problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
If $3 \in \{a, a^2 - 2a\}$, then the value of the real number $a$ is ______. | -1 |
Find all numbers that can be expressed in exactly $2010$ different ways as the sum of powers of two with non-negative exponents, each power appearing as a summand at most three times. A sum can also be made from just one summand. | 2010 |
All the positive integers greater than 1 are arranged in five columns (A, B, C, D, E) as shown. Continuing the pattern, in what column will the integer 800 be written?
[asy]
label("A",(0,0),N);
label("B",(10,0),N);
label("C",(20,0),N);
label("D",(30,0),N);
label("E",(40,0),N);
label("Row 1",(-10,-7),W);
label("2",(10,... | \text{B} |
Given an equilateral triangle \( ABC \). Point \( K \) is the midpoint of side \( AB \), and point \( M \) lies on side \( BC \) such that \( BM : MC = 1 : 3 \). A point \( P \) is chosen on side \( AC \) such that the perimeter of triangle \( PKM \) is minimized. In what ratio does point \( P \) divide side \( AC \)? | 2/3 |
Suppose that $a$ and $b$ are nonzero real numbers, and that the equation $x^2+ax+b=0$ has solutions $a$ and $b$. Find the ordered pair $(a,b).$ | (1,-2) |
A permutation $a_1,a_2,\cdots ,a_6$ of numbers $1,2,\cdots ,6$ can be transformed to $1,2,\cdots,6$ by transposing two numbers exactly four times. Find the number of such permutations. | 360 |
Julio goes fishing and can catch 7 fish every hour. By the 9th hour, how many fish does Julio have if he loses 15 fish in the process? | Julio catches a total of 7 fish/hour * 9 hours = <<7*9=63>>63 fish.
After losing some of the fish, Julio has 63 fish - 15 fish = <<63-15=48>>48 fish.
#### 48 |
The average (arithmetic mean) age of a group consisting of doctors and lawyers in 40. If the doctors average 35 and the lawyers 50 years old, then the ratio of the numbers of doctors to the number of lawyers is | 2: 1 |
Yann and Camille go to a restaurant. If there are 10 items on the menu, and each orders one dish, how many different combinations of meals can Yann and Camille order? (Note that they are allowed to order the same dish, and that it does matter who orders what.) | 100 |
Evaluate the following expression: $$ 0 - 1 -2 + 3 - 4 + 5 + 6 + 7 - 8 + ... + 2000 $$ The terms with minus signs are exactly the powers of two.
| 1996906 |
Given that cosα + 2cos(α + $$\frac{π}{3}$$) = 0, find tan(α + $$\frac{π}{6}$$). | 3\sqrt{3} |
One of the following four-digit numbers is not divisible by 4: 3544, 3554, 3564, 3572, 3576. What is the product of the units digit and the tens digit of that number? | 20 |
After evaluating his students on the final exams. Professor Oscar reviews all 10 questions on each exam. How many questions must he review if he has 5 classes with 35 students each? | For each class, he has to review the questions of 35 students, in total, he will review 10 * 35 = <<10*35=350>>350 questions.
For the 5 classes, he will review a total of 350 * 5 = <<350*5=1750>>1750 questions.
#### 1750 |
Adam has $15$ of a certain type of rare coin and is interested in knowing how much this collection is worth. He discovers that $5$ of these coins are worth $12$ dollars in total. Assuming that the value of each coin is the same, how many dollars is his entire collection worth? | 36 |
Mr. Sanchez's students were asked to add two positive integers. Juan subtracted by mistake and got 2. Maria mistakenly multiplied and got 120. What was the correct answer? | 22 |
If there are 8 slices in a large pizza, how many slices will remain if Mary orders 2 large pizzas and eats 7 slices? | The number of total slices of pizza is 8 slices per pizza * 2 large pizzas = <<8*2=16>>16 slices
If Mary eats 7 slices, then 16 - 7 = <<16-7=9>>9 remain
#### 9 |
Beth is looking at her book collection and is wondering how many comic books she owns. She has 120 books and 65% are novels. 18 are graphic novels. The rest are comic books. What percentage of comic books does she own? | The proportion of graphic novels is .15 because 18 / 120 = <<18/120=.15>>.15
The percentage of graphic novels is 15 because 100 x .15 = <<100*.15=15>>15
The percentage of comic books is 20 because 100 - 65 - 15 = <<100-65-15=20>>20
#### 20 |
In triangle $ABC$, $AB = 5$, $BC = 4$, and $CA = 3$.
[asy]
defaultpen(1);
pair C=(0,0), A = (0,3), B = (4,0);
draw(A--B--C--cycle);
label("\(A\)",A,N);
label("\(B\)",B,E);
label("\(C\)",C,SW);
[/asy]
Point $P$ is randomly selected inside triangle $ABC$. What is the probability that $P$ is closer to $C$ than it is ... | \frac{1}{2} |
There are 400 students at Pascal H.S., where the ratio of boys to girls is $3: 2$. There are 600 students at Fermat C.I., where the ratio of boys to girls is $2: 3$. What is the ratio of boys to girls when considering all students from both schools? | 12:13 |
A year ago, the total cost of buying a lawnmower was 2/5 times less than the cost it goes for now. If the cost was $1800 a year ago, calculate how much it would cost Mr. Lucian to buy 4 such lawnmowers. | If the price of a lawnmower was $1800 a year ago, 2/5 less than the cost right now, then it now costs 2/5*1800 = $720 more to buy one lawnmower.
The total cost of buying one lawnmower right now is $1800+$720 = $<<1800+720=2520>>2520
To purchase 4 such lawnmowers, Mr. Lucian will have to pay $2520*4 = $<<2520*4=10080>>1... |
Compute $$\lim _{A \rightarrow+\infty} \frac{1}{A} \int_{1}^{A} A^{\frac{1}{x}} \mathrm{~d} x$$ | 1 |
What is the value of $x$ in the diagram?
[asy]
import olympiad;
draw((0,0)--(sqrt(3),0)--(0,sqrt(3))--cycle);
draw((0,0)--(-1,0)--(0,sqrt(3))--cycle);
label("8",(-1/2,sqrt(3)/2),NW);
label("$x$",(sqrt(3)/2,sqrt(3)/2),NE);
draw("$45^{\circ}$",(1.5,0),NW);
draw("$60^{\circ}$",(-0.9,0),NE);
draw(rightanglemark((0... | 4\sqrt{6} |
When a positive integer $x$ is divided by a positive integer $y$, the quotient is $u$ and the remainder is $v$, where $u$ and $v$ are integers.
What is the remainder when $x+2uy$ is divided by $y$? | v |
Find the remainder when $x^{44} + x^{33} + x^{22} + x^{11} + 1$ is divided by $x^4 + x^3 + x^2 + x + 1.$ | 0 |
Let $ABC$ be a triangle with $AB=5, BC=4$ and $AC=3$. Let $\mathcal{P}$ and $\mathcal{Q}$ be squares inside $ABC$ with disjoint interiors such that they both have one side lying on $AB$. Also, the two squares each have an edge lying on a common line perpendicular to $AB$, and $\mathcal{P}$ has one vertex on $AC$ and $\... | \frac{144}{49} |
What is $3\cdot 9+4\cdot 10+11\cdot 3+3\cdot 8$? | 124 |
Given the real numbers $a$, $b$, $c$, $d$ that satisfy $$\frac {a-2e^{a}}{b}= \frac {2-c}{d-1}=1$$, where $e$ is the base of the natural logarithm, find the minimum value of $(a-c)^2+(b-d)^2$. | \frac{25}{2} |
Liz sold her car at 80% of what she originally paid. She uses the proceeds of that sale and needs only $4,000 to buy herself a new $30,000 car. How much cheaper is her new car versus what she originally paid for her old one? | If Liz needs only $4,000 to buy a new $30,000 car, that means she has $30,000-$4,000=$<<30000-4000=26000>>26,000 from the proceeds of selling her old car
If she sold her car at 80% of what she originally paid for and sold it for $26,000 then she originally paid $26,000/80% = $32,500 for her old car
If she paid $32,500 ... |
Given the function $f(x)= \overrightarrow{m} \cdot \overrightarrow{n} - \frac{1}{2}$, where $\overrightarrow{m}=( \sqrt{3}\sin x,\cos x)$ and $\overrightarrow{n}=(\cos x,-\cos x)$.
1. Find the range of the function $y=f(x)$ when $x\in[0, \frac{\pi}{2}]$.
2. In $\triangle ABC$, the sides opposite to angles $A$, $B$, and... | \sqrt{7} |
Compute
$3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3(1+3)))))))))$ | 88572 |
There are 7 people standing in a row. How many different arrangements are there according to the following requirements?
(1) Among them, A, B, and C cannot stand next to each other;
(2) Among them, A and B have exactly one person between them;
(3) A does not stand at the head of the row, and B does not stand at the end... | 3720 |
Bobby can deadlift 300 pounds at 13. When he is 18 he can deadlift 100 pounds more than 250% of his previous deadlift. How many pounds did he add per year? | His new deadlift is 300*2.5+100=<<300*2.5+100=850>>850 pounds
So he added 850-300=<<850-300=550>>550 pounds to his deadlift
This took 18-13=<<18-13=5>>5 years
So he added 550/5=<<550/5=110>>110 pounds per year
#### 110 |
Moving only south and east along the line segments, how many paths are there from $A$ to $B$? [asy]
import olympiad; size(250); defaultpen(linewidth(0.8)); dotfactor=4;
for(int i = 0; i <= 9; ++i)
if (i!=4 && i !=5)
draw((2i,0)--(2i,3));
for(int j = 0; j <= 3; ++j)
draw((0,j)--(18,j));
draw((2*4,0)--(2*4,1));
draw(... | 160 |
James hurt himself exercising. The pain subsided after 3 days, but he knew that the injury would take at least 5 times that long to fully heal. After that, he wanted to wait another 3 days before he started working out again. If he wants to wait 3 weeks after that to start lifting heavy again, how long until he can l... | The injury would be fully healed after 3*5=<<3*5=15>>15 days
So he would start exercising again in 15+3=<<15+3=18>>18 days
He would then wait for 3*7=<<3*7=21>>21 more days to start lifting heavy again
That means he needs to wait 18+21=<<18+21=39>>39 days
#### 39 |
Real numbers $x,$ $y,$ and $z$ satisfy the following equality:
\[4(x + y + z) = x^2 + y^2 + z^2.\]Let $M$ be the maximum value of $xy + xz + yz,$ and let $m$ be the minimum value of $xy + xz + yz.$ Find $M + 10m.$ | 28 |
A man bought a number of ping-pong balls where a 16% sales tax is added. If he did not have to pay tax, he could have bought 3 more balls for the same amount of money. If \( B \) is the total number of balls that he bought, find \( B \). | 18.75 |
At the 2013 Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $50, you get a second pair at a 40% discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What percentage ... | 30 |
Given that sequence {a_n} is an equal product sequence, with a_1=1, a_2=2, and a common product of 8, calculate the sum of the first 41 terms of the sequence {a_n}. | 94 |
A round table has radius $4$. Six rectangular place mats are placed on the table. Each place mat has width $1$ and length $x$ as shown. They are positioned so that each mat has two corners on the edge of the table, these two corners being end points of the same side of length $x$. Further, the mats are positioned so th... | $\frac{3\sqrt{7}-\sqrt{3}}{2}$ |
A polynomial with integer coefficients is of the form
\[12x^3 - 4x^2 + a_1x + 18 = 0.\]
Determine the number of different possible rational roots of this polynomial. | 20 |
Given a tetrahedron $P-ABC$ with its circumscribed sphere's center $O$ on $AB$, and $PO \perp$ plane $ABC$, $2AC = \sqrt{3}AB$. If the volume of the tetrahedron $P-ABC$ is $\frac{3}{2}$, find the volume of the sphere. | 4\sqrt{3}\pi |
Given that \(\log_8 2 = 0.2525\) in base 8 (to 4 decimal places), find \(\log_8 4\) in base 8 (to 4 decimal places). | 0.5050 |
Find the area in the plane contained by the graph of
\[|x + y| + |x - y| \le 4.\] | 16 |
A shopper buys a $100$ dollar coat on sale for $20\%$ off. An additional $5$ dollars are taken off the sale price by using a discount coupon. A sales tax of $8\%$ is paid on the final selling price. The total amount the shopper pays for the coat is | 81.00 |
In Zuminglish-Advanced, all words still consist only of the letters $M, O,$ and $P$; however, there is a new rule that any occurrence of $M$ must be immediately followed by $P$ before any $O$ can occur again. Also, between any two $O's$, there must appear at least two consonants. Determine the number of $8$-letter word... | 24 |
In triangle $\triangle ABC$, the sides opposite to the internal angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b=2\left(a\cos B-c\right)$. Find:<br/>
$(1)$ The value of angle $A$;<br/>
$(2)$ If $a\cos C=\sqrt{3}$ and $b=1$, find the value of $c$. | 2\sqrt{3} - 2 |
The United States flag has 50 stars, one for every state in the union, and 13 stripes, which represent the original 13 colonies. Pete, inspired by history, decided to make his own flag. He used circles, to represent how many scoops of ice cream he can consume in one sitting, and squares, to represent the number of br... | Half the number of stars on the US flag is 50/2=<<50/2=25>>25
Three less than half the number of stars on the US flag is 25-3=<<25-3=22>>22 circles.
Twice the number of stripes on the US flag is 13*2=<<13*2=26>>26.
Six more than twice the number of stripes on the US flag is 26+6=32 squares.
In total, Pete's flag has 22... |
Find the number of sets ${a,b,c}$ of three distinct positive integers with the property that the product of $a,b,$ and $c$ is equal to the product of $11,21,31,41,51,61$. | 728 |
In right triangle $ABC$, $\sin A = \frac{8}{17}$ and $\sin B = 1$. Find $\sin C$. | \frac{15}{17} |
If $\frac{1}{x} + \frac{1}{y} = 3$ and $\frac{1}{x} - \frac{1}{y} = -7$ what is the value of $x + y$? Express your answer as a common fraction. | -\frac{3}{10} |
Determine the slope \(m\) of the asymptotes for the hyperbola given by the equation
\[
\frac{y^2}{16} - \frac{x^2}{9} = 1.
\] | \frac{4}{3} |
Shift the graph of the function $y=3\sin (2x+ \frac {\pi}{6})$ to the graph of the function $y=3\cos 2x$ and determine the horizontal shift units. | \frac {\pi}{6} |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 3000$ with $S(n)$ odd, and let $b$ denote the number of positive integers $n \leq 3000... | 54 |
What is the value of $1234 + 2341 + 3412 + 4123$ | 11110 |
The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours? | 30 |
In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. What is the digit $A$? | 2 |
What is the smallest positive number that is prime and $10$ less than a perfect square? | 71 |
How many 9-digit numbers divisible by 2 can be formed by rearranging the digits of the number 131152152? | 3360 |
The point $A$ $(3,4)$ is reflected over the $x$-axis to $B$. Then $B$ is reflected over the line $y=x$ to $C$. What is the area of triangle $ABC$? | 28 |
In the final game of the basketball season, four players scored points. Chandra scored twice as many points as did Akiko. Akiko scored 4 more points than did Michiko, and Michiko scored half as many points as did Bailey. If Bailey scored 14 points, how many points in total did the team score in the final game of the... | If Bailey scored 14 points, and Michiko scored half as many points as Bailey, then Michiko scored 14/2=<<14/2=7>>7 points.
If Akiko scored 4 more points than did Michiko, then Akiko scored 7+4=11 points.
Since Chandra scored twice as many points as did Akiko, then Chandra scored 11*2=<<11*2=22>>22 points.
Thus, in tota... |
Points on a square with side length $ c$ are either painted blue or red. Find the smallest possible value of $ c$ such that how the points are painted, there exist two points with same color having a distance not less than $ \sqrt {5}$ . | $ \frac {\sqrt {10} }{2} $ |
An eight-digit integer is formed by repeating a positive four-digit integer. For example, 25,632,563 or 60,786,078 are integers of this form. What is the greatest common divisor of all eight-digit integers of this form? | 10001 |
Given \\(x \geqslant 0\\), \\(y \geqslant 0\\), \\(x\\), \\(y \in \mathbb{R}\\), and \\(x+y=2\\), find the minimum value of \\( \dfrac {(x+1)^{2}+3}{x+2}+ \dfrac {y^{2}}{y+1}\\). | \dfrac {14}{5} |
Given that $F\_1(-4,0)$ and $F\_2(4,0)$ are the two foci of the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$, and $P$ is a point on the ellipse such that the area of $\triangle PF\_1F\_2$ is $3\sqrt{3}$, find the value of $\cos\angle{F\_1PF\_2}$. | \frac{1}{2} |
James has 6 ounces of tea in a ten-ounce mug and 6 ounces of milk in a separate ten-ounce mug. He first pours one-third of the tea from the first mug into the second mug and stirs well. Then he pours one-fourth of the mixture from the second mug back into the first. What fraction of the liquid in the first mug is now m... | \frac{1}{4} |
In the expansion of $(x^{2}+1)^{2}(x-1)^{6}$, the coefficient of $x^{5}$ is ____. | -52 |
Given that $a > 2b$ ($a, b \in \mathbb{R}$), the range of the function $f(x) = ax^2 + x + 2b$ is $[0, +\infty)$. Determine the minimum value of $$\frac{a^2 + 4b^2}{a - 2b}$$. | \sqrt{2} |
A mathematical organization is producing a set of commemorative license plates. Each plate contains a sequence of five characters chosen from the four letters in AIME and the four digits in 2007. No character may appear in a sequence more times than it appears among the four letters in AIME or the four digits in 2007. ... | 372 |
An amoeba is placed in a puddle one day, and on that same day it splits into two amoebas. The next day, each new amoeba splits into two new amoebas, and so on, so that each day every living amoeba splits into two new amoebas. After one week, how many amoebas are in the puddle? (Assume the puddle has no amoebas before ... | 128 |
In the diagram, what is the value of $x$? [asy]
size(120);
draw(Circle((0,0),1));
draw((0,0)--(.5,sqrt(3)/2));
draw((0,0)--(sqrt(3)/2,.5));
draw((0,0)--(sqrt(3)/2,-.5));
draw((0,0)--(-1,0));
label("$4x^\circ$",(0,0),NNW); label("$5x^\circ$",(0,0),SSW);
label("$2x^\circ$",(.3,0));label("$x^\circ$",(.3,.3));
[/asy] | 30 |
The curve given by the equation \( y = 2^p x^2 + 5px - 2^{p^2} \) intersects the \( Ox \) axis at points \( A \) and \( B \), and the \( Oy \) axis at point \( C \). Find the sum of all values of the parameter \( p \) for which the center of the circle circumscribed around triangle \( ABC \) lies on the \( Ox \) axis. | -1 |
Find the number of ordered triples $(x,y,z)$ of real numbers such that $x + y = 2$ and $xy - z^2 = 1.$ | 1 |
For how many integers $a$ with $1 \leq a \leq 10$ is $a^{2014}+a^{2015}$ divisible by 5? | 4 |
Given that all terms are positive in the geometric sequence $\{a_n\}$, and the sum of the first $n$ terms is $S_n$, if $S_1 + 2S_5 = 3S_3$, then the common ratio of $\{a_n\}$ equals \_\_\_\_\_\_. | \frac{\sqrt{2}}{2} |
A regular hexagon is inscribed in an equilateral triangle. If the hexagon has an area of 12, what is the area of the equilateral triangle? | 18 |
Define $f(x) = \frac{3}{27^x + 3}.$ Calculate the sum
\[ f\left(\frac{1}{2001}\right) + f\left(\frac{2}{2001}\right) + f\left(\frac{3}{2001}\right) + \dots + f\left(\frac{2000}{2001}\right). \] | 1000 |
Lidia has a collection of books. Her collection is four times bigger than the collection which her friend Susan has. In total Susan and Lidia, both have 3000 books. How many books does Susan have in her collection? | Let x be the number of books in Susan's collection.
The collection of both Susan and Lidia would then be 4*x + x = 3000.
5*x = 3000
x = <<600=600>>600
#### 600 |
Find the largest positive integer $m$ such that an $m \times m$ square can be exactly divided into 7 rectangles with pairwise disjoint interiors, and the lengths of the 14 sides of these 7 rectangles are $1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14$. | 22 |
Amber worked for 12 hours last weekend. Armand worked one-third as long and Ella worked twice as long. How many hours did the 3 people work in total? | Armand = (1/3) * 12 = <<(1/3)*12=4>>4 hours
Ella = 2 * 12 = <<2*12=24>>24 hours
Total hours = 12 + 4 + 24 = <<12+4+24=40>>40 hours
They worked a total of 40 hours.
#### 40 |
In a particular state, the design of vehicle license plates was changed from an old format to a new one. Under the old scheme, each license plate consisted of two letters followed by three digits (e.g., AB123). The new scheme is made up of four letters followed by two digits (e.g., ABCD12). Calculate by how many times ... | \frac{26^2}{10} |
If $x$ and $y$ are positive integers less than $30$ for which $x + y + xy = 104$, what is the value of $x + y$? | 20 |
Given $x+x^{-1}=3$, calculate the value of $x^{ \frac {3}{2}}+x^{- \frac {3}{2}}$. | \sqrt{5} |
At football tryouts, the coach wanted to see who could throw the ball the farthest. Parker threw the ball 16 yards. Grant threw the ball 25 percent farther than Parker and Kyle threw the ball 2 times farther than Grant. Compared to Parker, how much farther did Kyle throw the ball? | Grant threw the ball 25% farther than Parker. If Parker threw the ball 16 yards, then Grant threw it 16*.25 = <<16*.25=4>>4 farther
In total, Grant threw the ball 16+4 = <<16+4=20>>20 yards
Kyle threw 2 times farther than Grant so Kyle threw the ball 2*20 = <<2*20=40>>40 yards
If Kyle threw the ball for 40 yards and Pa... |
Harper has 15 rubber bands. His brother has 6 fewer rubber bands than he has. How many rubber bands do they have together? | His brother has 15 rubber bands - 6 rubber bands = <<15-6=9>>9 rubber bands.
They have 15 rubber bands + 9 rubber bands = <<15+9=24>>24 rubber bands together.
#### 24 |
Let $O$ and $H$ be the circumcenter and orthocenter of triangle $ABC$, respectively. Let $a$, $b$, and $c$ denote the side lengths, and let $R$ denote the circumradius. Find $OH^2$ if $R = 7$ and $a^2 + b^2 + c^2 = 29$. | 412 |
Given $f(x)=x^{3}+3x^{2}+6x$, $f(a)=1$, $f(b)=-9$, the value of $a+b$ is \_\_\_\_\_\_. | -2 |
Find the smallest two-digit prime number such that reversing the digits of the number produces a composite number. | 19 |
Let $T = (a,b,c)$ be a triangle with sides $a,b$ and $c$ and area $\triangle$ . Denote by $T' = (a',b',c')$ the triangle whose sides are the altitudes of $T$ (i.e., $a' = h_a, b' = h_b, c' = h_c$ ) and denote its area by $\triangle '$ . Similarly, let $T'' = (a'',b'',c'')$ be the triangle formed from t... | 45 |
Mary had 89 stickers. She used 3 large stickers on the front page of her journal and 7 stickers each to 6 other pages of her journal. How many stickers does Mary have remaining? | Mary added a total of 7 stickers/page * 6 pages= <<7*6=42>>42 stickers to the 6 other pages.
In total, Mary added 3 large stickers + 42 stickers = <<3+42=45>>45 stickers to her journal.
Since she started with 89 stickers, she now has 89 - 45 = <<89-45=44>>44 stickers left.
#### 44 |
Arlene hiked 24 miles on Saturday, carrying a 60-pound backpack. She is exactly 5 feet tall. She spent 6 hours hiking. The temperature during the day was 60 degrees Fahrenheit. What was Arlene's average hiking pace, in miles per hour? | She hiked 24 miles.
She spent 6 hours hiking.
Her average pace was 24/6=<<24/6=4>>4 mph.
#### 4 |
The number \( N = 3^{16} - 1 \) has a divisor of 193. It also has some divisors between 75 and 85 inclusive. What is the sum of these divisors? | 247 |
Let $C(k)$ denotes the sum of all different prime divisors of a positive integer $k$. For example, $C(1)=0$, $C(2)=2, C(45)=8$. Find all positive integers $n$ such that $C(2^{n}+1)=C(n)$ | n=3 |
A standard 52-card deck contains cards of 4 suits and 13 numbers, with exactly one card for each pairing of suit and number. If Maya draws two cards with replacement from this deck, what is the probability that the two cards have the same suit or have the same number, but not both? | 15/52 |
The condition for three line segments to form a triangle is: the sum of the lengths of any two line segments is greater than the length of the third line segment. Now, there is a wire 144cm long, and it needs to be cut into $n$ small segments ($n>2$), each segment being no less than 1cm in length. If any three of these... | 10 |
A boat carrying 20 sheep, 10 cows and 14 dogs capsized. 3 of the sheep drowned. Twice as many cows drowned as did sheep. All of the dogs made it to shore. How many total animals made it to the shore? | The boat had 20 - 3 = <<20-3=17>>17 sheep make it to shore.
3 sheep drowned, so 2 * 3 = <<2*3=6>>6 cows drowned.
10 cows on the boat - 6 drowned = <<10-6=4>>4 cows made it to the shore.
17 sheep + 4 cows + 14 dogs = <<17+4+14=35>>35 animals made it to shore.
#### 35 |
If $2x - y = 5$ and $x + 2y = 5$, what is the value of $x$? | 3 |
Let $a \bowtie b = a + \sqrt{b + \sqrt{b + \sqrt{b + \ldots}}}$. If $5 \bowtie x = 12$, find the value of $x$. | 42 |
Compute
$$\sum_{k=1}^{1000} k(\lceil \log_{\sqrt{2}}{k}\rceil- \lfloor\log_{\sqrt{2}}{k} \rfloor).$$ | 499477 |
For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$ . Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$ . | \[
7
\] |
Grady distributed $x$ pieces of candy evenly among nine Halloween bags such that every bag received the greatest possible number of whole pieces of candy, but some candy was left over. What is the greatest possible number of pieces that could have been left over? | 8 |
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