problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
What is the nearest integer to $(3+\sqrt2)^6$? | 7414 |
In the staircase-shaped region below, all angles that look like right angles are right angles, and each of the eight congruent sides marked with a tick mark have length 1 foot. If the region has area 53 square feet, what is the number of feet in the perimeter of the region? [asy]
size(120);
draw((5,7)--(0,7)--(0,0)--(... | 32 |
Calculate the limit of the function:
$$\lim _{x \rightarrow 0} \frac{e^{4 x}-1}{\sin \left(\pi\left(\frac{x}{2}+1\right)\right)}$$ | -\frac{8}{\pi} |
An infinite geometric series has a first term of $15$ and a second term of $5$. A second infinite geometric series has the same first term of $15$, a second term of $5+n$, and a sum of three times that of the first series. Find the value of $n$. | 6.67 |
In an $8 \times 8$ grid filled with different natural numbers, where each cell contains only one number, if a cell's number is greater than the numbers in at least 6 other cells in its row and greater than the numbers in at least 6 other cells in its column, then this cell is called a "good cell". What is the maximum n... | 16 |
Gary is restocking the grocery produce section. He adds 60 bundles of asparagus at $3.00 each, 40 boxes of grapes at $2.50 each, and 700 apples at $0.50 each. How much is all the produce he stocked worth? | First find the total cost of the asparagus: 60 bundles * $3/bundle = $<<60*3=180>>180
Then find the total cost of the grapes: 40 boxes * $2.50/bundle = $<<40*2.5=100>>100
Then find the total cost of the apples: 700 apples * $0.50/apple = $<<700*0.5=350>>350
Then add the cost of each type of food to find the total cost ... |
The greatest common divisor of 30 and some number between 70 and 80 is 10. What is the number, if the least common multiple of these two numbers is also between 200 and 300? | 80 |
A store purchased a batch of New Year cards at a price of 21 cents each and sold them for a total of 14.57 yuan. If each card is sold at the same price and the selling price does not exceed twice the purchase price, how many cents did the store earn in total? | 470 |
When the fraction $\frac{49}{84}$ is expressed in simplest form, then the sum of the numerator and the denominator will be | 19 |
Akeno spent $2985 to furnish his apartment. Lev spent one-third of that amount on his apartment and Ambrocio spent $177 less than Lev. How many dollars more did Akeno spend than the other 2 people combined? | Lev = (1/3) * 2985 = $<<(1/3)*2985=995>>995
Ambrocio = 995 - 177 = $<<995-177=818>>818
2985 - (995 + 818) = $<<2985-(995+818)=1172>>1172
Akeno spent $1172 more than Lev and Ambrocio combined.
#### 1172 |
Given the sets $A=\{x|x^{2}-px-2=0\}$ and $B=\{x|x^{2}+qx+r=0\}$, if $A\cup B=\{-2,1,5\}$ and $A\cap B=\{-2\}$, find the value of $p+q+r$. | -14 |
The line \(y = M\) intersects the graph of the function \(y = x^{3} - 84x\) at points with abscissas \(a\), \(b\), and \(c\) (\(a < b < c\)). It is given that the distance between \(a\) and \(b\) is half the distance between \(b\) and \(c\). Find \(M\). | 160 |
Let $f(x)$ be a function defined on $\mathbb{R}$ with a period of 2. On the interval $[-1,1)$, $f(x)$ is given by
$$
f(x) = \begin{cases}
x+a & \text{for } -1 \leq x < 0,\\
\left| \frac{2}{5} - x \right| & \text{for } 0 \leq x < 1,
\end{cases}
$$
where $a \in \mathbb{R}$. If $f\left(-\frac{5}{2}\right) = f\left(\frac{9... | -\frac{2}{5} |
A polynomial product of the form
\[(1 - z)^{b_1} (1 - z^2)^{b_2} (1 - z^3)^{b_3} (1 - z^4)^{b_4} (1 - z^5)^{b_5} \dotsm (1 - z^{32})^{b_{32}},\]where the $b_k$ are positive integers, has the surprising property that if we multiply it out and discard all terms involving $z$ to a power larger than 32, what is left is jus... | 2^{27} - 2^{11} |
An ant starts from vertex \( A \) of rectangular prism \( ABCD-A_1B_1C_1D_1 \) and travels along the surface to reach vertex \( C_1 \) with the shortest distance being 6. What is the maximum volume of the rectangular prism? | 12\sqrt{3} |
For certain pairs $(m,n)$ of positive integers with $m\geq n$ there are exactly $50$ distinct positive integers $k$ such that $|\log m - \log k| < \log n$. Find the sum of all possible values of the product $mn$. | 125 |
Expand and simplify the expression $-(4-d)(d+3(4-d))$. What is the sum of the coefficients of the expanded form? | -30 |
Given vector $\vec{b}=(\frac{1}{2}, \frac{\sqrt{3}}{2})$, and $\vec{a}\cdot \vec{b}=\frac{1}{2}$, calculate the projection of vector $\vec{a}$ in the direction of vector $\vec{b}$. | \frac{1}{2} |
The area of this region formed by six congruent squares is 294 square centimeters. What is the perimeter of the region, in centimeters?
[asy]
draw((0,0)--(-10,0)--(-10,10)--(0,10)--cycle);
draw((0,10)--(0,20)--(-30,20)--(-30,10)--cycle);
draw((-10,10)--(-10,20));
draw((-20,10)--(-20,20));
draw((-20,20)--(-20,30)--(-40... | 98 |
A rug is made with three different colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of the two shaded regions is $1$ foot wide on all four sides. What is the length in feet of the inner rectangle? | 2 |
The sum of the heights on the two equal sides of an isosceles triangle is equal to the height on the base. Find the sine of the base angle. | $\frac{\sqrt{15}}{4}$ |
Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with its left and right foci being $F_1$ and $F_2$ respectively, and passing through the point $P(0, \sqrt{5})$, with an eccentricity of $\frac{2}{3}$, and $A$ being a moving point on the line $x=4$.
- (I) Find the equation of the ellipse $C$;
- (... | \sqrt{21} |
How many non-empty subsets $S$ of $\{1,2,3,\ldots,20\}$ have the following three properties?
$(1)$ No two consecutive integers belong to $S$.
$(2)$ If $S$ contains $k$ elements, then $S$ contains no number less than $k$.
$(3)$ $S$ contains at least one number greater than $10$. | 2526 |
If $x$ is a real number and $|x-4|+|x-3|<a$ where $a>0$, then: | a > 1 |
Consider a game involving two standard decks of 52 cards each mixed together, making a total of 104 cards. Each deck has 13 ranks and 4 suits, with the suits retaining their colors as in a standard deck. What is the probability that the top card of this combined deck is the King of $\diamondsuit$? | \frac{1}{52} |
Parallelepiped $ABCDEFGH$ is generated by vectors $\overrightarrow{AB},$ $\overrightarrow{AD},$ and $\overrightarrow{AE},$ as shown below.
[asy]
import three;
size(220);
currentprojection = orthographic(0.5,0.3,0.2);
triple I = (1,0,0), J = (0,1,0), K = (0,0,1), O = (0,0,0);
triple V = (-1,0.2,0.5), W = (0,3,0.7), U... | 4 |
Given the function $f(x) = x^3 - 3x - 1$, if for any $x_1$, $x_2$ in the interval $[-3,2]$, it holds that $|f(x_1) - f(x_2)| \leq t$, then the minimum value of the real number $t$ is ______. | 20 |
A rectangular floor that is $10$ feet wide and $17$ feet long is tiled with $170$ one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and the last tile, how many tiles does the bug visit? | 26 |
BoatWorks built 3 canoes in January of this year and then each subsequent calendar month they built twice the number of canoes they had built the previous month. How many total canoes were built by BoatWorks by the end of March of this year? | 21 |
On the first day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{2}$, meaning that if the length of the object was originally $x,$ then it is now $x + \frac{1}{2} x.$ On the second day he increased the object's longer length by $\frac{1}{3}$; on the third day he increased the object'... | 198 |
Emily just purchased 2 pairs of curtains for $30.00 each and 9 wall prints at $15.00 each. The store also offers an installation service. For $50.00 they will come to your house and professionally hang your curtains and prints. If Emily agrees to this service, how much will her entire order cost? | She buys 2 pair of curtains for $30.00 each so that's 2*30 = $<<2*30=60.00>>60.00
She buys 9 wall prints at $15.00 each so that's 9*15 = $<<9*15=135.00>>135.00
The curtains cost $60.00, the prints are $135.00 and the installation service is $50.00 for a total of 60+135+50 = $<<60+135+50=245.00>>245.00
#### 245 |
Let \( p, q, r, s, t, u, v, w \) be real numbers such that \( pqrs = 16 \) and \( tuvw = 25 \). Find the minimum value of
\[ (pt)^2 + (qu)^2 + (rv)^2 + (sw)^2. \] | 400 |
Given a positive sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, if both $\{a_n\}$ and $\{\sqrt{S_n}\}$ are arithmetic sequences with the same common difference, calculate $S_{100}$. | 2500 |
Out of 8 shots, 3 hit the target. The total number of ways in which exactly 2 hits are consecutive is: | 30 |
The function f is defined recursively by f(1)=f(2)=1 and f(n)=f(n-1)-f(n-2)+n for all integers n ≥ 3. Find the value of f(2018). | 2017 |
If $\log_6 (4x)=2$, find $\log_x 27$. Express your answer in simplest fractional form. | \frac32 |
Three distinct segments are chosen at random among the segments whose end-points are the vertices of a regular 12-gon. What is the probability that the lengths of these three segments are the three side lengths of a triangle with positive area? | \frac{223}{286} |
A rectangular prism with integer edge lengths is painted red on its entire surface and then cut into smaller cubes with edge length 1. Among these smaller cubes, 40 cubes have two red faces and 66 cubes have one red face. What is the volume of this rectangular prism? | 150 |
Let $P$ be a $2019-$ gon, such that no three of its diagonals concur at an internal point. We will call each internal intersection point of diagonals of $P$ a knot. What is the greatest number of knots one can choose, such that there doesn't exist a cycle of chosen knots? ( Every two adjacent knots in a cycle must... | 2018 |
Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How m... | 23 |
In the right triangle \(ABC\) with an acute angle of \(30^\circ\), an altitude \(CD\) is drawn from the right angle vertex \(C\). Find the distance between the centers of the inscribed circles of triangles \(ACD\) and \(BCD\), if the shorter leg of triangle \(ABC\) is 1. | \frac{\sqrt{3}-1}{\sqrt{2}} |
Given a positive real number \(\alpha\), determine the greatest real number \(C\) such that the inequality
$$
\left(1+\frac{\alpha}{x^{2}}\right)\left(1+\frac{\alpha}{y^{2}}\right)\left(1+\frac{\alpha}{z^{2}}\right) \geq C \cdot\left(\frac{x}{z}+\frac{z}{x}+2\right)
$$
holds for all positive real numbers \(x, y\), an... | 16 |
Big boxes contain 7 dolls each. Small boxes contain 4 dolls each. There are 5 big boxes and 9 small boxes. How many dolls are there in total? | There are 7*5=<<7*5=35>>35 dolls in the big boxes.
There are 4*9=<<4*9=36>>36 dolls in the small boxes.
There are 35+36=<<35+36=71>>71 dolls altogether.
#### 71 |
The energy stored by any pair of positive charges is inversely proportional to the distance between them, and directly proportional to their charges. Three identical point charges start at the vertices of an equilateral triangle, and this configuration stores 15 Joules of energy. How much more energy, in Joules, would ... | 10 |
Emmy has a collection of 14 iPods. She loses 6 out of the 14 she had but she still has twice as many as Rosa. How many iPods does Emmy and Rosa have together? | Emmy has 14-6 = <<14-6=8>>8 iPods left.
Rosa has 8/2 = <<8/2=4>>4 iPods.
Emmy and Rosa have 8+4 = <<8+4=12>>12 iPods together.
#### 12 |
The value of $x$ that satisfies $\log_{2^x} 3^{20} = \log_{2^{x+3}} 3^{2020}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. | 103 |
When the number $2^{1000}$ is divided by $13$, the remainder in the division is | 3 |
Out of 1200 people polled, $30\%$ do not like radio, and $10\%$ of the people who do not like radio also do not like music. How many people polled do not like both radio and music? | 36 |
Sonia and Joss are moving to their new house at the lake. They have too much stuff in their previous house and decide to make multiple trips to move it all to the new house. They spend 15 minutes filling the car with their stuff and spend 30 minutes driving from the previous house to the new house. Fortunately, they di... | They spend 15 minutes/trip + 30 minutes/trip = <<15+30=45>>45 minutes/trip filling the car and driving to the new house.
They had to take 6 trips, so they spend 45 minutes/trip x 6/trip = <<45*6=270>>270 minutes.
They also had to drive back to their previous house 5 times to complete the move: 30 minutes/trip x 5 trips... |
Rhombus $PQRS^{}_{}$ is inscribed in rectangle $ABCD^{}_{}$ so that vertices $P^{}_{}$, $Q^{}_{}$, $R^{}_{}$, and $S^{}_{}$ are interior points on sides $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$, respectively. It is given that $PB^{}_{}=15$, $BQ^{}_{}=20$, $PR^{}_{}=30$, and $QS^{}_{}=40$. ... | 677 |
Nedy can eat 8 packs of crackers from Monday to Thursday. If Nedy ate twice the amount on Friday, how many crackers did Nedy eat in all? | Nedy can eat 8 x 4 = <<8*4=32>>32 crackers from Monday to Thursday.
He can eat 8 x 2 = <<8*2=16>>16 packs of crackers on Friday.
Therefore, Nedy ate 32 + 16 = <<32+16=48>>48 crackers in all.
#### 48 |
The real numbers $x, y, z, w$ satisfy $$\begin{aligned} & 2 x+y+z+w=1 \\ & x+3 y+z+w=2 \\ & x+y+4 z+w=3 \\ & x+y+z+5 w=25 \end{aligned}$$ Find the value of $w$. | 11/2 |
In tetrahedron \(ABCD\), \(AB = 1\), \(BC = 5\), \(CD = 7\), \(DA = 5\), \(AC = 5\), \(BD = 2\sqrt{6}\). Find the distance between skew lines \(AC\) and \(BD\). | \frac{3\sqrt{11}}{10} |
The equation \( x y z 1 = 4 \) can be rewritten as \( x y z = 4 \). | 48 |
Moor has $2016$ white rabbit candies. He and his $n$ friends split the candies equally amongst themselves, and they find that they each have an integer number of candies. Given that $n$ is a positive integer (Moor has at least $1$ friend), how many possible values of $n$ exist? | 35 |
$O$ is the center of square $A B C D$, and $M$ and $N$ are the midpoints of $\overline{B C}$ and $\overline{A D}$, respectively. Points $A^{\prime}, B^{\prime}, C^{\prime}, D^{\prime}$ are chosen on $\overline{A O}, \overline{B O}, \overline{C O}, \overline{D O}$, respectively, so that $A^{\prime} B^{\prime} M C^{\prim... | 8634 |
Let \( a < b < c < d < e \) be real numbers. We calculate all the possible sums of two distinct numbers among these five numbers. The three smallest sums are 32, 36, and 37, and the two largest sums are 48 and 51. Find all possible values of \( e \). | \frac{55}{2} |
Mitch made 20 macarons. Joshua made 6 more macarons than Mitch but half as many macarons as Miles. Renz made 1 fewer than 3/4 as many macarons as Miles. If they will combine their macarons to give them to the kids on the street, how many kids would receive 2 macarons each? | Joshua made 20 + 6 = <<20+6=26>>26 macarons.
Miles made 26 x 2 = <<26*2=52>>52 macarons.
Three-fourths of Miles' macarons is 52 x 3/4 = <<52*3/4=39>>39.
So Renz made 39 - 1 = <<39-1=38>>38 macarons.
The four of them have a total of 20 + 26 + 52 + 38 = <<20+26+52+38=136>>136 macarons combined.
Thus, 136/2 = <<136/2=68>>... |
Consider a string of $n$ $7$s, $7777\cdots77,$ into which $+$ signs are inserted to produce an arithmetic expression. How many values of $n$ are possible if the inserted $+$ signs create a sum of $7350$ using groups of $7$s, $77$s, $777$s, and possibly $7777$s? | 117 |
Among all triangles $ABC,$ find the maximum value of $\cos A + \cos B \cos C.$ | \frac{1}{\sqrt{2}} |
Given a set $I=\{1,2,3,4,5\}$, select two non-empty subsets $A$ and $B$ such that the largest number in set $A$ is less than the smallest number in set $B$. The total number of different selection methods is $\_\_\_\_\_\_$. | 49 |
The point \( N \) is the center of the face \( ABCD \) of the cube \( ABCDEFGH \). Also, \( M \) is the midpoint of the edge \( AE \). If the area of \(\triangle MNH\) is \( 13 \sqrt{14} \), what is the edge length of the cube? | 2\sqrt{13} |
When $7$ fair standard $6$-sided dice are thrown, the probability that the sum of the numbers on the top faces is $10$ can be written as $\frac{n}{6^{7}}$, where $n$ is a positive integer. What is $n$? | 84 |
A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ ... | 4489 |
A regular polygon of $m$ sides is exactly enclosed (no overlaps, no gaps) by $m$ regular polygons of $n$ sides each. (Shown here for $m=4, n=8$.) If $m=10$, what is the value of $n$?
[asy] size(200); defaultpen(linewidth(0.8)); draw(unitsquare); path p=(0,1)--(1,1)--(1+sqrt(2)/2,1+sqrt(2)/2)--(1+sqrt(2)/2,2+sqrt(2)/2)... | 5 |
Compute $10^{-1}\pmod{1001}$. Express your answer as a residue from $0$ to $1000$, inclusive. | 901 |
What is the value of the following expression: $\frac{1}{3}-\frac{1}{9}+\frac{1}{27}-\frac{1}{81}+\frac{1}{243}$? Express your answer as a common fraction. | \frac{61}{243} |
Carl drove continuously from 7:30 a.m. until 2:15 p.m. of the same day and covered a distance of 234 miles. What was his average speed in miles per hour? | \frac{936}{27} |
$A B C D$ is a regular tetrahedron of volume 1. Maria glues regular tetrahedra $A^{\prime} B C D, A B^{\prime} C D$, $A B C^{\prime} D$, and $A B C D^{\prime}$ to the faces of $A B C D$. What is the volume of the tetrahedron $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$? | \frac{125}{27} |
Two identical rulers are placed together. Each ruler is exactly 10 cm long and is marked in centimeters from 0 to 10. The 3 cm mark on each ruler is aligned with the 4 cm mark on the other. The overall length is \( L \) cm. What is the value of \( L \)? | 13 |
On a $3 \times 3$ chessboard, each square contains a Chinese knight with $\frac{1}{2}$ probability. What is the probability that there are two Chinese knights that can attack each other? (In Chinese chess, a Chinese knight can attack any piece which is two squares away from it in a particular direction and one square a... | \frac{79}{256} |
The product $N$ of three positive integers is $6$ times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of $N$. | 336 |
Alex is at the candy store buying jellybeans. He wants to buy at least 100 jellybeans. He wants to buy the least amount such that he would have exactly $11$ leftover after dividing the jellybeans evenly among $13$ people. How many jellybeans should Alex buy? | 102 |
Six congruent circles form a ring with each circle externally tangent to the two circles adjacent to it. All six circles are internally tangent to a circle $\cal C$ with radius 30. Let $K$ be the area of the region inside $\cal C$ and outside all of the six circles in the ring. Find $\lfloor K\rfloor$. (The notation... | 942 |
Let $f(x)=2 x^{3}-2 x$. For what positive values of $a$ do there exist distinct $b, c, d$ such that $(a, f(a))$, $(b, f(b)),(c, f(c)),(d, f(d))$ is a rectangle? | $\left[\frac{\sqrt{3}}{3}, 1\right]$ |
Find the largest value of $c$ such that $-2$ is in the range of $f(x)=x^2+3x+c$. | \frac{1}{4} |
The town is having a race to see who can run around the town square 7 times the fastest. The town square is 3/4 of a mile long. The winner finishes the race in 42 minutes. Last year's winner finished in 47.25 minutes. How many minutes on average faster did this year's winner run one mile of the race compared to last ye... | The race is 5.25 miles long because 7 x .75 = <<7*.75=5.25>>5.25
They ran a mile in 8 minutes because 42 / 5.25 = <<42/5.25=8>>8
Last year's winner ran a mile in 9 minutes because 47.25 / 5.25 = <<47.25/5.25=9>>9
They ran 1 minute faster per mile compared to last year's winner.
#### 1 |
It takes Nissa 10 seconds to clip each of her cats' nails. 90 seconds to clean each of her ears, and 5 minutes to shampoo her. If the cat has four claws on each foot, how many seconds does grooming her cat take total? | First find the total number of claws the cat has: 4 claws/foot * 4 feet = <<4*4=16>>16 claws
Then find the total time Nissa spends cutting the claws: 16 claws * 10 seconds/claw = <<16*10=160>>160 seconds
Then find the total time she spends cleaning the cats' ears: 90 seconds/ear * 2 ears = 180 seconds
Then find how lon... |
In Pascal's Triangle, each number is the sum of the number just above it and to the left and the number just above it and to the right. So the middle number in Row 2 is $2$ because $1+1=2.$ What is the sum of the numbers in Row 8 of Pascal's Triangle?
\begin{tabular}{rccccccccccc}
Row 0:& & & & & & 1\\\noalign{\smalls... | 256 |
Find the non-zero value of $c$ for which there is exactly one positive value of $b$ for which there is one solution to the equation $x^2 + \left(b + \frac 1b\right)x + c = 0$. | 1 |
Adam earns $40 daily in his job. 10% of his money is deducted as taxes. How much money will Adam have earned after taxes after 30 days of work? | The amount of money deducted from Adam's daily pay is $40 / 10 = $<<40/10=4>>4.
So after deducting 10%, Adam’s daily pay is $40 – $4 = $<<40-4=36>>36.
This means that in 30 days he earns $36 * 30 = $<<36*30=1080>>1080.
#### 1,080 |
A driver travels 30 miles per hour for 3 hours and 25 miles per hour for 4 hours to deliver goods to a town every day from Monday to Saturday. How many miles does the driver travel in a week? | The driver travels (3 hours * 30 mph) + (25 mph * 4 hours) = <<(3*30)+(25*4)=190>>190 miles per day
From Monday to Saturday he travels in total 190 miles/day * 6 days = <<190*6=1140>>1,140 miles in a week
#### 1140 |
Given an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) with its left focus at F and the eccentricity $e = \frac{\sqrt{2}}{2}$, the line segment cut by the ellipse from the line passing through F and perpendicular to the x-axis has length $\sqrt{2}$.
(Ⅰ) Find the equation of the ellipse.
(Ⅱ) A line... | \frac{3}{2} |
Michael has never taken a foreign language class, but is doing a story on them for the school newspaper. The school offers French and Spanish. Michael has a list of all 25 kids in the school enrolled in at least one foreign language class. He also knows that 18 kids are in the French class and 21 kids are in the Span... | \frac{91}{100} |
The expression \(\frac{3}{10}+\frac{3}{100}+\frac{3}{1000}\) can be simplified by converting each fraction to a decimal, and then calculating the sum. | 0.333 |
A stacking of circles in the plane consists of a base, or some number of unit circles centered on the $x$-axis in a row without overlap or gaps, and circles above the $x$-axis that must be tangent to two circles below them (so that if the ends of the base were secured and gravity were applied from below, then nothing w... | 14 |
The diameter, in inches, of a sphere with twice the volume of a sphere of radius 9 inches can be expressed in the form $a\sqrt[3]{b}$ where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$. | 20 |
If $x$ varies as the cube of $y$, and $y$ varies as the fifth root of $z$, then $x$ varies as the nth power of $z$, where n is: | \frac{3}{5} |
There are eight points on a circle that divide the circumference equally. Count the number of acute-angled triangles or obtuse-angled triangles that can be formed with these division points as vertices. | 32 |
If $x=2$ and $y=3$, express the value of the following as a common fraction: $$
\frac
{~\frac{1}{y}~}
{\frac{1}{x}}
$$ | \frac{2}{3} |
Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<f(100)<5000(k+1)$ for some integer $k$. What is $k$? | 3 |
The value of $\frac{1}{16}a^0+\left (\frac{1}{16a} \right )^0- \left (64^{-\frac{1}{2}} \right )- (-32)^{-\frac{4}{5}}$ is: | 1 |
Given the function $f(x)= \begin{cases} kx^{2}+2x-1, & x\in (0,1] \\ kx+1, & x\in (1,+\infty) \end{cases}$ has two distinct zeros $x_{1}$ and $x_{2}$, then the maximum value of $\dfrac {1}{x_{1}}+ \dfrac {1}{x_{2}}$ is ______. | \dfrac {9}{4} |
Two years ago Pete was three times as old as his cousin Claire. Two years before that, Pete was four times as old as Claire. In how many years will the ratio of their ages be $2$ : $1$ ? | 4 |
Billy's age is twice Joe's age and the sum of their ages is 45. How old is Billy? | 30 |
Find the smallest natural number such that when multiplied by 9, the resulting number consists of the same digits but in some different order. | 1089 |
Given the function $f(x) = \frac{1}{2}x^2 - 2ax + b\ln(x) + 2a^2$ achieves an extremum of $\frac{1}{2}$ at $x = 1$, find the value of $a+b$. | -1 |
Find the sum of all integers $k$ such that $\binom{23}{4} + \binom{23}{5} = \binom{24}{k}$. | 24 |
How many triangles can be formed using the vertices of a regular pentadecagon (a 15-sided polygon), if no side of the triangle can be a side of the pentadecagon? | 440 |
A sports lottery stipulates that 7 numbers are drawn from a total of 36 numbers, ranging from 01 to 36, for a single entry, which costs 2 yuan. A person wants to select the lucky number 18 first, then choose 3 consecutive numbers from 01 to 17, 2 consecutive numbers from 19 to 29, and 1 number from 30 to 36 to form an ... | 2100 |
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