problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Consider a geometric sequence where the first term is $\frac{5}{8}$, and the second term is $25$. What is the smallest $n$ for which the $n$th term of the sequence, multiplied by $n!$, is divisible by one billion (i.e., $10^9$)? | 10 |
Each cell of a $3 \times 3$ grid is labeled with a digit in the set $\{1,2,3,4,5\}$. Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from 1 to 5 is recorded at least once. | 2664 |
Let the positive numbers \( x \) and \( y \) satisfy \( x^{3} + y^{3} = x - y \). Find the maximum value of the real number \( \lambda \) such that \( x^{2} + \lambda y^{2} \leq 1 \) always holds. | 2 + 2\sqrt{2} |
If $A=4-3i$, $M=-4+i$, $S=i$, and $P=2$, find $A-M+S-P$. | 6-3i |
Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$ . Let $R_1$ , $R_2$ , $R_3$ , $R_4$ be the circumradii of triangles $APB$ , $BPC$ , $CPD$ , $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$ , find $R_4$ . | 19 |
The price of electricity went up by 25%. John's old computer used 800 watts and his new computer uses 50% more. If the old price of electricity was 12 cents per kilowatt-hour, how much does his computer cost, in dollars, to run for 50 hours? | His new computer uses 800*.5=<<800*.5=400>>400 watts more than his old computer
So it users 800+400=<<800+400=1200>>1200 watts
That is 1200/1000=<<1200/1000=1.2>>1.2 kw
Electricity cost 12*.25=<<12*.25=3>>3 cents more than it used to
So it now cost 12+3=<<12+3=15>>15 cents per kwh
He used 1.2*50=<<1.2*50=60>>60 kwh
So ... |
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.
[i] | 2^{m+2}(m+1) |
Mike bought a DVD of his favorite movie. He paid $5 for it at the store. A friend of Mike's, Steve, saw this and also decided to buy a DVD of the movie, but it was already sold out. He needed to order it online, which cost him twice as much. And in addition, he needed to pay the shipping costs which were 80% of the pri... | The cost of the movie online was twice what Mike paid, so 2 * 5 = $<<2*5=10>>10.
The shopping costs were standing at 80/100 * 10 = $<<80/100*10=8>>8.
So in total Steve paid 10 + 8 = $<<10+8=18>>18.
#### 18 |
On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was 15 inches. | .34 |
Let $ABCDEF$ be a regular hexagon. Let $G$, $H$, $I$, $J$, $K$, and $L$ be the midpoints of sides $AB$, $BC$, $CD$, $DE$, $EF$, and $AF$, respectively. The segments $\overline{AH}$, $\overline{BI}$, $\overline{CJ}$, $\overline{DK}$, $\overline{EL}$, and $\overline{FG}$ bound a smaller regular hexagon. Let the ratio of ... | 11 |
Calculate $46_8 - 27_8$ and express your answer in base 8. | 17_8 |
Three faucets fill a 100-gallon tub in 6 minutes. How long, in seconds, does it take six faucets to fill a 25-gallon tub? Assume that all faucets dispense water at the same rate. | 45 |
Given that Bob was instructed to subtract 5 from a certain number and then divide the result by 7, but instead subtracted 7 and then divided by 5, yielding an answer of 47, determine what his answer would have been had he worked the problem correctly. | 33 |
Choose one digit from 0, 2, 4, and two digits from 1, 3, 5 to form a three-digit number without repeating digits. The total number of different three-digit numbers that can be formed is ( )
A 36 B 48 C 52 D 54 | 48 |
Given the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, passing through point $Q(\sqrt{2}, 1)$ and having the right focus at $F(\sqrt{2}, 0)$,
(I) Find the equation of the ellipse $C$;
(II) Let line $l$: $y = k(x - 1) (k > 0)$ intersect the $x$-axis, $y$-axis, and ellipse $C$ at points $C$, ... | \frac{\sqrt{42}}{2} |
Find the area of the region in the first quadrant \(x>0, y>0\) bounded above by the graph of \(y=\arcsin(x)\) and below by the graph of \(y=\arccos(x)\). | 2 - \sqrt{2} |
Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{2}{3} |
Given that $-9, a_1, a_2, -1$ form an arithmetic sequence and $-9, b_1, b_2, b_3, -1$ form a geometric sequence, find the value of $b_2(a_2 - a_1)$. | -8 |
A store transports fresh vegetables in crates and cartons using its delivery truck. One crate of vegetables weighs 4 kilograms and one carton 3 kilograms. How much will a load of 12 crates, and 16 cartons of vegetables weigh? | The weight of 12 crates of vegetables is 12 * 4 = <<12*4=48>>48 kilograms.
The weight of 16 cartons of vegetables is 16 * 3 = <<16*3=48>>48 kilograms.
So the total weight of all crates and cartons is 48 + 48 = <<48+48=96>>96 kilograms.
#### 96 |
In the diagram, $\triangle ABC$, $\triangle BCD$, and $\triangle CDE$ are right-angled at $B$, $C$, and $D$ respectively, with $\angle ACB=\angle BCD = \angle CDE = 45^\circ$, and $AB=15$. [asy]
pair A, B, C, D, E;
A=(0,15);
B=(0,0);
C=(10.6066,0);
D=(15,0);
E=(21.2132,0);
draw(A--B--C--D--E);
draw(B--C);
draw(C--D);
l... | \frac{15\sqrt{2}}{2} |
3000 people each go into one of three rooms randomly. What is the most likely value for the maximum number of people in any of the rooms? Your score for this problem will be 0 if you write down a number less than or equal to 1000. Otherwise, it will be $25-27 \frac{|A-C|}{\min (A, C)-1000}$. | 1019 |
Let $f(x) = 2a^{x} - 2a^{-x}$ where $a > 0$ and $a \neq 1$. <br/> $(1)$ Discuss the monotonicity of the function $f(x)$; <br/> $(2)$ If $f(1) = 3$, and $g(x) = a^{2x} + a^{-2x} - 2f(x)$, $x \in [0,3]$, find the minimum value of $g(x)$. | -2 |
Find all solutions $x$ (real and otherwise) to the equation
\[x^4+64=0.\]Enter all the solutions, separated by commas. | 2+2i,\,-2-2i,\,-2+2i,\,2-2i |
Determine all positive integers $n$ for which the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has an integer as a solution. | n=1 |
For how many positive integers \(n\) less than or equal to \(50\) is \(n!\) evenly divisible by \(1 + 2 + \cdots + n\)? | 34 |
Find a sequence of maximal length consisting of non-zero integers in which the sum of any seven consecutive terms is positive and that of any eleven consecutive terms is negative. | (-7,-7,18,-7,-7,-7,18,-7,-7,18,-7,-7,-7,18,-7,-7) |
If the program flowchart on the right is executed, determine the value of the output S. | 55 |
Seth and Max were selling candy bars for a school fundraiser. Seth sold 6 more candy bars than 3 times the number of candy bars that Max sold. If Max sold 24 candy bars, how many did Seth sell? | Let C be the number of candy bars Seth sold
3*24 +6=C
72+6=C
C=<<78=78>>78
#### 78 |
Before Ashley started a three-hour drive, her car's odometer reading was 29792, a palindrome. (A palindrome is a number that reads the same way from left to right as it does from right to left). At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of 75 miles per hou... | 70 \frac{1}{3} |
In a school's mentoring program, several first-grade students can befriend one sixth-grade student, while one sixth-grade student cannot befriend multiple first-grade students. It is known that $\frac{1}{3}$ of the sixth-grade students and $\frac{2}{5}$ of the first-grade students have become friends. What fraction of ... | $\frac{4}{11}$ |
Given a quadrilateral $ABCD$ inscribed in a circle with side $AB$ extended beyond $B$ to point $E$, if $\angle BAD=92^\circ$ and $\angle ADC=68^\circ$, find $\angle EBC$. | 68^\circ |
Let the set \( A = \{0, 1, 2, \ldots, 9\} \). The collection \( \{B_1, B_2, \ldots, B_k\} \) is a family of non-empty subsets of \( A \). When \( i \neq j \), the intersection \( B_i \cap B_j \) has at most two elements. Find the maximum value of \( k \). | 175 |
Find the quotient of the division $(3z^4-4z^3+5z^2-11z+2)/(2+3z)$. | z^3 -2z^2+3z-\frac{17}{3} |
Let \( A B C D \) be a quadrilateral and \( P \) the intersection of \( (A C) \) and \( (B D) \). Assume that \( \widehat{C A D} = 50^\circ \), \( \widehat{B A C} = 70^\circ \), \( \widehat{D C A} = 40^\circ \), and \( \widehat{A C B} = 20^\circ \). Calculate the angle \( \widehat{C P D} \). | 70 |
Earl and Bob start their new jobs on the same day. Earl's work schedule is to work for 3 days followed by 1 day off, while Bob's work schedule is to work for 7 days followed by 3 days off. In the first 1000 days, how many days off do they have in common? | 100 |
Pi Pi Lu wrote a 2020-digit number: \( 5368 \cdots \cdots \). If any four-digit number taken randomly from this multi-digit number is divisible by 11, what is the sum of the digits of this multi-digit number? | 11110 |
Find the minimum value of the function \( f(x) = \operatorname{tg}^{2} x - 4 \operatorname{tg} x - 12 \operatorname{ctg} x + 9 \operatorname{ctg}^{2} x - 3 \) on the interval \(\left( -\frac{\pi}{2}, 0 \right)\). | 3 + 8\sqrt{3} |
Let $x$ and $y$ be two positive real numbers such that $x + y = 35.$ Enter the ordered pair $(x,y)$ for which $x^5 y^2$ is maximized. | (25,10) |
Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Find the area of the shaded region. Express your answer in terms of $\pi$. | 40\pi |
Jolene and Phil have four children, each with the same birthday. They gave birth to their first child exactly 15 years ago. They gave birth to their second child exactly one year after the birth of their first child. They gave birth to their third child on the fourth birthday of their second child. Two years after t... | Their 1st child was born 15 years ago, and therefore is <<15=15>>15 years old.
Their 2nd child was born 1 year after their 15-year-old child, and therefore is 15-1=<<15-1=14>>14 years old.
Their 3rd child was born 4 years after their 14-year-old child, and therefore is 14-4=10 years old.
Their 4th child was born 2 year... |
If $3+x=5$ and $-3+y=5$, what is the value of $x+y$? | 10 |
How many of the divisors of $8!$ are larger than $7!$? | 7 |
The sides of a triangle have lengths of $13$, $84$, and $85$. Find the length of the shortest altitude. | 12.8470588235 |
In the diagram, the equilateral triangle has a base of $8$ m. What is the perimeter of the triangle? [asy]
size(100);
draw((0,0)--(8,0)--(4,4*sqrt(3))--cycle);
label("8 m",(4,0),S);
draw((4,-.2)--(4,.2));
draw((1.8,3.5)--(2.2,3.3));
draw((6.3,3.5)--(5.8,3.3));
[/asy] | 24 |
Jason spent 1/4 of his money and an additional $10 on some books. He then spent 2/5 of the remaining money and an additional $8 on some DVDs. If he was left with $130, how much money did he have at first? | Let X be the money Jason had at first.
Jason spent 1/4*X + 10 on books.
Jason spent 2/5*[X - (1/4*X + 10)] + 8 on some DVDs.
Jason has X - (1/4*X + 10) - {2/5*[X - (1/4*X + 10)] + 8} = $130 left.
X - 1/4*X - 10 - 2/5*X + 1/10*X + 4 - 8 = $130.
9/20*X - 14 = $130.
X = $<<320=320>>320.
#### 320 |
Given that $a > 0$, if $f(g(a)) = 18$, where $f(x) = x^2 + 10$ and $g(x) = x^2 - 6$, what is the value of $a$? | \sqrt{2\sqrt{2} + 6} |
An isosceles right triangle has a leg length of 36 units. Starting from the right angle vertex, an infinite series of equilateral triangles is drawn consecutively on one of the legs. Each equilateral triangle is inscribed such that their third vertices always lie on the hypotenuse, and the opposite sides of these vert... | 324 |
What is the coefficient of $x^2y^6$ in the expansion of $\left(\frac{3}{5}x-\frac{y}{2}\right)^8$? Express your answer as a common fraction. | \frac{63}{400} |
Given the function $f(x)=(a+ \frac {1}{a})\ln x-x+ \frac {1}{x}$, where $a > 0$.
(I) If $f(x)$ has an extreme value point in $(0,+\infty)$, find the range of values for $a$;
(II) Let $a\in(1,e]$, when $x_{1}\in(0,1)$, $x_{2}\in(1,+\infty)$, denote the maximum value of $f(x_{2})-f(x_{1})$ as $M(a)$, does $M(a)$ have... | \frac {4}{e} |
A green point and a purple point are chosen at random on the number line between 0 and 1. What is the probability that the purple point is greater than the green point but less than three times the green point? | \frac{1}{3} |
The figure shows two concentric circles. If the length of chord AB is 80 units and chord AB is tangent to the smaller circle, what is the area of the shaded region? Express your answer in terms of $\pi$.
[asy]
defaultpen(linewidth(.8pt));
dotfactor=4;
filldraw(circle((0,0),50),gray);
filldraw(circle((0,0),30),white);
... | 1600\pi |
A bug starts at a vertex of an equilateral triangle. On each move, it randomly selects one of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bug moves to its starting vertex on its tenth move is $m/n,$ where $m$ and $n$ are... | 683 |
Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perim... | 60 |
Two skaters, Allie and Billie, are at points $A$ and $B$, respectively, on a flat, frozen lake. The distance between $A$ and $B$ is $100$ meters. Allie leaves $A$ and skates at a speed of $8$ meters per second on a straight line that makes a $60^\circ$ angle with $AB$. At the same time Allie leaves $A$, Billie leaves $... | 160 |
A metallic weight has a mass of 20 kg and is an alloy of four metals. The first metal in this alloy is one and a half times the amount of the second metal. The mass of the second metal relates to the mass of the third metal as $3:4$, and the mass of the third metal to the mass of the fourth metal as $5:6$. Determine th... | 5.89 |
Paul, Amoura, and Ingrid were to go to a friend's party planned to start at 8:00 a.m. Paul arrived at 8:25. Amoura arrived 30 minutes later than Paul, and Ingrid was three times later than Amoura. How late, in minutes, was Ingrid to the party? | If the party was planned to start at 8:00 am, Paul was 8:25-8:00 = <<825-800=25>>25 minutes late.
If Paul was 25 minutes late, Amoura was 25+30 = <<25+30=55>>55 minutes late.
If Ingrid was three times late than Amoura was, she was 3*55 = <<3*55=165>>165 minutes late
#### 165 |
Given a moving circle $E$ that passes through the point $F(1,0)$, and is tangent to the line $l: x=-1$.
(Ⅰ) Find the equation of the trajectory $G$ for the center $E$ of the moving circle.
(Ⅱ) Given the point $A(3,0)$, if a line with a slope of $1$ intersects with the line segment $OA$ (not passing through the origin $... | \frac{32 \sqrt{3}}{9} |
Debbie works at a post office packing boxes to mail. Each large box takes 4 feet of packing tape to seal, each medium box takes 2 feet of packing tape to seal, and each small box takes 1 foot of packing tape to seal. Each box also takes 1 foot of packing tape to stick the address label on. Debbie packed two large boxes... | Debbie used 4 * 2 = <<4*2=8>>8 feet of tape for the large boxes.
She used 2 * 8 = <<2*8=16>>16 feet of tape for the medium boxes.
She used 1 * 5 = <<1*5=5>>5 feet of tape for the small boxes.
There were 2 + 8 + 5 = <<2+8+5=15>>15 boxes in all.
She used 1 * 15 = <<1*15=15>>15 feet of tape for the address labels.
Thus, s... |
Last week, a farmer shipped 10 boxes of pomelos which had 240 pomelos in all. This week, the farmer shipped 20 boxes. How many dozens of pomelos did the farmer ship in all? | There were 240/10 = <<240/10=24>>24 pomelos in a box.
This means that each box has 24/12 = <<24/12=2>>2 dozens of pomelos since 1 dozen is equal to 12.
The farmer shipped a total of 10 + 20 = <<10+20=30>>30 boxes.
Thus, the farmer shipped a total of 30 x 2 = <<30*2=60>>60 dozens of pomelos.
#### 60 |
Knot is ready to face Gammadorf in a card game. In this game, there is a deck with twenty cards numbered from 1 to 20. Each player starts with a five card hand drawn from this deck. In each round, Gammadorf plays a card in his hand, then Knot plays a card in his hand. Whoever played a card with greater value gets a poi... | 2982 |
What is the value of $m$ if Tobias downloads $m$ apps, each app costs $\$ 2.00$ plus $10 \%$ tax, and he spends $\$ 52.80$ in total on these $m$ apps? | 24 |
Given triangle $\triangle ABC$, $A=120^{\circ}$, $D$ is a point on side $BC$, $AD\bot AC$, and $AD=2$. Calculate the possible area of $\triangle ABC$. | \frac{8\sqrt{3}}{3} |
Given that $\sin x + \cos x = \frac{1}{2}$, where $x \in [0, \pi]$, find the value of $\sin x - \cos x$. | \frac{\sqrt{7}}{2} |
Let \(a\), \(b\), and \(c\) be real numbers such that \(9a^2 + 4b^2 + 25c^2 = 4\). Find the maximum value of
\[6a + 3b + 10c.\] | \sqrt{41} |
Let $m$ and $n$ be any two odd numbers, with $n$ less than $m$. The largest integer which divides all possible numbers of the form $m^2-n^2$ is:
$\textbf{(A)}\ 2\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 6\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$
| 8 |
Given the curve $\frac{y^{2}}{b} - \frac{x^{2}}{a} = 1 (a \cdot b \neq 0, a \neq b)$ and the line $x + y - 2 = 0$, the points $P$ and $Q$ intersect at the curve and line, and $\overrightarrow{OP} \cdot \overrightarrow{OQ} = 0 (O$ is the origin$), then the value of $\frac{1}{b} - \frac{1}{a}$ is $\_\_\_\_\_\_\_\_\_$. | \frac{1}{2} |
Lucas writes two distinct positive integers on a whiteboard. He decreases the smaller number by 20 and increases the larger number by 23 , only to discover the product of the two original numbers is equal to the product of the two altered numbers. Compute the minimum possible sum of the original two numbers on the boar... | 321 |
The rectangle $ABCD$ below has dimensions $AB = 12 \sqrt{3}$ and $BC = 13 \sqrt{3}$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $P$. If triangle $ABP$ is cut out and removed, edges $\overline{AP}$ and $\overline{BP}$ are joined, and the figure is then creased along segments $\overline{CP}$ and $\overlin... | 594 |
Square \( ABCD \) has a side length of 12 inches. A segment \( AE \) is drawn where \( E \) is on side \( DC \) and \( DE \) is 5 inches long. The perpendicular bisector of \( AE \) intersects \( AE, AD, \) and \( BC \) at points \( M, P, \) and \( Q \) respectively. The ratio of the segments \( PM \) to \( MQ \) is: | 5:19 |
Given that the terminal side of angle $\alpha$ is in the second quadrant and intersects the unit circle at point $P(m, \frac{\sqrt{15}}{4})$.
$(1)$ Find the value of the real number $m$;
$(2)$ Let $f(\alpha) = \frac{\cos(2\pi - \alpha) + \tan(3\pi + \alpha)}{\sin(\pi - \alpha) \cdot \cos(\alpha + \frac{3\pi}{2})}$.... | -\frac{4 + 16\sqrt{15}}{15} |
Two identical CDs regularly cost a total of $\$28.$ What is the cost in dollars of five of these CDs? | 70 |
Natural numbers \(a, b, c\) are chosen such that \(a < b < c\). It is also known that the system of equations \(2x + y = 2019\) and \(y = |x-a| + |x-b| + |x-c|\) has exactly one solution. Find the minimum possible value of \(c\). | 1010 |
In the sum shown, each letter represents a different digit with $T \neq 0$ and $W \neq 0$. How many different values of $U$ are possible?
\begin{tabular}{rrrrr}
& $W$ & $X$ & $Y$ & $Z$ \\
+ & $W$ & $X$ & $Y$ & $Z$ \\
\hline & $W$ & $U$ & $Y$ & $V$
\end{tabular} | 3 |
What value of $x$ will give the maximum value for $-x^2- 6x + 12$? | -3 |
A box contains 4 cards, each with one of the following functions defined on \\(R\\): \\(f_{1}(x)={x}^{3}\\), \\(f_{2}(x)=|x|\\), \\(f_{3}(x)=\sin x\\), \\(f_{4}(x)=\cos x\\). Now, if we randomly pick 2 cards from the box and multiply the functions on the cards to get a new function, the probability that the resulting f... | \dfrac{2}{3} |
What is the units digit of $\frac{20 \cdot 21 \cdot 22 \cdot 23 \cdot 24 \cdot 25}{1000}$? | 2 |
Given a set $A_n = \{1, 2, 3, \ldots, n\}$, define a mapping $f: A_n \rightarrow A_n$ that satisfies the following conditions:
① For any $i, j \in A_n$ with $i \neq j$, $f(i) \neq f(j)$;
② For any $x \in A_n$, if the equation $x + f(x) = 7$ has $K$ pairs of solutions, then the mapping $f: A_n \rightarrow A_n$ is said t... | 40 |
The solutions to the equation $x^2 = x$ are $x=0$ and $x=1$. | 0 or 1 |
Find the coefficient of the $x^2$ term in the expansion of the product $$(2x^2 +3x +4)(5x^2 +6x +7).$$ | 52 |
Given the sequence 1, 1+2, 2+3+4, 3+4+5+6, ..., the value of the 8th term in this sequence is: ______. | 84 |
Anne drops a mirror and breaks it into 60 pieces. She sweeps up half of them, then her cat steals 3 pieces and her boyfriend picks up 1/3 of the remaining pieces. How many pieces does her boyfriend pick up? | First find the number of pieces left after Anne sweeps up: 60 pieces / 2 = <<60/2=30>>30 pieces
Then subtract the number her cat takes: 30 pieces - 3 pieces = <<30-3=27>>27 pieces
Divide that number by 3 to find how many pieces her boyfriend picks up: 27 pieces / 3 = <<27/3=9>>9 pieces
#### 9 |
Jimmy notices $7$ oranges weigh the same as $5$ apples. If Jimmy has $28$ oranges, how many apples would Jimmy need to equal the weight of his $28$ oranges? | 20 |
If $z=3+4i$, find $z^2$. (Note that $i^2 = -1.$) | -7+24i |
John starts at an elevation of 400 feet. He travels downward at a rate of 10 feet down per minute for 5 minutes. What is his elevation now? | He traveled down 10*5=<<10*5=50>>50 feet.
So he is at an elevation of 400-50=<<400-50=350>>350 feet.
#### 350 |
John assembles widgets at a factory. He can make 20 widgets an hour and works for 8 hours a day 5 days a week. How many widgets does he make a week? | He works 8*5=<<8*5=40>>40 hours
So he made 20*40=<<20*40=800>>800 widgets
#### 800 |
Susy goes to a large school with 800 students, while Sarah goes to a smaller school with only 300 students. At the start of the school year, Susy had 100 social media followers. She gained 40 new followers in the first week of the school year, half that in the second week, and half of that in the third week. Sarah o... | After one week, Susy has 100+40 = <<100+40=140>>140 followers.
In the second week, Susy gains 40/2 = <<40/2=20>>20 new followers.
In the third week, Susy gains 20/2 = <<20/2=10>>10 new followers.
In total, Susy finishes the three weeks with 140+20+10 = <<140+20+10=170>>170 total followers.
After one week, Sarah has 50+... |
In triangle \(ABC\), the heights are given: \(h_{a} = \frac{1}{3}\), \(h_{b} = \frac{1}{4}\), \(h_{c} = \frac{1}{5}\). Find the ratio of the angle bisector \(CD\) to the circumradius. | \frac{24\sqrt{2}}{35} |
Given that $x_{1}$ is a root of the one-variable quadratic equation about $x$, $\frac{1}{2}m{x^2}+\sqrt{2}x+{m^2}=0$, and ${x_1}=\sqrt{a+2}-\sqrt{8-a}+\sqrt{-{a^2}}$ (where $a$ is a real number), find the values of $m$ and the other root of the equation. | 2\sqrt{2} |
The first term of a given sequence is 1, and each successive term is the sum of all the previous terms of the sequence. What is the value of the first term which exceeds 5000? | 8192 |
Simplify: $\sqrt{1\frac{7}{9}}=$______, $\sqrt[3]{(-3)^{3}}=\_\_\_\_\_\_$. | -3 |
Suppose that we have an 8-sided die with 4 red faces, 3 yellow faces, and a blue face. What is the probability of rolling a yellow face? | \dfrac38 |
Find the product of $0.\overline{6}$ and 6. | 4 |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\tau (1)=1$ and $\tau(6) =4.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote ... | 25 |
An iterative average of the numbers 1, 2, 3, 4, and 5 is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. What is... | \frac{17}{8} |
Each side of a cube has a stripe drawn diagonally from one vertex to the opposite vertex. The stripes can either go from the top-right to bottom-left or from top-left to bottom-right, chosen at random for each face. What is the probability that there is at least one continuous path following the stripes that goes from ... | \frac{3}{64} |
A valid license plate in Xanadu consists of two letters followed by three digits. How many valid license plates are possible? | 676,\!000 |
What is the length of the segment of the number line whose endpoints satisfy $|x-\sqrt[5]{16}|=3$? | 6 |
Ron is part of a book club that allows each member to take a turn picking a new book every week. The club is made up of three couples and five single people along with Ron and his wife. How many times a year does Ron get to pick a new book? | There are 3 couples in the club, so there are 3 * 2 = <<3*2=6>>6 people in couples in the club.
Along with the singles and Ron and his wife, there are 6 + 5 + 2 = <<6+5+2=13>>13 people in the book club.
There are 52 weeks in a year, so Ron gets to pick a new book 52 / 13 = <<52/13=4>>4 times a year.
#### 4 |
What is the remainder of $19^{1999}$ divided by 25? | 4 |
Given that a 4-digit positive integer has only even digits (0, 2, 4, 6, 8) and is divisible by 4, calculate the number of such integers. | 300 |
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