problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | difficulty float64 0.75 2.02k | difficulty_raw listlengths 3 8 |
|---|---|---|---|---|
Six rhombi of side length 1 are arranged as shown. What is the perimeter of this figure? | 14 | The first rhombus and the last rhombus each have three edges that form part of the exterior of the figure, and so they each contribute 3 to the perimeter. The inner four rhombi each have two edges that form part of the exterior of the figure, and so they each contribute 2 to the perimeter. Thus, the perimeter is $2 \ti... | 2.375 | [
2,
2,
2,
3,
2,
3,
3,
2
] |
For each positive integer $n$, define $S(n)$ to be the smallest positive integer divisible by each of the positive integers $1, 2, 3, \ldots, n$. How many positive integers $n$ with $1 \leq n \leq 100$ have $S(n) = S(n+4)$? | 11 | For each positive integer $n, S(n)$ is defined to be the smallest positive integer divisible by each of $1, 2, 3, \ldots, n$. In other words, $S(n)$ is the least common multiple (lcm) of $1, 2, 3, \ldots, n$. To calculate the lcm of a set of numbers, we determine the prime factorization of each number in the set, deter... | 6 | [
7,
5,
6,
6,
6,
6,
6,
6
] |
What is the perimeter of $\triangle UVZ$ if $UVWX$ is a rectangle that lies flat on a horizontal floor, a vertical semi-circular wall with diameter $XW$ is constructed, point $Z$ is the highest point on this wall, and $UV=20$ and $VW=30$? | 86 | The perimeter of $\triangle UVZ$ equals $UV+UZ+VZ$.
We know that $UV=20$. We need to calculate $UZ$ and $VZ$.
Let $O$ be the point on $XW$ directly underneath $Z$.
Since $Z$ is the highest point on the semi-circle and $XW$ is the diameter, then $O$ is the centre of the semi-circle.
We join $UO, VO, UZ$, and $VZ... | 4.25 | [
5,
4,
4,
4,
4,
4,
4,
5
] |
Lucas chooses one, two or three different numbers from the list $2, 5, 7, 12, 19, 31, 50, 81$ and writes down the sum of these numbers. (If Lucas chooses only one number, this number is the sum.) How many different sums less than or equal to 100 are possible? | 41 | If Lucas chooses 1 number only, there are 8 possibilities for the sum, namely the 8 numbers themselves: $2, 5, 7, 12, 19, 31, 50, 81$. To count the number of additional sums to be included when Lucas chooses two numbers, we make a table, adding the number on left to the number on top when it is less than the number on ... | 5.125 | [
5,
5,
5,
5,
5,
6,
5,
5
] |
In a photograph, Aristotle, David, Flora, Munirah, and Pedro are seated in a random order in a row of 5 chairs. If David is seated in the middle of the row, what is the probability that Pedro is seated beside him? | \frac{1}{2} | After David is seated, there are 4 seats in which Pedro can be seated, of which 2 are next to David. Thus, the probability that Pedro is next to David is $ rac{2}{4}$ or $ rac{1}{2}$. | 2.125 | [
2,
3,
2,
2,
2,
2,
2,
2
] |
How many points $(x, y)$, with $x$ and $y$ both integers, are on the line with equation $y=4x+3$ and inside the region bounded by $x=25, x=75, y=120$, and $y=250$? | 32 | We determine the number of integers $x$ with $25 \leq x \leq 75$ for which $120 \leq 4x+3 \leq 250$. When $x=30$, $4x+3=123$ and when $x=61$, $4x+3=247$. Therefore, $4x+3$ is between 120 and 250 exactly when $30 \leq x \leq 61$. There are $61-30+1=32$ such values of $x$. | 3.875 | [
4,
4,
4,
4,
4,
3,
4,
4
] |
If $2n + 5 = 16$, what is the value of the expression $2n - 3$? | 8 | Since $2n + 5 = 16$, then $2n - 3 = (2n + 5) - 8 = 16 - 8 = 8$. Alternatively, we could solve the equation $2n + 5 = 16$ to obtain $2n = 11$ or $n = \frac{11}{2}$. From this, we see that $2n - 3 = 2\left(\frac{11}{2}\right) - 3 = 11 - 3 = 8$. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
Last summer, Pat worked at a summer camp. For each day that he worked, he earned \$100 and he was not charged for food. For each day that he did not work, he was not paid and he was charged \$20 for food. After 70 days, the money that he earned minus his food costs equalled \$5440. On how many of these 70 days did Pat ... | 57 | Let \( x \) be the number of days on which Pat worked. On each of these days, he earned \$100 and had no food costs, so he earned a total of \( 100x \) dollars. Since Pat worked for \( x \) of the 70 days, then he did not work on \( 70-x \) days. On each of these days, he earned no money and was charged \$20 for food, ... | 3.125 | [
3,
3,
4,
3,
3,
3,
3,
3
] |
If $3+\triangle=5$ and $\triangle+\square=7$, what is the value of $\triangle+\Delta+\Delta+\square+\square$? | 16 | Since $3+\triangle=5$, then $\triangle=5-3=2$. Since $\triangle+\square=7$ and $\triangle=2$, then $\square=5$. Thus, $\triangle+\Delta+\Delta+\square+\square=3 \times 2+2 \times 5=6+10=16$. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
Four congruent rectangles and a square are assembled without overlapping to form a large square. Each of the rectangles has a perimeter of 40 cm. What is the total area of the large square? | 400 \mathrm{~cm}^{2} | Suppose that each of the smaller rectangles has a longer side of length $x \mathrm{~cm}$ and a shorter side of length $y \mathrm{~cm}$. Since the perimeter of each of the rectangles is 40 cm, then $2x+2y=40$ or $x+y=20$. But the side length of the large square is $x+y \mathrm{~cm}$. Therefore, the area of the large squ... | 2.875 | [
3,
3,
3,
3,
3,
2,
3,
3
] |
If \( 3x + 4 = x + 2 \), what is the value of \( x \)? | -1 | If \( 3x + 4 = x + 2 \), then \( 3x - x = 2 - 4 \) and so \( 2x = -2 \), which gives \( x = -1 \). | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
Five students play chess matches against each other. Each student plays three matches against each of the other students. How many matches are played in total? | 30 | We label the players as A, B, C, D, and E. The total number of matches played will be equal to the number of pairs of players that can be formed times the number of matches that each pair plays. The possible pairs of players are AB, AC, AD, AE, BC, BD, BE, CD, CE, and DE. There are 10 such pairs. Thus, the total number... | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
When three positive integers are added in pairs, the resulting sums are 998, 1050, and 1234. What is the difference between the largest and smallest of the three original positive integers? | 236 | Suppose that the three integers are $x, y$ and $z$ where $x+y=998$, $x+z=1050$, and $y+z=1234$. From the first two equations, $(x+z)-(x+y)=1050-998$ or $z-y=52$. Since $z+y=1234$ and $z-y=52$, then $(z+y)+(z-y)=1234+52$ or $2z=1286$ and so $z=643$. Since $z=643$ and $z-y=52$, then $y=z-52=643-52=591$. Since $x+y=998$ a... | 3.5 | [
3,
4,
3,
3,
4,
4,
3,
4
] |
If the line that passes through the points $(2,7)$ and $(a, 3a)$ has a slope of 2, what is the value of $a$? | 3 | Since the slope of the line through points $(2,7)$ and $(a, 3a)$ is 2, then $ rac{3a-7}{a-2}=2$. From this, $3a-7=2(a-2)$ and so $3a-7=2a-4$ which gives $a=3$. | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
Three $1 imes 1 imes 1$ cubes are joined side by side. What is the surface area of the resulting prism? | 14 | When three $1 imes 1 imes 1$ cubes are joined together as in the diagram, the resulting prism is $3 imes 1 imes 1$. This prism has four rectangular faces that are $3 imes 1$ and two rectangular faces that are $1 imes 1$. Therefore, the surface area is $4 imes(3 imes 1)+2 imes(1 imes 1)=4 imes 3+2 imes 1=12+... | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
A cube has six faces. Each face has some dots on it. The numbers of dots on the six faces are 2, 3, 4, 5, 6, and 7. Harry removes one of the dots at random, with each dot equally likely to be removed. When the cube is rolled, each face is equally likely to be the top face. What is the probability that the top face has ... | \frac{13}{27} | When a dot is removed from a face with an even number of dots, that face then has an odd number of dots. When a dot is removed from a face with an odd number of dots, that face then has an even number of dots. Initially, there are 3 faces with an even number of dots and 3 faces with an odd number of dots. If a dot is r... | 4.625 | [
4,
4,
5,
5,
5,
4,
5,
5
] |
If a bag contains only green, yellow, and red marbles in the ratio $3: 4: 2$ and 63 of the marbles are not red, how many red marbles are in the bag? | 18 | Since the ratio of green marbles to yellow marbles to red marbles is $3: 4: 2$, then we can let the numbers of green, yellow and red marbles be $3n, 4n$ and $2n$ for some positive integer $n$. Since 63 of the marbles in the bag are not red, then $3n+4n=63$ and so $7n=63$ or $n=9$, which means that the number of red mar... | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
In a factory, Erika assembles 3 calculators in the same amount of time that Nick assembles 2 calculators. Also, Nick assembles 1 calculator in the same amount of time that Sam assembles 3 calculators. How many calculators in total can be assembled by Nick, Erika, and Sam in the same amount of time as Erika assembles 9 ... | 33 | Erika assembling 9 calculators is the same as assembling three groups of 3 calculators. Since Erika assembles 3 calculators in the same amount of time that Nick assembles 2 calculators, then he assembles three groups of 2 calculators (that is, 6 calculators) in this time. Since Nick assembles 1 calculator in the same a... | 3.375 | [
4,
4,
3,
3,
3,
4,
3,
3
] |
Three friends are in the park. Bob and Clarise are standing at the same spot and Abe is standing 10 m away. Bob chooses a random direction and walks in this direction until he is 10 m from Clarise. What is the probability that Bob is closer to Abe than Clarise is to Abe? | \frac{1}{3} | We call Clarise's spot $C$ and Abe's spot $A$. Consider a circle centred at $C$ with radius 10 m. Since $A$ is 10 m from $C$, then $A$ is on this circle. Bob starts at $C$ and picks a direction to walk, with every direction being equally likely to be chosen. We model this by having Bob choose an angle $\theta$ between ... | 4.125 | [
4,
4,
5,
4,
4,
4,
4,
4
] |
The Athenas are playing a 44 game season. They have 20 wins and 15 losses so far. What is the smallest number of their remaining games that they must win to make the playoffs, given they must win at least 60% of all of their games? | 7 | In order to make the playoffs, the Athenas must win at least 60% of their 44 games. That is, they must win at least $0.6 \times 44=26.4$ games. Since they must win an integer number of games, then the smallest number of games that they can win to make the playoffs is the smallest integer larger than 26.4, or 27. Since ... | 2.375 | [
3,
2,
3,
2,
3,
2,
2,
2
] |
If the number of zeros in the integer equal to $(10^{100}) imes (100^{10})$ is sought, what is this number? | 120 | Since $100=10^{2}$, then $100^{10}=(10^{2})^{10}=10^{20}$. Therefore, $(10^{100}) imes (100^{10})=(10^{100}) imes (10^{20})=10^{120}$. When written out, this integer consists of a 1 followed by 120 zeros. | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
There are $n$ students in the math club. When grouped in 4s, there is one incomplete group. When grouped in 3s, there are 3 more complete groups than with 4s, and one incomplete group. When grouped in 2s, there are 5 more complete groups than with 3s, and one incomplete group. What is the sum of the digits of $n^{2}-n$... | 12 | Suppose that, when the $n$ students are put in groups of 2, there are $g$ complete groups and 1 incomplete group. Since the students are being put in groups of 2, an incomplete group must have exactly 1 student in it. Therefore, $n=2g+1$. Since the number of complete groups of 2 is 5 more than the number of complete gr... | 5.625 | [
6,
6,
6,
5,
5,
5,
6,
6
] |
How many words are there in a language that are 10 letters long and begin with a vowel, given that the language uses only the letters A, B, C, D, and E, where A and E are vowels, and B, C, and D are consonants, and a word does not include the same letter twice in a row or two vowels in a row? | 199776 | Using the given rules, the words that are 1 letter long are A, B, C, D, E. Using the given rules, the words that are 2 letters long are AB, AC, AD, BA, BC, BD, BE, CA, CB, CD, CE, DA, DB, DC, DE, EB, EC, ED. Let $v_{n}$ be the number of words that are $n$ letters long and that begin with a vowel. Note that $v_{1}=2$ an... | 6.25 | [
7,
7,
6,
7,
6,
6,
5,
6
] |
If $4x + 14 = 8x - 48$, what is the value of $2x$? | 31 | Since $4x + 14 = 8x - 48$, then $14 + 48 = 8x - 4x$ or $62 = 4x$. Dividing both sides of this equation by 2, we obtain $\frac{4x}{2} = \frac{62}{2}$ which gives $2x = 31$. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
The integers -5 and 6 are shown on a number line. What is the distance between them? | 11 | The distance between two numbers on the number line is equal to their positive difference. Here, this distance is $6-(-5)=11$. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
The top section of an 8 cm by 6 cm rectangular sheet of paper is folded along a straight line so that when the top section lies flat on the bottom section, corner $P$ lies on top of corner $R$. What is the length of the crease? | 7.5 | Suppose that the crease intersects $PS$ at $X$, $QR$ at $Y$, and the line $PR$ at $Z$. We want to determine the length of $XY$. Since $P$ folds on top of $R$, then line segment $PZ$ folds on top of line segment $RZ$, since after the fold $Z$ corresponds with itself and $P$ corresponds with $R$. This means that $PZ=RZ$ ... | 5.375 | [
5,
5,
6,
5,
6,
5,
5,
6
] |
What is the value of $x$ if $P Q S$ is a straight line and $\angle P Q R=110^{\circ}$? | 24 | Since $P Q S$ is a straight line and $\angle P Q R=110^{\circ}$, then $\angle R Q S=180^{\circ}-\angle P Q R=70^{\circ}$. Since the sum of the angles in $\triangle Q R S$ is $180^{\circ}$, then $70^{\circ}+(3 x)^{\circ}+(x+14)^{\circ} =180^{\circ}$. Solving, $4 x =96$ gives $x =24$. | 2.875 | [
3,
3,
3,
3,
3,
3,
3,
2
] |
The real numbers $x, y$ and $z$ satisfy the three equations $x+y=7$, $xz=-180$, and $(x+y+z)^{2}=4$. If $S$ is the sum of the two possible values of $y$, what is $-S$? | 42 | Since $x+y=7$, then $x+y+z=7+z$. Thus, the equation $(x+y+z)^{2}=4$ becomes $(7+z)^{2}=4$. Since the square of $7+z$ equals 4, then $7+z=2$ or $7+z=-2$. If $7+z=2$, then $z=-5$. In this case, since $xz=-180$, we get $x=\frac{-180}{-5}=36$ which gives $y=7-x=-29$. If $7+z=-2$, then $z=-9$. In this case, since $xz=-180$,... | 4.25 | [
4,
5,
4,
4,
4,
4,
5,
4
] |
How many candies were in the bag before the first day if a group of friends eat candies over five days as follows: On the first day, they eat \( \frac{1}{2} \) of the candies, on the second day \( \frac{2}{3} \) of the remaining, on the third day \( \frac{3}{4} \) of the remaining, on the fourth day \( \frac{4}{5} \) o... | 720 | We work backwards through the given information. At the end, there is 1 candy remaining. Since \( \frac{5}{6} \) of the candies are removed on the fifth day, this 1 candy represents \( \frac{1}{6} \) of the candies left at the end of the fourth day. Thus, there were \( 6 \times 1 = 6 \) candies left at the end of the f... | 5.625 | [
5,
5,
6,
5,
7,
5,
6,
6
] |
The integers \(a, b,\) and \(c\) satisfy the equations \(a + 5 = b\), \(5 + b = c\), and \(b + c = a\). What is the value of \(b\)? | -10 | Since \(a + 5 = b\), then \(a = b - 5\). Substituting \(a = b - 5\) and \(c = 5 + b\) into \(b + c = a\), we obtain \(b + (5 + b) = b - 5\). Simplifying, we get \(2b + 5 = b - 5\), which gives \(b = -10\). | 1.875 | [
2,
2,
2,
2,
2,
2,
1,
2
] |
How many interior intersection points are there on a 12 by 12 grid of squares? | 121 | A 12 by 12 grid of squares will have 11 interior vertical lines and 11 interior horizontal lines. Each of the 11 interior vertical lines intersects each of the 11 interior horizontal lines and creates an interior intersection point. Therefore, the number of interior intersection points is $11 imes 11=121$. | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
What is the value of $n$ if $2^{n}=8^{20}$? | 60 | Since $8=2 \times 2 \times 2=2^{3}$, then $8^{20}=\left(2^{3}\right)^{20}=2^{3 \times 20}=2^{60}$. Thus, if $2^{n}=8^{20}$, then $n=60$. | 1.875 | [
2,
2,
2,
2,
2,
2,
2,
1
] |
Shuxin begins with 10 red candies, 7 yellow candies, and 3 blue candies. After eating some of the candies, there are equal numbers of red, yellow, and blue candies remaining. What is the smallest possible number of candies that Shuxin ate? | 11 | For there to be equal numbers of each colour of candy, there must be at most 3 red candies and at most 3 yellow candies, since there are 3 blue candies to start. Thus, Shuxin ate at least 7 red candies and at least 4 yellow candies. This means that Shuxin ate at least $7+4=11$ candies. We note that if Shuxin eats 7 red... | 3.125 | [
3,
3,
3,
3,
3,
3,
4,
3
] |
Suppose that $x$ and $y$ are real numbers with $-4 \leq x \leq -2$ and $2 \leq y \leq 4$. What is the greatest possible value of $\frac{x+y}{x}$? | \frac{1}{2} | We note that $\frac{x+y}{x} = \frac{x}{x} + \frac{y}{x} = 1 + \frac{y}{x}$. The greatest possible value of $\frac{x+y}{x} = 1 + \frac{y}{x}$ thus occurs when $\frac{y}{x}$ is as great as possible. Since $x$ is always negative and $y$ is always positive, then $\frac{y}{x}$ is negative. Therefore, for $\frac{y}{x}$ to be... | 3 | [
3,
3,
3,
3,
3,
3,
3,
3
] |
What is the quantity equivalent to '2% of 1'? | \frac{2}{100} | The quantity $2 \%$ is equivalent to the fraction $\frac{2}{100}$, so '2% of 1' is equal to $\frac{2}{100}$. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
Eugene swam on Sunday, Monday, and Tuesday. On Monday, he swam for 30 minutes. On Tuesday, he swam for 45 minutes. His average swim time over the three days was 34 minutes. For how many minutes did he swim on Sunday? | 27 | Since Eugene swam three times and had an average swim time of 34 minutes, he swam for \( 3 \times 34 = 102 \) minutes in total. Since he swam for 30 minutes and 45 minutes on Monday and Tuesday, then on Sunday, he swam for \( 102 - 30 - 45 = 27 \) minutes. | 2.125 | [
2,
2,
2,
2,
3,
2,
2,
2
] |
Evaluate the expression $\sqrt{13+\sqrt{7+\sqrt{4}}}$. | 4 | We evaluate from the inside towards the outside: $\sqrt{13+\sqrt{7+\sqrt{4}}}=\sqrt{13+\sqrt{7+2}}=\sqrt{13+\sqrt{9}}=\sqrt{13+3}=\sqrt{16}=4$. | 2.125 | [
3,
2,
2,
2,
2,
2,
2,
2
] |
Positive integers $a$ and $b$ satisfy $a b=2010$. If $a>b$, what is the smallest possible value of $a-b$? | 37 | Note that $2010=10(201)=2(5)(3)(67)$ and that 67 is prime. Therefore, the positive divisors of 2010 are $1,2,3,5,6,10,15,30,67,134,201,335,402,670$, $1005,2010$. Thus, the possible pairs $(a, b)$ with $a b=2010$ and $a>b$ are $(2010,1),(1005,2),(670,3)$, $(402,5),(335,6),(201,10),(134,15),(67,30)$. Of these pairs, the ... | 2.75 | [
2,
3,
3,
2,
3,
3,
3,
3
] |
How many of the numbers in Grace's sequence, starting from 43 and each number being 4 less than the previous one, are positive? | 11 | We write out the numbers in the sequence until we obtain a negative number: $43,39,35,31,27,23,19,15,11,7,3,-1$. Since each number is 4 less than the number before it, then once a negative number is reached, every following number will be negative. Thus, Grace writes 11 positive numbers in the sequence. | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
How many of the 200 students surveyed said that their favourite food was sandwiches, given the circle graph results? | 20 | Since the angle in the sector representing cookies is $90^{\circ}$, then this sector represents $\frac{1}{4}$ of the total circle. Therefore, 25% of the students chose cookies as their favourite food. Thus, the percentage of students who chose sandwiches was $100\%-30\%-25\%-35\%=10\%$. Since there are 200 students in ... | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
If the perimeter of a square is 28, what is the side length of the square? | 7 | Since a square has four equal sides, the side length of a square equals one-quarter of the perimeter of the square. Thus, the side length of a square with perimeter 28 is $28 \div 4 = 7$. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
In triangle $XYZ$, $XY=XZ$ and $W$ is on $XZ$ such that $XW=WY=YZ$. What is the measure of $\angle XYW$? | 36^{\circ} | Let $\angle XYW=\theta$. Since $\triangle XYW$ is isosceles with $WX=WY$, then $\angle YXW=\angle XYW=\theta$. Since the sum of the angles in $\triangle XYW$ is $180^{\circ}$, then $\angle XWY=180^{\circ}-2\theta$. Since $\angle XWY+\angle ZWY=180^{\circ}$, then $\angle ZWY=180^{\circ}-(180^{\circ}-2\theta)=2\theta$. S... | 4.625 | [
5,
5,
4,
5,
4,
5,
4,
5
] |
If $(x+a)(x+8)=x^{2}+bx+24$ for all values of $x$, what is the value of $a+b$? | 14 | Since $(x+a)(x+8)=x^{2}+bx+24$ for all $x$, then $x^{2}+ax+8x+8a=x^{2}+bx+24$ or $x^{2}+(a+8)x+8a=x^{2}+bx+24$ for all $x$. Since the equation is true for all $x$, then the coefficients on the left side must match the coefficients on the right side. Therefore, $a+8=b$ and $8a=24$. The second equation gives $a=3$, from ... | 2.875 | [
3,
3,
3,
2,
3,
3,
3,
3
] |
A tetrahedron of spheres is formed with thirteen layers and each sphere has a number written on it. The top sphere has a 1 written on it and each of the other spheres has written on it the number equal to the sum of the numbers on the spheres in the layer above with which it is in contact. What is the sum of the number... | 772626 | First, we fill in the numbers on the top four layers. The top layer consists of only one sphere, labelled 1. In the second layer, each sphere touches only one sphere in the layer above. This sphere is labelled 1, so each sphere in the second layer is labelled 1. In the third layer, each of the corner spheres touches on... | 7 | [
7,
7,
8,
7,
7,
7,
7,
6
] |
A rectangular piece of paper $P Q R S$ has $P Q=20$ and $Q R=15$. The piece of paper is glued flat on the surface of a large cube so that $Q$ and $S$ are at vertices of the cube. What is the shortest distance from $P$ to $R$, as measured through the cube? | 18.4 | Since $P Q R S$ is rectangular, then $\angle S R Q=\angle S P Q=90^{\circ}$. Also, $S R=P Q=20$ and $S P=Q R=15$. By the Pythagorean Theorem in $\triangle S P Q$, since $Q S>0$, we have $Q S=\sqrt{S P^{2}+P Q^{2}}=\sqrt{15^{2}+20^{2}}=\sqrt{225+400}=\sqrt{625}=25$. Draw perpendiculars from $P$ and $R$ to $X$ and $Y$, r... | 6.25 | [
6,
6,
6,
7,
6,
6,
6,
7
] |
Storage space on a computer is measured in gigabytes (GB) and megabytes (MB), where $1 \mathrm{~GB} = 1024 \mathrm{MB}$. Julia has an empty 300 GB hard drive and puts 300000 MB of data onto it. How much storage space on the hard drive remains empty? | 7200 \mathrm{MB} | Since $1 \mathrm{~GB} = 1024 \mathrm{MB}$, then Julia's 300 GB hard drive has $300 \times 1024 = 307200 \mathrm{MB}$ of storage space. When Julia puts 300000 MB of data on the empty hard drive, the amount of empty space remaining is $307200 - 300000 = 7200 \mathrm{MB}$. | 2.125 | [
2,
2,
2,
2,
2,
3,
2,
2
] |
If $wxyz$ is a four-digit positive integer with $w \neq 0$, the layer sum of this integer equals $wxyz + xyz + yz + z$. If the layer sum of $wxyz$ equals 2014, what is the value of $w + x + y + z$? | 13 | The layer sum of $wxyz$ equals 2014. This means that the sum of the integer with digits $wxyz$, the integer with digits $xyz$, the integer with digits $yz$, and the integer $z$ is 2014. Note that the integer with digits $wxyz$ equals $1000w + 100x + 10y + z$, the integer with digits $xyz$ equals $100x + 10y + z$, and t... | 5.875 | [
6,
6,
5,
6,
6,
6,
6,
6
] |
The regular price for a bicycle is $\$320$. The bicycle is on sale for $20\%$ off. The regular price for a helmet is $\$80$. The helmet is on sale for $10\%$ off. If Sandra bought both items on sale, what is her percentage savings on the total purchase? | 18\% | Since the regular price for the bicycle is $\$320$ and the savings are $20\%$, then the amount of money that Sandra saves on the bicycle is $\$320 \times 20\%=\$320 \times 0.2=\$64$. Since the regular price for the helmet is $\$80$ and the savings are $10\%$, then the amount of money that Sandra saves on the helmet is ... | 3.125 | [
3,
3,
3,
3,
4,
3,
3,
3
] |
Suppose that $N = 3x + 4y + 5z$, where $x$ equals 1 or -1, and $y$ equals 1 or -1, and $z$ equals 1 or -1. How many of the following statements are true? - $N$ can equal 0. - $N$ is always odd. - $N$ cannot equal 4. - $N$ is always even. | 1 | When $N = 3x + 4y + 5z$ with each of $x, y$ and $z$ equal to either 1 or -1, there are 8 possible combinations of values for $x, y$ and $z$. From this information, $N$ cannot equal 0, $N$ is never odd, $N$ can equal 4, and $N$ is always even. Therefore, exactly one of the four given statements is true. | 3.625 | [
3,
4,
3,
4,
3,
4,
3,
5
] |
How many such nine-digit positive integers can Ricardo make if he wants to arrange three 1s, three 2s, two 3s, and one 4 with the properties that there is at least one 1 before the first 2, at least one 2 before the first 3, and at least one 3 before the 4, and no digit 2 can be next to another 2? | 254 | Case 1: $N$ begins 12. There are 10 possible pairs of positions for the 2s. There are 10 pairs of positions for the 1s. There are 2 orders for the 3s and 4. In this case, there are $10 \times 10 \times 2=200$ possible integers $N$. Case 2: $N$ begins 112. There are 6 possible pairs of positions for the 2s. There are 4 ... | 6.5 | [
7,
6,
6,
7,
6,
6,
7,
7
] |
If points $P, Q, R$, and $S$ are arranged in order on a line segment with $P Q=1, Q R=2 P Q$, and $R S=3 Q R$, what is the length of $P S$? | 9 | Since $P Q=1$ and $Q R=2 P Q$, then $Q R=2$. Since $Q R=2$ and $R S=3 Q R$, then $R S=3(2)=6$. Therefore, $P S=P Q+Q R+R S=1+2+6=9$. | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
Two different numbers are randomly selected from the set $\{-3, -1, 0, 2, 4\}$ and then multiplied together. What is the probability that the product of the two numbers chosen is 0? | \frac{2}{5} | There are ten possible pairs of numbers that can be chosen: -3 and $-1$; -3 and $0$; -3 and 2; -3 and $4$; -1 and $0$; -1 and $2$; -1 and $4$; 0 and $2$; 0 and $4$; 2 and 4. Each pair is equally likely to be chosen. Pairs that include 0 (4 pairs) have a product of 0; pairs that do not include 0 (6 of them) do not have ... | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
What is the number of positive integers $p$ for which $-1<\sqrt{p}-\sqrt{100}<1$? | 39 | If $-1<\sqrt{p}-\sqrt{100}<1$, then $-1<\sqrt{p}-10<1$ or $9<\sqrt{p}<11$. Since $\sqrt{p}$ is greater than 9, then $p$ is greater than $9^{2}=81$. Since $\sqrt{p}$ is less than 11, then $p$ is less than $11^{2}=121$. In other words, $81<p<121$. Since $p$ is a positive integer, then $82 \leq p \leq 120$. Therefore, the... | 3.125 | [
3,
3,
3,
3,
3,
4,
3,
3
] |
What is the value of $(5
abla 1)+(4
abla 1)$ if the operation $k
abla m$ is defined as $k(k-m)$? | 32 | From the definition, $(5
abla 1)+(4
abla 1)=5(5-1)+4(4-1)=5(4)+4(3)=20+12=32$. | 1.5 | [
1,
1,
2,
1,
2,
1,
2,
2
] |
Two numbers $a$ and $b$ with $0 \leq a \leq 1$ and $0 \leq b \leq 1$ are chosen at random. The number $c$ is defined by $c=2a+2b$. The numbers $a, b$ and $c$ are each rounded to the nearest integer to give $A, B$ and $C$, respectively. What is the probability that $2A+2B=C$? | \frac{7}{16} | By definition, if $0 \leq a<\frac{1}{2}$, then $A=0$ and if $\frac{1}{2} \leq a \leq 1$, then $A=1$. Similarly, if $0 \leq b<\frac{1}{2}$, then $B=0$ and if $\frac{1}{2} \leq b \leq 1$, then $B=1$. We keep track of our information on a set of axes, labelled $a$ and $b$. The area of the region of possible pairs $(a, b)$... | 5.625 | [
5,
6,
6,
6,
6,
6,
5,
5
] |
If $u=-6$ and $x= rac{1}{3}(3-4 u)$, what is the value of $x$? | 9 | Substituting, $x= rac{1}{3}(3-4 u)= rac{1}{3}(3-4(-6))= rac{1}{3}(3+24)= rac{1}{3}(27)=9$. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
What is the sum of all numbers $q$ which can be written in the form $q=\frac{a}{b}$ where $a$ and $b$ are positive integers with $b \leq 10$ and for which there are exactly 19 integers $n$ that satisfy $\sqrt{q}<n<q$? | 777.5 | Suppose that a number $q$ has the property that there are exactly 19 integers $n$ with $\sqrt{q}<n<q$. Suppose that these 19 integers are $m, m+1, m+2, \ldots, m+17, m+18$. Then $\sqrt{q}<m<m+1<m+2<\cdots<m+17<m+18<q$. This tells us that $q-\sqrt{q}>(m+18)-m=18$ because $q-\sqrt{q}$ is as small as possible when $q$ is ... | 6.875 | [
7,
7,
7,
6,
7,
7,
7,
7
] |
In $\triangle ABC$, points $D$ and $E$ lie on $AB$, as shown. If $AD=DE=EB=CD=CE$, what is the measure of $\angle ABC$? | 30^{\circ} | Since $CD=DE=EC$, then $\triangle CDE$ is equilateral, which means that $\angle DEC=60^{\circ}$. Since $\angle DEB$ is a straight angle, then $\angle CEB=180^{\circ}-\angle DEC=180^{\circ}-60^{\circ}=120^{\circ}$. Since $CE=EB$, then $\triangle CEB$ is isosceles with $\angle ECB=\angle EBC$. Since $\angle ECB+\angle CE... | 3.5 | [
3,
3,
3,
4,
4,
4,
3,
4
] |
Find the least possible area of a convex set in the plane that intersects both branches of the hyperbola $xy = 1$ and both branches of the hyperbola $xy = -1$. (A set $S$ in the plane is called \emph{convex} if for any two points in $S$ the line segment connecting them is contained in $S$.) | 4 | The minimum is 4, achieved by the square with vertices $(\pm 1, \pm 1)$.
\textbf{First solution:}
To prove that 4 is a lower bound, let $S$ be a convex set of the desired form. Choose $A,B,C,D \in S$ lying on the branches of the two hyperbolas, with $A$ in the upper right quadrant, $B$ in the upper left, $C$ in the lo... | 7.25 | [
7,
7,
8,
7,
6,
7,
8,
8
] |
Evaluate \[ \lim_{x \to 1^-} \prod_{n=0}^\infty \left(\frac{1 + x^{n+1}}{1 + x^n}\right)^{x^n}. \] | \frac{2}{e} | By taking logarithms, we see that the desired limit is $\exp(L)$, where $L = \lim_{x\to 1^-} \sum_{n=0}^{\infty} x^n \left( \ln(1+x^{n+1}) - \ln(1+x^n) \right)$. Now \begin{align*} &\sum_{n=0}^N x^n \left( \ln(1+x^{n+1}) - \ln(1+x^n) \right) \\ & = 1/x \sum_{n=0}^N x^{n+1} \ln(1+x^{n+1}) - \sum_{n=0}^N x^n\ln(1+x^n) \\... | 8.125 | [
8,
9,
8,
8,
8,
8,
8,
8
] |
For every real number $x$, what is the value of the expression $(x+1)^{2} - x^{2}$? | 2x + 1 | Expanding and simplifying, $(x+1)^{2} - x^{2} = (x^{2} + 2x + 1) - x^{2} = 2x + 1$. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
Suppose that $PQRS TUVW$ is a regular octagon. There are 70 ways in which four of its sides can be chosen at random. If four of its sides are chosen at random and each of these sides is extended infinitely in both directions, what is the probability that they will meet to form a quadrilateral that contains the octagon? | \frac{19}{35} | If the four sides that are chosen are adjacent, then when these four sides are extended, they will not form a quadrilateral that encloses the octagon. If the four sides are chosen so that there are exactly three adjacent sides that are not chosen and one other side not chosen, then when these four sides are extended, t... | 5.75 | [
6,
6,
6,
6,
5,
6,
6,
5
] |
Let $d_n$ be the determinant of the $n \times n$ matrix whose entries, from
left to right and then from top to bottom, are $\cos 1, \cos 2, \dots, \cos
n^2$. Evaluate
$\lim_{n\to\infty} d_n$. | 0 | The limit is $0$; we will show this by checking that $d_n = 0$ for all $n \geq 3$.
Starting from the given matrix, add the third column to the first column; this does not change the
determinant. However, thanks to the identity
$\cos x + \cos y = 2 \cos \frac{x+y}{2} \cos \frac{x-y}{2}$,
the resulting matrix has the for... | 7 | [
7,
7,
7,
8,
6,
7,
7,
7
] |
Calculate the value of $(3,1) \nabla (4,2)$ using the operation ' $\nabla$ ' defined by $(a, b) \nabla (c, d)=ac+bd$. | 14 | From the definition, $(3,1) \nabla (4,2)=(3)(4)+(1)(2)=12+2=14$. | 1.875 | [
2,
2,
2,
2,
2,
2,
1,
2
] |
The set $S$ consists of 9 distinct positive integers. The average of the two smallest integers in $S$ is 5. The average of the two largest integers in $S$ is 22. What is the greatest possible average of all of the integers of $S$? | 16 | Since the average of the two smallest integers in $S$ is 5, their sum is $2 \cdot 5=10$. Since the average of the two largest integers in $S$ is 22, their sum is $2 \cdot 22=44$. Suppose that the other five integers in the set $S$ are $p<q<r<t<u$. (Note that the integers in $S$ are all distinct.) The average of the nin... | 5.875 | [
6,
6,
6,
6,
5,
6,
6,
6
] |
If each of Bill's steps is $ rac{1}{2}$ metre long, how many steps does Bill take to walk 12 metres in a straight line? | 24 | Since each of Bill's steps is $ rac{1}{2}$ metre long, then 2 of Bill's steps measure 1 m. To walk 12 m, Bill thus takes $12 imes 2=24$ steps. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
Rectangle $W X Y Z$ has $W X=4, W Z=3$, and $Z V=3$. The rectangle is curled without overlapping into a cylinder so that sides $W Z$ and $X Y$ touch each other. In other words, $W$ touches $X$ and $Z$ touches $Y$. The shortest distance from $W$ to $V$ through the inside of the cylinder can be written in the form $\sqrt... | 18 | When the cylinder is created, $W$ and $X$ touch and $Z$ and $Y$ touch. This means that $W Y$ is vertical and so is perpendicular to the plane of the circular base of the cylinder. This means that $\triangle V Y W$ is right-angled at $Y$. By the Pythagorean Theorem, $W V^{2}=W Y^{2}+V Y^{2}$. Note that $W Y$ equals the ... | 6.875 | [
7,
8,
8,
6,
7,
7,
5,
7
] |
For how many positive integers $n$, with $n \leq 100$, is $n^{3}+5n^{2}$ the square of an integer? | 8 | For $n^{3}+5n^{2}$ to be the square of an integer, $\sqrt{n^{3}+5n^{2}}$ must be an integer. We know that $\sqrt{n^{3}+5n^{2}}=\sqrt{n^{2}(n+5)}=\sqrt{n^{2}} \sqrt{n+5}=n \sqrt{n+5}$. For $n \sqrt{n+5}$ to be an integer, $\sqrt{n+5}$ must be an integer. In other words, $n+5$ must be a perfect square. Since $n$ is betwe... | 3.625 | [
4,
3,
4,
4,
3,
4,
3,
4
] |
Cube $A B C D E F G H$ has edge length 100. Point $P$ is on $A B$, point $Q$ is on $A D$, and point $R$ is on $A F$, as shown, so that $A P=x, A Q=x+1$ and $A R=\frac{x+1}{2 x}$ for some integer $x$. For how many integers $x$ is the volume of triangular-based pyramid $A P Q R$ between $0.04 \%$ and $0.08 \%$ of the vol... | 28 | Triangular-based pyramid $A P Q R$ can be thought of as having triangular base $\triangle A P Q$ and height $A R$. Since this pyramid is built at a vertex of the cube, then $\triangle A P Q$ is right-angled at $A$ and $A R$ is perpendicular to the base. The area of $\triangle A P Q$ is $\frac{1}{2} \times A P \times A ... | 7 | [
7,
7,
7,
7,
7,
7,
7,
7
] |
How many foonies are in a stack that has a volume of $50 \mathrm{~cm}^{3}$, given that each foonie has a volume of $2.5 \mathrm{~cm}^{3}$? | 20 | Since the face of a foonie has area $5 \mathrm{~cm}^{2}$ and its thickness is 0.5 cm, then the volume of one foonie is $5 \times 0.5=2.5 \mathrm{~cm}^{3}$. If a stack of foonies has a volume of $50 \mathrm{~cm}^{3}$ and each foonie has a volume of $2.5 \mathrm{~cm}^{3}$, then there are $50 \div 2.5=20$ foonies in the s... | 1.875 | [
2,
2,
2,
2,
2,
1,
2,
2
] |
If $x=11, y=8$, and $2x+3z=5y$, what is the value of $z$? | 6 | Since $x=11, y=8$ and $2x+3z=5y$, then $2 \times 11+3z=5 \times 8$ or $3z=40-22$. Therefore, $3z=18$ and so $z=6$. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
If $x = 2y$ and $y \neq 0$, what is the value of $(x-y)(2x+y)$? | 5y^{2} | Since $x = 2y$, then $(x-y)(2x+y) = (2y-y)(2(2y)+y) = (y)(5y) = 5y^{2}$. | 1.625 | [
2,
2,
2,
2,
1,
1,
1,
2
] |
A sequence consists of 2010 terms. Each term after the first is 1 larger than the previous term. The sum of the 2010 terms is 5307. When every second term is added up, starting with the first term and ending with the second last term, what is the sum? | 2151 | We label the terms $x_{1}, x_{2}, x_{3}, \ldots, x_{2009}, x_{2010}$. Suppose that $S$ is the sum of the odd-numbered terms in the sequence; that is, $S=x_{1}+x_{3}+x_{5}+\cdots+x_{2007}+x_{2009}$. We know that the sum of all of the terms is 5307; that is, $x_{1}+x_{2}+x_{3}+\cdots+x_{2009}+x_{2010}=5307$. Next, we pai... | 4.5 | [
5,
4,
5,
4,
4,
5,
5,
4
] |
Alain and Louise are driving on a circular track with radius 25 km. Alain leaves the starting line first, going clockwise at 80 km/h. Fifteen minutes later, Louise leaves the same starting line, going counterclockwise at 100 km/h. For how many hours will Louise have been driving when they pass each other for the fourth... | \\frac{10\\pi-1}{9} | Since the track is circular with radius 25 km, then its circumference is $2 \pi(25)=50 \pi$ km. In the 15 minutes that Alain drives at 80 km/h, he drives a distance of $\frac{1}{4}(80)=20$ km (because 15 minutes is one-quarter of an hour). When Louise starts driving, she drives in the opposite direction to Alain. Suppo... | 6.25 | [
6,
6,
6,
6,
6,
7,
6,
7
] |
In a rectangle $P Q R S$ with $P Q=5$ and $Q R=3$, $P R$ is divided into three segments of equal length by points $T$ and $U$. What is the area of quadrilateral $S T Q U$? | 5 | Since $P Q R S$ is a rectangle, then $P Q$ is perpendicular to $Q R$. Therefore, the area of $\triangle P Q R$ is $\frac{1}{2}(P Q)(Q R)=\frac{1}{2}(5)(3)=\frac{15}{2}$. Since $P T=T U=U R$, then the areas of $\triangle P T Q, \triangle T U Q$ and $\triangle U R Q$ are equal. Therefore, the area of $\triangle T U Q$ is... | 2.875 | [
2,
3,
3,
3,
3,
3,
3,
3
] |
What is the average (mean) number of hamburgers eaten per student if 12 students ate 0 hamburgers, 14 students ate 1 hamburger, 8 students ate 2 hamburgers, 4 students ate 3 hamburgers, and 2 students ate 4 hamburgers? | 1.25 | The mean number of hamburgers eaten per student equals the total number of hamburgers eaten divided by the total number of students. 12 students each eat 0 hamburgers. This is a total of 0 hamburgers eaten. 14 students each eat 1 hamburger. This is a total of 14 hamburgers eaten. 8 students each eat 2 hamburgers. This ... | 1.25 | [
1,
1,
2,
1,
2,
1,
1,
1
] |
There are 20 students in a class. In total, 10 of them have black hair, 5 of them wear glasses, and 3 of them both have black hair and wear glasses. How many of the students have black hair but do not wear glasses? | 7 | Since 10 students have black hair and 3 students have black hair and wear glasses, then a total of $10-3=7$ students have black hair but do not wear glasses. | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
After a fair die with faces numbered 1 to 6 is rolled, the number on the top face is $x$. What is the most likely outcome? | x > 2 | With a fair die that has faces numbered from 1 to 6, the probability of rolling each of 1 to 6 is $\frac{1}{6}$. We calculate the probability for each of the five choices. There are 4 values of $x$ that satisfy $x>2$, so the probability is $\frac{4}{6}=\frac{2}{3}$. There are 2 values of $x$ that satisfy $x=4$ or $x=5$... | 2 | [
2,
2,
2,
2,
2,
2,
2,
2
] |
Find the minimum value of $| \sin x + \cos x + \tan x + \cot x + \sec x + \csc x |$ for real numbers $x$. | 2\sqrt{2} - 1 | \textbf{First solution:} Write \begin{align*} f(x) &= \sin x + \cos x + \tan x + \cot x + \sec x + \csc x \\ &= \sin x + \cos x + \frac{1}{\sin x \cos x} + \frac{\sin x + \cos x}{\sin x \cos x}. \end{align*} We can write $\sin x + \cos x = \sqrt{2} \cos(\pi/4 - x)$; this suggests making the substitution $y = \pi/4 - x$... | 6.875 | [
7,
7,
7,
7,
7,
6,
7,
7
] |
Suppose that $x$ and $y$ are positive numbers with $xy=\frac{1}{9}$, $x(y+1)=\frac{7}{9}$, and $y(x+1)=\frac{5}{18}$. What is the value of $(x+1)(y+1)$? | \frac{35}{18} | If we multiply the second and third equations together, we obtain $x(y+1)y(y+1)=\frac{7}{9} \cdot \frac{5}{18}$ or $xy(x+1)(y+1)=\frac{35}{162}$. From the first equation, $xy=\frac{1}{9}$. Therefore, $\frac{1}{9}(x+1)(y+1)=\frac{35}{162}$ or $(x+1)(y+1)=9\left(\frac{35}{162}\right)=\frac{35}{18}$. | 4.75 | [
4,
5,
4,
5,
5,
5,
5,
5
] |
A rectangle is divided into four smaller rectangles, labelled W, X, Y, and Z. The perimeters of rectangles W, X, and Y are 2, 3, and 5, respectively. What is the perimeter of rectangle Z? | 6 | Label the lengths of the vertical and horizontal segments as $a, b, c, d$. Rectangle W is $b$ by $c$, so its perimeter is $2 b+2 c$, which equals 2. Rectangle X is $b$ by $d$, so its perimeter is $2 b+2 d$, which equals 3. Rectangle Y is $a$ by $c$, so its perimeter is $2 a+2 c$, which equals 5. Rectangle Z is $a$ by $... | 3.5 | [
3,
4,
5,
3,
4,
3,
3,
3
] |
Let $S = \{1, 2, \dots, n\}$ for some integer $n > 1$. Say a permutation $\pi$ of $S$ has a \emph{local maximum} at $k \in S$ if \begin{enumerate} \item[(i)] $\pi(k) > \pi(k+1)$ for $k=1$; \item[(ii)] $\pi(k-1) < \pi(k)$ and $\pi(k) > \pi(k+1)$ for $1 < k < n$; \item[(iii)] $\pi(k-1) < \pi(k)$ for $k=n$. \end{enumerate... | \frac{n+1}{3} | \textbf{First solution:} By the linearity of expectation, the average number of local maxima is equal to the sum of the probability of having a local maximum at $k$ over $k=1,\dots, n$. For $k=1$, this probability is 1/2: given the pair $\{\pi(1), \pi(2)\}$, it is equally likely that $\pi(1)$ or $\pi(2)$ is bigger. Sim... | 6 | [
6,
6,
6,
6,
7,
5,
6,
6
] |
For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1, \sqrt{2k})$. Evaluate
\[
\sum_{k=1}^\infty (-1)^{k-1} \frac{A(k)}{k}.
\] | \frac{\pi^2}{16} | We will prove that the sum converges to $\pi^2/16$.
Note first that the sum does not converge absolutely, so we are not free to rearrange it arbitrarily. For that matter, the standard alternating sum test does not apply because the absolute values of the terms does not decrease to 0, so even the convergence of the sum ... | 7.5 | [
8,
8,
7,
8,
7,
8,
7,
7
] |
Snacks are purchased for 17 soccer players. Juice boxes come in packs of 3 and cost $2.00 per pack. Apples come in bags of 5 and cost $4.00 per bag. What is the minimum amount of money that Danny spends to ensure every player gets a juice box and an apple? | \$28.00 | Since juice boxes come in packs of 3, Danny needs to buy at least 6 packs for the 17 players. Since apples come in bags of 5, Danny needs to buy at least 4 bags. Therefore, the minimum amount that Danny can spend is $6 \cdot \$2.00 + 4 \cdot \$4.00 = \$28.00$. | 2.5 | [
2,
2,
2,
3,
3,
3,
2,
3
] |
If $n$ is a positive integer, the notation $n$! (read " $n$ factorial") is used to represent the product of the integers from 1 to $n$. That is, $n!=n(n-1)(n-2) \cdots(3)(2)(1)$. For example, $4!=4(3)(2)(1)=24$ and $1!=1$. If $a$ and $b$ are positive integers with $b>a$, what is the ones (units) digit of $b!-a$! that c... | 7 | The first few values of $n$! are
\begin{aligned}
& 1!=1 \\
& 2!=2(1)=2 \\
& 3!=3(2)(1)=6 \\
& 4!=4(3)(2)(1)=24 \\
& 5!=5(4)(3)(2)(1)=120
\end{aligned}
We note that
\begin{aligned}
& 2!-1!=1 \\
& 4!-1!=23 \\
& 3!-1!=5 \\
& 5!-1!=119
\end{aligned}
This means that if $a$ and $b$ are positive integers with $b>a$, then ... | 4.875 | [
5,
4,
4,
6,
6,
5,
4,
5
] |
Pablo has 27 solid $1 \times 1 \times 1$ cubes that he assembles in a larger $3 \times 3 \times 3$ cube. If 10 of the smaller cubes are red, 9 are blue, and 8 are yellow, what is the smallest possible surface area of the larger cube that is red? | 12 | The 27 small cubes that make up the larger $3 \times 3 \times 3$ can be broken into 4 categories: 1 small cube in the very centre of the larger cube (not seen in the diagram), 8 small cubes at the vertices of larger cube (an example is marked with $V$), 12 small cubes on the edges not at vertices (an example is marked ... | 4.5 | [
5,
5,
4,
4,
5,
4,
5,
4
] |
Find a nonzero polynomial $P(x,y)$ such that $P(\lfloor a \rfloor, \lfloor 2a \rfloor) = 0$ for all real numbers $a$. (Note: $\lfloor \nu \rfloor$ is the greatest integer less than or equal to $\nu$.) | (y-2x)(y-2x-1) | Take $P(x,y) = (y-2x)(y-2x-1)$. To see that this works, first note that if $m = \lfloor a \rfloor$, then $2m$ is an integer less than or equal to $2a$, so $2m \leq \lfloor 2a \rfloor$. On the other hand, $m+1$ is an integer strictly greater than $a$, so $2m+2$ is an integer strictly greater than $2a$, so $\lfloor 2a \r... | 6.125 | [
6,
7,
6,
6,
6,
6,
6,
6
] |
For all $n \geq 1$, let \[ a_n = \sum_{k=1}^{n-1} \frac{\sin \left( \frac{(2k-1)\pi}{2n} \right)}{\cos^2 \left( \frac{(k-1)\pi}{2n} \right) \cos^2 \left( \frac{k\pi}{2n} \right)}. \] Determine \[ \lim_{n \to \infty} \frac{a_n}{n^3}. \] | \frac{8}{\pi^3} | The answer is $\frac{8}{\pi^3}$.
By the double angle and sum-product identities for cosine, we have \begin{align*} 2\cos^2\left(\frac{(k-1)\pi}{2n}\right) - 2\cos^2 \left(\frac{k\pi}{2n}\right) &= \cos\left(\frac{(k-1)\pi}{n}\right) - \cos\left(\frac{k\pi}{n}\right) \\ &= 2\sin\left(\frac{(2k-1)\pi}{2n}\right) \sin\le... | 8.25 | [
9,
8,
9,
8,
8,
8,
8,
8
] |
For how many odd integers $k$ between 0 and 100 does the equation $2^{4m^{2}}+2^{m^{2}-n^{2}+4}=2^{k+4}+2^{3m^{2}+n^{2}+k}$ have exactly two pairs of positive integers $(m, n)$ that are solutions? | 18 | Step 1: Using parity and properties of powers of 2 to simplify the equation. We note that if $2^{x}=2^{y}$ for some real numbers $x$ and $y$, then $x=y$. We examine equations of the form $2^{a}+2^{b}=2^{c}+2^{d}$ where $a, b, c$, and $d$ are integers. We may assume without loss of generality that $a \leq b$ and $c \leq... | 6.75 | [
6,
7,
7,
6,
7,
7,
7,
7
] |
If $2^{x}=16$, what is the value of $2^{x+3}$? | 128 | Since $2^{x}=16$, then $2^{x+3}=2^{3} 2^{x}=8(16)=128$. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
Each of $a, b$ and $c$ is equal to a number from the list $3^{1}, 3^{2}, 3^{3}, 3^{4}, 3^{5}, 3^{6}, 3^{7}, 3^{8}$. There are $N$ triples $(a, b, c)$ with $a \leq b \leq c$ for which each of $\frac{ab}{c}, \frac{ac}{b}$ and $\frac{bc}{a}$ is equal to an integer. What is the value of $N$? | 86 | We write $a=3^{r}, b=3^{s}$ and $c=3^{t}$ where each of $r, s, t$ is between 1 and 8, inclusive. Since $a \leq b \leq c$, then $r \leq s \leq t$. Next, we note that $\frac{ab}{c}=\frac{3^{r} 3^{s}}{3^{t}}=3^{r+s-t}$, $\frac{ac}{b}=\frac{3^{r} 3^{t}}{3^{s}}=3^{r+t-s}$, and $\frac{bc}{a}=\frac{3^{s} 3^{t}}{3^{r}}=3^{s+t-... | 5.875 | [
6,
6,
7,
5,
6,
6,
6,
5
] |
Let $a$ and $b$ be positive integers for which $45a+b=2021$. What is the minimum possible value of $a+b$? | 85 | Since $a$ is a positive integer, $45 a$ is a positive integer. Since $b$ is a positive integer, $45 a$ is less than 2021. The largest multiple of 45 less than 2021 is $45 \times 44=1980$. (Note that $45 \cdot 45=2025$ which is greater than 2021.) If $a=44$, then $b=2021-45 \cdot 44=41$. Here, $a+b=44+41=85$. If $a$ is ... | 2.875 | [
3,
2,
3,
3,
3,
4,
3,
2
] |
Given a positive integer $n$, let $M(n)$ be the largest integer $m$ such that \[ \binom{m}{n-1} > \binom{m-1}{n}. \] Evaluate \[ \lim_{n \to \infty} \frac{M(n)}{n}. \] | \frac{3+\sqrt{5}}{2} | The answer is $\frac{3+\sqrt{5}}{2}$. Note that for $m > n+1$, both binomial coefficients are nonzero and their ratio is \[ {m\choose n-1}/{m-1\choose n} = \frac{m!n!(m-n-1)!}{(m-1)!(n-1)!(m-n+1)!} = \frac{mn}{(m-n+1)(m-n)}. \] Thus the condition ${m\choose{n-1}} > {{m-1}\choose n}$ is equivalent to $(m-n+1)(m-n)-mn < ... | 8 | [
8,
8,
8,
8,
8,
8,
8,
8
] |
A pizza is cut into 10 pieces. Two of the pieces are each \(\frac{1}{24}\) of the whole pizza, four are each \(\frac{1}{12}\), two are each \(\frac{1}{8}\), and two are each \(\frac{1}{6}\). A group of \(n\) friends share the pizza by distributing all of these pieces. They do not cut any of these pieces. Each of the \(... | 39 | Each of the \(n\) friends is to receive \(\frac{1}{n}\) of the pizza. Since there are two pieces that are each \(\frac{1}{6}\) of the pizza and these pieces cannot be cut, then each friend receives at least \(\frac{1}{6}\) of the pizza. This means that there cannot be more than 6 friends; that is, \(n \leq 6\). Therefo... | 5.375 | [
5,
5,
6,
5,
6,
4,
6,
6
] |
Four points are chosen uniformly and independently at random in the interior of a given circle. Find the probability that they are the vertices of a convex quadrilateral. | 1 - \frac{35}{12 \pi^2} | \textbf{First solution:} (by Daniel Kane) The probability is $1 - \frac{35}{12\pi^2}$. We start with some notation and simplifications. For simplicity, we assume without loss of generality that the circle has radius 1. Let $E$ denote the expected value of a random variable over all choices of $P,Q,R$. Write $[XYZ]$ for... | 8.125 | [
8,
8,
8,
9,
8,
8,
8,
8
] |
When a number is tripled and then decreased by 5, the result is 16. What is the original number? | 7 | To get back to the original number, we undo the given operations. We add 5 to 16 to obtain 21 and then divide by 3 to obtain 7. These are the 'inverse' operations of decreasing by 5 and multiplying by 3. | 1 | [
1,
1,
1,
1,
1,
1,
1,
1
] |
The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,\dots,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? | 61917364224 | The number of such colorings is $2^{20} 3^{10} = 61917364224$. Identify the three colors red, white, and blue with (in some order) the elements of the field \mathbb{F}_3 of three elements (i.e., the ring of integers mod 3). The set of colorings may then be identified with the \mathbb{F}_3-vector space \mathbb{F}_3^E ge... | 7.625 | [
8,
7,
8,
8,
8,
8,
7,
7
] |
On the number line, points $M$ and $N$ divide $L P$ into three equal parts. What is the value at $M$? | \frac{1}{9} | The difference between $\frac{1}{6}$ and $\frac{1}{12}$ is $\frac{1}{6}-\frac{1}{12}=\frac{2}{12}-\frac{1}{12}=\frac{1}{12}$, so $L P=\frac{1}{12}$. Since $L P$ is divided into three equal parts, then this distance is divided into three equal parts, each equal to $\frac{1}{12} \div 3=\frac{1}{12} \times \frac{1}{3}=\fr... | 2.25 | [
2,
3,
2,
2,
2,
2,
2,
3
] |
Point $P$ is on the $y$-axis with $y$-coordinate greater than 0 and less than 100. A circle is drawn through $P, Q(4,4)$ and $O(0,0)$. How many possible positions for $P$ are there so that the radius of this circle is an integer? | 66 | Suppose that $P$ has coordinates $P(0,2a)$ for some real number $a$. Since $P$ has $y$-coordinate greater than 0 and less than 100, then $0 < 2a < 100$ or $0 < a < 50$. We determine an expression for the radius of the circle in terms of $a$ and then determine how many values of $a$ give an integer radius. We determine ... | 6.125 | [
6,
6,
6,
6,
6,
7,
6,
6
] |
Find all positive integers $n, k_1, \dots, k_n$ such that $k_1 + \cdots + k_n = 5n-4$ and \[ \frac{1}{k_1} + \cdots + \frac{1}{k_n} = 1. \] | n = 1, k_1 = 1; n = 3, (k_1,k_2,k_3) = (2,3,6); n = 4, (k_1,k_2,k_3,k_4) = (4,4,4,4) | By the arithmetic-harmonic mean inequality or the Cauchy-Schwarz inequality, \[ (k_1 + \cdots + k_n)\left(\frac{1}{k_1} + \cdots + \frac{1}{k_n} \right) \geq n^2. \] We must thus have $5n-4 \geq n^2$, so $n \leq 4$. Without loss of generality, we may suppose that $k_1 \leq \cdots \leq k_n$. If $n=1$, we must have $k_1 ... | 6.625 | [
7,
7,
7,
6,
7,
7,
6,
6
] |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.