Found problems: 3632
2013 AMC 8, 6
The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6\times5$. What is the missing number in the top row?
[asy]
unitsize(0.8cm);
draw((-1,0)--(1,0)--(1,-2)--(-1,-2)--cycle);
draw((-2,0)--(0,0)--(0,2)--(-2,2)--cycle);
draw((0,0)--(2,0)--(2,2)--(0,2)--cycle);
draw((-3,2)--(-1,2)--(-1,4)--(-3,4)--cycle);
draw((-1,2)--(1,2)--(1,4)--(-1,4)--cycle);
draw((1,2)--(1,4)--(3,4)--(3,2)--cycle);
label("600",(0,-1));
label("30",(-1,1));
label("6",(-2,3));
label("5",(0,3));
[/asy]
$\textbf{(A)}\ 2 \qquad \textbf{(B)}\ 3 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 5 \qquad \textbf{(E)}\ 6$
2010 AMC 12/AHSME, 3
A ticket to a school play costs $ x$ dollars, where $ x$ is a whole number. A group of 9th graders buys tickets costing a total of $ \$48$, and a group of 10th graders buys tickets costing a total of $ \$64$. How many values of $ x$ are possible?
$ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$
2007 AMC 10, 9
Real numbers $ a$ and $ b$ satisfy the equations $ 3^{a} \equal{} 81^{b \plus{} 2}$ and $ 125^{b} \equal{} 5^{a \minus{} 3}$. What is $ ab$?
$ \textbf{(A)} \minus{} \!60 \qquad \textbf{(B)} \minus{} \!17 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 60$
2011 AMC 12/AHSME, 23
A bug travels in the coordinate plane, moving only along the lines that are parallel to the $x$-axis or $y$-axis. Let $A=(-3, 2)$ and $B=(3, -2)$. Consider all possible paths of the bug from $A$ to $B$ of length at most $20$. How many points with integer coordinates lie on at least one of these paths?
$ \textbf{(A)}\ 161 \qquad
\textbf{(B)}\ 185 \qquad
\textbf{(C)}\ 195 \qquad
\textbf{(D)}\ 227 \qquad
\textbf{(E)}\ 255 $
2014 AMC 10, 1
What is $10 \cdot \left(\tfrac{1}{2} + \tfrac{1}{5} + \tfrac{1}{10}\right)^{-1}?$
${ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ \frac{25}{2}\qquad\textbf{(D)}}\ \frac{170}{3}\qquad\textbf{(E)}\ 170$
2009 AMC 12/AHSME, 19
For each positive integer $ n$, let $ f(n)\equal{}n^4\minus{}360n^2\plus{}400$. What is the sum of all values of $ f(n)$ that are prime numbers?
$ \textbf{(A)}\ 794\qquad
\textbf{(B)}\ 796\qquad
\textbf{(C)}\ 798\qquad
\textbf{(D)}\ 800\qquad
\textbf{(E)}\ 802$
2013 NIMO Problems, 8
Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$. Suppose that $AZ-AX=6$, $BX-BZ=9$, $AY=12$, and $BY=5$. Find the greatest integer not exceeding the perimeter of quadrilateral $OXYZ$, where $O$ is the midpoint of $AB$.
[i]Proposed by Evan Chen[/i]
2011 USAMO, 3
In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$. Furthermore $AB=DE$, $BC=EF$, and $CD=FA$. Prove that diagonals $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent.
1968 AMC 12/AHSME, 20
The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^\circ$ and the largest angle is $160^\circ$, then $n$ equals:
$\textbf{(A)}\ 9\qquad
\textbf{(B)}\ 10\qquad
\textbf{(C)}\ 12\qquad
\textbf{(D)}\ 16\qquad
\textbf{(E)}\ 32 $
1993 AMC 8, 19
$(1901+1902+1903+\cdots + 1993) - (101+102+103+\cdots + 193) = $
$\text{(A)}\ 167,400 \qquad \text{(B)}\ 172,050 \qquad \text{(C)}\ 181,071 \qquad \text{(D)}\ 199,300 \qquad \text{(E)}\ 362,142$
2023 AMC 10, 9
A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date?
$\textbf{(A)}~5\qquad\textbf{(B)}~6\qquad\textbf{(C)}~7\qquad\textbf{(D)}~8\qquad\textbf{(E)}~9$
2005 Romania National Olympiad, 1
Let $ABCD$ be a parallelogram. The interior angle bisector of $\angle ADC$ intersects the line $BC$ in $E$, and the perpendicular bisector of the side $AD$ intersects the line $DE$ in $M$. Let $F= AM \cap BC$. Prove that:
a) $DE=AF$;
b) $AD\cdot AB = DE\cdot DM$.
[i]Daniela and Marius Lobaza, Timisoara[/i]
2017 AMC 12/AHSME, 2
The sum of two nonzero real numbers is $4$ times their product. What is the sum of the reciprocals of the two numbers?
$\textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 4\qquad\textbf{(D)}\ 8\qquad\textbf{(E)}\ 12$
2002 AMC 12/AHSME, 16
Tina randomly selects two distinct numbers from the set $ \{1,2,3,4,5\}$ and Sergio randomly selects a number from the set $ \{1,2,...,10\}$. The probability that Sergio's number is larger than the sum of the two numbers chosen by Tina is
$ \textbf{(A)}\ 2/5 \qquad \textbf{(B)}\ 9/20 \qquad \textbf{(C)}\ 1/2\qquad \textbf{(D)}\ 11/20 \qquad \textbf{(E)}\ 24/25$
2016 AIME Problems, 3
Let $x,y$ and $z$ be real numbers satisfying the system \begin{align*}
\log_2(xyz-3+\log_5 x) &= 5 \\
\log_3(xyz-3+\log_5 y) &= 4 \\
\log_4(xyz-3+\log_5 z) &= 4.
\end{align*} Find the value of $|\log_5 x|+|\log_5 y|+|\log_5 z|$.
2006 AMC 10, 7
The $ 8\times 18$ rectangle $ ABCD$ is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is $ y$?
[asy] unitsize(2mm); defaultpen(fontsize(10pt)+linewidth(.8pt)); dotfactor=4; draw((0,4)--(18,4)--(18,-4)--(0,-4)--cycle); draw((6,4)--(6,0)--(12,0)--(12,-4)); label("$D$",(0,4),NW); label("$C$",(18,4),NE); label("$B$",(18,-4),SE); label("$A$",(0,-4),SW); label("$y$",(9,1)); [/asy]$ \textbf{(A) } 6\qquad \textbf{(B) } 7\qquad \textbf{(C) } 8\qquad \textbf{(D) } 9\qquad \textbf{(E) } 10$
1992 AIME Problems, 9
Trapezoid $ABCD$ has sides $AB=92$, $BC=50$, $CD=19$, and $AD=70$, with $AB$ parallel to $CD$. A circle with center $P$ on $AB$ is drawn tangent to $BC$ and $AD$. Given that $AP=\frac mn$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.
1967 AMC 12/AHSME, 14
Let $f(t)=\frac{t}{1-t}$, $t \not= 1$. If $y=f(x)$, then $x$ can be expressed as
$\textbf{(A)}\ f\left(\frac{1}{y}\right)\qquad
\textbf{(B)}\ -f(y)\qquad
\textbf{(C)}\ -f(-y)\qquad
\textbf{(D)}\ f(-y)\qquad
\textbf{(E)}\ f(y)$
2005 AMC 12/AHSME, 14
A circle having center $ (0,k)$, with $ k > 6$, is tangent to the lines $ y \equal{} x, y \equal{} \minus{} x$ and $ y \equal{} 6$. What is the radius of this circle?
$ \textbf{(A)}\ 6 \sqrt 2 \minus{} 6\qquad
\textbf{(B)}\ 6\qquad
\textbf{(C)}\ 6 \sqrt 2\qquad
\textbf{(D)}\ 12\qquad
\textbf{(E)}\ 6 \plus{} 6 \sqrt 2$
1979 AMC 12/AHSME, 25
If $q_1 ( x )$ and $r_ 1$ are the quotient and remainder, respectively, when the polynomial $x^ 8$ is divided by $x + \tfrac{1}{2}$ , and if $q_ 2 ( x )$ and $r_2$ are the quotient and remainder, respectively, when $q_ 1 ( x )$ is divided by $x + \tfrac{1}{2}$, then $r_2$ equals
$\textbf{(A) }\frac{1}{256}\qquad\textbf{(B) }-\frac{1}{16}\qquad\textbf{(C) }1\qquad\textbf{(D) }-16\qquad\textbf{(E) }256$
2014 Contests, 3
Isabella had a week to read a book for a school assignment. She read an average of $36$ pages per day for the first three days and an average of $44$ pages per day for the next three days. She then finished the book by reading $10$ pages on the last day. How many pages were in the book?
$\textbf{(A) }240\qquad\textbf{(B) }250\qquad\textbf{(C) }260\qquad\textbf{(D) }270\qquad \textbf{(E) }280$
1997 AMC 8, 5
There are many two-digit multiples of 7, but only two of the multiples have a digit sum of 10. The sum of these two multiples of 7 is
$\textbf{(A)}\ 119 \qquad \textbf{(B)}\ 126 \qquad \textbf{(C)}\ 140 \qquad \textbf{(D)}\ 175 \qquad \textbf{(E)}\ 189$
1971 AMC 12/AHSME, 3
If the point $(x,-4)$ lies on the straight line joining the points $(0,8)$ and $(-4,0)$ in the xy-plane, then $x$ is equal to
$\textbf{(A) }-2\qquad\textbf{(B) }2\qquad\textbf{(C) }-8\qquad\textbf{(D) }6\qquad \textbf{(E) }-6$
2015 Switzerland Team Selection Test, 2
Let $a$, $b$, $c$ be real numbers greater than or equal to $1$. Prove that
\[ \min \left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )\leq abc. \]
2009 AIME Problems, 5
Triangle $ ABC$ has $ AC \equal{} 450$ and $ BC \equal{} 300$. Points $ K$ and $ L$ are located on $ \overline{AC}$ and $ \overline{AB}$ respectively so that $ AK \equal{} CK$, and $ \overline{CL}$ is the angle bisector of angle $ C$. Let $ P$ be the point of intersection of $ \overline{BK}$ and $ \overline{CL}$, and let $ M$ be the point on line $ BK$ for which $ K$ is the midpoint of $ \overline{PM}$. If $ AM \equal{} 180$, find $ LP$.