf{(C)}\ \frac{4}{7} \qquad\textbf{(D)}\ \frac{3}{5} \qquad\textbf{(E)}\ \frac{2}{3} $
2012 AMC 8, 24
A circle of radius 2 is cut into four congruent arcs. The four arcs are joined to form the star figure shown. What is the ratio of the area of the star figure to the area of the original circle?
[asy]
size(0,50);
draw((-1,1)..(-2,2)..(-3,1)..(-2,0)..cycle);
dot((-1,1));
dot((-2,2));
dot((-3,1));
dot((-2,0));
draw((1,0){up}..{left}(0,1));
dot((1,0));
dot((0,1));
draw((0,1){right}..{up}(1,2));
dot((1,2));
draw((1,2){down}..{right}(2,1));
dot((2,1));
draw((2,1){left}..{down}(1,0));
[/asy]
$\textbf{(A)}\hspace{.05in}\dfrac{4-\pi}\pi \qquad \textbf{(B)}\hspace{.05in}\dfrac1\pi \qquad \textbf{(C)}\hspace{.05in}\dfrac{\sqrt2}{\pi} \qquad \textbf{(D)}\hspace{.05in}\dfrac{\pi-1}\pi \qquad \textbf{(E)}\hspace{.05in}\dfrac3\pi $
1971 AMC 12/AHSME, 32
If $s=(1+2^{-\frac{1}{32}})(1+2^{-\frac{1}{16}})(1+2^{-\frac{1}{8}})(1+2^{-\frac{1}{4}})(1+2^{-\frac{1}{2}})$, then $s$ is equal to
$\textbf{(A) }\textstyle{\frac{1}{2}}(1-2^{-\frac{1}{32}})^{-1}\qquad\textbf{(B) }(1-2^{-\frac{1}{32}})^{-1}\qquad\textbf{(C) }1-2^{-\frac{1}{32}}\qquad$
$\textbf{(D) }\textstyle{\frac{1}{2}}(1-2^{-\frac{1}{32}})\qquad \textbf{(E) }\frac{1}{2}$
1993 AMC 8, 10
This line graph represents the price of a trading card during the first $6$ months of $1993$.
[asy]
unitsize(18);
for (int a = 0; a <= 6; ++a)
{
draw((4*a,0)--(4*a,10));
}
for (int a = 0; a <= 5; ++a)
{
draw((0,2*a)--(24,2*a));
}
draw((0,5)--(4,4)--(8,8)--(12,3)--(16,9)--(20,6)--(24,2),linewidth(1.5));
label("$Jan$",(2,0),S);
label("$Feb$",(6,0),S);
label("$Mar$",(10,0),S);
label("$Apr$",(14,0),S);
label("$May$",(18,0),S);
label("$Jun$",(22,0),S);
label("$\textbf{1993 PRICES FOR A TRADING CARD}$",(12,10),N);
label("$\begin{tabular}{c}\textbf{P} \\ \textbf{R} \\ \textbf{I} \\ \textbf{C} \\ \textbf{E} \end{tabular}$",(-2,5),W);
label("$1$",(0,2),W);
label("$2$",(0,4),W);
label("$3$",(0,6),W);
label("$4$",(0,8),W);
label("$5$",(0,10),W);
[/asy]
The greatest monthly drop in price occurred during
$\text{(A)}\ \text{January} \qquad \text{(B)}\ \text{March} \qquad \text{(C)}\ \text{April} \qquad \text{(D)}\ \text{May} \qquad \text{(E)}\ \text{June}$
1978 USAMO, 5
Nine mathematicians meet at an international conference and discover that among any three of them, at least two speak a common language. If each of the mathematicians speak at most three languages, prove that there are at least three of the mathematicians who can speak the same language.
1986 USAMO, 4
Two distinct circles $K_1$ and $K_2$ are drawn in the plane. They intersect at points $A$ and $B$, where $AB$ is the diameter of $K_1$. A point $P$ on $K_2$ and inside $K_1$ is also given.
Using only a "T-square" (i.e. an instrument which can produce a straight line joining two points and the perpendicular to a line through a point on or off the line), find a construction for two points $C$ and $D$ on $K_1$ such that $CD$ is perpendicular to $AB$ and $\angle CPD$ is a right angle.
1975 AMC 12/AHSME, 26
In acute triangle $ABC$ the bisector of $\measuredangle A$ meets side $BC$ at $D$. The circle with center $B$ and radius $BD$ intersects side $AB$ at $M$; and the circle with center $C$ and radius $CD$ intersects side $AC$ at $N$. Then it is always true that
$ \textbf{(A)}\ \measuredangle CND+\measuredangle BMD-\measuredangle DAC =120^{\circ} \qquad\textbf{(B)}\ AMDN\ \text{is a trapezoid} \qquad\textbf{(C)}\ BC\ \text{is parallel to}\ MN \\ \qquad\textbf{(D)}\ AM-AN=\frac{3(DB-DC)}{2} \qquad\textbf{(E)}\ AB-AC=\frac{3(DB-DC)}{2}$
2022 AMC 12/AHSME, 20
Let $P(x)$ be a polynomial with rational coefficients such that when $P(x)$ is divided by the polynomial $x^2 + x + 1$, the remainder is $x + 2$, and when $P(x)$ is divided by the polynomial $x^2 + 1$, the remainder is $2x + 1$. There is a unique polynomial of least degree with these two properties. What is the sum of the squares of the coefficients of that polynomial?
$\textbf{(A) } 10 \qquad \textbf{(B) } 13 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 20 \qquad \textbf{(E) } 23$
2005 IMAR Test, 1
Let $a,b,c$ be positive real numbers such that $abc\geq 1$. Prove that \[ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1. \]
[hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5[/url].
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