problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Given that $0 < α < \dfrac {π}{2}$, and $\cos ( \dfrac {π}{3}+α)= \dfrac {1}{3}$, find the value of $\cos α$. | \dfrac {2 \sqrt {6}+1}{6} |
Find the smallest period of the function \( y = \cos^{10} x + \sin^{10} x \). | \frac{\pi}{2} |
One dimension of a cube is increased by $1$, another is decreased by $1$, and the third is left unchanged. The volume of the new rectangular solid is $5$ less than that of the cube. What was the volume of the cube? | 125 |
Find the number of six-digit palindromes. | 9000 |
Suppose $\alpha,\beta,\gamma\in\{-2,3\}$ are chosen such that
\[M=\max_{x\in\mathbb{R}}\min_{y\in\mathbb{R}_{\ge0}}\alpha x+\beta y+\gamma xy\]
is finite and positive (note: $\mathbb{R}_{\ge0}$ is the set of nonnegative real numbers). What is the sum of the possible values of $M$ ? | 13/2 |
What is the ratio of the area of a square inscribed in a semicircle with radius $r$ to the area of a square inscribed in a circle with radius $r$? Express your answer as a common fraction. | \dfrac{2}{5} |
Sammy can eat 15 pickle slices. His twin sister Tammy can eat twice as much as Sammy. Their older brother Ron eats 20% fewer pickles slices than Tammy. How many pickle slices does Ron eat? | Tammy can eat twice as many pickle slices as Sammy who eats 15 so Tammy can eat 15*2 = <<15*2=30>>30 pickle slices
Ron eats 20% fewer pickle slices than Tammy who eats 30 pickle slices so that's .20*30 = <<20*.01*30=6>>6 fewer slices than Tammy
Tammy eats 30 slices and Ron eats 6 less, then Ron eats 30-6 = <<30-6=24>>24 pickle slices
#### 24 |
A permutation of a finite set $S$ is a one-to-one function from $S$ to $S$ . A permutation $P$ of the set $\{ 1, 2, 3, 4, 5 \}$ is called a W-permutation if $P(1) > P(2) < P(3) > P(4) < P(5)$ . A permutation of the set $\{1, 2, 3, 4, 5 \}$ is selected at random. Compute the probability that it is a W-permutation. | 2/15 |
If I have a $5\times 5$ chess board, in how many ways can I place five distinct pawns on the board such that each column and row of the board contains no more than one pawn? | 14400 |
In triangle $DEF$, we have $\angle D = 90^\circ$ and $\sin E = \frac{3}{5}$. Find $\cos F$. | \frac{3}{5} |
The incircle $\omega$ of triangle $ABC$ is tangent to $\overline{BC}$ at $X$. Let $Y \neq X$ be the other intersection of $\overline{AX}$ with $\omega$. Points $P$ and $Q$ lie on $\overline{AB}$ and $\overline{AC}$, respectively, so that $\overline{PQ}$ is tangent to $\omega$ at $Y$. Assume that $AP = 3$, $PB = 4$, $AC = 8$, and $AQ = \dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Diagram
[asy] size(200); import olympiad; defaultpen(linewidth(1)+fontsize(12)); pair A,B,C,P,Q,Wp,X,Y,Z; B=origin; C=(6.75,0); A=IP(CR(B,7),CR(C,8)); path c=incircle(A,B,C); Wp=IP(c,A--C); Z=IP(c,A--B); X=IP(c,B--C); Y=IP(c,A--X); pair I=incenter(A,B,C); P=extension(A,B,Y,Y+dir(90)*(Y-I)); Q=extension(A,C,P,Y); draw(A--B--C--cycle, black+1); draw(c^^A--X^^P--Q); pen p=4+black; dot("$A$",A,N,p); dot("$B$",B,SW,p); dot("$C$",C,SE,p); dot("$X$",X,S,p); dot("$Y$",Y,dir(55),p); dot("$W$",Wp,E,p); dot("$Z$",Z,W,p); dot("$P$",P,W,p); dot("$Q$",Q,E,p); MA("\beta",C,X,A,0.3,black); MA("\alpha",B,A,X,0.7,black); [/asy] | 227 |
What is the slope of a line parallel to the line $2x - 4y = 9$? Express your answer as a common fraction. | \frac{1}{2} |
Paint is to be mixed so that the ratio of red paint to white paint is 3 to 2. If Ben wants to make 30 cans of the mixture and all cans hold the same volume of paint, how many cans of red paint will he need? | 18 |
Taipei 101 in Taiwan is 1,673 feet tall with 101 floors. Suppose the first to 100th floors have height each equal to 16.5 feet, how high is the 101st floor? | The total height of the first to 100th floor is 100 x 16.5 = <<100*16.5=1650>>1650 feet.
Hence, the 101st floor is 1673 - 1650 = <<1673-1650=23>>23 feet high.
#### 23 |
A two-digit integer $AB$ equals $\frac{1}{9}$ of the three-digit integer $CCB$, where $C$ and $B$ represent distinct digits from 1 to 9. What is the smallest possible value of the three-digit integer $CCB$? | 225 |
Let $f$ be the function defined by $f(x)=ax^2-\sqrt{2}$ for some positive $a$. If $f(f(\sqrt{2}))=-\sqrt{2}$ then $a=$ | \frac{\sqrt{2}}{2} |
Rectangle $PQRS$ is inscribed in a semicircle with diameter $\overline{GH}$, such that $PR=20$, and $PG=SH=12$. Determine the area of rectangle $PQRS$.
A) $120\sqrt{6}$
B) $150\sqrt{6}$
C) $160\sqrt{6}$
D) $180\sqrt{6}$
E) $200\sqrt{6}$ | 160\sqrt{6} |
How many different four-digit numbers can be formed by arranging the four digits in 2004? | 6 |
Papi Calot prepared his garden to plant potatoes. He planned to plant 7 rows of 18 plants each. But he still has a bit of room left, so he’s thinking about adding 15 additional potato plants. How many plants does Papi Calot have to buy? | Let’s first calculate the number of plants that Papi had planned: 7 rows * 18 plants/row = <<7*18=126>>126 plants
Now let’s add the other plants: 126 plants + 15 plants = <<126+15=141>>141 plants. So Papi has to buy 141 plants.
#### 141 |
A circle inscribed in triangle \( ABC \) divides median \( BM \) into three equal parts. Find the ratio \( BC: CA: AB \). | 5:10:13 |
Let \( M = 42 \cdot 43 \cdot 75 \cdot 196 \). Find the ratio of the sum of the odd divisors of \( M \) to the sum of the even divisors of \( M \). | \frac{1}{14} |
In the figure shown, segment $AB$ is parallel to segment $YZ$. If $AZ = 42$ units, $BQ = 12$ units, and $QY = 24$ units, what is the length of segment $QZ$? [asy]
import olympiad; import geometry; size(150); defaultpen(linewidth(0.8));
pair Y = (0,0), Z = (16,0), A = (0,8), B = (6,8);
draw(A--B--Y--Z--cycle);
label("$A$",A,W); label("$B$",B,E); label("$Y$",Y,W); label("$Z$",Z,E);
pair Q = intersectionpoint(A--Z,B--Y);
label("$Q$",Q,E);
[/asy] | 28 |
Mary is writing a story, and wants her 60 characters to be split according to their initials. Half of her characters have the initial A, and half of this amount have the initial C. Mary wants the rest of her characters to have the initials D and E, but she wants there to be twice as many characters with the initial D as there are characters with the initial E. How many of Mary’s characters have the initial D? | Half the characters have the initial A, which is a total of 60 / 2 = <<60/2=30>>30 characters.
This means that 30 / 2 = <<30/2=15>>15 characters have the initial C.
Mary wants the remaining 60 – 30 – 15 = <<60-30-15=15>>15 characters to have the initials D or E.
For there to be twice as many characters with the initial D, the characters are divided into 2 + 1 = <<2+1=3>>3 parts.
Dividing by this means there are 15 / 3 = <<15/3=5>>5 characters with the initial E.
This leaves 15 – 5 = <<15-5=10>>10 characters with the initial D.
#### 10 |
We know the following to be true:
$\bullet$ 1. $Z$ and $K$ are integers with $500 < Z < 1000$ and $K > 1;$
$\bullet$ 2. $Z$ = $K \times K^2.$
What is the value of $K$ for which $Z$ is a perfect square? | 9 |
Let $m$ and $n$ be positive integers satisfying the conditions
$\quad\bullet\ \gcd(m+n,210)=1,$
$\quad\bullet\ m^m$ is a multiple of $n^n,$ and
$\quad\bullet\ m$ is not a multiple of $n.$
Find the least possible value of $m+n.$
| 407 |
Vicky has an excellent internet connection. She can download up to 50 MB/second. She has to download a new program to finish a college assignment. The program’s size is 360GB. If the internet connection is at maximum speed, how many hours does she have to wait until the program is fully downloaded? (There are 1000 MB per GB.) | We have to know how many MB are 360GB, 360GB x 1,000MB/1GB = <<360000=360000>>360,000 MB
The total seconds downloading will be 360,000MB ÷ 50MB/second = <<360000/50=7200>>7,200 seconds
So she has to wait 7,200 seconds ÷ 3,600 seconds/1 hour = <<7200/3600=2>>2 hours
#### 2 |
In the Cartesian coordinate system, with the origin as the pole and the positive x-axis as the polar axis, the polar equation of line $l$ is $$ρ\cos(θ+ \frac {π}{4})= \frac { \sqrt {2}}{2}$$, and the parametric equation of curve $C$ is $$\begin{cases} x=5+\cos\theta \\ y=\sin\theta \end{cases}$$, (where $θ$ is the parameter).
(Ⅰ) Find the Cartesian equation of line $l$ and the general equation of curve $C$;
(Ⅱ) Curve $C$ intersects the x-axis at points $A$ and $B$, with $x_A < x_B$, $P$ is a moving point on line $l$, find the minimum perimeter of $\triangle PAB$. | 2+ \sqrt {34} |
Let the triangle $ABC$ have area $1$ . The interior bisectors of the angles $\angle BAC,\angle ABC, \angle BCA$ intersect the sides $(BC), (AC), (AB) $ and the circumscribed circle of the respective triangle $ABC$ at the points $L$ and $G, N$ and $F, Q$ and $E$ . The lines $EF, FG,GE$ intersect the bisectors $(AL), (CQ) ,(BN)$ respectively at points $P, M, R$ . Determine the area of the hexagon $LMNPR$ . | 1/2 |
The sum of two positive numbers is $5$ times their difference. What is the ratio of the larger number to the smaller number? | \frac{3}{2} |
What is the tenth number in the row of Pascal's triangle that has 100 numbers? | \binom{99}{9} |
Mr. Mendez awards extra credit on quizzes to his students with quiz grades that exceed the class mean. Given that 107 students take the same quiz, what is the largest number of students who can be awarded extra credit? | 106 |
If $$\sin\alpha= \frac {4}{7} \sqrt {3}$$ and $$\cos(\alpha+\beta)=- \frac {11}{14}$$, and $\alpha$, $\beta$ are acute angles, then $\beta= \_\_\_\_\_\_$. | \frac {\pi}{3} |
Simplify $(2-3z) - (3+4z)$. | -1-7z |
A deep-sea monster rises from the waters once every hundred years to feast on a ship and sate its hunger. Over three hundred years, it has consumed 847 people. Ships have been built larger over time, so each new ship has twice as many people as the last ship. How many people were on the ship the monster ate in the first hundred years? | Let S be the number of people on the first hundred years’ ship.
The second hundred years’ ship had twice as many as the first, so it had 2S people.
The third hundred years’ ship had twice as many as the second, so it had 2 * 2S = <<2*2=4>>4S people.
All the ships had S + 2S + 4S = 7S = 847 people.
Thus, the ship that the monster ate in the first hundred years had S = 847 / 7 = <<847/7=121>>121 people on it.
#### 121 |
Let $L$ be the number formed by $2022$ digits equal to $1$, that is, $L=1111\dots 111$.
Compute the sum of the digits of the number $9L^2+2L$. | 4044 |
How many orbitals contain one or more electrons in an isolated ground state iron atom (Z = 26)? | 15 |
A rectangle has a perimeter of 64 inches and each side has an integer length. How many non-congruent rectangles meet these criteria? | 16 |
For $\{1, 2, 3, \ldots, 10\}$ and each of its non-empty subsets, a unique alternating sum is defined as follows: Arrange the numbers in the subset in decreasing order and then, beginning with the largest, alternately add and subtract successive numbers. Find the sum of all such alternating sums for $n=10$. | 5120 |
John with his five friends ordered 3 pizzas. Each pizza had 8 slices. If they all finished and ate the same amount of pizzas, how many slices did each person eat? | There were 3 x 8 = <<3*8=24>>24 slices of pizza.
Since there were 6 persons, then each person ate 24/6 = <<24/6=4>>4 slices
#### 4 |
James used a calculator to find the product $0.005 \times 3.24$. He forgot to enter the decimal points, and the calculator showed $1620$. If James had entered the decimal points correctly, what would the answer have been?
A) $0.00162$
B) $0.0162$
C) $0.162$
D) $0.01620$
E) $0.1620$ | 0.0162 |
The novelty shop on the Starship Conundrum sells magazines and chocolates. The cost of four chocolate bars is equal to the cost of 8 magazines. If one magazine costs $1, how much does a dozen chocolate bars cost, in dollars? | If one magazine costs $1, then 8 magazines cost 8*$1=$<<8*1=8>>8.
Let's define "x" as the cost of one chocolate bar.
The cost of four chocolate bars is equal to the cost of 8 magazines and 4*x=$8.
Dividing each side by 4, we get the cost of one chocolate bar x=$2.
Therefore, a dozen chocolate bars would cost 12*$2=$<<12*2=24>>24.
#### 24 |
The quadratic $4x^2 - 40x + 100$ can be written in the form $(ax+b)^2 + c$, where $a$, $b$, and $c$ are constants. What is $2b-3c$? | -20 |
For a rational number $r$ , its *period* is the length of the smallest repeating block in its decimal expansion. for example, the number $r=0.123123123...$ has period $3$ . If $S$ denotes the set of all rational numbers of the form $r=\overline{abcdefgh}$ having period $8$ , find the sum of all elements in $S$ . | 50000000 |
Kat gets a 95% on her first math test and an 80% on her second math test. If she wants an average grade of at least 90% in her math class, what does she need to get on her third final math test? | Let x = the percent needed on her third math test
(95 + 80 + x)/3 = 90
(95 + 80 + x) = 270
x = 270 - 95 - 80
x = <<95=95>>95
#### 95 |
Someone says that 7 times their birth year divided by 13 gives a remainder of 11, and 13 times their birth year divided by 11 gives a remainder of 7. How old will this person be in the year 1954? | 86 |
Bill can buy mags, migs, and mogs for $\$3$, $\$4$, and $\$8$ each, respectively. What is the largest number of mogs he can purchase if he must buy at least one of each item and will spend exactly $\$100$? | 10 |
For how many integer values of $a$ does the equation $$x^2 + ax + 8a = 0$$ have integer solutions for $x$? | 8 |
John takes a pill every 6 hours. How many pills does he take a week? | He takes 24/6=<<24/6=4>>4 pills a day
So he takes 4*7=<<4*7=28>>28 pills a week
#### 28 |
Ali had a stock of 800 books in his Room. He sold 60 on Monday, 10 on Tuesday, 20 on Wednesday, 44 on Thursday and 66 on Friday. How many books were not sold? | We look first for the total number of books that were sold: 60 + 10 + 20 + 44 + 66 = <<60+10+20+44+66=200>>200 books.
So the total number of books that were not sold is: 800 – 200 = <<800-200=600>>600 books.
#### 600 |
Given two lines $l_1: x+3y-3m^2=0$ and $l_2: 2x+y-m^2-5m=0$ intersect at point $P$ ($m \in \mathbb{R}$).
(1) Express the coordinates of the intersection point $P$ of lines $l_1$ and $l_2$ in terms of $m$.
(2) For what value of $m$ is the distance from point $P$ to the line $x+y+3=0$ the shortest? And what is the shortest distance? | \sqrt{2} |
Compute \((1+i^{-100}) + (2+i^{-99}) + (3+i^{-98}) + \cdots + (101+i^0) + (102+i^1) + \cdots + (201+i^{100})\). | 20302 |
Neil baked 20 cookies. He gave 2/5 of the cookies to his friend. How many cookies are left for Neil? | Neil gave 20 x 2/5 = <<20*2/5=8>>8 cookies
So he is left with 20 - 8 = <<20-8=12>>12 cookies.
#### 12 |
If $a \div b = 2$ and $b \div c = \frac{3}{4}$, what is the value of $c \div a$? Express your answer as a common fraction. | \frac{2}{3} |
A gives B as many cents as B has and C as many cents as C has. Similarly, B then gives A and C as many cents as each then has. C, similarly, then gives A and B as many cents as each then has. After this, each person gives half of what they have to each other person. If each finally has 24 cents, calculate the number of cents A starts with. | 24 |
Given complex numbers $w$ and $z$ such that $|w+z|=3$ and $|w^2+z^2|=18,$ find the smallest possible value of $|w^3+z^3|.$ | \frac{81}{2} |
Let $a_1, a_2, \ldots$ be a sequence determined by the rule $a_n = \frac{a_{n-1}}{2}$ if $a_{n-1}$ is even and $a_n = 3a_{n-1} + 1$ if $a_{n-1}$ is odd. For how many positive integers $a_1 \le 3000$ is it true that $a_1$ is less than each of $a_2$, $a_3$, $a_4$, and $a_5$? | 750 |
Butch and Sundance need to get out of Dodge. To travel as quickly as possible, each alternates walking and riding their only horse, Sparky, as follows. Butch begins by walking while Sundance rides. When Sundance reaches the first of the hitching posts that are conveniently located at one-mile intervals along their route, he ties Sparky to the post and begins walking. When Butch reaches Sparky, he rides until he passes Sundance, then leaves Sparky at the next hitching post and resumes walking, and they continue in this manner. Sparky, Butch, and Sundance walk at $6,$ $4,$ and $2.5$ miles per hour, respectively. The first time Butch and Sundance meet at a milepost, they are $n$ miles from Dodge, and they have been traveling for $t$ minutes. Find $n + t$. | 279 |
Let $AC$ and $CE$ be two diagonals of a regular hexagon $ABCDEF$. Points $M$ and $N$ divide $AC$ and $CE$ internally such that $AM:AC = CN:CE = r$. If $B$, $M$, and $N$ are collinear, find $r$. | \frac{1}{\sqrt{3}} |
In the star shown, the sum of the four integers along each straight line is to be the same. Five numbers have been entered. The five missing numbers are 19, 21, 23, 25, and 27. Which number is represented by \( q \)? | 27 |
Given that $x \sim N(-1,36)$ and $P(-3 \leqslant \xi \leqslant -1) = 0.4$, calculate $P(\xi \geqslant 1)$. | 0.1 |
Find the quadratic polynomial $p(x)$ such that $p(-3) = 10,$ $p(0) = 1,$ and $p(2) = 5.$ | x^2 + 1 |
$R$ varies directly as $S$ and inversely as $T$. When $R = \frac{4}{3}$ and $T = \frac{9}{14}$, $S = \frac{3}{7}$. Find $S$ when $R = \sqrt{48}$ and $T = \sqrt{75}$. | 30 |
Tammy has 10 orange trees from which she can pick 12 oranges each day. Tammy sells 6-packs of oranges for $2. How much money will Tammy have earned after 3 weeks if she sells all her oranges? | Tammy picks 10 trees x 12 oranges/tree = <<10*12=120>>120 oranges per day.
Tammy sells 120 oranges/day / 6 oranges/pack = <<120/6=20>>20 6-packs of oranges a day
Tammy sells 20 packs/day x 7 days/week = <<20*7=140>>140 6-packs a week
In 3 weeks Tammy sells 140 packs/week x 3 weeks = <<140*3=420>>420 6 -packs
Tammy makes 420 packs x $2/pack = $<<420*2=840>>840
#### 840 |
Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of 3000 km from $A$ and with $B$ a distance of 3250 km from $A$. To take a bus, it costs Piravena $\$0.15$ per kilometer. To take an airplane, it costs her a $\$100$ booking fee, plus $\$0.10$ per kilometer. [asy]
pair A, B, C;
C=(0,0);
B=(0,1250);
A=(3000,0);
draw(A--B--C--A);
label("A", A, SE);
label("B", B, NW);
label("C", C, SW);
label("3000 km", (A+C)/2, S);
label("3250 km", (A+B)/2, NE);
draw((0,125)--(125,125)--(125,0));
[/asy]
Piravena chose the least expensive way to travel between cities. What was the total cost? | \$1012.50 |
Let $f : \mathbb{R} \to \mathbb{R}$ be a function such that
\[f(xf(y) + x) = xy + f(x)\]for all $x,$ $y.$
Let $n$ be the number of possible values of $f(2),$ and let $s$ be the sum of all possible values of $f(2).$ Find $n \times s.$ | 0 |
A pen and pencil have a total cost of $6. If the pen costs twice as much as the pencil, what is the cost of the pen? | Let x be the cost of the pencil.
If the pen costs 2 times the cost of the pencil, then it costs 2x.
Adding the cost of the pen and pencil we get 2x + x = 3x
Since the total cost is $6 then 3x = $6 therefore x = $6 / 3 = $2
One pen is equal to 2 * x which is 2 * $2 = $4
#### 4 |
How many ways can one color the squares of a $6 \times 6$ grid red and blue such that the number of red squares in each row and column is exactly 2? | 67950 |
Determine how many ordered pairs of positive integers $(x, y)$ where $x < y$, such that the harmonic mean of $x$ and $y$ is equal to $24^{10}$. | 619 |
Find the distance between the vertices of the hyperbola $9x^2 + 54x - y^2 + 10y + 55 = 0.$ | \frac{2}{3} |
If $p(x) = x^4 - 3x + 2$, then find the coefficient of the $x^3$ term in the polynomial $(p(x))^3$. | -27 |
A regular hexagon and an equilateral triangle have equal areas. What is the ratio of the length of a side of the triangle to the length of a side of the hexagon? | \sqrt{6} |
Let $\triangle ABC$ be an acute scalene triangle with circumcircle $\omega$. The tangents to $\omega$ at $B$ and $C$ intersect at $T$. Let $X$ and $Y$ be the projections of $T$ onto lines $AB$ and $AC$, respectively. Suppose $BT = CT = 16$, $BC = 22$, and $TX^2 + TY^2 + XY^2 = 1143$. Find $XY^2$.
| 717 |
LaKeisha is mowing lawns to raise money for a collector set of books. She charges $.10 for every square foot of lawn. The book set costs $150. If she has already mowed three 20 x 15 foot lawns, how many more square feet does she have to mow to earn enough for the book set? | She has mowed 900 square feet because 3 x 20 x 15 = <<3*20*15=900>>900
She has earned $90 because 900 x .1 = <<900*.1=90>>90
She has to earn $60 more dollars because 150 - 90 = <<150-90=60>>60
She has to mow 600 more square feet of lawns because 60 / .1 = <<60/.1=600>>600
#### 600 |
In a six-digit decimal number $\overline{a_{1} a_{2} a_{3} a_{4} a_{5} a_{6}}$, each digit $a_{i}(1 \leqslant i \leqslant 6)$ is an odd number, and the digit 1 is not allowed to appear consecutively (for example, 135131 and 577797 satisfy the conditions, while 311533 does not satisfy the conditions). Find the total number of such six-digit numbers. $\qquad$ . | 13056 |
Find the minimum value of
\[f(x) = x + \frac{1}{x} + \frac{1}{x + \frac{1}{x}}\]for $x > 0.$ | \frac{5}{2} |
Given that $2x + y = 4$ and $x + 2y = 5$, find $5x^2 + 8xy + 5y^2$. | 41 |
For the power of _n_ of a natural number _m_ greater than or equal to 2, the following decomposition formula is given:
2<sup>2</sup> = 1 + 3, 3<sup>2</sup> = 1 + 3 + 5, 4<sup>2</sup> = 1 + 3 + 5 + 7…
2<sup>3</sup> = 3 + 5, 3<sup>3</sup> = 7 + 9 + 11…
2<sup>4</sup> = 7 + 9…
According to this pattern, the third number in the decomposition of 5<sup>4</sup> is ______. | 125 |
Let $S$ be a set of $6$ integers taken from $\{1,2,\dots,12\}$ with the property that if $a$ and $b$ are elements of $S$ with $a<b$, then $b$ is not a multiple of $a$. What is the least possible value of an element in $S$? | 4 |
The angle bisector of the acute angle formed at the origin by the graphs of the lines $y = x$ and $y=3x$ has equation $y=kx.$ What is $k?$ | \frac{1+\sqrt{5}}{2} |
Inside triangle $ABC$, there are 1997 points. Using the vertices $A, B, C$ and these 1997 points, into how many smaller triangles can the original triangle be divided? | 3995 |
Simplify first, then evaluate: $(a-2b)(a^2+2ab+4b^2)-a(a-5b)(a+3b)$, where $a$ and $b$ satisfy $a^2+b^2-2a+4b=-5$. | 120 |
Avery puts 4 shirts in the donation box. He adds twice as many pants as shirts in the box. He throws in half as many shorts as pants in the box. In total, how many pieces of clothes is Avery donating? | The total number of shirts in the donation box is <<4=4>>4.
The total number of pants in the donation box is 4 x 2 = <<4*2=8>>8.
The total number of shorts in the donation box is 8 / 2 = <<8/2=4>>4.
The total number of clothes donated is 4 + 8 + 4 = <<4+8+4=16>>16 pieces.
#### 16 |
A certain tour group checked the weather conditions on the day of the outing. A weather forecasting software predicted that the probability of rain during the time periods $12:00$ to $13:00$ and $13:00$ to $14:00$ on the day of the outing are $0.5$ and $0.4$ respectively. Then, the probability of rain during the time period $12:00$ to $14:00$ on the day of the outing for this tour group is ______. (Provide your answer in numerical form) | 0.7 |
Let $\omega$ be a complex number such that $\omega^7 = 1$ and $\omega \ne 1.$ Compute
\[\omega^{16} + \omega^{18} + \omega^{20} + \dots + \omega^{54}.\] | -1 |
At a hypothetical school, there are three departments in the faculty of sciences: biology, physics and chemistry. Each department has three male and one female professor. A committee of six professors is to be formed containing three men and three women, and each department must be represented by two of its members. Every committee must include at least one woman from the biology department. Find the number of possible committees that can be formed subject to these requirements. | 27 |
If \( y+4=(x-2)^{2} \) and \( x+4=(y-2)^{2} \), and \( x \neq y \), then the value of \( x^{2}+y^{2} \) is: | 15 |
Let $x^2-mx+24$ be a quadratic with roots $x_1$ and $x_2$. If $x_1$ and $x_2$ are integers, how many different values of $m$ are possible? | 8 |
Find the four-digit number that is a perfect square, where the thousands digit is the same as the tens digit, and the hundreds digit is 1 greater than the units digit. | 8281 |
Given that $\sqrt{51.11}\approx 7.149$ and $\sqrt{511.1}\approx 22.608$, determine the value of $\sqrt{511100}$. | 714.9 |
The total cost of staying at High Services Hotel is $40 per night per person. Jenny and two of her friends went to the hotel and stayed for three nights. What's the total amount of money they all paid together? | If each of them stayed 3 nights, the total cost per person is $40/night * 3 nights = $<<40*3=120>>120
Since they were three, the combined cost is $120 + $120 + $120 = $<<120+120+120=360>>360
#### 360 |
If $\log_2 x^2 + \log_{1/2} x = 5,$ compute $x.$ | 32 |
Given the curve $C$: $y^{2}=4x$ with a focus at point $F$, a line $l$ passes through point $F$ and intersects curve $C$ at points $P$ and $Q$. If the relationship $\overrightarrow{FP}+2\overrightarrow{FQ}=\overrightarrow{0}$ holds, calculate the area of triangle $OPQ$. | \frac{3\sqrt{2}}{2} |
Given the function $f(x)$ and its derivative $f''(x)$ on $\mathbb{R}$, and for any real number $x$, it satisfies $f(x)+f(-x)=2x^{2}$, and for $x < 0$, $f''(x)+1 < 2x$, find the minimum value of the real number $a$ such that $f(a+1) \leqslant f(-a)+2a+1$. | -\dfrac {1}{2} |
Let $P_1$ be a regular $r~\mbox{gon}$ and $P_2$ be a regular $s~\mbox{gon}$ $(r\geq s\geq 3)$ such that each interior angle of $P_1$ is $\frac{59}{58}$ as large as each interior angle of $P_2$. What's the largest possible value of $s$?
| 117 |
Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$ . Find the value of $2^{-(1+\log_23)x}$ | 216 |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy: $|\overrightarrow{a}| = |\overrightarrow{b}| = 1$, and $|k\overrightarrow{a} + \overrightarrow{b}| = \sqrt{3}|\overrightarrow{a} - k\overrightarrow{b}| (k > 0)$. Find the maximum value of the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$. | \frac{\pi}{3} |
Let $a,$ $b,$ $c$ be real numbers such that $9a^2 + 4b^2 + 25c^2 = 1.$ Find the maximum value of
\[8a + 3b + 5c.\] | \frac{\sqrt{373}}{6} |
How many one-thirds are in one-sixth? | \frac{1}{2} |
Given that the sequence $\{a_{n}\}$ is an arithmetic sequence with a non-zero common difference, and $a_{1}+a_{10}=a_{9}$, find $\frac{{a}_{1}+{a}_{2}+…+{a}_{9}}{{a}_{10}}$. | \frac{27}{8} |
Lydia has 80 plants. 40% of her plants are flowering plants. Lydia wants to place a fourth of her flowering plants on the porch. If each flowering plant produces 5 flowers, how many flowers are there in total on the porch? | Lydia has 80 * 0.40 = <<80*0.40=32>>32 flowering plants
Lydia places 32 / 4 = <<32/4=8>>8 of the flowering plants on the porch
There are 8 * 5 = <<8*5=40>>40 flowers on the porch
#### 40 |
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