Found problems: 74
$A, B, C$ can walk at $5$ km/hr. They have a car that can accomodate any two of them whch travels at $50$ km/hr. Can they reach a point $62$ km away in less than $3$ hrs?
In the centre of a square swimming pool is a boy, while his teacher (who cannot swim) is standing at one corner of the pool. The teacher can run three times as fast as the boy can swim, but the boy can run faster than the teacher . Can the boy escape from the teacher?
Some math team members decide to study at Cary Library after school. They walk $6$ blocks north, then $8$ blocks west to get there. If they walk $n$ blocks east from the library, they can buy boba from CoCo's. If CoCo's is the same distance from school as it is from the library, find $n$.
Find the number of ordered pairs $(x,y)$, where $x$ and $y$ are both integers between $1$ and $9$, inclusive, such that the product $x\times y$ ends in the digit $5$.
[i]Proposed by Andrew Wen[/i]
Find the number of ways to split the numbers from $1$ to $12$ into $4$ non-intersecting sets of size $3$ such that each set has sum divisible by $3$.
If positive real numbers $x,y,z$ satisfy the following system of equations, compute $x+y+z$.
$$xy+yz = 30$$
$$yz+zx = 36$$
$$zx+xy = 42$$
[i]Proposed by Nathan Xiong[/i]
There are 12 students in Mr. DoBa's math class. On the final exam, the average score of the top 3 students was 8 more than the average score of the other students, and the average score of the entire class was 85. Compute the average score of the top $3$ students.
[i]Proposed by Yifan Kang[/i]
Let $ABCD$ be a unit square in the plane. Points $X$ and $Y$ are chosen independently and uniformly at random on the perimeter of $ABCD$. If the expected value of the area of triangle $\triangle AXY$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, compute $m+n$.
[i]Proposed by Nathan Xiong[/i]
Suppose $(a,b)$ is an ordered pair of integers such that the three numbers $a$, $b$, and $ab$ form an arithmetic progression, in that order. Find the sum of all possible values of $a$.
[i]Proposed by Nathan Xiong[/i]
Find the median of the positive divisors of $6^4-1$.
In three houses $A,B$ and $C$, forming a right triangle with the legs $AC=30$ and $CB=40$, live three beetles $a,b$ and $c$, capable of moving at speeds of $2, 3$ and $4$, respectively. Suppose that you simultaneously release these bugs from point $M$ and mark the time after which beetles reach their homes. Find on the plane such a point $M$, where is the last time to reach the house a bug would be minimal.
Find the value of \[(2+0+2+4)+\left(2^0+2^4\right)+\left(2^{0^{2^4}}\right).\]
Let $a$, $b$, and $c$ be real numbers such that $0\le a,b,c\le 5$ and $2a + b + c = 10$. Over all possible values of $a$, $b$, and $c$, determine the maximum possible value of $a + 2b + 3c$.
[i]Proposed by Andrew Wen[/i]
Triangle $\triangle ABC$ has $\angle{A}=90^\circ$ with $BC=12$. Square $BCDE$ is drawn such that $A$ is in its interior. The line through $A$ tangent to the circumcircle of $\triangle ABC$ intersects $CD$ and $BE$ at $P$ and $Q$, respectively. If $PA=4\cdot QA$, and the area of $\triangle ABC$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, then compute $m+n$.
[i]Proposed by Andy Xu[/i]
The graph of the function $ f$ is shown below. How many solutions does the equation $ f(f(x)) \equal{} 6$ have?
[asy]size(220);
defaultpen(fontsize(10pt)+linewidth(.8pt));
dotfactor=4;
pair P1=(-7,-4), P2=(-2,6), P3=(0,0), P4=(1,6), P5=(5,-6);
real[] xticks={-7,-6,-5,-4,-3,-2,-1,1,2,3,4,5,6};
real[] yticks={-6,-5,-4,-3,-2,-1,1,2,3,4,5,6};
draw(P1--P2--P3--P4--P5);
dot("(-7, -4)",P1);
dot("(-2, 6)",P2,LeftSide);
dot("(1, 6)",P4);
dot("(5, -6)",P5);
xaxis("$x$",-7.5,7,Ticks(xticks),EndArrow(6));
yaxis("$y$",-6.5,7,Ticks(yticks),EndArrow(6));[/asy]$ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$
If $a \diamondsuit b = \vert a - b \vert \cdot \vert b - a \vert$ then find the value of $1 \diamondsuit (2 \diamondsuit (3 \diamondsuit (4 \diamondsuit 5)))$.
[i]Proposed by Muztaba Syed[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{9}$
$a\diamondsuit b = (a-b)^2$. This gives us an answer of $\boxed{9}$.
[/hide]
Eddie has a study block that lasts $1$ hour. It takes Eddie $25$ minutes to do his homework and $5$ minutes to play a game of Clash Royale. He can’t do both at the same time. How many games can he play in this study block while still completing his homework?
[i]Proposed by Edwin Zhao[/i]
[hide=Solution]
[i]Solution.[/i] $\boxed{7}$
Study block lasts 60 minutes, thus he has 35 minutes to play Clash Royale, during which he can play $\frac{35}{5}=\boxed{7}$ games.
[/hide]
The radius $OM$ of a circle rotates uniformly at a rate of $360/n$ degrees per second , where $n$ is a positive integer . The initial radius is $OM_0$. After $1$ second the radius is $OM_1$ , after two more seconds (i.e. after three seconds altogether) the radius is $OM_2$ , after $3$ more seconds (after $6$ seconds altogether) the radius is $OM_3$, ..., after $n - 1$ more seconds its position is $OM_{n-1}$. For which values of $n$ do the points $M_0, M_1 , ..., M_{n-1}$ divide the circle into $n$ equal arcs?
(a) Is it true that the powers of $2$ are such values?
(b) Does there exist such a value which is not a power of $2$?
(V. V. Proizvolov , Moscow)
Circles $\omega_1$ and $\omega_2$ intersect at points $X$ and $Y$. The common external tangent of the two circles closer to $X$ intersects $\omega_1$ and $\omega_2$ at $A$ and $B,$ respectively. Given that $AB=6,$ the radius of $\omega_1$ is $3,$ and $AY$ is tangent to $\omega_2,$ find $XY^2$.
There are 12 students in Mr. DoBa's math class. On the final exam, the average score of the top 3 students was 8 more than the average score of the other students, and the average score of the entire class was 85. Compute the average score of the top $3$ students.
[i]Proposed by Yifan Kang[/i]
Let $a$ and $b$ be two-digit positive integers. Find the greatest possible value of $a+b$, given that the greatest common factor of $a$ and $b$ is $6$.
[i]Proposed by Jacob Xu[/i]
[hide=Solution][i]Solution[/i]. $\boxed{186}$
We can write our two numbers as $6x$ and $6y$. Notice that $x$ and $y$ must be relatively prime. Since $6x$ and $6y$ are two digit numbers, we just need to check values of $x$ and $y$ from $2$ through $16$ such that $x$ and $y$ are relatively prime. We maximize the sum when $x = 15$ and $y = 16$, since consecutive numbers are always relatively prime. So the sum is $6 \cdot (15+16) = \boxed{186}$.[/hide]
Isabella assigns a distinct integer from $1$ to $6$ to each row and column of a $3\times 3$ grid. In each entry, she writes either the sum or the product of the values assigned to the corresponding row and column. Find the maximum possible value of the sum of all entries in the grid.
Find the sum of the distinct prime factors of $512512$.
Blue rolls a fair $n$-sided die that has sides its numbered with the integers from $1$ to $n$, and then he flips a coin. Blue knows that the coin is weighted to land heads either $\dfrac{1}{3}$ or $\dfrac{2}{3}$ of the time. Given that the probability of both rolling a $7$ and flipping heads is $\dfrac{1}{15}$, find $n$.
[i]Proposed by Jacob Xu[/i]
[hide=Solution][i]Solution[/i]. $\boxed{10}$
The chance of getting any given number is $\dfrac{1}{n}$
, so the probability of getting $7$ and heads is either $\dfrac{1}{n} \cdot \dfrac{1}{3}=\dfrac{1}{3n}$ or $\dfrac{1}{n} \cdot \dfrac{2}{3}=\dfrac{2}{3n}$. We get that either $n = 5$ or $n = 10$, but since rolling a $7$ is possible, only $n = \boxed{10}$ is a solution.[/hide]