Found problems: 15
2014 IPhOO, 7
A uniform solid semi-circular disk of radius $R$ and negligible thickness rests on its diameter as shown. It is then tipped over by some angle $\gamma$ with respect to the table. At what minimum angle $\gamma$ will the disk lose balance and tumble over? Express your answer in degrees, rounded to the nearest integer.
[asy]
draw(arc((2,0), 1, 0,180));
draw((0,0)--(4,0));
draw((0,-2.5)--(4,-2.5));
draw(arc((3-sqrt(2)/2, -4+sqrt(2)/2+1.5), 1, -45, 135));
draw((3-sqrt(2), -4+sqrt(2)+1.5)--(3, -4+1.5));
draw(anglemark((3-sqrt(2), -4+sqrt(2)+1.5), (3, -4+1.5), (0, -4+1.5)));
label("$\gamma$", (2.8, -3.9+1.5), WNW, fontsize(8));
[/asy]
[i]Problem proposed by Ahaan Rungta[/i]
Cono Sur Shortlist - geometry, 2012.G4.2
2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.
2014 NIMO Problems, 8
Three of the below entries, with labels $a$, $b$, $c$, are blatantly incorrect (in the United States).
What is $a^2+b^2+c^2$?
041. The Gentleman's Alliance Cross
042. Glutamine (an amino acid)
051. Grant Nelson and Norris Windross
052. A compact region at the center of a galaxy
061. The value of \verb+'wat'-1+. (See \url{https://www.destroyallsoftware.com/talks/wat}.)
062. Threonine (an amino acid)
071. Nintendo Gamecube
072. Methane and other gases are compressed
081. A prank or trick
082. Three carbons
091. Australia's second largest local government area
092. Angoon Seaplane Base
101. A compressed archive file format
102. Momordica cochinchinensis
111. Gentaro Takahashi
112. Nat Geo
121. Ante Christum Natum
122. The supreme Siberian god of death
131. Gnu C Compiler
132. My TeX Shortcut for $\angle$.
2003 Rioplatense Mathematical Olympiad, Level 3, 2
Triangle $ABC$ is inscribed in the circle $\Gamma$. Let $\Gamma_a$ denote the circle internally tangent to $\Gamma$ and also tangent to sides $AB$ and $AC$. Let $A'$ denote the point of tangency of $\Gamma$ and $\Gamma_a$. Define $B'$ and $C'$ similarly. Prove that $AA'$, $BB'$ and $CC'$ are concurrent.
2007 Harvard-MIT Mathematics Tournament, 6
There are three video game systems: the Paystation, the WHAT, and the ZBoz2$\pi$, and none of these systems will play games for the other systems. Uncle Riemann has three nephews: Bernoulli, Galois, and Dirac. Bernoulli owns a Paystation and a WHAT, Galois owns a WHAT and a ZBoz2$\pi$, and Dirac owns a ZBoz2$\pi$ and a Paystation. A store sells $4$ different games for the Paystation, $6$ different games for the WHAT, and $10$ different games for the ZBoz2$\pi$. Uncle Riemann does not understand the difference between the systems, so he walks into the store and buys $3$ random games (not necessarily distinct) and randomly hands them to his nephews. What is the probability that each nephew receives a game he can play?
2014 AMC 10, 22
In rectangle $ABCD$, $AB=20$ and $BC=10$. Let $E$ be a point on $\overline{CD}$ such that $\angle CBE=15^\circ$. What is $AE$?
$ \textbf{(A)}\ \dfrac{20\sqrt3}3\qquad\textbf{(B)}\ 10\sqrt3\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 11\sqrt3\qquad\textbf{(E)}\ 20 $
2008 ITest, 58
Finished with rereading Isaac Asimov's $\textit{Foundation}$ series, Joshua asks his father, "Do you think somebody will build small devices that run on nuclear energy while I'm alive?"
"Honestly, Josh, I don't know. There are a lot of very different engineering problems involved in designing such devices. But technology moves forward at an amazing pace, so I won't tell you we can't get there in time for you to see it. I $\textit{did}$ go to a graduate school with a lady who now works on $\textit{portable}$ nuclear reactors. They're not small exactly, but they aren't nearly as large as most reactors. That might be the first step toward a nuclear-powered pocket-sized video game.
Hannah adds, "There are already companies designing batteries that are nuclear in the sense that they release energy from uranium hydride through controlled exoenergetic processes. This process is not the same as the nuclear fission going on in today's reactors, but we can certainly call it $\textit{nuclear energy}$."
"Cool!" Joshua's interest is piqued.
Hannah continues, "Suppose that right now in the year $2008$ we can make one of these nuclear batteries in a battery shape that is $2$ meters $\textit{across}$. Let's say you need that size to be reduced to $2$ centimeters $\textit{across}$, in the same proportions, in order to use it to run your little video game machine. If every year we reduce the necessary volume of such a battery by $1/3$, in what year will the batteries first get small enough?"
Joshua asks, "The battery shapes never change? Each year the new batteries are similar in shape - in all dimensions - to the bateries from previous years?"
"That's correct," confirms Joshua's mother. "Also, the base $10$ logarithm of $5$ is about $0.69897$ and the base $10$ logarithm of $3$ is around $0.47712$." This makes Joshua blink. He's not sure he knows how to use logarithms, but he does think he can compute the answer. He correctly notes that after $13$ years, the batteries will already be barely more than a sixth of their original width.
Assuming Hannah's prediction of volume reduction is correct and effects are compounded continuously, compute the first year that the nuclear batteries get small enough for pocket video game machines. Assume also that the year $2008$ is $7/10$ complete.
2013 AMC 8, 21
Samantha lives 2 blocks west and 1 block south of the southwest corner of City Park. Her school is 2 blocks east and 2 blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?
$\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 18$
2007 Today's Calculation Of Integral, 177
On $xy$plane the parabola $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$ intersects with the line $y=x$ at the point $P$ that is different from the origin.
Assumed that the circle $C$ is touched to $K$ at $P$ and $y$ axis at the point $Q.$
Let $S_{1}$ be the area of the region surrounded by the line passing through two points $P,\ Q$ and $K,$ or $S_{2}$ be the area of the region surrounded by the line which is passing through $P$ and parallel to $x$ axis and $K.$ Find the value of $\frac{S_{1}}{S_{2}}.$
1963 AMC 12/AHSME, 27
Six straight lines are drawn in a plane with no two parallel and no three concurrent. The number of regions into which they divide the plane is:
$\textbf{(A)}\ 16 \qquad
\textbf{(B)}\ 20\qquad
\textbf{(C)}\ 22 \qquad
\textbf{(D)}\ 24 \qquad
\textbf{(E)}\ 26$
2012 Cono Sur Olympiad, 2
2. In a square $ABCD$, let $P$ be a point in the side $CD$, different from $C$ and $D$. In the triangle $ABP$, the altitudes $AQ$ and $BR$ are drawn, and let $S$ be the intersection point of lines $CQ$ and $DR$. Show that $\angle ASB=90$.
2011 Math Prize For Girls Problems, 3
The figure below shows a triangle $ABC$ with a semicircle on each of its three sides.
[asy]
unitsize(5);
pair A = (0, 20 * 21) / 29.0;
pair B = (-20^2, 0) / 29.0;
pair C = (21^2, 0) / 29.0;
draw(A -- B -- C -- cycle);
label("$A$", A, S);
label("$B$", B, S);
label("$C$", C, S);
filldraw(arc((A + C)/2, C, A)--cycle, gray);
filldraw(arc((B + C)/2, C, A)--cycle, white);
filldraw(arc((A + B)/2, A, B)--cycle, gray);
filldraw(arc((B + C)/2, A, B)--cycle, white);
[/asy]
If $AB = 20$, $AC = 21$, and $BC = 29$, what is the area of the shaded region?
2012 AMC 12/AHSME, 21
Square $AXYZ$ is inscribed in equiangular hexagon $ABCDEF$ with $X$ on $\overline{BC}$, $Y$ on $\overline{DE}$, and $Z$ on $\overline{EF}$. Suppose that $AB=40$, and $EF=41(\sqrt{3}-1)$. What is the side-length of the square?
[asy]
size(200);
defaultpen(linewidth(1));
pair A=origin,B=(2.5,0),C=B+2.5*dir(60), D=C+1.75*dir(120),E=D-(3.19,0),F=E-1.8*dir(60);
pair X=waypoint(B--C,0.345),Z=rotate(90,A)*X,Y=rotate(90,Z)*A;
draw(A--B--C--D--E--F--cycle);
draw(A--X--Y--Z--cycle,linewidth(0.9)+linetype("2 2"));
dot("$A$",A,W,linewidth(4));
dot("$B$",B,dir(0),linewidth(4));
dot("$C$",C,dir(0),linewidth(4));
dot("$D$",D,dir(20),linewidth(4));
dot("$E$",E,dir(100),linewidth(4));
dot("$F$",F,W,linewidth(4));
dot("$X$",X,dir(0),linewidth(4));
dot("$Y$",Y,N,linewidth(4));
dot("$Z$",Z,W,linewidth(4));
[/asy]
$ \textbf{(A)}\ 29\sqrt{3} \qquad\textbf{(B)}\ \frac{21}{2}\sqrt{2}+\frac{41}{2}\sqrt{3}\qquad\textbf{(C)}\ 20\sqrt{3}+16$
$\textbf{(D)}\ 20\sqrt{2}+13\sqrt{3}
\qquad\textbf{(E)}\ 21\sqrt{6}$
1999 USAMTS Problems, 3
The figure on the right shows the map of Squareville, where each city block is of the same length. Two friends, Alexandra and Brianna, live at the corners marked by $A$ and $B$, respectively. They start walking toward each other's house, leaving at the same time, walking with the same speed, and independently choosing a path to the other's house with uniform distribution out of all possible minimum-distance paths [that is, all minimum-distance paths are equally likely]. What is the probability they will meet?
[asy]
size(200);
defaultpen(linewidth(0.8));
for(int i=0;i<=2;++i) {
for(int j=0;j<=4;++j) {
draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle);
}
}
for(int i=3;i<=4;++i) {
for(int j=3;j<=6;++j) {
draw((i,j)--(i+1,j)--(i+1,j+1)--(i,j+1)--cycle);
}
}
label("$A$",origin,SW);
label("$B$",(5,7),SE);
[/asy]
2003 AMC 10, 15
What is the probability that an integer in the set $ \{1,2,3,\ldots,100\}$ is divisible by $ 2$ and not divisible by $ 3$?
$ \textbf{(A)}\ \frac{1}{6} \qquad
\textbf{(B)}\ \frac{33}{100} \qquad
\textbf{(C)}\ \frac{17}{50} \qquad
\textbf{(D)}\ \frac{1}{2} \qquad
\textbf{(E)}\ \frac{18}{25}$