problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
What is the value of $x$ if a cube's volume is $5x$ cubic units and its surface area is $x$ square units? | 5400 |
A line parallel to leg \(AC\) of right triangle \(ABC\) intersects leg \(BC\) at point \(K\) and the hypotenuse \(AB\) at point \(N\). On leg \(AC\), a point \(M\) is chosen such that \(MK = MN\). Find the ratio \(\frac{AM}{MC}\) if \(\frac{BK}{BC} = 14\). | 27 |
Find the product of the least common multiple (LCM) of $8$ and $6$ and the greatest common divisor (GCD) of $8$ and $6$. | 48 |
In the acute-angled triangle \(ABC\), it is known that \(\sin (A+B)=\frac{3}{5}\), \(\sin (A-B)=\frac{1}{5}\), and \(AB=3\). Find the area of \(\triangle ABC\). | \frac{6 + 3\sqrt{6}}{2} |
Suppose \(A, B\) are the foci of a hyperbola and \(C\) is a point on the hyperbola. Given that the three sides of \(\triangle ABC\) form an arithmetic sequence, and \(\angle ACB = 120^\circ\), determine the eccentricity of the hyperbola. | 7/2 |
Determine the time in hours it will take to fill a 32,000 gallon swimming pool using three hoses that deliver 3 gallons of water per minute. | 59 |
There are three novel series Peter wishes to read. Each consists of 4 volumes that must be read in order, but not necessarily one after the other. Let \( N \) be the number of ways Peter can finish reading all the volumes. Find the sum of the digits of \( N \). (Assume that he must finish a volume before reading a new one.) | 18 |
Jerry's favorite number is $97$ . He knows all kinds of interesting facts about $97$ :
- $97$ is the largest two-digit prime.
- Reversing the order of its digits results in another prime.
- There is only one way in which $97$ can be written as a difference of two perfect squares.
- There is only one way in which $97$ can be written as a sum of two perfect squares.
- $\tfrac1{97}$ has exactly $96$ digits in the [smallest] repeating block of its decimal expansion.
- Jerry blames the sock gnomes for the theft of exactly $97$ of his socks.
A repunit is a natural number whose digits are all $1$ . For instance, \begin{align*}&1,&11,&111,&1111,&\vdots\end{align*} are the four smallest repunits. How many digits are there in the smallest repunit that is divisible by $97?$ | 96 |
Frank is practicing a new dance move. It starts with him take 5 steps back, and then 10 steps forward, and then 2 steps back, and then double that amount forward. How many steps forward is Frank from his original starting point? | Frank stars by taking 10 steps forward from a position 5 steps behind where he started, so he's 10-5= <<10-5=5>>5 steps forward in total
Frank then takes 2 steps back, meaning he's now 5-2=<<5-2=3>>3 steps forward in total from where he began
He then takes double the previous amount of 2 steps forward, so he takes 2*2= <<2*2=4>>4 steps forward
Since Frank was 3 steps forward before that, that means Frank finishes 3+4=<<3+4=7>>7 steps forward
#### 7 |
In square $EFGH$, $EF$ is 8 centimeters, and $N$ is the midpoint of $\overline{GH}$. Let $P$ be the intersection of $\overline{EC}$ and $\overline{FN}$, where $C$ is a point on segment $GH$ such that $GC = 6$ cm. What is the area ratio of triangle $EFP$ to triangle $EPG$? | \frac{2}{3} |
Let's call a number palindromic if it reads the same left to right as it does right to left. For example, the number 12321 is palindromic.
a) Write down any five-digit palindromic number that is divisible by 5.
b) How many five-digit palindromic numbers are there that are divisible by 5? | 100 |
If $\sum_{n = 0}^{\infty}\cos^{2n}\theta = 5$, what is the value of $\cos{2\theta}$? | \frac{3}{5} |
An $11 \times 11 \times 11$ wooden cube is formed by gluing together $11^3$ unit cubes. What is the greatest number of unit cubes that can be seen from a single point? | 331 |
Let $x$ be chosen at random from the interval $(0,1)$. What is the probability that $\lfloor\log_{10}4x\rfloor - \lfloor\log_{10}x\rfloor = 0$? Here $\lfloor x\rfloor$ denotes the greatest integer that is less than or equal to $x$. | \frac{1}{6} |
Let \( f(x) \) be a function defined on \( \mathbf{R} \). If \( f(x) + x^{2} \) is an odd function, and \( f(x) + 2^{x} \) is an even function, then the value of \( f(1) \) is ______. | -\frac{7}{4} |
Find the number of real solutions of the equation
\[\frac{x}{100} = \sin x.\] | 63 |
How many positive four-digit integers of the form $\_\_45$ are divisible by 45? | 10 |
Two puppies, two kittens, and three parakeets were for sale at the pet shop. The puppies were three times more expensive than the parakeets, and the parakeets were half as expensive as the kittens. If the cost of one parakeet was $10, what would it cost to purchase all of the pets for sale at the pet shop, in dollars? | Puppies cost three times more than parakeets, or 3*$10=$<<3*10=30>>30 per puppy.
Parakeets were half as expensive as the kittens, or 2*$10=$<<2*10=20>>20 per kitten.
Two puppies cost 2*$30=$<<2*30=60>>60
Two kittens cost 2*$20=$<<2*20=40>>40.
And three parakeets cost 3*$10=$<<3*10=30>>30.
Thus, the cost to purchase all of the pets for sale at the pet shop is $60+$40+$30=$<<60+40+30=130>>130.
#### 130 |
The polynomial $$P(x)=(1+x+x^2+\ldots+x^{17})^2-x^{17}$$has 34 complex zeros of the form $z_k=r_k\left[\cos(2\pi\alpha_k)
+i\sin(2\pi\alpha_k)\right]$, $k=1,2,3,\ldots,34$, with $0<\alpha_1\le\alpha_2\le\alpha_3\le\dots\le\alpha_{34}<1$ and $r_k>0$. Find $\alpha_1+\alpha_2+\alpha_3+\alpha_4+\alpha_5.$ | \frac{159}{323} |
If the function $$f(x)=(2m+3)x^{m^2-3}$$ is a power function, determine the value of $m$. | -1 |
Find the greatest positive integer $A$ with the following property: For every permutation of $\{1001,1002,...,2000\}$ , the sum of some ten consecutive terms is great than or equal to $A$ . | 10055 |
There are 10 books on the table. Two-fifths of them are reading books and three-tenths of them are math books. There is one fewer science book than math books and the rest are history books. How many history books are there? | Out of the 10 books, 10 x 2/5 = <<10*2/5=4>>4 are reading books.
While 10 x 3/10 = <<10*3/10=3>>3 are math books.
There are 3 - 1 = <<3-1=2>>2 science books.
So, the total number of reading, math, and science books is 4 + 3 + 2 = <<4+3+2=9>>9.
Thus, there is only 10 - 9 = <<10-9=1>>1 history book.
#### 1 |
What is $\dbinom{n}{1}$ for any positive integer $n$? | n |
Hannah wants to save $80 for five weeks. In the first week, she saved $4 and she plans to save twice as much as her savings as the previous week. How much will she save in the fifth week to reach her goal? | In the second week, Hannah will save $4 x 2 = $<<4*2=8>>8.
In the third week, she will save $8 x 2 = $<<8*2=16>>16.
In the fourth week, she will save $16 x 2 = $<<16*2=32>>32.
Her total savings for a month will be $4 + $8 + $16 + $32 = $<<4+8+16+32=60>>60.
Hannah needs to save $80 - $60 = $<<80-60=20>>20 in the fifth week.
#### 20 |
For how many integers $n$ between 1 and 20 (inclusive) is $\frac{n}{18}$ a repeating decimal? | 18 |
How many integers $m \neq 0$ satisfy the inequality $\frac{1}{|m|}\geq \frac{1}{8}$? | 16 |
Find $2^{-1} \pmod{185}$, as a residue modulo 185. (Give an answer between 0 and 184, inclusive.) | 93 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, $C$ are denoted as $a$, $b$, $c$ respectively, and $a^{2}$, $b^{2}$, $c^{2}$ form an arithmetic sequence. Calculate the maximum value of $\sin B$. | \dfrac{ \sqrt {3}}{2} |
10 - 1.05 ÷ [5.2 × 14.6 - (9.2 × 5.2 + 5.4 × 3.7 - 4.6 × 1.5)] = ? | 9.93 |
Ben makes a sandwich that has 1250 calories total that has two strips of bacon with 125 calories each. What percentage of the sandwich's total calories come from bacon? | First find the total number of calories in the bacon: 125 calories + 125 calories = <<125+125=250>>250 calories
Then divide the number of bacon calories by the total number of calories and multiply by 100% to express the answer as a percentage: 250 calories / 1250 calories * 100% = 20%
#### 20 |
For a nonnegative integer $n$, let $r_9(n)$ stand for the remainder left when $n$ is divided by $9.$ For example, $r_9(25)=7.$
What is the $22^{\text{nd}}$ entry in an ordered list of all nonnegative integers $n$ that satisfy $$r_9(5n)\le 4~?$$(Note that the first entry in this list is $0$.) | 38 |
In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs? | 2 |
What is the residue of $9^{2010}$, modulo 17? | 13 |
In a grain storage facility, the following are the amounts of grain (in tons) that were received or dispatched over a period of 6 days (where "+" indicates received and "-" indicates dispatched):
+26, -32, -15, +34, -38, -20.
(1) After these 6 days, did the amount of grain in the storage increase or decrease? By how much?
(2) After these 6 days, when the manager did the settlement, it was found that there were still 480 tons of grain in storage. How much grain was there in the storage 6 days ago?
(3) If the loading and unloading fee is 5 yuan per ton, how much would the loading and unloading fees be for these 6 days? | 825 |
A 2 by 2003 rectangle consists of unit squares as shown below. The middle unit square of each row is shaded. If a rectangle from the figure is chosen at random, what is the probability that the rectangle does not include a shaded square? Express your answer as a common fraction. [asy]
size(7cm);
defaultpen(linewidth(0.7));
dotfactor=4;
int i,j;
fill((6,0)--(7,0)--(7,2)--(6,2)--cycle,gray);
for(i=0;i<=3;++i)
{
draw((i,0)--(i,2));
draw((i+5,0)--(i+5,2));
draw((i+10,0)--(i+10,2));
}
for(j=0;j<=2;++j)
{
draw((0,j)--(3.3,j));
draw((0,j)--(3.3,j));
draw((4.7,j)--(8.3,j));
draw((4.7,j)--(8.3,j));
draw((9.7,j)--(13,j));
draw((9.7,j)--(13,j));
}
real x;
for(x=3.7;x<=4.3;x=x+0.3)
{
dot((x,0));
dot((x,2));
dot((x+5,0));
dot((x+5,2));
}[/asy] | \dfrac{1001}{2003} |
In the diagram, the side \(AB\) of \(\triangle ABC\) is divided into \(n\) equal parts (\(n > 1990\)). Through the \(n-1\) division points, lines parallel to \(BC\) are drawn intersecting \(AC\) at points \(B_i, C_i\) respectively for \(i=1, 2, 3, \cdots, n-1\). What is the ratio of the area of \(\triangle AB_1C_1\) to the area of the quadrilateral \(B_{1989} B_{1990} C_{1990} C_{1989}\)? | 1: 3979 |
In $\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). The ratio $PA:AB$ is:
$\textbf{(A)}\ 1:3 \qquad \textbf{(B)}\ 3:4 \qquad \textbf{(C)}\ 4:3 \qquad \textbf{(D)}\ 3:1 \qquad \textbf{(E)}\ 7:1$
| 3:1 |
Let $A$ be a subset of $\{1,2,\ldots,2020\}$ such that the difference of any two distinct elements in $A$ is not prime. Determine the maximum number of elements in set $A$ . | 505 |
Given that the four vertices of the tetrahedron $P-ABC$ are all on the surface of a sphere with radius $3$, and $PA$, $PB$, $PC$ are mutually perpendicular, find the maximum value of the lateral surface area of the tetrahedron $P-ABC$. | 18 |
Belinda can throw a ball at a speed of 20 feet/second. If the ball flies for 8 seconds before hitting the ground, and Belinda's border collie can run 5 feet/second, how many seconds will it take the border collie to catch up to the ball? | First find the total distance the ball flies: 20 feet/second * 8 seconds = <<20*8=160>>160 feet
Then divide that distance by the dog's speed to find how long it takes the dog to catch up to the ball: 160 feet / 5 feet/second = <<160/5=32>>32 seconds
#### 32 |
Points \( M \) and \( N \) divide side \( AC \) of triangle \( ABC \) into three equal parts, each of which is 5, with \( AB \perp BM \) and \( BC \perp BN \). Find the area of triangle \( ABC \). | \frac{75 \sqrt{3}}{4} |
In the Cartesian coordinate system $xoy$, point $P(0, \sqrt{3})$ is given. The parametric equation of curve $C$ is $\begin{cases} x = \sqrt{2} \cos \varphi \\ y = 2 \sin \varphi \end{cases}$ (where $\varphi$ is the parameter). A polar coordinate system is established with the origin as the pole and the positive half-axis of $x$ as the polar axis. The polar equation of line $l$ is $\rho = \frac{\sqrt{3}}{2\cos(\theta - \frac{\pi}{6})}$.
(Ⅰ) Determine the positional relationship between point $P$ and line $l$, and explain the reason;
(Ⅱ) Suppose line $l$ intersects curve $C$ at two points $A$ and $B$, calculate the value of $\frac{1}{|PA|} + \frac{1}{|PB|}$. | \sqrt{14} |
A five-digit palindrome is a positive integer with respective digits $abcba$, where $a$ is non-zero. Let $S$ be the sum of all five-digit palindromes. What is the sum of the digits of $S$? | 45 |
For a wedding, chairs were arranged in 7 rows with 12 chairs in each row. Some people arrived late and grabbed 11 extra chairs, placing them at the back. How many chairs were there altogether? | At first, there were 7*12=<<7*12=84>>84 chairs.
Later, there were 84+11=<<84+11=95>>95 chairs altogether.
#### 95 |
The distances from one end of the diameter of a circle to the ends of a chord parallel to this diameter are 5 and 12. Find the radius of the circle. | 6.5 |
Two boys, Ben and Leo, are fond of playing marbles. Ben has 56 marbles, while Leo has 20 more marbles than Ben. They put the marbles in one jar. How many marbles in the jar? | Leo has 56 + 20 = <<56+20=76>>76 marbles.
Therefore, there are 56 + 76 = <<56+76=132>>132 marbles in the jar.
#### 132 |
Mr. Wang drives from his home to location $A$. On the way there, he drives the first $\frac{1}{2}$ of the distance at a speed of 50 km/h and increases his speed by $20\%$ for the remaining distance. On the way back, he drives the first $\frac{1}{3}$ of the distance at a speed of 50 km/h and increases his speed by $32\%$ for the remaining distance. The return trip takes 31 minutes less than the trip to $A$. What is the distance in kilometers between Mr. Wang's home and location $A$? | 330 |
Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads, and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \tfrac{1}{2}$? | \frac{7}{16} |
The value of $\frac{2^4 - 2}{2^3 - 1}$ is? | 2 |
A child builds towers using identically shaped cubes of different colors. Determine the number of different towers with a height of 6 cubes that can be built with 3 yellow cubes, 3 purple cubes, and 2 orange cubes (Two cubes will be left out). | 350 |
Bill is buying healthcare on an exchange. The normal monthly price of the plan he wants is $500. The government will pay for part of this cost depending on Bill's income: 90% if he makes less than $10,000, 50% if he makes between $10,001 and $40,000, and 20% if he makes more than $50,000. Bill earns $25/hour and works 30 hours per week, four weeks per month. How much will Bill spend for health insurance in a year? | First find how much money Bill makes every week by multiplying his hourly rate by the number of hours he works each week: $25/hour * 30 hours/week = $<<25*30=750>>750/week
Then multiply that number by the number of weeks per month to find his monthly earnings: $750/week * 4 weeks/month = $<<750*4=3000>>3,000/month
Then multiply his monthly earnings by the number of months in a year to find his annual income: $3,000/month * 12 months/year = $<<3000*12=36000>>36,000/year.
This income means Bill gets a 50% monthly healthcare subsidy from the government. Multiply the cost of the premium by 50% to find the monthly cost Bill pays: $500/month * .5 = $<<500*.5=250>>250/month
Finally, multiply Bill's monthly cost by the number of months in a year to find his annual cost: $250/month * 12 months/year = $<<250*12=3000>>3,000/year
#### 3000 |
James decides to sell 80% of his toys. He bought them for $20 each and sells them for $30 each. If he had 200 toys how much more money did he have compared to before he bought them? | He bought all the toys for 20*200=$<<20*200=4000>>4000
He sold 200*.8=<<200*.8=160>>160 toys
He made 160*30=$<<160*30=4800>>4800 from selling them
So he profits 4800-4000=$<<4800-4000=800>>800 profit
#### 800 |
Divers extracted a certain number of pearls, not exceeding 1000. The distribution of the pearls happens as follows: each diver in turn approaches the heap of pearls and takes either exactly half or exactly one-third of the remaining pearls. After all divers have taken their share, the remainder of the pearls is offered to the sea god. What is the maximum number of divers that could have participated in the pearl extraction? | 12 |
What is the product of the [real](https://artofproblemsolving.com/wiki/index.php/Real) [roots](https://artofproblemsolving.com/wiki/index.php/Root) of the [equation](https://artofproblemsolving.com/wiki/index.php/Equation) $x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}$? | 20 |
The expression below has six empty boxes. Each box is to be fi lled in with a number from $1$ to $6$ , where all six numbers are used exactly once, and then the expression is evaluated. What is the maximum possible final result that can be achieved? $$ \dfrac{\frac{\square}{\square}+\frac{\square}{\square}}{\frac{\square}{\square}} $$ | 14 |
Diane is twice as old as her brother, Will. If Will was 4 years old 3 years ago, what will the sum of their ages be in 5 years? | Since Will was 4 years old 3 years ago, then he is 4+3 = <<4+3=7>>7 years old now
Dina is twice as old as he is so she is 2*7 = <<2*7=14>>14 years old now
In 5 years, Will will be 7+5 = <<7+5=12>>12 years old
In 5 years, Diane will be 14+5 = 19 years old
The sum of their ages at that time will be 12+19 = <<12+19=31>>31 years
#### 31 |
Thomas buys a weight vest. It weighed 60 pounds and worked well for him in the beginning but after a bit of training he decides he wants to increase the weight by 60%. The weights come in 2-pound steel ingots. Each ingot cost $5 and if you buy more than 10 you get a 20% discount. How much does it cost to get the weight he needs? | He needs to buy 60*.6=<<60*.6=36>>36 pounds
So he needs to buy 36/2=<<36/2=18>>18 weight plates
That would cost 18*5=$<<18*5=90>>90
Since he bought more than 10 he gets a discount so he gets 90*.2=$<<90*.2=18>>18 off.
That means he paid 90-18=$<<90-18=72>>72
#### 72 |
John bought a tennis racket. He also bought sneakers that cost $200 and a sports outfit that cost $250. He spent a total of $750 for all those items. What was the price of the racket? | Total amount spent for athletic shoes and sportswear is $200 + $250 = $<<200+250=450>>450.
The price of the racket is $750 - $450 = $<<750-450=300>>300.
#### 300 |
What is the least positive integer with exactly five distinct positive factors? | 16 |
Bruce can make 15 batches of pizza dough using a sack of flour. If he uses 5 sacks of flour per day, how many pizza doughs can he make in a week? | He can make 15 x 5 = <<15*5=75>>75 batches of pizza dough per day.
Therefore, Bruce can make 75 x 7 days in a week = <<75*7=525>>525 batches of pizza dough in a week.
#### 525 |
What is the degree measure of an interior angle of a regular pentagon? | 108\text{ degrees} |
The regular tetrahedron, octahedron, and icosahedron have equal surface areas. How are their edges related? | 2 \sqrt{10} : \sqrt{10} : 2 |
Here is a fairly simple puzzle: EH is four times greater than OY. AY is four times greater than OH. Find the sum of all four. | 150 |
If $A=4-3i$, $M=-4+i$, $S=i$, and $P=2$, find $A-M+S-P$. | 6-3i |
Let $ABCD$ be a parallelogram with area $15$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ See the figure, which also shows the relative locations of these points.
Suppose $PQ=6$ and $RS=8,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Then $d^2$ can be written in the form $m+n\sqrt p,$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime. What is $m+n+p?$ | 81 |
Two mathematicians, Kelly and Jason, play a cooperative game. The computer selects some secret positive integer $n<60$ (both Kelly and Jason know that $n<60$, but that they don't know what the value of $n$ is). The computer tells Kelly the unit digit of $n$, and it tells Jason the number of divisors of $n$. Then, Kelly and Jason have the following dialogue: Kelly: I don't know what $n$ is, and I'm sure that you don't know either. However, I know that $n$ is divisible by at least two different primes. Jason: Oh, then I know what the value of $n$ is. Kelly: Now I also know what $n$ is. Assuming that both Kelly and Jason speak truthfully and to the best of their knowledge, what are all the possible values of $n$? | 10 |
What is $11111111_2+111111_2$? Write your answer in base $10$. | 318 |
Rocco stores his coins in piles of 10 coins each. He has 4 piles of quarters, 6 piles of dimes, 9 piles of nickels, and 5 piles of pennies. How much money does Rocco have? | Each pile has 10 coins, so he has:
10 * 4 * $.25 = $<<10*4*.25=10.00>>10.00 worth of quarters,
10 * 6 * $.10 = $<<10*6*.10=6.00>>6.00 worth of dimes,
10 * 9 * $.05 = $<<10*9*.05=4.50>>4.50 worth of nickels, and
10 * 5 * $.01 = $<<10*5*.01=0.50>>0.50 worth of pennies.
Altogether he has $10.00 + $6.00 + $4.50 + $0.50 = $<<10+6+4.5+0.5=21.00>>21.00
#### 21 |
Determine the number of functions $f: \{1, 2, 3\} \rightarrow \{1, 2, 3\}$ satisfying the property $f(f(x)) = f(x)$. | 10 |
In the diagram, a rectangle has a perimeter of $40$, and a triangle has a height of $40$. If the rectangle and the triangle have the same area, what is the value of $x?$ Assume the length of the rectangle is twice its width. [asy]
draw((0,0)--(3,0)--(3,1)--(0,1)--cycle);
draw((4,0)--(7,0)--(7,5)--cycle);
draw((6.8,0)--(6.8,.2)--(7,.2));
label("$x$",(5.5,0),S);
label("40",(7,2.5),E);
[/asy] | 4.4445 |
There exists a scalar $c$ so that
\[\mathbf{i} \times (\mathbf{v} \times \mathbf{i}) + \mathbf{j} \times (\mathbf{v} \times \mathbf{j}) + \mathbf{k} \times (\mathbf{v} \times \mathbf{k}) = c \mathbf{v}\]for all vectors $\mathbf{v}.$ Find $c.$ | 2 |
The function \[f(x) = \left\{ \begin{aligned} x-2 & \quad \text{ if } x < 4 \\ \sqrt{x} & \quad \text{ if } x \ge 4 \end{aligned} \right.\]has an inverse $f^{-1}.$ Find the value of $f^{-1}(-5) + f^{-1}(-4) + \dots + f^{-1}(4) + f^{-1}(5).$ | 54 |
In right triangle $MNO$, $\tan{M}=\frac{5}{4}$, $OM=8$, and $\angle O = 90^\circ$. Find $MN$. Express your answer in simplest radical form. | 2\sqrt{41} |
What is the sum of all positive integers $n$ that satisfy $$\mathop{\text{lcm}}[n,100] = \gcd(n,100)+450~?$$ | 250 |
A right rectangular prism has integer side lengths $a$ , $b$ , and $c$ . If $\text{lcm}(a,b)=72$ , $\text{lcm}(a,c)=24$ , and $\text{lcm}(b,c)=18$ , what is the sum of the minimum and maximum possible volumes of the prism?
*Proposed by Deyuan Li and Andrew Milas* | 3024 |
A set containing three real numbers can be represented as $\{a,\frac{b}{a},1\}$, or as $\{a^{2}, a+b, 0\}$. Find the value of $a^{2023}+b^{2024}$. | -1 |
Suppose $a, b, c, d$, and $e$ are objects that we can multiply together, but the multiplication doesn't necessarily satisfy the associative law, i.e. ( $x y) z$ does not necessarily equal $x(y z)$. How many different ways are there to interpret the product abcde? | 14 |
What is the arithmetic mean of the integers from -4 through 5, inclusive? Express your answer as a decimal to the nearest tenth. | 0.5 |
There exist $r$ unique nonnegative integers $n_1 > n_2 > \cdots > n_r$ and $r$ unique integers $a_k$ ($1\le k\le r$) with each $a_k$ either $1$ or $- 1$ such that \[a_13^{n_1} + a_23^{n_2} + \cdots + a_r3^{n_r} = 1025.\]
Find $n_1 + n_2 + \cdots + n_r$. | 17 |
A sequence consists of $2010$ terms. Each term after the first is 1 larger than the previous term. The sum of the $2010$ terms is $5307$. When every second term is added up, starting with the first term and ending with the second last term, what is the sum? | 2151 |
A fair coin is flipped 9 times. What is the probability that at least 6 of the flips result in heads? | \frac{65}{256} |
What is the sum of the integers that are both greater than 3 and less than 12? | 60 |
The on-time arrival rate of bus No. 101 in a certain city is 90%. Calculate the probability that the bus arrives on time exactly 4 times out of 5 rides for a person. | 0.32805 |
Given that \( M \) is the midpoint of the height \( D D_{1} \) of a regular tetrahedron \( ABCD \), find the dihedral angle \( A-M B-C \) in radians. | \frac{\pi}{2} |
Let the function \( f(x) = x^3 + a x^2 + b x + c \) (where \( a, b, c \) are all non-zero integers). If \( f(a) = a^3 \) and \( f(b) = b^3 \), then the value of \( c \) is | 16 |
Simplify first, then evaluate: $\left(\frac{{a}^{2}-1}{a-3}-a-1\right) \div \frac{a+1}{{a}^{2}-6a+9}$, where $a=3-\sqrt{2}$. | -2\sqrt{2} |
A box of 100 personalized pencils costs $\$30$. How many dollars does it cost to buy 2500 pencils? | \$750 |
9 pairs of table tennis players participate in a doubles match, their jersey numbers are 1, 2, …, 18. The referee is surprised to find that the sum of the jersey numbers of each pair of players is exactly a perfect square. The player paired with player number 1 is . | 15 |
Twelve students in Mrs. Stephenson's class have brown eyes. Twenty students in the class have a lunch box. Of Mrs. Stephenson's 30 students, what is the least possible number of students who have brown eyes and a lunch box? | 2 |
After a track and field event, each athlete shook hands once with every athlete from every other team, but not with their own team members. Afterwards, two coaches arrived, each only shaking hands with each athlete from their respective teams. If there were a total of 300 handshakes at the event, what is the fewest number of handshakes the coaches could have participated in? | 20 |
For each value of $x$, $g(x)$ is defined to be the minimum value of the three numbers $3x + 3$, $\frac{1}{3}x + 1$, and $-\frac{2}{3}x + 8$. Find the maximum value of $g(x)$. | \frac{10}{3} |
If $\displaystyle\frac{a}{b} = 4$, $\displaystyle\frac{b}{c} = \frac{1}{3}$, and $\displaystyle \frac{c}{d} = 6$, then what is $\displaystyle\frac{d}{a}$? | \frac{1}{8} |
Roman and Remy took separate showers. Remy used 1 more gallon than 3 times the number of gallons that Roman used for his shower. Together the boys used 33 gallons of water. How many gallons did Remy use? | Let R = Roman's gallons
Remy = 3R + 1
4R + 1 = 33
4R = <<32=32>>32
R = <<8=8>>8
Remy used 25 gallons of water for his shower.
#### 25 |
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from each of the three departments. Find the number of possible committees that can be formed subject to these requirements. | 88 |
Define a positive integer $n$ to be a factorial tail if there is some positive integer $m$ such that the decimal representation of $m!$ ends with exactly $n$ zeroes. How many positive integers less than $2500$ are not factorial tails? | 500 |
An 8-foot by 10-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/2 foot centered at each corner of the tile. The remaining portion of the tile is shaded. How many square feet of the floor are shaded?
[asy]
fill((5,5)--(5,-5)--(-5,-5)--(-5,5)--cycle,gray(0.7));
fill(Circle((-5,5),5),white);
fill(Circle((5,5),5),white);
fill(Circle((-5,-5),5),white);
fill(Circle((5,-5),5),white);
draw((-5,5)--(-5,-5)--(5,-5)--(5,5)--cycle);
[/asy] | 80 - 20\pi |
Find \(\cos \frac{\alpha - \beta}{2}\), given \(\sin \alpha + \sin \beta = -\frac{27}{65}\), \(\tan \frac{\alpha + \beta}{2} = \frac{7}{9}\), \(\frac{5}{2} \pi < \alpha < 3 \pi\) and \(-\frac{\pi}{2} < \beta < 0\). | \frac{27}{7 \sqrt{130}} |
In triangle $A B C$ with altitude $A D, \angle B A C=45^{\circ}, D B=3$, and $C D=2$. Find the area of triangle $A B C$. | 15 |
A frog lays 800 eggs a year. 10 percent dry up, and 70 percent are eaten. 1/4 of the remaining eggs end up hatching, how many frogs hatch out of the 800? | Dried up:800(.10)=80 eggs
800(.70)=560 eggs
Total left:800-80-560=<<800-80-560=160>>160 eggs
160/4=<<160/4=40>>40 eggs survive to hatch
#### 40 |
Find the number of complex solutions to
\[\frac{z^3 - 1}{z^2 + z - 2} = 0.\] | 2 |
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