problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Count the number of functions $f: \mathbb{Z} \rightarrow\{$ 'green', 'blue' $\}$ such that $f(x)=f(x+22)$ for all integers $x$ and there does not exist an integer $y$ with $f(y)=f(y+2)=$ 'green'. | 39601 |
Every card in a deck has a picture of one shape - circle, square, or triangle, which is painted in one of the three colors - red, blue, or green. Furthermore, each color is applied in one of three shades - light, medium, or dark. The deck has 27 cards, with every shape-color-shade combination represented. A set of three cards from the deck is called complementary if all of the following statements are true:
i. Either each of the three cards has a different shape or all three of the cards have the same shape.
ii. Either each of the three cards has a different color or all three of the cards have the same color.
iii. Either each of the three cards has a different shade or all three of the cards have the same shade.
How many different complementary three-card sets are there? | 117 |
A certain number is written in the base-12 numeral system. For which divisor \( m \) is the following divisibility rule valid: if the sum of the digits of the number is divisible by \( m \), then the number itself is divisible by \( m \)? | 11 |
Simplify $(x+15)+(100x+15)$. | 101x+30 |
Given that x > 0, y > 0, and x + 2y = 4, find the minimum value of $$\frac {(x+1)(2y+1)}{xy}$$. | \frac {9}{2} |
Find $\left(\frac{1}{2}\right)^{8} \cdot \left(\frac{3}{4}\right)^{-3}$. | \frac{1}{108} |
The Yellers are coached by Coach Loud. The Yellers have 15 players, but three of them, Max, Rex, and Tex, refuse to play together in any combination. How many starting lineups (of 5 players) can Coach Loud make, if the starting lineup can't contain any two of Max, Rex, and Tex together? | 2277 |
The points $(9, -5)$ and $(-3, -1)$ are the endpoints of a diameter of a circle. What is the sum of the coordinates of the center of the circle? | 0 |
Find the value of $x$ such that $\sqrt{x - 2} = 8$. | 66 |
Let the base areas of two cylinders be $S_1$ and $S_2$, and their volumes be $\upsilon_1$ and $\upsilon_2$, respectively. If their lateral areas are equal, and $$\frac {S_{1}}{S_{2}}= \frac {16}{9},$$ then the value of $$\frac {\upsilon_{1}}{\upsilon_{2}}$$ is \_\_\_\_\_\_. | \frac {4}{3} |
How many integers between 1000 and 3000 have all three of the numbers 18, 24, and 36 as factors? | 28 |
A cylinder with a volume of 9 is inscribed in a cone. The plane of the top base of this cylinder cuts off a frustum from the original cone, with a volume of 63. Find the volume of the original cone. | 64 |
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$ , where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$ , $2+2$ , $2+1+1$ , $1+2+1$ , $1+1+2$ , and $1+1+1+1$ . Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd. | \[ 2047 \] |
Teacher Zhang led the students of class 6 (1) to plant trees. The students can be divided into 5 equal groups. It is known that each teacher and student plants the same number of trees, with a total of 527 trees planted. How many students are there in class 6 (1)? | 30 |
The Grammar club has 20 members: 10 boys and 10 girls. A 4-person committee is chosen at random. What is the probability that the committee has at least 1 boy and at least 1 girl? | \dfrac{295}{323} |
In $\triangle ABC$, $A=\frac{\pi}{4}, B=\frac{\pi}{3}, BC=2$.
(I) Find the length of $AC$;
(II) Find the length of $AB$. | 1+ \sqrt{3} |
Hydras consist of heads and necks (each neck connects exactly two heads). With one sword strike, all the necks emanating from a head \( A \) of the hydra can be cut off. However, a new neck immediately grows from head \( A \) to every head that \( A \) was not previously connected to. Hercules wins if he manages to split the hydra into two disconnected parts. Find the smallest \( N \) for which Hercules can defeat any 100-headed hydra by making no more than \( N \) strikes. | 10 |
A bag contains ten balls, some of which are red and the rest of which are yellow. When two balls are drawn at random at the same time, the probability that both balls are red is $\frac{1}{15}$. How many balls in the bag are red? | 3 |
At a school cafeteria, Sam wants to buy a lunch consisting of one main course, one beverage, and one snack. The table below lists Sam's options available in the cafeteria. How many different lunch combinations can Sam choose from?
\begin{tabular}{ |c | c | c | }
\hline \textbf{Main Courses} & \textbf{Beverages} & \textbf{Snacks} \\ \hline
Burger & Water & Apple \\ \hline
Pasta & Soda & Banana \\ \hline
Salad & Juice & \\ \hline
Tacos & & \\ \hline
\end{tabular} | 24 |
What is the smallest positive integer $n$ such that $\frac{1}{n}$ is a terminating decimal and $n$ contains the digit 9? | 4096 |
In how many ways can George choose two out of seven colors to paint his room? | 21 |
Determine the maximum possible value of \[\frac{\left(x^2+5x+12\right)\left(x^2+5x-12\right)\left(x^2-5x+12\right)\left(-x^2+5x+12\right)}{x^4}\] over all non-zero real numbers $x$ .
*2019 CCA Math Bonanza Lightning Round #3.4* | 576 |
Calculate the value of the function $f(x)=3x^{6}-2x^{5}+x^{3}+1$ at $x=2$ using the Horner's method (also known as the Qin Jiushao algorithm) to determine the value of $v_{4}$. | 34 |
Ali and Ernie lined up boxes to make circles. Ali used 8 boxes to make each of his circles and Ernie used 10 for his. If they had 80 boxes to begin with and Ali makes 5 circles, how many circles can Ernie make? | Ali made 5 circles with 8 boxes each so he used 5*8 = <<5*8=40>>40 boxes
There were 80 boxes to start with so now there are 80-40 = <<80-40=40>>40 boxes left
Ernie used 10 boxes to make one circle so with 40 boxes he can make 40/10 = <<40/10=4>>4 circles
#### 4 |
In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, $c$, $\left(a+c\right)\sin A=\sin A+\sin C$, $c^{2}+c=b^{2}-1$. Find:<br/>
$(1)$ $B$;<br/>
$(2)$ Given $D$ is the midpoint of $AC$, $BD=\frac{\sqrt{3}}{2}$, find the area of $\triangle ABC$. | \frac{\sqrt{3}}{2} |
This pattern is made from toothpicks. If the pattern is continued by adding two toothpicks to the previous stage, how many toothpicks are used to create the figure for the $15^{th}$ stage?
[asy]draw((0,0)--(7.5,13)--(-7.5,13)--cycle);
draw((0,0)--(-15,0)--(-7.5,13)--cycle);
label("stage 2",(-4,0),S);
draw((-23,0)--(-30.5,13)--(-38,0)--cycle);
label("stage 1",(-30,0),S);
draw((12,0)--(19.5,13)--(27,0)--cycle);
draw((19.5,13)--(34.5,13)--(27,0)--cycle);
draw((34.5,13)--(27,0)--(42,0)--cycle);
label("stage 3",(27,0),S);
[/asy] | 31 |
Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $a b c=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\sum_{\substack{a b c=2310 \\ a, b, c \in \mathbb{N}}}(a+b+c)$$ where $\mathbb{N}$ denotes the positive integers. | 49140 |
Given the equation $5^{12} = \frac{5^{90/x}}{5^{50/x} \cdot 25^{30/x}}$, find the value of $x$ that satisfies this equation. | -\frac{5}{3} |
Given that acute angles $\alpha$ and $\beta$ satisfy $\sin\alpha=\frac{4}{5}$ and $\cos(\alpha+\beta)=-\frac{12}{13}$, determine the value of $\cos \beta$. | -\frac{16}{65} |
If $a + 4b = 33$ and $6a + 3b = 51$, what is the value of $a + b$? | 12 |
Cassie is trimming her pet's nails. She has four dogs and eight parrots. Each dog has four nails on each foot, and each parrot has three claws on each leg, except for one parrot who has an extra toe. How many nails does Cassie need to cut? | First, find the number of dog nails by multiplying the number of dogs by the number of nails on each leg and the number of legs on each dog: 4 dogs * 4 legs per dog * 4 nails per leg = <<4*4*4=64>>64 nails.
Then do the same for parrots, ignoring the one parrot's extra toe for now: 8 parrots * 2 legs per parrot * 3 nails per leg = <<8*2*3=48>>48 nails
Now add one to the previous number because one parrot has an extra toe: 48 + 1 = <<48+1=49>>49 nails.
Now add the number of nails from the parrots and the dog: 49 nails + 64 nails = <<49+64=113>>113 nails.
#### 113 |
Let $A B C D$ be a quadrilateral with $A=(3,4), B=(9,-40), C=(-5,-12), D=(-7,24)$. Let $P$ be a point in the plane (not necessarily inside the quadrilateral). Find the minimum possible value of $A P+B P+C P+D P$. | 16 \sqrt{17}+8 \sqrt{5} |
In rectangle $ABCD$, angle $C$ is trisected by $\overline{CF}$ and $\overline{CE}$, where $E$ is on $\overline{AB}$, $F$ is on $\overline{AD}$, $BE=8$, and $AF=4$. Find the area of $ABCD$. | 192\sqrt{3}-96 |
From the 10 numbers $0, 1, 2, \cdots, 9$, select 3 such that their sum is an even number not less than 10. How many different ways are there to make such a selection? | 51 |
Given 500 points inside a convex 1000-sided polygon, along with the polygon's vertices (a total of 1500 points), none of which are collinear, the polygon is divided into triangles with these 1500 points as the vertices of the triangles. There are no other vertices apart from these. How many triangles is the convex 1000-sided polygon divided into? | 1998 |
Lily goes to the supermarket. She has $60. She needs to buy 6 items: celery, cereal, bread, milk, potatoes, and coffee. She finds a stalk of celery for $5, cereal for 50% off $12, bread for $8, and milk for 10% off $10. She finds potatoes for $1 each and buys 6. How many dollars does she have left to spend on coffee? | Lily spends $5+$8=$<<5+8=13>>13 on celery and bread.
Lily spends $12*(50/100)=$<<12*(50/100)=6>>6 on cereal.
Lily gets 10% off the milk, so she pays 100% - 10% = 90% of its original price.
Lily spends $10*90%=$<<10*90*.01=9>>9 on milk.
She spends $1*6=$<<1*6=6>>6 on potatoes.
Thus, she has $60-$13-$6-$9-$6=$<<60-13-6-9-6=26>>26 left over for coffee.
#### 26 |
Aziz's parents moved to America in 1982. The year is 2021 and Aziz just celebrated his 36th birthday. How many years had his parents been living in America before Aziz was born? | Aziz was born in the year 2021 - 36 years = <<2021-36=1985>>1985.
Therefore, Aziz's parents had been living in America for 1985 - 1982 = <<1985-1982=3>>3 years.
#### 3 |
The distance between Robin's house and the city center is 500 meters. He leaves the house to go to the city center. After he walks 200 meters he realizes that he forgot his bag. So he returns to his house, then goes back to the city center. How many meters does he walk in total? | He walks 200 meters first then he returns to the house. So it makes 200 m + 200 m = <<200+200=400>>400 m
The city center is 500 meters away from his house so the total distance is 400 m + 500 m = <<500+400=900>>900 m
#### 900 |
Human beings have discovered a habitable planet and soon after, they find 10 more habitable planets. Of these 11, only 5 are deemed ``Earth-like'' in their resources and the rest are deemed ``Mars-like'' since they lack many important resources. Assume that planets like Earth take up 2 units of colonization, while those like Mars take up only 1. If humanity mobilizes 12 total units of colonies, how many different combinations of planets can be occupied if the planets are all different from each other? | 100 |
Triangle $A B C$ has $A B=1, B C=\sqrt{7}$, and $C A=\sqrt{3}$. Let $\ell_{1}$ be the line through $A$ perpendicular to $A B, \ell_{2}$ the line through $B$ perpendicular to $A C$, and $P$ the point of intersection of $\ell_{1}$ and $\ell_{2}$. Find $P C$. | 3 |
A standard die is rolled consecutively two times. Calculate the probability that the face-up numbers are adjacent natural numbers. | \frac{5}{18} |
Let $a$ , $b$ , $c$ , $d$ , $e$ , $f$ and $g$ be seven distinct positive integers not bigger than $7$ . Find all primes which can be expressed as $abcd+efg$ | 179 |
Sarah's age is equal to three times Mark's age minus 4. Mark is four years older than Billy. Billy is half Ana's age. If Ana will be 15 in 3 years, how old is Sarah? | First figure out how old Ana is right now by subtracting 3 from 15: 15 - 3 = <<15-3=12>>12
Next, figure out how old Billy is by dividing Ana's age by 2: 12 / 2 = <<6=6>>6
Now figure out how old Mark is by adding 4 to Billy's age: 6 + 4 = <<6+4=10>>10
Now, to figure out Sarah's age, first multiply Mark's age by 3: 10 * 3 = <<10*3=30>>30
Finally, subtract 4 from triple Mark's age to find Sarah's age: 30 - 4 = <<30-4=26>>26
#### 26 |
Mary used 15 gallons of fuel this week. Last week she used 20% less. How much fuel did she use in total for the two weeks? | Last week, she used 15 gallons * 0.2 = <<15*0.2=3>>3 gallons less fuel than she used this week.
This means that she used 15 gallons – 3 gallons = <<15-3=12>>12 gallons of fuel last week.
In total, Mary used 15 gallons + 12 gallons = <<15+12=27>>27 gallons of fuel.
#### 27 |
John is twice as old as Mary and half as old as Tonya. If Tanya is 60, what is their average age? | John is 30 because 60 / 2 = <<60/2=30>>30
Mary is fifteen because 30 / 2 = <<30/2=15>>15
Their total age is 105 because 60 + 30 + 15 = <<60+30+15=105>>105
Their average age is 35 because 105 / 3 = <<105/3=35>>35
#### 35 |
A biologist sequentially placed 150 beetles into ten jars. In each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is no less than half the number of beetles in the tenth jar. How many beetles are in the sixth jar? | 16 |
In the diagram, \(O\) is the center of a circle with radii \(OA=OB=7\). A quarter circle arc from \(A\) to \(B\) is removed, creating a shaded region. What is the perimeter of the shaded region? | 14 + 10.5\pi |
Ali has a small flower shop. He sold 4 flowers on Monday, 8 flowers on Tuesday and on Friday, he sold double the number of flowers he sold on Monday. How many flowers does Ali sell? | Combining Tuesday and Monday, Ali sold 4 + 8 = <<4+8=12>>12 flowers.
On Friday, he sold 2 * 4 = <<2*4=8>>8 flowers.
Altogether, Ali sold 12 + 8 = <<12+8=20>>20 flowers.
#### 20 |
In trapezoid $ABCD$, the parallel sides $AB$ and $CD$ have lengths of 8 and 20 units, respectively, and the altitude is 12 units. Points $E$ and $F$ are the midpoints of sides $AD$ and $BC$, respectively. What is the area of quadrilateral $EFCD$ in square units? | 102 |
Given the inequality $(|x|-1)^2+(|y|-1)^2<2$, determine the number of lattice points $(x, y)$ that satisfy it. | 16 |
Find the number of 10-tuples $(x_1, x_2, \dots, x_{10})$ of real numbers such that
\[(1 - x_1)^2 + (x_1 - x_2)^2 + (x_2 - x_3)^2 + \dots + (x_9 - x_{10})^2 + x_{10}^2 = \frac{1}{11}.\] | 1 |
What is the first digit (from left to right) of the base $8$ representation of $473_{10}$? | 7 |
Let $x = (2 + \sqrt{3})^{1000},$ let $n = \lfloor x \rfloor,$ and let $f = x - n.$ Find
\[x(1 - f).\] | 1 |
Given that $\binom{23}{3}=1771$, $\binom{23}{4}=8855$, and $\binom{23}{5}=33649$, find $\binom{25}{5}$. | 53130 |
In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively, and
$$
a=5, \quad b=4, \quad \cos(A-B)=\frac{31}{32}.
$$
Find the area of $\triangle ABC$. | \frac{15 \sqrt{7}}{4} |
Given the function $f(x) = 2x^3 - 3ax^2 + 3a - 2$ ($a \in \mathbb{R}$).
$(1)$ If $a=1$, determine the intervals of monotonicity for the function $f(x)$.
$(2)$ If the maximum value of $f(x)$ is $0$, find the value of the real number $a$. | \dfrac{2}{3} |
Martin owns a farm with hens. 10 hens do lay 80 eggs in 10 days. Martin decided to buy 15 more hens. How many eggs will all hens lay in 15 days? | If 10 hens lay 80 eggs, then 1 hen lays 80 / 10 = <<80/10=8>>8 eggs in 10 days.
That means one hen is laying 8 / 2 = <<8/2=4>>4 eggs in 5 days.
So in 15 days, 1 hen will lay 3 times more eggs, so 4 * 3 = 12 eggs.
All hens in total will then lay 25 * 12 = <<25*12=300>>300 eggs in 15 days.
#### 300 |
Mrs. Riley recorded this information from a recent test taken by all of her students. Using the data, what was the average percent score for these $100$ students?
\begin{tabular}{|c|c|}
\multicolumn{2}{c}{}\\\hline
\textbf{$\%$ Score}&\textbf{Number of Students}\\\hline
100&7\\\hline
90&18\\\hline
80&35\\\hline
70&25\\\hline
60&10\\\hline
50&3\\\hline
40&2\\\hline
\end{tabular} | 77 |
If $f(x) = 2$ for all real numbers $x$, what is the value of $f(x + 2)$? | 2 |
Suppose we flip five coins simultaneously: a penny, a nickel, a dime, a quarter, and a half dollar. What is the probability that at least 25 cents worth of coins come up heads? | \frac{3}{4} |
In the numbers from $1$ to $2002$, the number of positive integers that contain exactly one digit $0$ is: | 414 |
In how many ways can you form 5 quartets from 5 violinists, 5 violists, 5 cellists, and 5 pianists? | (5!)^3 |
What will be the length of the strip if a cubic kilometer is cut into cubic meters and laid out in a single line? | 1000000 |
Find the smallest positive integer whose cube ends in $888$. | 192 |
Solve for $r$ in the equation $19-3=2+r$. | 14 |
Find $160\div \left(10+11\cdot 2\right)$. | 5 |
In triangle $ABC$, we have $\angle C = 90^\circ$, $AB = 26$, and $BC = 10$. What is $\sin A$? | \frac{5}{13} |
Billy made 49 sandwiches; Katelyn made 47 more than that. Chloe made a quarter of the amount that Katelyn made. How many sandwiches did they make in all? | Katelyn made 49+47 = <<49+47=96>>96 sandwiches.
Chloe made 96/4 = <<96/4=24>>24 sandwiches.
Billy, Katelyn, and Chloe made 49+96+24 = <<49+96+24=169>>169 sandwiches in all.
#### 169 |
Consider a $4 \times 4$ grid of squares, each of which are originally colored red. Every minute, Piet can jump on one of the squares, changing the color of it and any adjacent squares (two squares are adjacent if they share a side) to blue. What is the minimum number of minutes it will take Piet to change the entire grid to blue? | 4 |
There are 920 deer in a park. 10% of the deer have 8 antlers, and a quarter of that number also have albino fur. How many albino 8-antlered deer are there? | First find how many deer have 8 antlers: 920 deer * 10% = <<920*10*.01=92>>92 deer
Then divide that number by 4 to find how many deer meet both criteria: 92 deer / 4 = <<92/4=23>>23 deer
#### 23 |
Nala found 5 seashells at the beach. The next day, she found another 7, and the following day, she found twice the number she got from the first two days. How many seashells does Nala have? | On the first and second days, Nala found 5 + 7 = <<5+7=12>>12 seashells.
On the third day, she found 12 x 2 = <<12*2=24>>24 seashells.
Altogether, she found 12 + 24 = <<12+24=36>>36 seashells.
#### 36 |
From 6 students, 4 are to be selected to undertake four different tasks labeled A, B, C, and D. If two of the students, named A and B, cannot be assigned to task A, calculate the total number of different assignment plans. | 240 |
Find the number of five-digit positive integers, $n$, that satisfy the following conditions:
(a) the number $n$ is divisible by $5,$
(b) the first and last digits of $n$ are equal, and
(c) the sum of the digits of $n$ is divisible by $5.$ | 200 |
Let $b_1, b_2, \ldots$ be a sequence determined by the rule $b_n= \frac{b_{n-1}}{2}$ if $b_{n-1}$ is even and $b_n=3b_{n-1}+1$ if $b_{n-1}$ is odd. For how many positive integers $b_1 \le 1000$ is it true that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$? | 250 |
Carter has a jar with 20 green M&Ms and 20 red M&Ms. He eats 12 of the green M&Ms, then his sister comes and eats half the red M&Ms and adds 14 yellow M&Ms. If Carter picks an M&M at random now, what is the percentage chance he'll get a green M&M? | First find the final number of green M&Ms by subtracting 12 from the starting number: 20 M&Ms - 12 M&Ms = <<20-12=8>>8 green M&Ms
Then find the final number of red M&Ms by dividing the starting number by half: 20 M&Ms / 2 = <<20/2=10>>10 red M&Ms
Then find the total number of M&Ms at the end by adding the number of red, green and yellow M&Ms: 8 M&Ms + 10 M&Ms + 14 M&Ms = <<8+10+14=32>>32 M&Ms
Then divide the number of green M&Ms by the total number of M&Ms and multiply by 100 to find the percentage chance Carter gets a green M&M: 8 M&Ms / 32 M&Ms * 100 = <<8/32*100=25>>25%
#### 25 |
How many of the 343 smallest positive integers written in base 7 use 4 or 5 (or both) as a digit? | 218 |
Nancy is crafting clay pots to sell. She creates 12 clay pots on Monday, twice as many on Tuesday, a few more on Wednesday, then ends the week with 50 clay pots. How many did she create on Wednesday? | On Tuesday, Nancy creates 12 clay pots * 2 = <<12*2=24>>24 clay plots.
So she created a combined total of 12 + 24 = <<12+24=36>>36 clay pots on Monday and Tuesday.
Subtracting this from the number of clay pots she ended the week with means she must have created 50 – 36 = <<50-36=14>>14 clay pots on Wednesday.
#### 14 |
An isosceles triangle $ABC$, with $AB=AC$, is fixed in the plane. A point $P$ is randomly placed within the triangle. The length of each equal side $AB$ and $AC$ is 6 units and the base $BC$ is 8 units. What is the probability that the area of triangle $PBC$ is more than one-third of the area of triangle $ABC$? | \frac{1}{3} |
Simplify $15 \cdot \frac{7}{10} \cdot \frac{1}{9}$. | \frac{7}{6} |
In a certain country, the airline system is arranged so that each city is connected by airlines to no more than three other cities, and from any city, it's possible to reach any other city with no more than one transfer. What is the maximum number of cities that can exist in this country? | 10 |
Evaluate or simplify:
\\((1)\\dfrac{\\sqrt{1-2\\sin {15}^{\\circ}\\cos {15}^{\\circ}}}{\\cos {15}^{\\circ}-\\sqrt{1-\\cos^2 {165}^{\\circ}}}\\)；
\\((2)\\)Given \\(| \\vec{a} |=4\\), \\(| \\vec{b} |=2\\), and the angle between \\(\\vec{a}\\) and \\(\\vec{b}\\) is \\(\\dfrac{2\\pi }{3}\\), find the value of \\(| \\vec{a} + \\vec{b} |\\). | 2\\sqrt{3} |
A neighbor bought a certain quantity of beef at two shillings a pound, and the same quantity of sausages at eighteenpence a pound. If she had divided the same money equally between beef and sausages, she would have gained two pounds in the total weight. Determine the exact amount of money she spent. | 168 |
Anička received a rectangular cake for her birthday. She cut the cake with two straight cuts. The first cut was made such that it intersected both longer sides of the rectangle at one-third of their length. The second cut was made such that it intersected both shorter sides of the rectangle at one-fifth of their length. Neither cut was parallel to the sides of the rectangle, and at each corner of the rectangle, there were either two shorter segments or two longer segments of the divided sides joined.
Anička ate the piece of cake marked in grey. Determine what portion of the cake this was. | 2/15 |
Find the minimum value of the function
$$
f(x)=x^{2}+(x-2)^{2}+(x-4)^{2}+\ldots+(x-104)^{2}
$$
If the result is a non-integer, round it to the nearest integer. | 49608 |
A batch of disaster relief supplies is loaded into 26 trucks. The trucks travel at a constant speed of \( v \) kilometers per hour directly to the disaster area. If the distance between the two locations is 400 kilometers and the distance between every two trucks must be at least \( \left(\frac{v}{20}\right)^{2} \) kilometers, how many hours will it take to transport all the supplies to the disaster area? | 10 |
A solid rectangular block is created using $N$ congruent 1-cm cubes adhered face-to-face. When observing the block to maximize visibility of its surfaces, exactly $252$ of the 1-cm cubes remain hidden from view. Determine the smallest possible value of $N.$ | 392 |
Given that the function f(x) defined on the set of real numbers ℝ satisfies f(x+1) = 1/2 + √(f(x) - f^2(x)), find the maximum value of f(0) + f(2017). | 1+\frac{\sqrt{2}}{2} |
Four years ago you invested some money at $10\%$ interest. You now have $\$439.23$ in the account. If the interest was compounded yearly, how much did you invest 4 years ago? | 300 |
A square and four circles, each with a radius of 8 inches, are arranged as in the previous problem. What is the area, in square inches, of the square? | 1024 |
For $1 \le n \le 200$, how many integers are there such that $\frac{n}{n+1}$ is a repeating decimal? | 182 |
Janina spends $30 each day for rent and uses $12 worth of supplies daily to run her pancake stand. If she sells each pancake for $2, how many pancakes must Janina sell each day to cover her expenses? | Janina spends $30+$12=$<<30+12=42>>42 each day for her pancake stand.
She must sell $42/$2=<<42/2=21>>21 pancakes each day to cover her expenses.
#### 21 |
Polly and Peter play chess. Polly takes an average of 28 seconds per move, while Peter takes an average of 40 seconds per move. The match ends after 30 moves. How many minutes did the match last? | Both Polly and Peter played 30 / 2 =<<30/2=15>>15 moves each one
With an average of 28 seconds per move, Polly's moves lasted a total of 15 * 28 = <<15*28=420>>420 seconds.
With an average of 40 seconds per move, Peter's moves lasted a total of 15*40=<<15*40=600>>600 seconds.
In total, the match lasted 420 + 600 = <<420+600=1020>>1020 seconds.
In minutes, the match lasted 1020 / 60 = <<1020/60=17>>17 minutes
#### 17 |
Let
\[f(a,b) = \left\{
\renewcommand{\arraystretch}{3}
\begin{array}{cl}
\dfrac{ab - a + 2}{2a} & \text{if $a + b \le 3$}, \\
\dfrac{ab - b - 2}{-2b} & \text{if $a + b > 3$}.
\end{array}
\renewcommand{\arraystretch}{1}
\right.\]Find $f(2,1) + f(2,4).$ | \frac{1}{4} |
There are $2024$ people, which are knights and liars and some of them are friends. Every person is asked for the number of their friends and the answers were $0,1, \ldots, 2023$ . Every knight answered truthfully, while every liar changed the real answer by exactly $1$ . What is the minimal number of liars? | 1012 |
Find the maximum value of the function $y=\frac{x}{{{e}^{x}}}$ on the interval $[0,2]$.
A) When $x=1$, $y=\frac{1}{e}$
B) When $x=2$, $y=\frac{2}{{{e}^{2}}}$
C) When $x=0$, $y=0$
D) When $x=\frac{1}{2}$, $y=\frac{1}{2\sqrt{e}}$ | \frac{1}{e} |
Given the total number of stations is 6 and 3 of them are selected for getting off, calculate the probability that person A and person B get off at different stations. | \frac{2}{3} |
Owen bought 12 boxes of face masks that cost $9 per box. Each box has 50 pieces of masks. He repacked 6 of these boxes and sold them for $5 per 25 pieces. He sold the remaining 300 face masks in baggies at the rate of 10 pieces of mask for $3. How much profit did he make? | Owen bought 12 boxes for a total of $9/box x 12 boxes = $<<9*12=108>>108.
Six boxes contains 6 boxes x 50 masks/box = <<6*50=300>>300 face masks.
Owen made 300 masks / 25 masks/repack = <<300/25=12>>12 repacks.
So, his revenue for selling those repacked face masks is $5 x 12 repacks = $<<5*12=60>>60.
There are 300 masks / 10 masks/baggy = <<300/10=30>>30 baggies face masks.
So, his revenue for that is $3/baggy x 30 baggies = $<<3*30=90>>90.
Hence, his total revenue for all the face masks is $60 + $90 = $<<60+90=150>>150.
This gives him a profit of $150 - $108 = $<<150-108=42>>42.
#### 42 |
Find the sum of the digits of the number $\underbrace{44 \ldots 4}_{2012 \text { times}} \cdot \underbrace{99 \ldots 9}_{2012 \text { times}}$. | 18108 |
Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $ ( x, y ) \in \mathbf{Z}^2$ with positive integers for which:
[list]
[*] only finitely many distinct labels occur, and
[*] for each label $i$, the distance between any two points labeled $i$ is at least $c^i$.
[/list]
[i] | c < \sqrt{2} |
What is the smallest possible perimeter of a triangle whose side lengths are all squares of distinct positive integers? | 77 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.