problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Simplify the expression $\frac{5^5 + 5^3 + 5}{5^4 - 2\cdot5^2 + 5}$. | \frac{651}{116} |
The greatest common divisor of two positive integers is $(x+6)$ and their least common multiple is $x(x+6)$, where $x$ is a positive integer. If one of the integers is 36, what is the smallest possible value of the other one? | 24 |
A trapezoid is given where the ratio of the lengths of its bases is 2. If the trapezoid is rotated $360^\circ$ around the longer base, the resulting solid is $\Phi_1$. If the trapezoid is rotated $360^\circ$ around the shorter base, the resulting solid is $\Phi_2$. Find the ratio of the volumes of the solids $\Phi_1$ a... | 5/4 |
An $8$ by $2\sqrt{2}$ rectangle has the same center as a circle of radius $2$. The area of the region common to both the rectangle and the circle is | 2\pi+4 |
Sylvie is feeding her turtles. Each turtle needs 1 ounce of food per 1/2 pound of body weight. She has 30 pounds of turtles. Each jar of food contains 15 ounces and costs $2. How much does it cost to feed the turtles? | She needs 60 ounces of food because 30 / .5 = <<30/.5=60>>60
She needs 4 jars of food because 60 / 15 = <<60/15=4>>4
It will cost $8 to feed them because 4 x 2 = <<4*2=8>>8
#### 8 |
Given that θ is an acute angle and $\sqrt {2}$sinθsin($θ+ \frac {π}{4}$)=5cos2θ, find the value of tanθ. | \frac {5}{6} |
Jack Sparrow needed to distribute 150 piastres into 10 purses. After putting some amount of piastres in the first purse, he placed more in each subsequent purse than in the previous one. As a result, the number of piastres in the first purse was not less than half the number of piastres in the last purse. How many pias... | 15 |
In 2023, a special international mathematical conference is held. Let $A$, $B$, and $C$ be distinct positive integers such that the product $A \cdot B \cdot C = 2023$. What is the largest possible value of the sum $A+B+C$? | 297 |
Mike and Ted planted tomatoes. In the morning, Mike planted 50 tomato seeds while Ted planted twice as much as Mike. In the afternoon, Mike planted 60 tomato seeds while Ted planted 20 fewer tomato seeds than Mike. How many tomato seeds did they plant altogether? | Ted planted 2 x 50 = <<2*50=100>>100 tomato seeds.
So, Mike and Ted planted 50 + 100 = <<50+100=150>>150 tomato seeds in the morning.
In the afternoon, Ted planted 60 - 20 = <<60-20=40>>40 seeds.
Thus, Mike and Ted planted 60 + 40 = <<60+40=100>>100 tomato seeds in the afternoon.
Therefore, they planted 150 + 100 = <<1... |
A running competition on an unpredictable distance is conducted as follows. On a circular track with a length of 1 kilometer, two points \( A \) and \( B \) are randomly selected (using a spinning arrow). The athletes then run from \( A \) to \( B \) along the shorter arc. Find the median value of the length of this ar... | 0.25 |
From time $t=0$ to time $t=1$ a population increased by $i\%$, and from time $t=1$ to time $t=2$ the population increased by $j\%$. Therefore, from time $t=0$ to time $t=2$ the population increased by | $\left(i+j+\frac{ij}{100}\right)\%$ |
2022 knights and liars are lined up in a row, with the ones at the far left and right being liars. Everyone except the ones at the extremes made the statement: "There are 42 times more liars to my right than to my left." Provide an example of a sequence where there is exactly one knight. | 48 |
A positive integer divisor of $12!$ is chosen at random. The probability that the divisor chosen is a perfect square can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? | 23 |
A figure is constructed from unit cubes. Each cube shares at least one face with another cube. What is the minimum number of cubes needed to build a figure with the front and side views shown? [asy]
/* AMC8 2003 #15 Problem */
draw((0,0)--(2,0)--(2,1)--(1,1)--(1,2)--(0,2)--cycle);
draw((0,1)--(1,1)--(1,0));
draw((4,... | 4 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $c\sin A= \sqrt {3}a\cos C$ and $(a-c)(a+c)=b(b-c)$, consider the function $f(x)=2\sin x\cos ( \frac {π}{2}-x)- \sqrt {3}\sin (π+x)\cos x+\sin ( \frac {π}{2}+x)\cos x$.
(1) Find the period and the equation... | \frac {5}{2} |
Barry, Thomas and Emmanuel are to share a jar of 200 jelly beans. If Thomas takes 10%, and Barry and Emmanuel are to share the remainder in the ratio 4:5 respectively, how many jelly beans will Emmanuel get? | 10% of 200 jelly beans is (10/100)*200 = <<200*10/100=20>>20 jelly beans
Thomas takes 20 jelly beans leaving 200-20 = <<200-20=180>>180 jelly beans
180 jelly beans are to be shared in the ratio 4:5 so each share is 180/(4+5) = <<180/(4+5)=20>>20 jelly beans
Emmanuel gets 5 shares which is 5*20 = <<5*20=100>>100
#### 10... |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given $a(4-2 \sqrt {7}\cos B)=b(2 \sqrt {7}\cos A-5)$, find the minimum value of $\cos C$. | -\frac{1}{2} |
Calculate: $(128)^{\frac{7}{3}}$ | 65536 \cdot \sqrt[3]{2} |
The Jacksonville Walmart normally gets 120 customer complaints per day. That number increases by 1/3rd when they're short-staffed and this increases by another 20% when the self-checkout is broken. If the store was short-staffed and the self-checkout was broken for 3 days, how many complaints does the store get? | First find the number of complaints per day when the store is short-staffed: 120 complaints/day * 4/3 = <<120*4/3=160>>160 complaints/day
Then find the number of complaints per day when the self-checkout is broken and they're short-staffed: 160 complaints/day * 1.2 = <<160*1.2=192>>192 complaints/day
Then multiply the ... |
Let $S_n$ be the sum of the reciprocals of the non-zero digits of the integers from $1$ to $10^n$ inclusive. Find the smallest positive integer $n$ for which $S_n$ is an integer.
| 63 |
A literary and art team went to a nursing home for a performance. Originally, there were 6 programs planned, but at the request of the elderly, they decided to add 3 more programs. However, the order of the original six programs remained unchanged, and the added 3 programs were neither at the beginning nor at the end. ... | 210 |
The sum of three different numbers is 75. The two larger numbers differ by 5 and the two smaller numbers differ by 4. What is the value of the largest number? | \frac{89}{3} |
The sides of a triangle have lengths \( 13, 17, \) and \( k, \) where \( k \) is a positive integer. For how many values of \( k \) is the triangle obtuse? | 14 |
Find the smallest composite number that has no prime factors less than 15. | 289 |
Let $x_1,$ $x_2,$ $x_3,$ $\dots,$ $x_{100}$ be positive real numbers such that $x_1^2 + x_2^2 + x_3^2 + \dots + x_{100}^2 = 1.$ Find the minimum value of
\[\frac{x_1}{1 - x_1^2} + \frac{x_2}{1 - x_2^2} + \frac{x_3}{1 - x_3^2} + \dots + \frac{x_{100}}{1 - x_{100}^2}.\] | \frac{3 \sqrt{3}}{2} |
Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$. | 6 |
Bob's password consists of a non-negative single-digit number followed by a letter and another non-negative single-digit number (which could be the same as the first one). What is the probability that Bob's password consists of an odd single-digit number followed by a letter and a positive single-digit number? | \frac{9}{20} |
Find the distance between the points $(2,2)$ and $(-1,-1)$. | 3\sqrt{2} |
Let $\{b_k\}$ be a sequence of integers such that $b_1 = 2$ and $b_{m+n} = b_m + b_n + mn + 1$, for all positive integers $m$ and $n$. Find $b_{12}$. | 101 |
Billy wants to watch something fun on YouTube but doesn't know what to watch. He has the website generate 15 suggestions but, after watching each in one, he doesn't like any of them. Billy's very picky so he does this a total of 5 times before he finally finds a video he thinks is worth watching. He then picks the 5... | Billy has 15 suggestions generate for him 5 times for 15*5=<<15*5=75>>75 videos.
There are 15-5 = <<15-5=10>>10 shows in the final set of suggestions he doesn't consider
This means he picks the 75 - 10 = <<75-10=65>>65th show YouTube suggests
#### 65 |
At a certain university, the division of mathematical sciences consists of the departments of mathematics, statistics, and computer science. There are two male and two female professors in each department. A committee of six professors is to contain three men and three women and must also contain two professors from ea... | 88 |
In the figure below, $N$ congruent semicircles lie on the diameter of a large semicircle, with their diameters covering the diameter of the large semicircle with no overlap. Let $A$ be the combined area of the small semicircles and $B$ be the area of the region inside the large semicircle but outside the semicircles. T... | 19 |
At Rosa's Rose Shop, a bouquet containing a dozen roses costs $\$20$. If the price of a bouquet is directly proportional to the number of roses it contains, how many dollars will a bouquet of 39 roses cost? | 65 |
The Hoopers, coached by Coach Loud, have 15 players. George and Alex are the two players who refuse to play together in the same lineup. Additionally, if George plays, another player named Sam refuses to play. How many starting lineups of 6 players can Coach Loud create, provided the lineup does not include both George... | 3795 |
650 students were surveyed about their pasta preferences. The choices were lasagna, manicotti, ravioli and spaghetti. The results of the survey are displayed in the bar graph. What is the ratio of the number of students who preferred spaghetti to the number of students who preferred manicotti? | \frac{5}{2} |
Given a triangle $ABC$ with sides opposite angles $A$, $B$, $C$ denoted by $a$, $b$, $c$ respectively, and with $b = 3$, $c = 2$, and the area $S_{\triangle ABC} = \frac{3\sqrt{3}}{2}$:
1. Determine the value of angle $A$;
2. When angle $A$ is obtuse, find the height from vertex $B$ to side $BC$. | \frac{3\sqrt{57}}{19} |
A circle of radius 5 is inscribed in a rectangle as shown. The ratio of the length of the rectangle to its width is 2:1. What is the area of the rectangle? | 200 |
Crackers contain 15 calories each and cookies contain 50 calories each. If Jimmy eats 7 cookies, how many crackers does he need to eat to have consumed a total of 500 calories? | He has consumed 7 * 50 = <<7*50=350>>350 calories from the cookies.
He needs to consume 500 - 350 = <<500-350=150>>150 more calories.
He should eat 150 / 15 = <<150/15=10>>10 crackers.
#### 10 |
Let $a$, $b$, and $c$ be the roots of $x^3 - 20x^2 + 18x - 7 = 0$. Compute \[(a+b)^2 + (b+c)^2 + (c+a)^2.\] | 764 |
Express \( 0.3\overline{45} \) as a common fraction. | \frac{83}{110} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and $c\sin\frac{A+C}{2}=b\sin C$.
$(1)$ Find angle $B$;
$(2)$ Let $BD$ be the altitude from $B$ to side $AC$, and $BD=1$, $b=\sqrt{3}$. Find the perimeter of $\triangle ABC$. | 3 + \sqrt{3} |
Johan has a large number of identical cubes. He has made a structure by taking a single cube and then sticking another cube to each face. He wants to make an extended structure in the same way so that each face of the current structure will have a cube stuck to it. How many extra cubes will he need to complete his exte... | 18 |
Let O be the center of the square ABCD. If 3 points are chosen from O, A, B, C, and D at random, find the probability that the 3 points are collinear. | \frac{1}{5} |
David is taking a data analytics course that lasts for 24 weeks. The course consists of 2 three-hour classes and 1 four-hour class each week. In addition, David must spend 4 hours each week working on small group homework assignments. How many hours will he spend on this course? | Each week, David will spend 3 hours + 3 hours + 4 hours = <<3+3+4=10>>10 hours in class.
Add to that his 4 hours of small group work, 4 + 10 = <<4+10=14>>14 hours a week.
He will spend 14 hours x 24 weeks = <<14*24=336>>336 hours on the course.
#### 336 |
In a game, \(N\) people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let \(m\) be the maximum possible ... | 34 |
Let $x,$ $y,$ $z$ be positive real numbers such that $xyz = 8.$ Find the minimum value of $x + 2y + 4z.$ | 12 |
A farmer has three trucks to carry water to his farm. Each truck uses three tanks with a capacity of 150 liters of water. How many liters of water in total can the farmer carry in his trucks? | 1 truck uses 3 tanks of 150 liters, so a truck can carry 3 tanks per truck * 150 liters = <<3*150=450>>450 liters per truck.
The total amount of water the 3 trucks carry is 3 trucks * 450 liters per truck= <<3*450=1350>>1350 liters.
#### 1350 |
What percent of square $EFGH$ is shaded? All angles in the diagram are right angles. [asy]
import graph;
defaultpen(linewidth(0.7));
xaxis(0,8,Ticks(1.0,NoZero));
yaxis(0,8,Ticks(1.0,NoZero));
fill((0,0)--(2,0)--(2,2)--(0,2)--cycle);
fill((3,0)--(5,0)--(5,5)--(0,5)--(0,3)--(3,3)--cycle);
fill((6,0)--(7,0)--(7,7)--(0,... | 67\% |
Let $T=TNFTPP$ . Points $A$ and $B$ lie on a circle centered at $O$ such that $\angle AOB$ is right. Points $C$ and $D$ lie on radii $OA$ and $OB$ respectively such that $AC = T-3$ , $CD = 5$ , and $BD = 6$ . Determine the area of quadrilateral $ACDB$ .
[asy]
draw(circle((0,0),10));
draw((0,10)... | 44 |
Pentagon $ABCDE$ is inscribed in a circle such that $ACDE$ is a square with area $12$. Determine the largest possible area of pentagon $ABCDE$. | 9 + 3\sqrt{2} |
Let $x,$ $y,$ and $z$ be positive real numbers such that $x + y + z = 1.$ Find the minimum value of
\[\frac{x + y}{xyz}.\] | 16 |
Suppose the point $(1,2)$ is on the graph of $y=\frac{f(x)}2$. Then there is one point which must be on the graph of $y=\frac{f^{-1}(x)}{2}$. What is the sum of that point's coordinates? | \frac 92 |
Find the least positive integer $m$ such that $m^2 - m + 11$ is a product of at least four not necessarily distinct primes. | 132 |
Two numbers are independently selected from the set of positive integers less than or equal to 7. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | \frac{36}{49} |
How many numbers are in the list $-50, -44, -38, \ldots, 68, 74$? | 22 |
On a complex plane map of a fictional continent, city A is located at the origin $0$, city B is at $3900i$, and city C is at $1170 + 1560i$. Calculate the distance from city C to city A on this plane. | 1950 |
What number should go in the $\square$ to make the equation $\frac{3}{4}+\frac{4}{\square}=1$ true? | 16 |
Determine the value of
\[2023 + \frac{1}{2} \left( 2022 + \frac{1}{2} \left( 2021 + \dots + \frac{1}{2} \left( 4 + \frac{1}{2} \cdot (3 + 1) \right) \right) \dotsb \right).\] | 4044 |
In triangle $ABC$ , find the smallest possible value of $$ |(\cot A + \cot B)(\cot B +\cot C)(\cot C + \cot A)| $$ | \frac{8\sqrt{3}}{9} |
Semicircle $\widehat{AB}$ has center $C$ and radius $1$. Point $D$ is on $\widehat{AB}$ and $\overline{CD} \perp \overline{AB}$. Extend $\overline{BD}$ and $\overline{AD}$ to $E$ and $F$, respectively, so that circular arcs $\widehat{AE}$ and $\widehat{BF}$ have $B$ and $A$ as their respective centers. Circular arc $\w... | $2\pi-\pi \sqrt{2}-1$ |
The volume of a given sphere is \(72\pi\) cubic inches. Find the surface area of the sphere. Express your answer in terms of \(\pi\). | 36\pi 2^{2/3} |
Suppose that $wz = 12-8i$, and $|w| = \sqrt{13}$. What is $|z|$? | 4 |
Find the smallest positive real number $x$ such that
\[\lfloor x^2 \rfloor - x \lfloor x \rfloor = 10.\] | \frac{131}{11} |
Find the digits left and right of the decimal point in the decimal form of the number \[ (\sqrt{2} + \sqrt{3})^{1980}. \] | 7.9 |
If \(a\), \(b\), and \(c\) are distinct positive integers such that \(abc = 16\), then the largest possible value of \(a^b - b^c + c^a\) is: | 263 |
What is the remainder when $2^{2001}$ is divided by $2^{7}-1$ ? | 64 |
Mrs. Crocker made 11 pieces of fried chicken for Lyndee and her friends. If Lyndee only ate one piece but each of her friends got to eat 2 pieces, how many friends did Lyndee have over? | After Lyndee ate, there were 11 - 1 = <<11-1=10>>10 pieces of chicken left.
Lyndee had 10/2 = <<10/2=5>>5 friends over.
#### 5 |
If $y=x+\frac{1}{x}$, then $x^4+x^3-4x^2+x+1=0$ becomes: | $x^2(y^2+y-6)=0$ |
Simplify
\[\tan x + 2 \tan 2x + 4 \tan 4x + 8 \cot 8x.\]The answer will be a trigonometric function of some simple function of $x,$ like "$\cos 2x$" or "$\sin (x^3)$". | \cot x |
Let $a,b,c,d$ be real numbers such that $a^2+b^2+c^2+d^2=1$. Determine the minimum value of $(a-b)(b-c)(c-d)(d-a)$ and determine all values of $(a,b,c,d)$ such that the minimum value is achived. | -\frac{1}{8} |
What is the maximum number of checkers that can be placed on a $6 \times 6$ board such that no three checkers (specifically, the centers of the cells they occupy) lie on the same straight line (at any angle)? | 12 |
Let the function $f(x)=\tan \frac {x}{4}\cdot \cos ^{2} \frac {x}{4}-2\cos ^{2}\left( \frac {x}{4}+ \frac {\pi}{12}\right)+1$.
(Ⅰ) Find the domain of $f(x)$ and its smallest positive period;
(Ⅱ) Find the maximum and minimum values of $f(x)$ in the interval $[-\pi,0]$. | -\frac{\sqrt{3}}{2} |
Factor the expression $3x(x+1) + 7(x+1)$. | (3x+7)(x+1) |
James makes potatoes for a group. Each person eats 1.5 pounds of potatoes. He makes potatoes for 40 people. A 20-pound bag of potatoes costs $5. How much does it cost? | He needs 1.5*40=<<1.5*40=60>>60 pounds of potatoes
So he needs to buy 60/20=<<60/20=3>>3 bags of potatoes
So it cost 3*5=$<<3*5=15>>15
#### 15 |
In the adjoining figure, $CD$ is the diameter of a semicircle with center $O$. Point $A$ lies on the extension of $DC$ past $C$; point $E$ lies on the semicircle, and $B$ is the point of intersection (distinct from $E$) of line segment $AE$ with the semicircle. If length $AB$ equals length $OD$, and the measure of $\... | 15^\circ |
Maryann spends seven times as long doing accounting as calling clients. If she worked 560 minutes today, how many minutes did she spend calling clients? | Let a be the amount of time Maryann spends doing accounting and c be the amount of time she spends calling clients: We know that a + c = 560 and a = 7c.
Substituting the second equation into the first equation, we get 7c + c = 560
Combining like terms, we get 8c = 560
Dividing both sides by 8, we get c = 70
#### 70 |
Triangle $PQR$ has positive integer side lengths with $PQ=PR$. Let $J$ be the intersection of the bisectors of $\angle Q$ and $\angle R$. Suppose $QJ=10$. Find the smallest possible perimeter of $\triangle PQR$. | 416 |
Let $ a$, $ b$, $ c$, $ x$, $ y$, and $ z$ be real numbers that satisfy the three equations
\begin{align*}
13x + by + cz &= 0 \\
ax + 23y + cz &= 0 \\
ax + by + 42z &= 0.
\end{align*}Suppose that $ a \ne 13$ and $ x \ne 0$. What is the value of
\[ \frac{a}{a - 13} + \frac{b}{b - 23} + \frac{c}{c - 42} \, ?... | 1 |
The Annual Interplanetary Mathematics Examination (AIME) is written by a committee of five Martians, five Venusians, and five Earthlings. At meetings, committee members sit at a round table with chairs numbered from $1$ to $15$ in clockwise order. Committee rules state that a Martian must occupy chair $1$ and an Earthl... | 346 |
The sum of all the positive factors of integer $x$ is 24. If one of the factors is 3, what is the value of $x$? | 15 |
Which number in the array below is both the largest in its column and the smallest in its row? (Columns go up and down, rows go right and left.)
\[\begin{tabular}{ccccc} 10 & 6 & 4 & 3 & 2 \\ 11 & 7 & 14 & 10 & 8 \\ 8 & 3 & 4 & 5 & 9 \\ 13 & 4 & 15 & 12 & 1 \\ 8 & 2 & 5 & 9 & 3 \end{tabular}\] | 7 |
John orders food for a massive restaurant. He orders 1000 pounds of beef for $8 per pound. He also orders twice that much chicken at $3 per pound. How much did everything cost? | The beef cost $8 * 1000 = $<<8*1000=8000>>8000
He buys 1000 * 2 = <<1000*2=2000>>2000 pounds of chicken
So the chicken cost 2000 * $3 = $<<2000*3=6000>>6000
So the total cost is $8000 + $6000 = $<<8000+6000=14000>>14,000
#### 14000 |
Let \( x \neq y \), and suppose the two sequences \( x, a_{1}, a_{2}, a_{3}, y \) and \( b_{1}, x, b_{2}, b_{3}, y, b_{1} \) are both arithmetic sequences. Determine the value of \( \frac{b_{4}-b_{3}}{a_{2}-a_{1}} \). | 8/3 |
When three standard dice are tossed, the numbers $a,b,c$ are obtained. Find the probability that $$(a-1)(b-1)(c-1) \neq 0$$ | \frac{125}{216} |
What is $\frac{1357_{9}}{100_{4}}-2460_{8}+5678_{9}$? Express your answer in base 10. | 2938 |
Let\[S=\sqrt{1+\dfrac1{1^2}+\dfrac1{2^2}}+\sqrt{1+\dfrac1{2^2}+\dfrac1{3^2}}+\cdots+\sqrt{1+\dfrac1{2007^2}+\dfrac1{2008^2}}.\]Compute $\lfloor S^2\rfloor$.
| 4032062 |
Given the function $y=2\sin \left(x+ \frac {\pi}{6}\right)\cos \left(x+ \frac {\pi}{6}\right)$, determine the horizontal shift required to obtain its graph from the graph of the function $y=\sin 2x$. | \frac{\pi}{6} |
When $n$ standard 6-sided dice are rolled, find the smallest possible value of $S$ such that the probability of obtaining a sum of 2000 is greater than zero and is the same as the probability of obtaining a sum of $S$. | 338 |
Given the ellipse $\frac{{x}^{2}}{3{{m}^{2}}}+\frac{{{y}^{2}}}{5{{n}^{2}}}=1$ and the hyperbola $\frac{{{x}^{2}}}{2{{m}^{2}}}-\frac{{{y}^{2}}}{3{{n}^{2}}}=1$ share a common focus, find the eccentricity of the hyperbola ( ). | \frac{\sqrt{19}}{4} |
If $5! \cdot 3! = n!$, what is the value of $n$? | 6 |
Find the number of six-digit palindromes. | 9000 |
What is the smallest positive integer $n$ which cannot be written in any of the following forms? - $n=1+2+\cdots+k$ for a positive integer $k$. - $n=p^{k}$ for a prime number $p$ and integer $k$ - $n=p+1$ for a prime number $p$. - $n=p q$ for some distinct prime numbers $p$ and $q$ | 40 |
Daniel practices basketball for 15 minutes each day during the school week. He practices twice as long each day on the weekend. How many minutes does he practice during a whole week? | Daniel practices for a total of 15 x 5 = <<15*5=75>>75 minutes during the school week.
He practices 15 x 2 = <<15*2=30>>30 minutes each day on the weekend.
So, he practices a total of 30 x 2 = <<30*2=60>>60 minutes on the weekend.
Thus, Daniel practices for a total of 75 + 60 = <<75+60=135>>135 minutes during a whole w... |
The average age of the $6$ people in Room A is $40$. The average age of the $4$ people in Room B is $25$. If the two groups are combined, what is the average age of all the people? | 34 |
Connor sleeps 6 hours a night. His older brother Luke sleeps 2 hours longer than Connor. Connor’s new puppy sleeps twice as long as Luke. How long does the puppy sleep? | Luke sleeps 2 hours longer than Connor who sleeps 6 hours so that’s 2+6 = <<2+6=8>>8 hours
The new puppy sleeps twice as long as Luke, who sleeps 8 hours, so the puppy sleeps 2*8 = <<2*8=16>>16 hours
#### 16 |
The Amaco Middle School bookstore sells pencils costing a whole number of cents. Some seventh graders each bought a pencil, paying a total of $1.43$ dollars. Some of the $30$ sixth graders each bought a pencil, and they paid a total of $1.95$ dollars. How many more sixth graders than seventh graders bought a pencil? | 4 |
Amy biked 12 miles yesterday. If she biked 3 miles less than twice as far as yesterday, how many miles did she bike in total in the two days? | Twice the distance Amy biked yesterday is 12*2 = <<12*2=24>>24 miles
If Amy biked 3 miles less than twice as far as yesterday, she biked 24-3 = <<24-3=21>>21 miles today.
The total distance she biked in the two days is 21+12 = <<21+12=33>>33 miles
#### 33 |
A square grid $100 \times 100$ is tiled in two ways - only with dominoes and only with squares $2 \times 2$. What is the least number of dominoes that are entirely inside some square $2 \times 2$? | 100 |
Sam spends his days walking around the following $2 \times 2$ grid of squares. Say that two squares are adjacent if they share a side. He starts at the square labeled 1 and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal... | 167 |
Four princesses each guessed a two-digit number, and Ivan guessed a four-digit number. After they wrote their numbers in a row in some order, they got the sequence 132040530321. Find Ivan's number. | 5303 |
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