problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Select 4 out of 7 different books to distribute to 4 students, one book per student, with the restriction that books A and B cannot be given to student C, and calculate the number of different distribution methods. | 600 |
In a sign pyramid a cell gets a "+" if the two cells below it have the same sign, and it gets a "-" if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a "+" at the top of the pyramid? | 8 |
Find the largest integer less than 74 that leaves a remainder of 3 when divided by 7. | 73 |
How much money does Roman give Dale if Roman wins a contest with a prize of $\$ 200$, gives $30 \%$ of the prize to Jackie, and then splits $15 \%$ of what remains equally between Dale and Natalia? | \$ 10.50 |
Find the smallest four-digit number that is equal to the square of the sum of the numbers formed by its first two digits and its last two digits. | 2025 |
Given that line $l_{1}$ passes through points $A(m,1)$ and $B(-3,4)$, and line $l_{2}$ passes through points $C(1,m)$ and $D(-1,m+1)$, for what value of $m$ do the lines $l_{1}$ and $l_{2}$
$(1)$ intersect at a right angle;
$(2)$ are parallel;
$(3)$ the angle of inclination of $l_{1}$ is $45^{\circ}$ | -6 |
Two real numbers are selected independently at random from the interval $[-20, 10]$. What is the probability that the product of those numbers is greater than zero? | \frac{5}{9} |
A running track is the ring formed by two concentric circles. If the circumferences of the two circles differ by $10\pi $ feet, how wide is the track in feet?
[asy]size(100); path g=scale(2)*unitcircle;
filldraw(unitcircle^^g,evenodd+grey,black);
[/asy] | 5 |
We call a pair $(a,b)$ of positive integers, $a<391$ , *pupusa* if $$ \textup{lcm}(a,b)>\textup{lcm}(a,391) $$ Find the minimum value of $b$ across all *pupusa* pairs.
Fun Fact: OMCC 2017 was held in El Salvador. *Pupusa* is their national dish. It is a corn tortilla filled with cheese, meat, etc. | 18 |
How many distinct terms are in the expansion of \[(a+b+c+d)(e+f+g+h+i)\] assuming that terms involving the product of $a$ and $e$, and $b$ and $f$ are identical and combine into a single term? | 19 |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% |
A polynomial $P$ of degree 2015 satisfies the equation $P(n)=\frac{1}{n^{2}}$ for $n=1,2, \ldots, 2016$. Find \lfloor 2017 P(2017)\rfloor. | -9 |
Let the odd function $f(x)$ defined on $\mathbb{R}$ satisfy $f(x+3) = -f(1-x)$. If $f(3) = 2$, then $f(2013) = \ $. | -2 |
Given that 70% of the light bulbs are produced by Factory A with a pass rate of 95%, and 30% are produced by Factory B with a pass rate of 80%, calculate the probability of buying a qualified light bulb produced by Factory A from the market. | 0.665 |
Suppose that $x, y, z$ are real numbers such that $x=y+z+2$, $y=z+x+1$, and $z=x+y+4$. Compute $x+y+z$. | -7 |
In $\triangle A B C, A B=2019, B C=2020$, and $C A=2021$. Yannick draws three regular $n$-gons in the plane of $\triangle A B C$ so that each $n$-gon shares a side with a distinct side of $\triangle A B C$ and no two of the $n$-gons overlap. What is the maximum possible value of $n$? | 11 |
For a positive number $x$, define $f(x)=\frac{2x}{x+1}$. Calculate: $f(\frac{1}{101})+f(\frac{1}{100})+f(\frac{1}{99})+\ldots +f(\frac{1}{3})+f(\frac{1}{2})+f(1)+f(2)+f(3)+\ldots +f(99)+f(100)+f(101)$. | 201 |
Let $z$ be a complex number satisfying $z^2 = 4z - 19 + 8i$. Given that $|z|$ is an integer, find $z.$ | 3 + 4i |
Real numbers \(a, b, c\) satisfy the following system of equations:
\[ \left\{ \begin{array}{l} a^{2}+a b+b^{2}=11 \\ b^{2}+b c+c^{2}=11 \end{array} \right. \]
(a) What is the minimum value that the expression \(c^{2}+c a+a^{2}\) can take?
(b) What is the maximum value that the expression \(c^{2}+c a+a^{2}\) can take? | 44 |
Given the quadratic function $f(x)=ax^{2}+bx+c$, where $a$, $b$, and $c$ are constants, if the solution set of the inequality $f(x) \geqslant 2ax+b$ is $\mathbb{R}$, find the maximum value of $\frac{b^{2}}{a^{2}+c^{2}}$. | 2\sqrt{2}-2 |
A box is 8 inches in height, 10 inches in width, and 12 inches in length. A wooden building block is 3 inches in height, 2 inches in width, and 4 inches in length. How many building blocks can fit into the box? | The volume of the box is 8 x 10 x 12 = <<8*10*12=960>>960 cu in.
The volume of a wooden block is 3 x 2 x 4 = <<3*2*4=24>>24 cu in.
960/24 = <<960/24=40>>40 wooden blocks can fit into the box.
#### 40 |
A third of all the cats in the village have spots. A quarter of the spotted cats in the village are fluffy. If there are 120 cats in the village, how many of the cats are spotted and fluffy? | If a third of the cats have spots and a quarter of those are fluffy (1 / 3) x (1 / 4) = 1/ 12 cats are both fluffy and spotted.
Out of 120 cats, 120 cats / 12 = <<120/12=10>>10 cats are spotted and fluffy.
#### 10 |
What is the inverse of $f(x)=4-5x$? | \frac{4-x}{5} |
Solve for $x$: $0.05x - 0.09(25 - x) = 5.4$. | 54.6428571 |
Solve for $y$: $4+2.3y = 1.7y - 20$ | -40 |
Let the sequence $\{a_n\}$ have a sum of the first $n$ terms denoted as $S_n$, and define $T_n=\frac{S_1+S_2+\cdots +S_n}{n}$ as the "ideal number" of the sequence $a_1,a_2,\cdots,a_n$. It is known that the "ideal number" of the sequence $a_1,a_2,\cdots,a_{504}$ is 2020. Determine the "ideal number" of the sequence $2,a_1,a_2,\cdots,a_{504}$. | 2018 |
The divisors of a natural number \( n \) (including \( n \) and 1) which has more than three divisors, are written in ascending order: \( 1 = d_{1} < d_{2} < \ldots < d_{k} = n \). The differences \( u_{1} = d_{2} - d_{1}, u_{2} = d_{3} - d_{2}, \ldots, u_{k-1} = d_{k} - d_{k-1} \) are such that \( u_{2} - u_{1} = u_{3} - u_{2} = \ldots = u_{k-1} - u_{k-2} \). Find all such \( n \). | 10 |
Determine the largest prime factor of the sum \(\sum_{k=1}^{11} k^{5}\). | 263 |
Given an arithmetic-geometric sequence {$a_n$} with the first term as $\frac{4}{3}$ and a common ratio of $- \frac{1}{3}$. The sum of its first n terms is represented by $S_n$. If $A ≤ S_{n} - \frac{1}{S_{n}} ≤ B$ holds true for any n∈N*, find the minimum value of B - A. | \frac{59}{72} |
The World Cup football tournament is held in Brazil, and the host team Brazil is in group A. In the group stage, the team plays a total of 3 matches. The rules stipulate that winning one match scores 3 points, drawing one match scores 1 point, and losing one match scores 0 points. If the probability of Brazil winning, drawing, or losing each match is 0.5, 0.3, and 0.2 respectively, then the probability that the team scores no less than 6 points is______. | 0.5 |
A triangular region is bounded by the two coordinate axes and the line given by the equation $2x + y = 6$. What is the area of the region, in square units? | 9 |
Compute
\[\frac{\lfloor \sqrt[4]{1} \rfloor \cdot \lfloor \sqrt[4]{3} \rfloor \cdot \lfloor \sqrt[4]{5} \rfloor \dotsm \lfloor \sqrt[4]{2015} \rfloor}{\lfloor \sqrt[4]{2} \rfloor \cdot \lfloor \sqrt[4]{4} \rfloor \cdot \lfloor \sqrt[4]{6} \rfloor \dotsm \lfloor \sqrt[4]{2016} \rfloor}.\] | \frac{5}{16} |
Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$ , for integers $i,j$ with $0\leq i,j\leq n$ , such that:
$\bullet$ for all $0\leq i,j\leq n$ , the set $S_{i,j}$ has $i+j$ elements; and
$\bullet$ $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$ . | \[
(2n)! \cdot 2^{n^2}
\] |
What is the sum of the digits of the decimal representation of $2^{2005} \times 5^{2007} \times 3$? | 12 |
Find the number of four-digit numbers with distinct digits, formed using the digits 0, 1, 2, ..., 9, such that the absolute difference between the units and hundreds digit is 8. | 154 |
Selena and Josh were running in a race. Together they ran 36 miles. Josh ran half of the distance that Selena ran. How many miles did Selena run? | Let J = Josh's distance
2J = Selena's distance
J + 2J = 36
3J = <<36=36>>36 miles
J = <<12=12>>12
2J = <<24=24>>24 miles
Selena ran 24 miles.
#### 24 |
Suppose
\[\frac{1}{x^3 - 3x^2 - 13x + 15} = \frac{A}{x-1} + \frac{B}{x-3} + \frac{C}{(x-3)^2}\]
where $A$, $B$, and $C$ are real constants. What is $A$? | \frac{1}{4} |
The white rabbit can hop 15 meters in one minute. The brown rabbit hops 12 meters per minute. What is the total distance the two rabbits will hop in 5 minutes? | White rabbit + brown rabbit = 15 + 12 = <<15+12=27>>27
5 minutes * 27 = <<5*27=135>>135 meters
The two rabbits will hop 135 meters in 5 minutes.
#### 135 |
The fraction $\frac{2(\sqrt2+\sqrt6)}{3\sqrt{2+\sqrt3}}$ is equal to | \frac43 |
Evaluate the expression $\log_{10} 60 + \log_{10} 80 - \log_{10} 15$. | 2.505 |
Given the function $f(x)=\sin (2x+ \frac {π}{3})- \sqrt {3}\sin (2x- \frac {π}{6})$
(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;
(2) When $x\in\[- \frac {π}{6}, \frac {π}{3}\]$, find the maximum and minimum values of $f(x)$, and write out the values of the independent variable $x$ when the maximum and minimum values are obtained. | -\sqrt {3} |
Wendy just started working at an Italian restaurant. She polished 50 small glasses and 10 more large glasses than small glasses. How many glasses did she polish? | She polished 50 + 10 = <<50+10=60>>60 large glasses.
Therefore, Wendy polished 50 + 60 = <<50+60=110>>110 glasses.
#### 110 |
In regular hexagon $ABCDEF$, points $W$, $X$, $Y$, and $Z$ are chosen on sides $\overline{BC}$, $\overline{CD}$, $\overline{EF}$, and $\overline{FA}$ respectively, so lines $AB$, $ZW$, $YX$, and $ED$ are parallel and equally spaced. What is the ratio of the area of hexagon $WCXYFZ$ to the area of hexagon $ABCDEF$? | \frac{11}{27} |
A square has sides of length 10, and a circle centered at one of its vertices has radius 10. What is the area of the union of the regions enclosed by the square and the circle? Express your answer in terms of $\pi$. | 100+75\pi |
Find the sum of $231_5 + 414_5 + 123_5$. Express your answer in base $5$. | 1323_5 |
A geometric progression with 10 terms starts with the first term as 2 and has a common ratio of 3. Calculate the sum of the new geometric progression formed by taking the reciprocal of each term in the original progression.
A) $\frac{29523}{59049}$
B) $\frac{29524}{59049}$
C) $\frac{29525}{59049}$
D) $\frac{29526}{59049}$ | \frac{29524}{59049} |
The area of a circle is \( 64\pi \) square units. Calculate both the radius and the circumference of the circle. | 16\pi |
Given $x^{2}-5x-2006=0$, evaluate the algebraic expression $\dfrac {(x-2)^{3}-(x-1)^{2}+1}{x-2}$. | 2010 |
Determine the remainder when $$2^{\frac{1 \cdot 2}{2}}+2^{\frac{2 \cdot 3}{2}}+\cdots+2^{\frac{2011 \cdot 2012}{2}}$$ is divided by 7. | 1 |
The cards in a stack of $2n$ cards are numbered consecutively from 1 through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaining cards form pile $B.$ The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A,$ respectively. In this process, card number $(n+1)$ becomes the bottom card of the new stack, card number 1 is on top of this card, and so on, until piles $A$ and $B$ are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named magical. For example, eight cards form a magical stack because cards number 3 and number 6 retain their original positions. Find the number of cards in the magical stack in which card number 131 retains its original position.
| 392 |
Given a square with four vertices and its center, find the probability that the distance between any two of these five points is less than the side length of the square. | \frac{2}{5} |
Given vectors $\overrightarrow{a} = (\sin x, \cos x)$, $\overrightarrow{b} = (\sin x, \sin x)$, and $f(x) = \overrightarrow{a} \cdot \overrightarrow{b}$
(1) If $x \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, find the range of the function $f(x)$.
(2) Let the sides opposite the acute angles $A$, $B$, and $C$ of triangle $\triangle ABC$ be $a$, $b$, and $c$, respectively. If $f(B) = 1$, $b = \sqrt{2}$, and $c = \sqrt{3}$, find the value of $a$. | \frac{\sqrt{6} + \sqrt{2}}{2} |
Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon.
[asy]
size(8cm);
defaultpen(fontsize(10pt));
pair A_2=(-0.4382971011,5.15554989475), B_4=(-2.1182971011,-0.0149584477027), B_5=(-4.8365942022,8.3510997895), A_3=(0.6,8.3510997895), B_1=(2.28,13.521608132), A_4=(3.96,8.3510997895), B_2=(9.3965942022,8.3510997895), A_5=(4.9982971011,5.15554989475), B_3=(6.6782971011,-0.0149584477027), A_1=(2.28,3.18059144705);
filldraw(A_2--A_5--B_2--B_5--cycle,rgb(.8,.8,.8));
draw(B_1--A_4^^A_4--B_2^^B_2--A_5^^A_5--B_3^^B_3--A_1^^A_1--B_4^^B_4--A_2^^A_2--B_5^^B_5--A_3^^A_3--B_1,linewidth(1.2)); draw(A_1--A_2--A_3--A_4--A_5--cycle);
pair O = (A_1+A_2+A_3+A_4+A_5)/5;
label(" $A_1$ ",A_1, 2dir(A_1-O));
label(" $A_2$ ",A_2, 2dir(A_2-O));
label(" $A_3$ ",A_3, 2dir(A_3-O));
label(" $A_4$ ",A_4, 2dir(A_4-O));
label(" $A_5$ ",A_5, 2dir(A_5-O));
label(" $B_1$ ",B_1, 2dir(B_1-O));
label(" $B_2$ ",B_2, 2dir(B_2-O));
label(" $B_3$ ",B_3, 2dir(B_3-O));
label(" $B_4$ ",B_4, 2dir(B_4-O));
label(" $B_5$ ",B_5, 2dir(B_5-O));
[/asy] | \frac{1}{2} |
Define a sequence recursively by $t_1 = 20$, $t_2 = 21$, and\[t_n = \frac{5t_{n-1}+1}{25t_{n-2}}\]for all $n \ge 3$. Then $t_{2020}$ can be expressed as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p+q$. | 626 |
In the Cartesian coordinate system $xoy$, the parametric equation of curve $C_1$ is $$\begin{cases} x=a\cos t+ \sqrt {3} \\ y=a\sin t\end{cases}$$ (where $t$ is the parameter, $a>0$). In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the equation of curve $C_2$ is $$\rho^{2}=2\rho\sin\theta+6$$.
(1) Identify the type of curve $C_1$ and convert its equation into polar coordinates;
(2) Given that $C_1$ and $C_2$ intersect at points $A$ and $B$, and line segment $AB$ passes through the pole, find the length of segment $AB$. | 3 \sqrt {3} |
What is the sum of the roots of $x^2 - 4x + 3 = 0$? | 4 |
In a group of 21 persons, every two person communicate with different radio frequency. It's possible for two person to not communicate (means there's no frequency occupied to connect them). Only one frequency used by each couple, and it's unique for every couple. In every 3 persons, exactly two of them is not communicating to each other. Determine the maximum number of frequency required for this group. Explain your answer. | 110 |
Determine the value of the expression
\[\log_2 (27 + \log_2 (27 + \log_2 (27 + \cdots))),\]assuming it is positive. | 5 |
A string has 150 beads of red, blue, and green colors. It is known that among any six consecutive beads, there is at least one green bead, and among any eleven consecutive beads, there is at least one blue bead. What is the maximum number of red beads that can be on the string? | 112 |
Marcia wants to buy some fruit. Apples cost $2, bananas cost $1, and oranges cost $3. If Marcia buys 12 apples, 4 bananas and 4 oranges, what is the average cost of each piece of fruit in dollars? | Find the price of each group of fruit
12 apples * $2 per apple = $<<12*2=24>>24
4 bananas * $1 per banana = $<<4*1=4>>4
3 oranges * $4 per orange = $<<3*4=12>>12
$24 + $4 + $12 = $<<24+4+12=40>>40 in total that was spent
12 + 4 + 4 = <<12+4+4=20>>20 total fruits bought
$40 / 20 fruits = $<<40/20=2>>2 per fruit
#### 2 |
Let square $WXYZ$ have sides of length $8$. An equilateral triangle is drawn such that no point of the triangle lies outside $WXYZ$. Determine the maximum possible area of such a triangle. | 16\sqrt{3} |
The route from $A$ to $B$ is traveled by a passenger train 3 hours and 12 minutes faster than a freight train. In the time it takes the freight train to travel from $A$ to $B$, the passenger train travels 288 km more. If the speed of each train is increased by $10 \mathrm{km} / h$, the passenger train would travel from $A$ to $B$ 2 hours and 24 minutes faster than the freight train. Determine the distance from $A$ to $B$. | 360 |
By solving the inequality \(\sqrt{x^{2}+3 x-54}-\sqrt{x^{2}+27 x+162}<8 \sqrt{\frac{x-6}{x+9}}\), find the sum of its integer solutions within the interval \([-25, 25]\). | 310 |
What integer value of $n$ will satisfy $n + 10 > 11$ and $-4n > -12$? | 2 |
The product of the positive integer divisors of a positive integer \( n \) is 1024, and \( n \) is a perfect power of a prime. Find \( n \). | 1024 |
Dr. Math's four-digit house number $ABCD$ contains no zeroes and can be split into two different two-digit primes ``$AB$'' and ``$CD$''. Moreover, both these two-digit primes are greater than 50 but less than 100. Find the total number of possible house numbers for Dr. Math. | 90 |
Calculate the product of all prime numbers between 1 and 20. | 9699690 |
The product of the base seven numbers $24_7$ and $30_7$ is expressed in base seven. What is the base seven sum of the digits of this product? | 6 |
Given that the function $f(x)$ is an odd function defined on $\mathbb{R}$ and $f(x+ \frac{5}{2})=-\frac{1}{f(x)}$, and when $x \in [-\frac{5}{2}, 0]$, $f(x)=x(x+ \frac{5}{2})$, find $f(2016)=$ \_\_\_\_\_\_. | \frac{3}{2} |
Mitya is 11 years older than Shura. When Mitya was as old as Shura is now, he was twice as old as she was. How old is Mitya? | 27.5 |
Six students taking a test sit in a row of seats with aisles only on the two sides of the row. If they finish the test at random times, what is the probability that some student will have to pass by another student to get to an aisle? | \frac{43}{45} |
A class has 54 students, and there are 4 tickets for the Shanghai World Expo to be distributed among the students using a systematic sampling method. If it is known that students with numbers 3, 29, and 42 have already been selected, then the student number of the fourth selected student is ▲. | 16 |
Mark has two dozen eggs to split with his three siblings. How many eggs does each person get to eat if they all eat the same amount? | There are 24 eggs because 2 x 12 = <<2*12=24>>24
There are four eaters because 1 + 3 = <<1+3=4>>4
They get 6 eggs each because 24 / 4 = <<24/4=6>>6
#### 6 |
Acme T-Shirt Company charges a $\$50$ set-up fee plus $\$9$ for each shirt printed. Beta T-shirt Company has no set up fee, but charges $\$14$ per shirt. What is the minimum number of shirts for which a customer saves money by using Acme? | 11 |
Simplify: $|{-3^2+4}|$ | 5 |
Lighters cost $1.75 each at the gas station, or $5.00 per pack of twelve on Amazon. How much would Amanda save by buying 24 lighters online instead of at the gas station? | First find how many packs Amanda would have to buy on Amazon: 24 lighters / 12 lighters/pack = <<24/12=2>>2 packs
Then multiply that number by the cost per pack to find the total cost from Amazon: 2 packs * $5/pack = $<<2*5=10>>10
Then multiply the total number of lighters Amanda buys by the cost per lighter at the gas station: 24 lighters * $1.75/lighter = $<<24*1.75=42>>42
Then subtract the total Amazon cost from the total gas station cost to find the savings: $42 - $10 = $<<42-10=32>>32
#### 32 |
A circle with a radius of 3 is inscribed in a right trapezoid, where the shorter base is 4. Find the length of the longer base of the trapezoid. | 12 |
Lucy plans to purchase potato chips for a party. Ten people will be at the party, including Lucy. The potato chips cost 25 cents per pound. How much will Lucy pay (in dollars) for the potato chips if she wants each person to get 1.2 pounds? | Lucy needs to purchase 10 x 1.2 = <<10*1.2=12>>12 pounds of potato chips.
So, Lucy will pay 12 x 25 = <<12*25=300>>300 cents for it.
Since there are 100 cents in $1, thus, Lucy will pay 300/100 = <<300/100=3>>3 dollars.
#### 3 |
Given that six coal freight trains are organized into two groups of three trains, with trains 'A' and 'B' in the same group, determine the total number of different possible departure sequences for the six trains. | 144 |
For the opening home game of the baseball season, the Madd Batters minor league baseball team offered the following incentives to its fans:
Every 75th fan who entered the stadium got a coupon for a free hot dog.
Every 30th fan who entered the stadium got a coupon for a free cup of soda.
Every 50th fan who entered the stadium got a coupon for a free bag of popcorn.
The stadium holds 4000 fans and was completely full for this game. How many of the fans at the game were lucky enough to receive all three free items? | 26 |
Given vectors $\overrightarrow{O A} \perp \overrightarrow{O B}$, and $|\overrightarrow{O A}|=|\overrightarrow{O B}|=24$. Find the minimum value of $|t \overrightarrow{A B}-\overrightarrow{A O}|+\left|\frac{5}{12} \overrightarrow{B O}-(1-t) \overrightarrow{B A}\right|$ for $t \in[0,1]$. | 26 |
A rectangular box has width $12$ inches, length $16$ inches, and height $\frac{m}{n}$ inches, where $m$ and $n$ are relatively prime positive integers. Three faces of the box meet at a corner of the box. The center points of those three faces are the vertices of a triangle with an area of $30$ square inches. Find $m+n$. | 41 |
On the first day, 1 bee brings back 5 companions. On the second day, 6 bees (1 from the original + 5 brought back on the first day) fly out, each bringing back 5 companions. Determine the total number of bees in the hive after the 6th day. | 46656 |
A car license plate contains three letters and three digits, for example, A123BE. The allowed letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters) and all digits except the combination 000. Kira considers a license plate lucky if the second letter is a vowel, the second digit is odd, and the third digit is even (other symbols have no restrictions). How many license plates does Kira consider lucky? | 359999 |
In the diagram, triangles $ABC$ and $CBD$ are isosceles. The perimeter of $\triangle CBD$ is $19,$ the perimeter of $\triangle ABC$ is $20,$ and the length of $BD$ is $7.$ What is the length of $AB?$ [asy]
size(7cm);
defaultpen(fontsize(11));
pair b = (0, 0);
pair d = 7 * dir(-30);
pair a = 8 * dir(-140);
pair c = 6 * dir(-90);
draw(a--b--d--c--cycle);
draw(b--c);
label("$y^\circ$", a, 2 * (E + NE));
label("$y^\circ$", b, 2 * (S + SW));
label("$x^\circ$", b, 2 * (S + SE));
label("$x^\circ$", d, 2 * (2 * W));
label("$A$", a, W);
label("$B$", b, N);
label("$D$", d, E);
label("$C$", c, S);
[/asy] | 8 |
In how many different ways can André form exactly \( \$10 \) using \( \$1 \) coins, \( \$2 \) coins, and \( \$5 \) bills? | 10 |
What is $\frac{2^2 \cdot 2^{-3}}{2^3 \cdot 2^{-2}}$? | \frac{1}{4} |
Emily and John each solved three-quarters of the homework problems individually and the remaining one-quarter together. Emily correctly answered 70% of the problems she solved alone, achieving an overall accuracy of 76% on her homework. John had an 85% success rate with the problems he solved alone. Calculate John's overall percentage of correct answers. | 87.25\% |
A segment of length $1$ is divided into four segments. Then there exists a quadrilateral with the four segments as sides if and only if each segment is: | x < \frac{1}{2} |
If $\theta \in (0^\circ, 360^\circ)$ and the terminal side of angle $\theta$ is symmetric to the terminal side of the $660^\circ$ angle with respect to the x-axis, and point $P(x, y)$ is on the terminal side of angle $\theta$ (not the origin), find the value of $$\frac {xy}{x^{2}+y^{2}}.$$ | \frac {\sqrt {3}}{4} |
Find the largest possible value of $k$ for which $3^{13}$ is expressible as the sum of $k$ consecutive positive integers. | 1458 |
Mr. Zhang knows that there are three different levels of bus service from location A to location B in the morning: good, average, and poor. However, he does not know their exact schedule. His plan is as follows: He will not board the first bus he sees but will take the second one if it's more comfortable than the first one; otherwise, he will wait for the third bus. What are the probabilities that Mr. Zhang ends up on a good bus and on a poor bus, respectively? | \frac{1}{6} |
Ket $f(x) = x^{2} +ax + b$ . If for all nonzero real $x$ $$ f\left(x + \dfrac{1}{x}\right) = f\left(x\right) + f\left(\dfrac{1}{x}\right) $$ and the roots of $f(x) = 0$ are integers, what is the value of $a^{2}+b^{2}$ ? | 13 |
Harris feeds his dog 1 large organic carrot over the course of 1 day. There are 5 carrots in a 1 pound bag and each bag costs $2.00. In one year, how much will Harris spend on carrots? | His dog gets 1 carrot a day and there are 365 days in a year, so his dog eats 1*365 = <<1*365=365>>365 carrots
There are 5 carrots per bag and he will need 365 carrots, so that's 365/5 = 73 bags of carrots
The bags cost $2.00 and he will need 73 bags so that's $2*73 = $<<2*73=146.00>>146.00 worth of organic carrots
#### 146 |
My school's math club has 6 boys and 8 girls. I need to select a team to send to the state math competition. We want 6 people on the team. In how many ways can I select the team to have 3 boys and 3 girls? | 1120 |
The solutions to the equation $x(5x+2) = 6(5x+2)$ are ___. | -\frac{2}{5} |
Four cars $A, B, C$ and $D$ start simultaneously from the same point on a circular track. $A$ and $B$ drive clockwise, while $C$ and $D$ drive counter-clockwise. All cars move at constant (but pairwise different) speeds. Exactly 7 minutes after the race begins, $A$ meets $C$ for the first time, and at that same moment, $B$ meets $D$ for the first time. After another 46 minutes, $A$ and $B$ meet for the first time. When will all the cars meet for the first time after the race starts? | 371 |
On Wednesday, 37 students played kickball. On Thursday, 9 fewer students played kickball. How many students played kickball on Wednesday and Thursday? | 37 - 9 = <<37-9=28>>28 students played kickball on Thursday.
So, 37 + 28 = <<37+28=65>>65 students played kickball on Wednesday and Thursday.
#### 65 |
Lucy begins a sequence with the first term at 4. Each subsequent term is generated as follows: If a fair coin flip results in heads, she triples the previous term and then adds 3. If it results in tails, she subtracts 3 and divides the result by 3. What is the probability that the fourth term in Lucy's sequence is an integer?
A) $\frac{1}{4}$
B) $\frac{1}{2}$
C) $\frac{3}{4}$
D) $\frac{7}{8}$ | \frac{3}{4} |
What two-digit positive integer is one more than a multiple of 2, 3, 4, 5 and 6? | 61 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.