problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Determine the probability that a 4 × 4 square grid becomes a single uniform color (all white or all black) after rotation. | \frac{1}{32768} |
Given that \(\triangle ABC\) has sides \(a\), \(b\), and \(c\) corresponding to angles \(A\), \(B\), and \(C\) respectively, and knowing that \(a + b + c = 16\), find the value of \(b^2 \cos^2 \frac{C}{2} + c^2 \cos^2 \frac{B}{2} + 2bc \cos \frac{B}{2} \cos \frac{C}{2} \sin \frac{A}{2}\). | 64 |
The diagram below shows a $4\times4$ rectangular array of points, each of which is $1$ unit away from its nearest neighbors.
[asy] unitsize(0.25inch); defaultpen(linewidth(0.7)); int i, j; for(i = 0; i < 4; ++i) for(j = 0; j < 4; ++j) dot(((real)i, (real)j)); [/asy]
Define a growing path to be a sequence of distinc... | 240 |
The expressions \[A=1 \times 2 + 3 \times 4 + 5 \times 6 + \cdots + 37 \times 38 + 39\]and \[B = 1 + 2 \times 3 + 4 \times 5 + \cdots + 36 \times 37 + 38 \times 39\]are obtained by writing multiplication and addition operators in an alternating pattern between successive integers. Find the positive difference between i... | 722 |
George bought some food for his trip: a bottle of juice, a sandwich, and a bottle of milk. The sandwich was for $4, and the juice was two times more expensive. The bottle of milk cost was 75% of the total cost of the sandwich and juice. How much did George pay for his food? | The juice was two times more expensive than the sandwich, so it was 4 * 2 = $<<2*4=8>>8.
The juice and the sandwich in total were a cost of 4 + 8 = $<<4+8=12>>12.
So the cost of one bottle of milk was 75/100 * 12 = $<<75/100*12=9>>9.
In total for all the food, George paid 12 + 9 = $<<12+9=21>>21.
#### 21 |
Given sets \( A = \{1, 2, 3\} \) and \( B = \{4, 5, 6, 7\} \), with a mapping \( f: A \rightarrow B \), if for \( x \in A \), the expression \( x + f(x) + x \cdot f(x) \) is odd, then the number of such mappings \( f \) is: | 32 |
Given the function $f(x) = x^2 + (a+8)x + a^2 + a - 12$ ($a < 0$), and $f(a^2 - 4) = f(2a - 8)$, determine the minimum value of $\frac {f(n)-4a}{n+1}$ for $n \in \mathbb{N}^{+}$. | \frac {37}{4} |
The numbers 1, 2, ..., 2016 are written on a board. You are allowed to erase any two numbers and write their arithmetic mean instead. How must you proceed so that the number 1000 remains on the board? | 1000 |
What is $\frac{1}{(-7^{3})^{3}}\cdot(-7)^{10}$? | -7 |
A supermarket has 2355 loaves of bread at the start of the day. By afternoon, 629 loaves are sold, and a further 489 loaves are delivered to the supermarket by their supplier in the evening. How many loaves of bread are there at the end of the day? | In the afternoon there are 2355 - 629 = <<2355-629=1726>>1726 loaves of bread left.
By the evening there is a delivery and there are now 1726 + 489 = <<1726+489=2215>>2215 loaves of bread.
#### 2215 |
In trapezoid \(ABCD\), \(\overrightarrow{AB} = 2 \overrightarrow{DC}\), \(|\overrightarrow{BC}| = 6\). Point \(P\) is a point in the plane of trapezoid \(ABCD\) and satisfies \(\overrightarrow{AP} + \overrightarrow{BP} + 4 \overrightarrow{DP} = 0\). Additionally, \(\overrightarrow{DA} \cdot \overrightarrow{CB} = |\over... | \frac{4 \sqrt{2}}{3} |
Suppose that $f(x)$ and $g(x)$ are functions which satisfy $f(g(x)) = x^2$ and $g(f(x)) = x^3$ for all $x \ge 1.$ If $g(16) = 16,$ then compute $[g(4)]^3.$ | 16 |
Compute: $\left(\frac{1}{2} \right)^{3} \cdot \left(\frac{1}{7} \right)$. | \frac{1}{56} |
Evaluate the expression \(\dfrac{\sqrt[4]{7}}{\sqrt[6]{7}}\). | 7^{\frac{1}{12}} |
A circle with center $O$ has radius $8$ units and circle $P$ has radius $2$ units. The circles are externally tangent to each other at point $Q$. Segment $TS$ is the common external tangent to circle $O$ and circle $P$ at points $T$ and $S$, respectively. What is the length of segment $OS$? Express your answer in s... | 8\sqrt{2} |
It took an alligator named Paul 4 hours to walk from his home at the River Nile to the Nile Delta. On the return journey, Paul traveled with six other alligators, the journey taking 2 more hours than Paul took to walk to the Nile Delta, to reach their home at the River Nile. What's the combined time the alligators wal... | If on the return journey to its home, the alligator took 2 more hours than it took to travel to the Nile Delta, Paul took 4+2=<<4+2=6>>6 hours to travel back to its home.
The total time Paul took to travel from its home to the Nile Delta and back is 6+4=<<6+4=10>>10
On his return journey, Paul traveled with six other a... |
When someone is at home, the probability of a phone call being answered at the first ring is 0.1, at the second ring is 0.2, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability that the phone call is answered within the first four rings. | 0.8 |
The sequence $(a_{n})$ is defined by the following relations: $a_{1}=1$, $a_{2}=3$, $a_{n}=a_{n-1}-a_{n-2}+n$ (for $n \geq 3$). Find $a_{1000}$. | 1002 |
The bowl of fruit contains apples, pears, and bananas. There are two more pears than apples, and three more bananas than pears. If the bowl contains 19 pieces of fruit, how many bananas does it contain? | Let A be the number of apples.
There are A+2 pears.
There are (A+2)+3 pears.
Then A+(A+2)+((A+2)+3)=19.
3A+7=19
3A=12
A=4
There are 4 apples.
There are (4+2)+3=<<4+2+3=9>>9 bananas.
#### 9 |
Katie's mother wants to get granola bars for all the kids to eat after Katie's soccer game. There will be 30 kids playing soccer, including Katie. Katie's mother wants to get 2 granola bars for each kid, since the bars are not that big and she figures the kids will be very hungry. Each box of granola bars has 12 bars i... | There will be 30 kids playing soccer and 2 granola bars for each kid, 30 x 2 = <<30*2=60>>60 granola bars total.
Each box of granola bars has 12 in it, so Katie's mother needs 60 granola bars / 12 bars each box = <<60/12=5>>5 boxes.
#### 5 |
In a factory that employed 852 people, 25% more workers have just been hired. How many employees are there now in the factory? | A quarter of the workers equals 852 * 25 / 100 = <<852*25/100=213>>213 workers.
There are now 852 + 213 = <<852+213=1065>>1065 employees in the factory.
#### 1065 |
What is the remainder when $6x^3-15x^2+21x-23$ is divided by $3x-6$? | 7 |
Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$ , while $BD$ and $CE$ meet at $Q$ . Find the area of $APQD$ . | 1/2 |
Let $ABCD$ be a trapezoid with $AB \parallel CD$. The bisectors of $\angle CDA$ and $\angle DAB$ meet at $E$, the bisectors of $\angle ABC$ and $\angle BCD$ meet at $F$, the bisectors of $\angle BCD$ and $\angle CDA$ meet at $G$, and the bisectors of $\angle DAB$ and $\angle ABC$ meet at $H$. Quadrilaterals $EABF$ and ... | \frac{256}{7} |
Given a polynomial $(a+b+c+d+e+1)^N$, when expanded and like terms are combined, the expression contains exactly 2002 terms that include all five variables $a, b, c, d, e$, each to some positive power, determine the value of $N$. | 17 |
The brothers found a treasure of gold and silver. They divided it so that each got 100 kg. The eldest got the most gold - 25 kg - and one-eighth of all the silver. How much gold was in the treasure? | 100 |
If $a$ is a constant such that $4x^2 - 12x + a$ is the square of a binomial, then what is $a$? | 9 |
It costs $.10 to print one page. Jenny wants to print 7 copies of her 25-page essay. She also wants to buy 7 pens that each cost $1.50. If she pays the store with 2 twenty dollar bills, how much money should she get in change? | Jenny needs to print 7*25= <<7*25=175>>175 pages.
This will cost 175*$.10 =$<<175*.10=17.5>>17.5.
The pens will cost 7*$1.50=$<<7*1.5=10.50>>10.50.
In total, she will spend $17.5+$10.5=$<<17.5+10.5=28>>28.
Since she pays with 2 twenty dollar bills she pays with 2*20=<<2*20=40>>40 dollars.
Thus, she will get $40-$28=$<<... |
Below is a portion of the graph of a function, $y=h(x)$:
[asy]
import graph; size(8cm); real lsf=0.5; pen dps=linewidth(0.7)+fontsize(10); defaultpen(dps); pen ds=black; real xmin=-4.25,xmax=4.25,ymin=-7.25,ymax=6.25;
pen cqcqcq=rgb(0.75,0.75,0.75);
/*grid*/ pen gs=linewidth(0.7)+cqcqcq+linetype("2 2"); real gx=1,gy... | 4 |
Form a five-digit number without repeating digits using the numbers 0, 1, 2, 3, 4, where exactly one even number is sandwiched between two odd numbers. How many such five-digit numbers are there? | 28 |
A rectangular grid consists of 5 rows and 6 columns with equal square blocks. How many different squares can be traced using the lines in the grid? | 70 |
In triangle \(ABC\), point \(P\) is located one-third of the way along segment \(AB\) closer to point \(A\). Point \(R\) is one-third of the way along segment \(PB\) closer to point \(P\), and point \(Q\) lies on segment \(BC\) such that angles \(PCB\) and \(RQB\) are congruent.
Determine the ratio of the areas of tri... | 9:2 |
The points \( M \) and \( N \) are chosen on the angle bisector \( AL \) of a triangle \( ABC \) such that \( \angle ABM = \angle ACN = 23^\circ \). \( X \) is a point inside the triangle such that \( BX = CX \) and \( \angle BXC = 2\angle BML \). Find \( \angle MXN \). | 46 |
The first six rows of Pascal's triangle are shown below, beginning with row zero. Except for the $1$ at each end, row $4$ consists of only even numbers, as does row $2.$ How many of the first $20$ rows have this property? (Don't include row $0$ or row $1$). \begin{tabular}{ccccccccccc}
&&&&&1&&&&&\\
&&&&1&&1&&&&\\
&&&1... | 4 |
Five food companies sponsored a local food bank. Foster Farms donated 45 dressed chickens; American Summits donated twice the number of bottled water than the number of dressed chicken donated by Foster Farms; Hormel donated three times the number of dressed chickens that Foster Farms donated; Boudin Butchers donated o... | American Summits donated 45 x 2 = <<45*2=90>>90 bottled waters.
Hormel donated 45 x 3 = <<45*3=135>>135 spams.
Boudin Bakery donated 135 x 1/3 = <<135*1/3=45>>45 sourdoughs.
Del Monte Foods donated 90 - 30 = <<90-30=60>>60 canned fruits.
Therefore, a total of 45 + 90 + 135 + 45 + 60 = <<45+90+135+45+60=375>>375 differe... |
Given that the math test scores, X, follow a normal distribution with a mean of 100 and a standard deviation of δ, and the number of students scoring between 80 and 120 points accounted for about 3/4 of the total 1600 students, calculate the number of students who scored no less than 80 points in this final exam. | 1400 |
Determine the smallest constant $n$, such that for any positive real numbers $x$, $y$, and $z$,
\[\sqrt{\frac{x}{y + 2z}} + \sqrt{\frac{y}{2x + z}} + \sqrt{\frac{z}{x + 2y}} > n.\] | \sqrt{3} |
The diagram shows 28 lattice points, each one unit from its nearest neighbors. Segment $AB$ meets segment $CD$ at $E$. Find the length of segment $AE$.
[asy]
unitsize(0.8cm);
for (int i=0; i<7; ++i) {
for (int j=0; j<4; ++j) {
dot((i,j));
};}
label("$A$",(0,3),W);
label("$B$",(6,0),E);
label("$D$",(2,0),S);
label("$E$... | \frac{5\sqrt{5}}{3} |
Seth lost 17.5 pounds. Jerome lost three times that many pounds and Veronica lost 1.5 pounds more than Seth. How many pounds did the 3 people lose in total? | Seth = <<17.5=17.5>>17.5 pounds
Jerome = 3 * 17.5 = <<3*17.5=52.5>>52.5 pounds
Veronica = 17.5 + 1.5 = <<17.5+1.5=19>>19 pounds
Total = 17.5 + 52.5 + 19 = <<17.5+52.5+19=89>>89 pounds
The 3 people lost a total of 89 pounds.
#### 89 |
Let $A B C$ be a triangle with $A B=13, B C=14$, and $C A=15$. We construct isosceles right triangle $A C D$ with $\angle A D C=90^{\circ}$, where $D, B$ are on the same side of line $A C$, and let lines $A D$ and $C B$ meet at $F$. Similarly, we construct isosceles right triangle $B C E$ with $\angle B E C=90^{\circ}$... | -\frac{5}{13} |
Tim used to run 3 times a week but decided to add an extra 2 days a week. She runs 1 hour in the morning and 1 in the evening every day she runs. How many hours a week does she run now? | He now runs 3+2=<<3+2=5>>5 times a week
Each day he runs for 1+1=<<1+1=2>>2 hours
So he runs 2*5=<<2*5=10>>10 hours a week
#### 10 |
Given 40 feet of fencing, what is the greatest possible number of square feet in the area of a rectangular pen enclosed by the fencing? | 100 |
Given the function $f\left( x \right)=\sin\left( 2x+\varphi \right)+\sqrt{3}\cos\left( 2x+\varphi \right)$ $\left( 0 < \varphi < \pi \right)$, its graph is shifted left by $\frac{\pi }{4}$ units, and the shifted graph is symmetric about the point $\left( \frac{\pi }{2},0 \right)$. Find the minimum value of the function... | \frac{1}{2} |
Edmund is buying a new computer and needs to save up $75 before he has enough. He convinces his parents to pay him for extra chores. He normally has to do 12 chores a week. His parents agree to pay him $2 for every extra chore he does during the week. If he does 4 chores a day for two weeks, how much does he earn? | In two weeks, he has to do 24 chores because 2 x 12 = <<2*12=24>>24
He now does 28 chores a week because 7 x 4 = <<7*4=28>>28
He does 56 chores in two weeks because 2 x 28 = <<2*28=56>>56
He has done 32 extra chores because 56 - 24 = <<56-24=32>>32
He earns $64 because 2 x 32 = <<2*32=64>>64
#### 64 |
Solve for $p$: $\frac 56 = \frac n{72} = \frac {m+n}{84}= \frac {p - m}{120}$. | 110 |
In a room containing $N$ people, $N > 3$, at least one person has not shaken hands with everyone else in the room.
What is the maximum number of people in the room that could have shaken hands with everyone else? | $N-1$ |
Given that the coordinate of one focus of the ellipse $3x^{2} + ky^{2} = 1$ is $(0, 1)$, determine its eccentricity. | \frac{\sqrt{2}}{2} |
Add $81.76$ to $34.587$ and round your answer to the nearest tenth. | 116.3 |
For certain real values of $p, q, r,$ and $s,$ the equation $x^4+px^3+qx^2+rx+s=0$ has four non-real roots. The product of two of these roots is $17 + 2i$ and the sum of the other two roots is $2 + 5i,$ where $i^2 = -1.$ Find $q.$ | 63 |
A particle moves so that it is at $(2t + 7, 4t - 13)$ at time $t.$ Find the speed of the particle, measured in unit of distance per unit of time. | 2 \sqrt{5} |
In a stationery store, there are three kinds of pencils. A pencil with an eraser, which costs $0.8 each, a regular pencil for $0.5 each, and a short pencil for $0.4 each. This store was able to sell 200 pencils with an eraser, 40 regular pencils, and 35 short pencils. How much money did the store make from these sales? | For pencils with erasers, the store received a total of 200 * 0.8 = $<<200*0.8=160>>160.
For regular pencils, the store received 40 * 0.5 = $<<40*0.5=20>>20.
And for short pencils only 35 * 0.4 = $<<35*0.4=14>>14.
In total, the store made 160 + 20 + 14 = $<<160+20+14=194>>194 from selling these pencils.
#### 194 |
Compute the following sum:
\[
\frac{1}{2^{2024}} \sum_{n = 0}^{1011} (-3)^n \binom{2024}{2n}.
\] | -\frac{1}{2} |
Leo and Ryan together have $48. Ryan owns 2/3 of the amount. Leo remembered that Ryan owed him $10 but he also owed Ryan $7. After the debts had been settled, how much money does Leo have? | Ryan owns $48 x 2/3 = $<<48*2/3=32>>32 of the $48.
While Leo owns $48 - $32 = $<<48-32=16>>16 of the $48.
Leo had $16 + $10 = $<<16+10=26>>26 after Ryan paid him $10.
But since he owed Ryan $7, then Leo has $26 - $7 = $<<26-7=19>>19.
#### 19 |
Find $y$ if $\dfrac{y^2 - 9y + 8}{y-1} + \dfrac{3y^2 +16y-12 }{3y -2} = -3$. | -\frac{1}{2} |
Let $M$ be the number of positive integers that are less than or equal to $2050$ and whose base-$2$ representation has more $1$'s than $0$'s. Find the remainder when $M$ is divided by $1000$. | 374 |
There are two concentric spheres of radii 3 units and 6 units. What is the volume, in cubic units, of the region within the larger sphere and not within the smaller sphere? Express your answer in terms of $\pi$. | 252\pi |
Find the smallest integer $k \geq 2$ such that for every partition of the set $\{2, 3,\hdots, k\}$ into two parts, at least one of these parts contains (not necessarily distinct) numbers $a$, $b$ and $c$ with $ab = c$. | 32 |
In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $c=a\cos B+b\sin A$.
(I) Find angle $A$.
(II) If $a=2$, find the maximum area of $\triangle ABC$. | \sqrt {2}+1 |
Given that the area of $\triangle ABC$ is $S$, and $\overrightarrow{AB} \cdot \overrightarrow{AC} = S$.
(1) Find the values of $\sin A$, $\cos A$, and $\tan 2A$.
(2) If $B = \frac{\pi}{4}, \; |\overrightarrow{CA} - \overrightarrow{CB}| = 6$, find the area $S$ of $\triangle ABC$. | 12 |
The vertex of the parabola described by the equation $y=-3x^2-30x-81$ is $(m,n)$. What is $n$? | -6 |
If the line $y=mx+1$ intersects the ellipse $x^2+4y^2=1$ exactly once, then the value of $m^2$ is | \frac{3}{4} |
The students in Mr. Neatkin's class took a penmanship test. Two-thirds of the boys and $\frac{3}{4}$ of the girls passed the test, and an equal number of boys and girls passed the test. What is the minimum possible number of students in the class? | 17 |
A rectangle with dimensions 100 cm by 150 cm is tilted so that one corner is 20 cm above a horizontal line, as shown. To the nearest centimetre, the height of vertex $Z$ above the horizontal line is $(100+x) \mathrm{cm}$. What is the value of $x$? | 67 |
Find an $n$ such that $n!-(n-1)!+(n-2)!-(n-3)!+\cdots \pm 1$ ! is prime. Be prepared to justify your answer for $\left\{\begin{array}{c}n, \\ {\left[\frac{n+225}{10}\right],}\end{array} n \leq 25\right.$ points, where $[N]$ is the greatest integer less than $N$. | 3, 4, 5, 6, 7, 8, 10, 15, 19, 41, 59, 61, 105, 160 |
Find the number of pairs of integers $(x, y)$ such that $x^{2}+2y^{2}<25$. | 55 |
A row consists of 10 chairs, but chair #5 is broken and cannot be used. Mary and James each sit in one of the available chairs, choosing their seats at random from the remaining chairs. What is the probability that they don't sit next to each other? | \frac{5}{6} |
Suppose $f(x)$ is a rational function such that $3f\left(\frac{1}{x}\right) + \frac{2f(x)}{x} = x^{2}$ for $x \neq 0$. Find $f(-2)$. | \frac{67}{20} |
Find the ratio of the volume of the cone to the volume of the cylinder. Express your answer as a common fraction.
[asy]
import solids; size(150); import three; defaultpen(linewidth(0.8)); currentprojection = orthographic(5,0,3);
revolution c = cylinder((0,0,0), 1, 3);
revolution c2 = cone((0,0,0), 1,1.5);
draw(c,black... | \frac{1}{6} |
Given $\sin (30^{\circ}+\alpha)= \frac {3}{5}$, and $60^{\circ} < \alpha < 150^{\circ}$, solve for the value of $\cos \alpha$. | \frac{3-4\sqrt{3}}{10} |
A class has 32 students and they all need to present their projects. Every period is 40 minutes long. How many periods will it take for every student to present their project if they have 5 minutes to do so? | All the students will present their projects in 32 * 5 = <<32*5=160>>160 minutes.
It will take 160 / 40 = <<160/40=4>>4 periods for all the students to present their projects.
#### 4 |
Find the smallest six-digit number that is divisible by 3, 7, and 13 without a remainder. | 100191 |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $a\cos(B-C)+a\cos A=2\sqrt{3}b\sin C\cos A$.
$(1)$ Find angle $A$;
$(2)$ If the perimeter of $\triangle ABC$ is $8$ and the radius of the circumcircle is $\sqrt{3}$, find the area of $\triangle AB... | \frac{4\sqrt{3}}{3} |
A circle is tangent to both branches of the hyperbola $x^{2}-20y^{2}=24$ as well as the $x$-axis. Compute the area of this circle. | 504\pi |
If $\log_5 (x+4)=3$, find $\log_{11} x$. | 2 |
Kennedy grew tomatoes from three plants she had in her backyard. The first tomato plant produced two dozen tomatoes. The second plant produced 5 more than half as many tomatoes as the first plant. And the third plant produced two more tomatoes than the second plant. How many tomatoes did the three plants produce? | Half as many as the first plant is 24/2=<<24/2=12>>12 tomatoes.
But the second plant produced 5 more than half as many tomatoes as the first plant., or a total of 12+5=<<12+5=17>>17 tomatoes.
And the third plant produced two more tomatoes than the second plant, or 17+2=<<17+2=19>>19 tomatoes.
In total, the three plants... |
James buys 20 pounds of beef and half that much pork. He uses 1.5 pounds of meat to make meals at his restaurant. Each meal sells for $20. How much money did he make? | He bought 20/2=<<20/2=10>>10 pounds of pork
So he has 20+10=<<20+10=30>>30 pounds of meat
That means he can make 30/1.5=<<30/1.5=20>>20 meals
So he makes 20*20=$<<20*20=400>>400
#### 400 |
Jennifer bought twice as many candies as Emily and three times as many as Bob bought. If Emily bought 6 candies, how many candies did Bob buy? | Jennifer bought twice as many as Emily (who bought 6 candies) which is 6*2 = <<6*2=12>>12 candies
12 candies (that Jennifer bought) is three times as much as what Bob bought which means 3*what Bob bought = 12
Dividing both sides of the equation by 3 gives: Number of candies Bob bought = 12/3 = <<12/3=4>>4 candies
#### ... |
My club has 25 members. In how many ways can I choose members to form a 4-person executive committee? | 12,\!650 |
Given that the asymptotes of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ and the axis of the parabola $x^{2} = 4y$ form a triangle with an area of $2$, calculate the eccentricity of the hyperbola. | \frac{\sqrt{5}}{2} |
Let $a_1 = \sqrt 7$ and $b_i = \lfloor a_i \rfloor$ , $a_{i+1} = \dfrac{1}{b_i - \lfloor b_i \rfloor}$ for each $i\geq i$ . What is the smallest integer $n$ greater than $2004$ such that $b_n$ is divisible by $4$ ? ( $\lfloor x \rfloor$ denotes the largest integer less than or equal to $x$ ) | 2005 |
Let $N=\overline{5 A B 37 C 2}$, where $A, B, C$ are digits between 0 and 9, inclusive, and $N$ is a 7-digit positive integer. If $N$ is divisible by 792, determine all possible ordered triples $(A, B, C)$. | $(0,5,5),(4,5,1),(6,4,9)$ |
The midpoint of a line segment is located at $(1, -2)$. If one of the endpoints is $(4, 5)$, what is the other endpoint? Express your answer as an ordered pair. | (-2,-9) |
Find the largest real number \( p \) such that all three roots of the equation below are positive integers:
\[
5x^{3} - 5(p+1)x^{2} + (71p-1)x + 1 = 66p .
\] | 76 |
Calculate the areas of the regions bounded by the curves given in polar coordinates.
$$
r=\cos 2 \phi
$$ | \frac{\pi}{2} |
Given four points $O,\ A,\ B,\ C$ on a plane such that $OA=4,\ OB=3,\ OC=2,\ \overrightarrow{OB}\cdot \overrightarrow{OC}=3.$ Find the maximum area of $\triangle{ABC}$ . | 2\sqrt{7} + \frac{3\sqrt{3}}{2} |
A child spends their time drawing pictures of Native Americans (referred to as "Indians") and Eskimos. Each drawing depicts either a Native American with a teepee or an Eskimo with an igloo. However, the child sometimes makes mistakes and draws a Native American with an igloo.
A psychologist noticed the following:
1.... | 7/8 |
Given that children enter at a discounted rate, half that of an adult ticket, and the total cost for $6$ adult tickets and $5$ child tickets amounts to $32.50$, calculate the total cost for $10$ adult tickets and $8$ child tickets. | 53.50 |
A product of five primes is of the form $A B C, A B C$, where $A, B$, and $C$ represent digits. If one of the primes is 491, find the product $A B C, A B C$. | 982,982 |
Let \( O \) be the origin, \( A_1, A_2, A_3, \ldots \) be points on the curve \( y = \sqrt{x} \) and \( B_1, B_2, B_3, \ldots \) be points on the positive \( x \)-axis such that the triangles \( O B_1 A_1, B_1 B_2 A_2, B_2 B_3 A_3, \ldots \) are all equilateral, with side lengths \( l_1, l_2, l_3, \ldots \) respectivel... | 4022030/3 |
How many integers between 100 and 300 have both 11 and 8 as factors? | 2 |
If the price of a stamp is 33 cents, what is the maximum number of stamps that could be purchased with $\$32$? | 96 |
A Ferris wheel can accommodate 70 people in 20 minutes. If the Ferris wheel is open from 1:00 pm until 7:00 pm, how many people will get to ride? | The Ferris wheel will be open for 7:00 - 1:00 = 6 hours.
In one hour the wheel can ride 60 minutes / 20 minutes = <<60/20=3>>3 times the number of riders.
The total per hour riders is 3 * 70 people = <<3*70=210>>210 people.
If the ride is open 6 hours * 210 people = <<6*210=1260>>1260 people can ride.
#### 1260 |
Lola and Dora combined their allowance of $9 each to buy a deck of playing cards for $10, they also bought $2 boxes of stickers and split the boxes evenly. How many packs of stickers did Dora get? | Lola and Dora together have $9/person x 2 people = $<<9*2=18>>18
After buying the deck of cards they had $18 - $10 = $<<18-10=8>>8
With 8 dollars they bought $8 / $2/pack = <<8/2=4>>4 packs of stickers
Dora received 4 packs / 2 people = <<4/2=2>>2 packs of stickers
#### 2 |
Let $n$ be the number of pairs of values of $b$ and $c$ such that $3x+by+c=0$ and $cx-2y+12=0$ have the same graph. Then $n$ is: | 2 |
A rectangular prism has dimensions 10 inches by 3 inches by 30 inches. If a cube has the same volume as this prism, what is the surface area of the cube, in square inches? | 6 \times 900^{2/3} |
What is the sum of the squares of the lengths of the medians of a triangle whose side lengths are $13, 13,$ and $10$? | 432 |
A zoo is arranging the layout of its animal enclosures. There are 4 tiger enclosures in a row and behind each of these are 2 zebra enclosures. There are three times as many giraffe enclosures as zebra enclosures scattered around the zoo. The tiger enclosures hold 4 tigers, the zebra enclosures hold 10 zebras and the gi... | There are twice as many zebra enclosures, so there must be 4 tiger enclosures * 2 = <<4*2=8>>8 zebra enclosures.
This means there are 8 zebra enclosures * 3 = <<8*3=24>>24 giraffe enclosures.
In the tiger enclosures, there are 4 tiger enclosures * 4 = <<4*4=16>>16 tigers.
In the zebra enclosures, there are 8 zebra encl... |
A club has 15 members and needs to choose 2 members to be co-presidents. In how many ways can the club choose its co-presidents? | 105 |
Three baskets A, B and C contain different numbers of differently colored marbles. Basket A contains 4 red marbles and 2 yellow marbles. Basket B contains 6 green marbles and 1 yellow marble. Basket C contains 3 white marbles and 9 yellow marbles. What is the difference between the number of each color of marble in the... | In basket A the number difference between differently colored marbles is 4-2 = <<4-2=2>>2
In basket B the difference is 6-1 = <<6-1=5>>5
In basket C the difference is 9-3 = <<9-3=6>>6
Comparing 2, 5 and 6 shows that 6 is the greatest difference
#### 6 |
For how many values of $k$ is $60^{10}$ the least common multiple of the positive integers $10^{10}$, $12^{12}$, and $k$? | 231 |
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