problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Juan bought T-shirts for his employees. He bought shirts for men and women. Women's t-shirts are $5 cheaper than men's t-shirts of the same color. His company has 2 sectors, one in white t-shirts and the other in black t-shirts. He paid $20 for white men's t-shirts and $18 for black men's t-shirts. The 2 sectors have t... | For each type of shirt, he needed to buy, 40 employees / 4 = <<40/4=10>>10 shirts.
White men's T-shirts cost $20/shirt * 10 shirts = $<<20*10=200>>200.
White women's T-shirts cost $5 less, or ($20/shirt - $5/shirt) * 10 shirts = $15/shirt * 10 shirts = $150.
Black men's t-shirts cost, $18/shirt * 10 shirts = $<<18*10=1... |
John has a party and invites 30 people. Of the people he invited 20% didn't show up. 75% of the people who show up get steak and the rest get chicken. How many people ordered chicken? | There were 30*.2=<<30*.2=6>>6 people who didn't show up
So 30-6=<<30-6=24>>24 people showed up
Of the people who showed up, 24*.75=<<24*.75=18>>18 got steak
So 24-18=<<24-18=6>>6 got chicken
#### 6 |
At the end of the first quarter, the winning team had double the points of the losing team. At the end of the second quarter, the winning team had 10 more points than it started with. At the end of the third quarter, the winning team had 20 more points than the number it had in the second quarter. If the total points t... | At the end of the first quarter, the winning team had double the points of the losing team, meaning the winning team had already scored 10*2=<<10*2=20>>20 points.
At the end of the second quarter, the winning team had 10 more points than it started with, a total of 20+10=30 points.
At the end of the third quarter, the ... |
A cylindrical barrel with radius $4$ feet and height $10$ feet is full of water. A solid cube with side length $8$ feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is $v$ cubic feet. Find $v^2$.
[asy] import three; import solids; size(5cm); currentprojection=o... | 384 |
The average age of 8 people in Room C is 35. The average age of 6 people in Room D is 30. Calculate the average age of all people when the two groups are combined. | \frac{460}{14} |
In how many ways is it possible to arrange the digits of 1150 to get a four-digit multiple of 5? | 5 |
Buying a toaster requires an insurance plan that is 20% of the MSRP, plus a mandatory state tax rate of 50% after the insurance plan calculation. Jon chooses to buy a toaster that costs $30 at MSRP. What is the total he must pay? | The insurance plan costs $30*0.2=$<<30*0.2=6>>6.
The pre-state tax total is $30+$6=$<<30+6=36>>36.
State tax will be $36*0.5=$<<36*0.5=18>>18.
Jon needs to pay $18+$36=$<<18+36=54>>54 for his new toaster.
#### 54 |
We write one of the numbers $0$ and $1$ into each unit square of a chessboard with $40$ rows and $7$ columns. If any two rows have different sequences, at most how many $1$ s can be written into the unit squares? | 198 |
Define the operation: \(a \quad b = \frac{a \times b}{a + b}\). Calculate the result of the expression \(\frac{20102010 \cdot 2010 \cdots 2010 \cdot 201 \text{ C}}{\text{ 9 " "}}\). | 201 |
A marathon is $26$ miles and $385$ yards. One mile equals $1760$ yards.
Leila has run ten marathons in her life. If the total distance Leila covered in these marathons is $m$ miles and $y$ yards, where $0\le y<1760$, what is the value of $y$? | 330 |
Given vectors $\overrightarrow{a}, \overrightarrow{b}$ that satisfy $|\overrightarrow{a}|^2, \overrightarrow{a}\cdot \overrightarrow{b} = \frac{3}{2}, |\overrightarrow{a}+ \overrightarrow{b}| = 2\sqrt{2}$, find $|\overrightarrow{b}| = \_\_\_\_\_\_\_.$ | \sqrt{5} |
The sequence $\{a_n\}$ satisfies $a_1 = 1$ and $5^{a_{n + 1} - a_n} - 1 = \frac {1}{n + \frac {2}{3}}$ for $n \geq 1$. Find the least integer $k$ greater than $1$ for which $a_k$ is an integer. | 41 |
Each third-grade classroom at Pearl Creek Elementary has $18$ students and $2$ pet rabbits. How many more students than rabbits are there in all $4$ of the third-grade classrooms? | 64 |
Rob planned on spending three hours reading in preparation for his literature exam. If he ends up spending only three-quarters of this time reading, and he reads a page every fifteen minutes, how many pages did he read in this time? | 1 hour is 60 minutes so 3 hours is 3*60 = <<3*60=180>>180 minutes
Three-quarters of 180 minutes is (3/4)*180 = <<(3/4)*180=135>>135 minutes
He spends 15 minutes on 1 page so he will spend 135 minutes on 135/15 = <<135/15=9>>9 pages
#### 9 |
Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality:
\[ x\plus{}af(y)\leq y\plus{}f(f(x))
\]
for all $ x,y\in\mathbb{R}$ | a < 0 \text{ or } a = 1 |
A dormitory is installing a shower room for 100 students. How many shower heads are economical if the boiler preheating takes 3 minutes per shower head, and it also needs to be heated during the shower? Each group is allocated 12 minutes for showering. | 20 |
Quadrilateral $A B C D$ satisfies $A B=8, B C=5, C D=17, D A=10$. Let $E$ be the intersection of $A C$ and $B D$. Suppose $B E: E D=1: 2$. Find the area of $A B C D$. | 60 |
The arithmetic sequence \( a, a+d, a+2d, a+3d, \ldots, a+(n-1)d \) has the following properties:
- When the first, third, fifth, and so on terms are added, up to and including the last term, the sum is 320.
- When the first, fourth, seventh, and so on, terms are added, up to and including the last term, the sum is 224... | 608 |
Set $S = \{1, 2, 3, ..., 2005\}$ . If among any $n$ pairwise coprime numbers in $S$ there exists at least a prime number, find the minimum of $n$ . | 15 |
The function $\lfloor x\rfloor$ is defined as the largest integer less than or equal to $x$. For example, $\lfloor 5.67\rfloor = 5$, $\lfloor -\tfrac 14\rfloor = -1$, and $\lfloor 8\rfloor = 8$.
What is the range of the function $$f(x) = \lfloor x\rfloor - x~?$$Express your answer in interval notation. | (-1,0] |
How many three-digit numbers exist that are 5 times the product of their digits? | 175 |
A company has calculated that investing x million yuan in project A will yield an economic benefit y that satisfies the relationship: $y=f(x)=-\frac{1}{4}x^{2}+2x+12$. Similarly, the economic benefit y from investing in project B satisfies the relationship: $y=h(x)=-\frac{1}{3}x^{2}+4x+1$.
(1) If the company has 10 mil... | 6.5 |
Compute $\sin 150^\circ$. | \frac{1}{2} |
Triangle $ABC$ is an obtuse, isosceles triangle. Angle $A$ measures 20 degrees. What is number of degrees in the measure of the largest interior angle of triangle $ABC$?
[asy]
draw((-20,0)--(0,8)--(20,0)--cycle);
label("$20^{\circ}$",(-13,-0.7),NE);
label("$A$",(-20,0),W);
label("$B$",(0,8),N);
label("$C$",(20,0),E);
... | 140 |
In the decimal representation of an even number \( M \), only the digits \( 0, 2, 4, 5, 7, \) and \( 9 \) are used, and the digits may repeat. It is known that the sum of the digits of the number \( 2M \) equals 39, and the sum of the digits of the number \( M / 2 \) equals 30. What values can the sum of the digits of ... | 33 |
What is the largest integer that is a divisor of \[
(n+1)(n+3)(n+5)(n+7)(n+9)
\]for all positive even integers $n$? | 15 |
In the complex plane, let $A$ be the set of solutions to $z^{3}-8=0$ and let $B$ be the set of solutions to $z^{3}-8z^{2}-8z+64=0.$ What is the greatest distance between a point of $A$ and a point of $B?$ | $2\sqrt{21}$ |
A certain real estate property is holding a lottery for homebuyers, with the following rules: For homeowners who purchase the property, they can randomly draw 2 balls from box $A$, which contains 2 red balls and 2 white balls, and 2 balls from box $B$, which contains 3 red balls and 2 white balls. If all 4 balls drawn ... | 3,675 |
Let \(\Gamma_{1}\) and \(\Gamma_{2}\) be two circles externally tangent to each other at \(N\) that are both internally tangent to \(\Gamma\) at points \(U\) and \(V\), respectively. A common external tangent of \(\Gamma_{1}\) and \(\Gamma_{2}\) is tangent to \(\Gamma_{1}\) and \(\Gamma_{2}\) at \(P\) and \(Q\), respec... | \frac{\left(Rr_{1}+Rr_{2}-2r_{1}r_{2}\right)2\sqrt{r_{1}r_{2}}}{\left|r_{1}-r_{2}\right|\sqrt{\left(R-r_{1}\right)\left(R-r_{2}\right)}} |
The probability of rain tomorrow is $\frac{3}{10}$. What is the probability that it will not rain tomorrow? Express your answer as a common fraction. | \frac{7}{10} |
What is the smallest possible median for the six number set $\{x, 2x, 3, 2, 5, 4x\}$ if $x$ can be any positive integer? | 2.5 |
Let's call an integer "extraordinary" if it has exactly one even divisor other than 2. How many extraordinary numbers exist in the interval $[1 ; 75]$? | 11 |
If $a$, $b$, and $c$ are positive numbers such that $ab = 24\sqrt{3}$, $ac = 30\sqrt{3}$, and $bc = 40\sqrt{3}$, find the value of $abc$. | 120\sqrt{6} |
Given that $\tan \alpha +\tan \beta -\tan \alpha \tan \beta +1=0$, and $\alpha ,\beta \in \left(\frac{\pi }{2},\pi \right)$, calculate $\alpha +\beta$. | \frac{7\pi}{4} |
The maximum and minimum values of the function $y=2x^{3}-3x^{2}-12x+5$ on the interval $[0,3]$ need to be determined. | -15 |
Jeanne wants to ride the Ferris wheel, the roller coaster, and the bumper cars. The Ferris wheel costs 5 tickets, the roller coaster costs 4 tickets and the bumper cars cost 4 tickets. Jeanne has 5 tickets. How many more tickets should Jeanne buy? | The total number of tickets needed is 5 tickets + 4 tickets + 4 tickets = <<5+4+4=13>>13 tickets.
Jeanne needs 13 tickets - 5 tickets = <<13-5=8>>8 tickets.
#### 8 |
Given an ellipse $\frac {x^{2}}{a^{2}} + \frac {y^{2}}{b^{2}} = 1$ ($a > b > 0$) and a line $l: y = -\frac { \sqrt {3}}{3}x + b$ intersect at two distinct points P and Q. The distance from the origin to line $l$ is $\frac { \sqrt {3}}{2}$, and the eccentricity of the ellipse is $\frac { \sqrt {6}}{3}$.
(Ⅰ) Find the eq... | -\frac {7}{6} |
Determine the tens digit of $17^{1993}$. | 3 |
Given that $\alpha$ and $\beta$ are acute angles, $\cos\alpha=\frac{{\sqrt{5}}}{5}$, $\cos({\alpha-\beta})=\frac{3\sqrt{10}}{10}$, find the value of $\cos \beta$. | \frac{\sqrt{2}}{10} |
Fiona has a deck of cards labelled $1$ to $n$, laid out in a row on the table in order from $1$ to $n$ from left to right. Her goal is to arrange them in a single pile, through a series of steps of the following form:
[list]
[*]If at some stage the cards are in $m$ piles, she chooses $1\leq k<m$ and arranges the car... | 2^{n-2} |
Let \(a\), \(b\), \(c\) be distinct complex numbers such that
\[
\frac{a+1}{2 - b} = \frac{b+1}{2 - c} = \frac{c+1}{2 - a} = k.
\]
Find the sum of all possible values of \(k\). | 1.5 |
Expand the following expression: $3(8x^2-2x+1)$. | 24x^2-6x+3 |
Kelly, Brittany, and Buffy went swimming at Salt Rock Lake and held a contest to see who could hold their breath underwater for the longest amount of time. Kelly held her breath underwater for 3 minutes. Brittany held her breath underwater for 20 seconds less time than than Kelly did, and Buffy held her breath underw... | Kelly held her breath for 3 minutes, or 3*60=<<3*60=180>>180 seconds.
Brittany held her breath for 20 seconds less time than Kelly did, or 180-20=<<180-20=160>>160 seconds
And Buffy held her breath for 40 seconds less time than Brittany did, or for 160-40=<<160-40=120>>120 seconds.
#### 120 |
Find the phase shift of the graph of $y = \sin (3x - \pi).$ | -\frac{\pi}{3} |
Given the function $f\left(x\right)=\cos 2x+\sin x$, if $x_{1}$ and $x_{2}$ are the abscissas of the maximum and minimum points of $f\left(x\right)$, then $\cos (x_{1}+x_{2})=$____. | \frac{1}{4} |
Let $n$ be the number of ordered quadruples $(x_1,x_2,x_3,x_4)$ of positive odd integers that satisfy $\sum_{i = 1}^4 x_i = 98.$ Find $\frac n{100}.$
| 196 |
Matias is a salesman in a bookstore. He sold 7 books on Tuesday and three times as many on Wednesday. If the sales from Wednesday were tripled on Thursday, how many books did Matias sell during these three days combined? | Matias sold 7 * 3 = <<7*3=21>>21 books on Wednesday, as he sold three times more books than on Tuesday.
The sales from Wednesday were tripled on Thursday, which means Matias sold then 21 * 3 = <<21*3=63>>63 books.
That means in total Matias sold 7 + 21 + 63 = <<7+21+63=91>>91 books during these three days.
#### 91 |
For what value of $x$ does $10^{x} \cdot 100^{2x}=1000^{5}$? | 3 |
The four circles in the diagram intersect to divide the interior into 8 parts. Fill these 8 parts with the numbers 1 through 8 such that the sum of the 3 numbers within each circle is equal. Calculate the maximum possible sum and provide one possible configuration. | 15 |
If $3(-2) = \nabla +2$, then what does $\nabla$ equal? | -8 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given vectors $\vec{m}=(a,c)$ and $\vec{n}=(\cos C,\cos A)$.
1. If $\vec{m}\parallel \vec{n}$ and $a= \sqrt {3}c$, find angle $A$;
2. If $\vec{m}\cdot \vec{n}=3b\sin B$ and $\cos A= \frac {3}{5}$, find the value of $\... | \frac {4-6 \sqrt {2}}{15} |
In square $ABCD$, points $E$ and $H$ lie on $\overline{AB}$ and $\overline{DA}$, respectively, so that $AE=AH.$ Points $F$ and $G$ lie on $\overline{BC}$ and $\overline{CD}$, respectively, and points $I$ and $J$ lie on $\overline{EH}$ so that $\overline{FI} \perp \overline{EH}$ and $\overline{GJ} \perp \overline{EH}$. ... | 8-4\sqrt{2} |
At exactly noon, Anna Kuzminichna looked out the window and saw Klava, the village shop clerk, going on a break. Two minutes past twelve, Anna Kuzminichna looked out the window again, and no one was at the closed store. Klava was absent for exactly 10 minutes, and when she returned, she found that Ivan and Foma were wa... | 1/2 |
A refrigerator is offered at sale at $250.00 less successive discounts of 20% and 15%. The sale price of the refrigerator is: | 77\% of 250.00 |
For any positive integer $n,$ let $\langle n \rangle$ denote the closest integer to $\sqrt{n}.$ Evaluate
\[\sum_{n = 1}^\infty \frac{2^{\langle n \rangle} + 2^{-\langle n \rangle}}{2^n}.\] | 3 |
There is a ten-digit number. From left to right:
- Its first digit indicates how many zeros are in the number.
- Its second digit indicates how many ones are in the number.
- Its third digit indicates how many twos are in the number.
- $\cdots \cdots$
- Its tenth digit indicates how many nines are in the number.
Find ... | 6210001000 |
Compute the circumradius of cyclic hexagon $A B C D E F$, which has side lengths $A B=B C=$ $2, C D=D E=9$, and $E F=F A=12$. | 8 |
In the diagram below, lines $k$ and $\ell$ are parallel. Find the measure of angle $x$ in degrees.
[asy]
size(200);
import markers;
pair A = dir(-22)*(0,0);
pair B = dir(-22)*(4,0);
pair C = dir(-22)*(4,2);
pair D = dir(-22)*(0,2);
pair F = dir(-22)*(0,1.3);
pair G = dir(-22)*(4,1.3);
pair H = dir(-22)*(2,1);
//mark... | 60^\circ |
Find all positive integers $n>1$ for which $\frac{n^{2}+7 n+136}{n-1}$ is the square of a positive integer. | 5, 37 |
For a special event, the five Vietnamese famous dishes including Phở, (Vietnamese noodle), Nem (spring roll), Bún Chả (grilled pork noodle), Bánh cuốn (stuffed pancake), and Xôi gà (chicken sticky rice) are the options for the main courses for the dinner of Monday, Tuesday, and Wednesday. Every dish must be used exact... | 150 |
In the figure, $PA$ is tangent to semicircle $SAR$, $PB$ is tangent to semicircle $RBT$, and $SRT$ is a straight line. If arc $AS$ is $58^\circ$ and arc $BT$ is $37^\circ$, then find $\angle APB$, in degrees.
[asy]
import graph;
unitsize(1.5 cm);
pair A, B, P, R, S, T;
pair[] O;
real[] r;
r[1] = 1;
r[2] = 0.8;
S ... | 95^\circ |
Consider a rectangle $ABCD$ with $BC = 2 \cdot AB$ . Let $\omega$ be the circle that touches the sides $AB$ , $BC$ , and $AD$ . A tangent drawn from point $C$ to the circle $\omega$ intersects the segment $AD$ at point $K$ . Determine the ratio $\frac{AK}{KD}$ .
*Proposed by Giorgi Arabidze, Georgia* | 1/2 |
An electric car is charged 3 times per week for 52 weeks. The cost to charge the car each time is $0.78. What is the total cost to charge the car over these 52 weeks? | \$121.68 |
Find $c$ such that $\lfloor c \rfloor$ satisfies
\[3x^2 - 9x - 30 = 0\]
and $\{ c \} = c - \lfloor c \rfloor$ satisfies
\[4x^2 - 8x + 1 = 0.\] | 6 - \frac{\sqrt{3}}{2} |
Given positive integers \( N \) and \( k \), we counted how many different ways the number \( N \) can be written in the form \( a + b + c \), where \( 1 \leq a, b, c \leq k \), and the order of the summands matters. Could the result be 2007? | 2007 |
Pentagon $ANDD'Y$ has $AN \parallel DY$ and $AY \parallel D'N$ with $AN = D'Y$ and $AY = DN$ . If the area of $ANDY$ is 20, the area of $AND'Y$ is 24, and the area of $ADD'$ is 26, the area of $ANDD'Y$ can be expressed in the form $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$ . ... | 71 |
Twelve standard 6-sided dice are rolled. What is the probability that exactly two of the dice show a 1? Express your answer as a decimal rounded to the nearest thousandth. | 0.298 |
Find the largest integer $x$ such that the number
$$
4^{27} + 4^{1000} + 4^{x}
$$
is a perfect square. | 1972 |
If each of Bill's steps is $ rac{1}{2}$ metre long, how many steps does Bill take to walk 12 metres in a straight line? | 24 |
Approximate the number $0.00356$ to the nearest ten-thousandth: $0.00356 \approx$____. | 0.0036 |
Marcia's hair is 24" long at the beginning of the school year. She cuts half of her hair off and lets it grow out 4 more inches. She then cuts off another 2" of hair. How long is her hair? | Her hair was 24" long and she cut half off so she cut 24/2 = <<24/2=12>>12 inches
Her hair was 24" long and she cut 12" off so she has 24-12 = <<24-12=12>>12 inches left
She has 12" of hair and lets it grow 4 more inches for a total of 12+4 = <<12+4=16>>16 inches
She has 16" and cuts off 2 more inches which leaves her ... |
Mark does a gig every other day for 2 weeks. For each gig, he plays 3 songs. 2 of the songs are 5 minutes long and the last song is twice that long. How many minutes did he play? | He played 7*2 = <<7*2=14>>14 gigs
The long song was 5*2=<<5*2=10>>10 minutes
So all the gigs were 5+5+10=<<5+5+10=20>>20 minutes
So he played 14*20=140 minutes
#### 280 |
Now a ball is launched from a vertex of an equilateral triangle with side length 5. It strikes the opposite side after traveling a distance of $\sqrt{19}$. Find the distance from the ball's point of first contact with a wall to the nearest vertex. | 2 |
How many triangles with positive area can be formed where each vertex is at point $(i,j)$ in the coordinate grid, with integers $i$ and $j$ ranging from $1$ to $4$ inclusive? | 516 |
The solutions to the equation $(z-4)^6 = 64$ are connected in the complex plane to form a convex regular polygon, three of whose vertices are labelled $D, E,$ and $F$. What is the least possible area of triangle $DEF$? | \sqrt{3} |
Given a set of data is multiplied by 2 and then reduced by 80 for each data point, resulting in a new set of data with an average of 1.2 and a variance of 4.4, determine the average and variance of the original data. | 1.1 |
Given that $0 < β < \dfrac{π}{2} < α < π$, and $\cos (α- \dfrac{β}{2} )= \dfrac{5}{13} $, $\sin ( \dfrac{α}{2}-β)= \dfrac{3}{5} $. Find the values of:
$(1) \tan (α- \dfrac{β}{2} )$
$(2) \cos ( \dfrac{α+β}{2} )$ | \dfrac{56}{65} |
In rectangle $ABCD$, $AB=7$ and $BC=4$. Points $J$ and $K$ are on $\overline{CD}$ such that $DJ = 2$ and $KC=3$. Lines $AJ$ and $BK$ intersect at point $L$. Find the area of $\triangle ABL$. Visualize as follows:
[asy]
pair A,B,C,D,L,J,K;
A=(0,0);
B=(7,0);
C=(7,4);
D=(0,4);
J=(2,4);
K=(4,4);
L=(3.5,6);
draw(A--B--C--D-... | \frac{98}{5} |
What is the base four equivalent of $123_{10}$? | 1323_{4} |
Let $p, q, r,$ and $s$ be the roots of the polynomial $3x^4 - 8x^3 - 15x^2 + 10x - 2 = 0$. Find $pqrs$. | \frac{2}{3} |
$A$ and $B$ travel around a circular track at uniform speeds in opposite directions, starting from diametrically opposite points. If they start at the same time, meet first after $B$ has travelled $100$ yards, and meet a second time $60$ yards before $A$ completes one lap, then the circumference of the track in yards i... | 480 |
In a revised game of Deal or No Deal, participants choose a box at random from a set of $30$, each containing one of the following values:
\[
\begin{array}{|c|c|}
\hline
\$0.50 & \$50,000 \\
\hline
\$5 & \$100,000 \\
\hline
\$20 & \$150,000 \\
\hline
\$50 & \$200,000 \\
\hline
\$100 & \$250,000 \\
\hline
\$250 & \$300,... | 20 |
A pen and its ink refill together cost $\;\$1.10$. The pen costs $\;\$1$ more than the ink refill. What is the cost of the pen in dollars? | 1.05 |
On a spherical planet with diameter $10,000 \mathrm{~km}$, powerful explosives are placed at the north and south poles. The explosives are designed to vaporize all matter within $5,000 \mathrm{~km}$ of ground zero and leave anything beyond $5,000 \mathrm{~km}$ untouched. After the explosives are set off, what is the ne... | 100,000,000 \pi |
A cowboy is 6 miles south of a stream which flows due east. He is also 12 miles west and 10 miles north of his cabin. Before returning to his cabin, he wishes to fill his water barrel from the stream and also collect firewood 5 miles downstream from the point directly opposite his starting point. Find the shortest dist... | 11 + \sqrt{305} |
What is the greatest integer less than $-\frac{15}4$? | -4 |
How many parallelograms with sides 1 and 2, and angles $60^{\circ}$ and $120^{\circ}$ can be placed at most inside a regular hexagon with side length 3? | 12 |
A street has 20 houses on each side, for a total of 40 houses. The addresses on the south side of the street form an arithmetic sequence, as do the addresses on the north side of the street. On the south side, the addresses are 4, 10, 16, etc., and on the north side they are 3, 9, 15, etc. A sign painter paints house n... | 84 |
Given \( x, y, z \in (0, +\infty) \) and \(\frac{x^2}{1+x^2} + \frac{y^2}{1+y^2} + \frac{z^2}{1+z^2} = 2 \), find the maximum value of \(\frac{x}{1+x^2} + \frac{y}{1+y^2} + \frac{z}{1+z^2}\). | \sqrt{2} |
Given four numbers $101010_{(2)}$, $111_{(5)}$, $32_{(8)}$, and $54_{(6)}$, the smallest among them is \_\_\_\_\_\_. | 32_{(8)} |
Compute the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, dx}{(1+\cos x+\sin x)^{2}}
$$ | \ln 2 - \frac{1}{2} |
Let the function \( f(x) \) satisfy the following conditions:
(i) If \( x > y \), and \( f(x) + x \geq w \geq f(y) + y \), then there exists a real number \( z \in [y, x] \), such that \( f(z) = w - z \);
(ii) The equation \( f(x) = 0 \) has at least one solution, and among the solutions of this equation, there exists ... | 2004 |
James has 7 more than 4 times the number of Oreos Jordan has. If there are 52 Oreos total, how many does James have? | Let x be the number of Oreos Jordan has
James has 7+4x Oreos
Total: 7+4x+x=52
5x+7=52
5x=45
x=<<9=9>>9
James has 7+4(9)=43 Oreos
#### 43 |
How many of the natural numbers from 1 to 800, inclusive, contain the digit 7 at least once? | 152 |
In convex quadrilateral \(WXYZ\), \(\angle W = \angle Y\), \(WZ = YX = 150\), and \(WX \ne ZY\). The perimeter of \(WXYZ\) is 520. Find \(\cos W\). | \frac{11}{15} |
Given \(\omega = -\frac{1}{2} + \frac{1}{2}i\sqrt{3}\), representing a cube root of unity, specifically \(\omega = e^{2\pi i / 3}\). Let \(T\) denote all points in the complex plane of the form \(a + b\omega + c\omega^2\), where \(0 \leq a \leq 2\), \(0 \leq b \leq 1\), and \(0 \leq c \leq 1\). Determine the area of \(... | 2\sqrt{3} |
Julio has two cylindrical candles with different heights and diameters. The two candles burn wax at the same uniform rate. The first candle lasts 6 hours, while the second candle lasts 8 hours. He lights both candles at the same time and three hours later both candles are the same height. What is the ratio of their ori... | 5:4 |
A triangle with side lengths in the ratio 3:4:5 is inscribed in a circle of radius 3. What is the area of the triangle? Provide your answer as a decimal rounded to the nearest hundredth. | 8.64 |
Juan is measuring the diameter of a large rather ornamental plate to cover it with a decorative film. Its actual diameter is 30cm, but his measurement tool has an error of up to $30\%$. Compute the largest possible percent error, in percent, in Juan's calculated area of the ornamental plate. | 69 |
What is $35_8-74_8?$ Express your answer in base 8. | -37_8 |
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