problem
stringlengths 18
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|---|---|---|
Given the quadratic equation $3x^2 - mx + 2x + 6 = 0$, find the values of $m$ for which the quadratic equation will have real and equal roots.
|
2 - 6\sqrt{2} \text{ and } 2 + 6\sqrt{2}
| 0.75 |
Find the hundreds digit of $(30! - 25!)$
|
0
| 0.833333 |
If $5(2y + 3\sqrt{3}) = S$, calculate the value of $10(4y + 6\sqrt{3})$.
|
4S
| 0.916667 |
Given that $n$ is the smallest positive integer such that $n$ is divisible by $12$, $n^2$ is a perfect cube, $n^3$ is a perfect square, and $n^4$ is a perfect fifth power, find the number of digits of $n$.
|
24
| 0.333333 |
What is the maximum value of $\frac{(2^t - 5t)t}{4^t}$ for real values of $t$?
|
\frac{1}{20}
| 0.083333 |
Four red beads, two white beads, and two blue beads are placed in line in random order. Calculate the probability that no two beads of the same color are neighbors.
|
\frac{1}{14}
| 0.25 |
A biased coin has a probability of $\frac{3}{4}$ of landing heads and $\frac{1}{4}$ of landing tails on each toss. The outcomes of the tosses are independent. The probability of winning Game C, where the player tosses the coin four times and wins if either all four outcomes are heads or all four are tails, can be compared to the probability of winning Game D, where the player tosses the coin five times and wins if the first two tosses are the same, the third toss is different from the first two, and the last two tosses are the same as the first two. Determine the difference in the probabilities of winning Game C and Game D.
|
\frac{61}{256}
| 0.75 |
Given a two-digit number, determine how many numbers satisfy the condition that the unit digit of the number formed by subtracting the sum of its digits is divisible by 7.
|
10
| 0.916667 |
A pentagon is inscribed in a circle. Find the sum of the angles inscribed in the five arcs cut off by the sides of the pentagon.
|
180^{\circ}
| 0.916667 |
Given that a medium bottle can hold $60$ milliliters and a large bottle can hold $720$ milliliters, and there are small bottles that hold $40$ or $45$ milliliters, determine the total number of small bottles that Jasmine must buy to completely fill the large bottle.
|
16
| 0.583333 |
Given that Linda has 30 coins consisting of nickels and dimes, and that if her nickels were dimes and her dimes were nickels, she would have 90 cents more, determine the total value of her coins in dollars.
|
\$1.80
| 0.916667 |
A 4x4x4 cube is made of $64$ normal dice, each with opposite faces summing to $7$. Find the smallest possible sum of all the values visible on the $6$ faces of the large cube.
|
144
| 0.083333 |
Given that 20% of the students scored 60 points, 25% scored 75 points, 40% scored 85 points, and the remainder scored 95 points, find the difference between the mean and median score of the students' scores in this competition.
|
6
| 0.583333 |
Given rectangle $EFGH$ with $EF=10$ and $EG=15$, and point $N$ on $\overline{EH}$ such that $EN=6$, calculate the area of $\triangle ENG$.
|
30
| 0.833333 |
Evaluate $(x^x)^{(x^x)}$ at $x = 3$.
|
27^{27}
| 0.916667 |
For the even number 144, determine the largest possible difference between two prime numbers whose sum is 144.
|
134
| 0.916667 |
Given the four-digit number $2552$, determine the total number of different arrangements of its digits.
|
6
| 0.916667 |
Given two poles, one 30 ft high and the other 70 ft high, are 150 ft apart, calculate the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole.
|
21
| 0.25 |
Given $S = i^{n+k} + i^{-(n+k)}$ where $i = \sqrt{-1}$, $n$ is any integer and $k \in \{1, 2\}$, determine the total number of distinct values for $S$.
|
3
| 0.75 |
If three plus the reciprocal of $(2-y)$ equals twice the reciprocal of $(2-y)$, find $y$.
|
\frac{5}{3}
| 0.916667 |
Let $N = 36 \cdot 42 \cdot 49 \cdot 280$. Find the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$.
|
\frac{1}{126}
| 0.583333 |
Calculate Glenda's average speed in miles per hour given that the odometer reading on her scooter was a palindrome of $1221$ and after riding for $5$ hours and $7$ hours, the odometer showed another palindrome of $1881$.
|
55
| 0.416667 |
Given four primes $p, q, s$, and $r$ satisfy $p + q + s = r$ and $2 < p < q < s$, determine the smallest value of $p$.
|
3
| 0.75 |
Positive integers $a$ and $b$ are such that $a < 6$ and $b \leq 7$. Calculate the smallest possible value for $2 \cdot a - a \cdot b$.
|
-25
| 0.916667 |
Evaluate the expression: $\frac{1}{1+\frac{1}{2+\frac{1}{3+\frac{1}{4}}}}$
|
\frac{30}{43}
| 0.916667 |
Given that the ratio of $2x - 5$ to $y + 20$ is constant, and $y = 6$ when $x = 7$, find the value of $x$ when $y = 21$.
|
\frac{499}{52}
| 0.916667 |
A point is chosen at random within a square in the coordinate plane whose vertices are $(0, 0), (2500, 0), (2500, 2500),$ and $(0, 2500)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{7}{16}$. Find the value of $d$ to the nearest tenth.
|
0.4
| 0.083333 |
Given the expressions \(\frac{x + a}{y}\) and \(\frac{y - b}{x}\), where \(x \neq 0\) and \(y \neq 0\), evaluate the arithmetic mean of these expressions.
|
\frac{x^2 + ax + y^2 - by}{2xy}
| 0.75 |
A grocer stacks apples in a cubic-like stack where the base cube consists of $4$ apples by $4$ apples by $4$ apples. Each apple above the first level rests in a pocket formed by four apples below, with the next layer reducing by one apple in each dimension. The stack is completed by a single apple. Find the total number of apples in the stack.
|
100
| 0.333333 |
Determine the number of real values of \( x \) that satisfy the equation \((2^{5x+2})(4^{2x+4}) = 8^{3x+7}\).
|
0
| 0.916667 |
Evaluate the expression $3 + 2\sqrt{3} + \frac{1}{3 + 2\sqrt{3}} + \frac{1}{2\sqrt{3} - 3}$.
|
3 + \frac{10\sqrt{3}}{3}
| 0.75 |
What is the greatest possible sum of the digits in the base-eight representation of a positive integer less than $1728$?
|
23
| 0.083333 |
If $\alpha$, $\beta$, $p$, and $q$ are positive numbers, and $q<50$, then the number obtained by increasing $\alpha \times \beta$ by $p\%$ followed by a decrease by $q\%$ exceeds $\alpha \times \beta$ if and only if...
A) $p > q$
B) $p > \dfrac{q}{100-q}$
C) $p > \dfrac{q}{1-q}$
D) $p > \dfrac{100q}{100+q}$
E) $p > \dfrac{100q}{100-q}$
|
p > \dfrac{100q}{100-q}
| 0.916667 |
Yesterday, Han drove 2 hours longer than Ian at an average speed 10 miles per hour faster than Ian. Jan drove 3 hours longer than Ian at an average speed 15 miles per hour faster than Ian. Han drove 100 miles more than Ian. Given this information, calculate the difference in the distances driven by Jan and Ian.
|
165
| 0.916667 |
How many whole numbers are between $\sqrt{18}$ and $\sqrt{200}$?
|
10
| 0.833333 |
Consider an isosceles triangle $ABC$ where $CA = CB = 15$ inches and $AB = 24$ inches. A circle with radius $15$ inches has its center at $C$ and passes through vertices $A$ and $B$. Extend $AB$ through $B$ to intersect the circle at point $E$. Determine the number of degrees in angle $AEC$.
|
90
| 0.333333 |
Consider an arithmetic progression where the sum of the first 15 terms is three times the sum of the first 8 terms, find the ratio of the first term to the common difference.
|
\frac{7}{3}
| 0.833333 |
How many 4-digit numbers greater than 1000 are there that use the four digits 2, 0, 3, and 3?
|
9
| 0.916667 |
What is the hundreds digit of $(25! - 20!)$?
|
0
| 0.916667 |
A triangle is inscribed in a circle. Calculate the sum of three angles, each inscribed in one of the three segments outside the triangle.
|
360^\circ
| 0.416667 |
Given Jane reads $\frac{1}{4}$ of the pages plus $10$ additional pages on the first day, $\frac{1}{5}$ of the remaining pages plus $20$ pages on the second day, and $\frac{1}{2}$ of what's left plus $25$ pages on the third day, calculate the total number of pages in Jane's book if there are $75$ pages remaining.
|
380
| 0.75 |
A box contains $35$ red balls, $25$ green balls, $22$ yellow balls, $15$ blue balls, $12$ white balls, and $10$ black balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $20$ balls of a single color will be drawn.
|
95
| 0.666667 |
Let \( x = 1729 \). Determine the value of \(\Bigg\vert\Big\vert |x| + x\Big\vert + |x|\Bigg\vert + x\).
|
6916
| 0.916667 |
In a botanical garden, there are three overlapping flower beds. Bed X contains 600 plants, bed Y contains 500 plants, and bed Z contains 400 plants. Beds X and Y share 70 plants, beds X and Z share 80 plants, and beds Y and Z share 60 plants. Additionally, 30 plants are shared among all three beds. Calculate the total number of distinct plants in the garden.
|
1320
| 0.75 |
Kiana has two older twin brothers. The product of their three ages is 162. What is the sum of their three ages?
|
20
| 0.166667 |
A rectangle $ABCD$ has a length of 8 inches and a width of 6 inches. Diagonal $AC$ is divided into four equal segments by points $P$, $Q$, and $R$. Calculate the area of the triangle $BPQ$.
|
6
| 0.583333 |
Evaluate the sum of the sequence: $1342 + 2431 + 3124 + 4213$, then calculate the product of the sum and $3$.
|
33330
| 0.833333 |
The sum of the first n positive odd integers from 1 to 2021, minus the sum of the first n positive even integers from 2 to 2020.
|
1011
| 0.666667 |
What is the maximum value of $\frac{(3^t - 5t)t}{9^t}$ for real values of $t$?
|
\frac{1}{20}
| 0.083333 |
A circle of radius 3 is centered at point $A$. An equilateral triangle with side 6 has a vertex at $A$. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle.
|
9(\pi - \sqrt{3})
| 0.166667 |
The circumference of the Earth at this latitude is 3000 \times 2 \pi, and the jet flies at 600 miles per hour, so calculate the number of hours the flight will take.
|
10\pi
| 0.666667 |
Calculate the sum of the numbers $1357 + 7531 + 3175 + 5713$.
|
17776
| 0.833333 |
Ms. Linda teaches mathematics to 22 students. Before she graded Eric's test, the average score for the class was 84. After grading Eric's test, the class average rose to 85. Determine Eric's score on the test.
|
106
| 0.083333 |
Given that ten points are chosen on the surface of a sphere, and lines are drawn connecting every pair of points, such that no three lines intersect in a single point inside the sphere, calculate the number of tetrahedrons with all four vertices in the interior of the sphere.
|
210
| 0.833333 |
Using the new operation \( x \otimes y = x^2 + y^2 \), simplify the expression \( x \otimes (x \otimes x) \).
|
x^2 + 4x^4
| 0.25 |
In $\bigtriangleup ABC$, $E$ is a point on side $\overline{AB}$, and $D$ is a point on side $\overline{BC}$ such that $BD=DE=EC$. Let $\angle BDE$ be $90^{\circ}$. Determine the degree measure of $\angle AED$.
|
45^\circ
| 0.166667 |
Given the permutation $(a_1, a_2, a_3, a_4, a_5, a_6)$ of $(1, 2, 3, 4, 5, 6)$, calculate the number of such permutations that satisfy the condition $a_1 + a_2 + a_3 < a_4 + a_5 + a_6$.
|
360
| 0.833333 |
How many 3-digit whole numbers, whose digit-sum is 27, are divisible by 3 and even?
|
0
| 0.666667 |
The dimensions of a rectangular box are all positive integers and the volume of the box is $3003$ in$^3$. Find the minimum possible sum of the three dimensions.
|
45
| 0.583333 |
Given Laura is adding two distinct three-digit positive integers, where all six digits in these numbers are different, and the sum, $S$, is a four-digit number, calculate the smallest possible value for the sum of the digits of $S$.
|
1
| 0.25 |
Alice needs to take one blue pill and one red pill each day for three weeks. A blue pill costs $2 more than a red pill, and Alice's pills cost a total of $945 for the three weeks. Find the cost of one blue pill.
|
\$23.5
| 0.75 |
A point is chosen at random within the square in the coordinate plane whose vertices are (0, 0), (4040, 0), (4040, 4040), and (0, 4040). The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{4}$. What is $d$ to the nearest tenth?
|
0.3
| 0.166667 |
Consider the expression $(2xy-1)^2 + (x-y)^2$. What is the least possible value of this expression for real numbers $x$ and $y$?
|
0
| 0.666667 |
Sophia's age is $S$ years, which is thrice the combined ages of her two children. Her age $M$ years ago was four times the sum of their ages at that time. Find the value of $S/M$.
|
21
| 0.833333 |
Given a set of 36 square blocks arranged into a 6 × 6 square, determine the number of different combinations of 4 blocks that can be selected from that set so that no two are in the same row or column.
|
5400
| 0.833333 |
Given triangle $PQR$, where $P = (0,0)$, $R = (8,0)$, and $Q$ is in the first quadrant with $\angle QRP = 90^\circ$ and $\angle QPR = 45^\circ$. Suppose that $PQ$ is rotated $90^\circ$ counterclockwise about $P$. Find the coordinates of the image of $Q$.
|
(-8, 8)
| 0.083333 |
Point $O$ is the center of the regular octagon $ABCDEFGH$, $X$ is the midpoint of side $\overline{AB}$, and $Y$ is the midpoint of side $\overline{CD}$. Calculate the fraction of the area of the octagon that is shaded if the shaded region includes the full area of triangles $\triangle BCO$, $\triangle CDO$, $\triangle DEO$, $\triangle EFO$, and half the area each of $\triangle ABO$ and $\triangle CDO$.
A) $\frac{3}{8}$
B) $\frac{7}{16}$
C) $\frac{13}{32}$
D) $\frac{1}{2}$
E) $\frac{5}{8}$
|
\frac{5}{8}
| 0.833333 |
Angie, Bridget, Carlos, Diego, and Edwin are seated at random around a circular table, one person to each chair. Calculate the probability that Angie and Carlos are seated in positions that are exactly two seats apart.
|
\frac{1}{2}
| 0.583333 |
Determine the number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values satisfying the equation $x^4y^4 - 16x^2y^2 + 15 = 0$.
|
1
| 0.333333 |
Calculate the sum of the series given by the expression:
\[3\left(1-\dfrac{1}{3}\right) + 4\left(1-\dfrac{1}{4}\right) + 5\left(1-\dfrac{1}{5}\right) + \cdots + 12\left(1-\dfrac{1}{12}\right)\]
|
65
| 0.25 |
A box contains $30$ red balls, $25$ green balls, $22$ yellow balls, $15$ blue balls, $12$ white balls, and $6$ black balls. Calculate the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $18$ balls of a single color will be drawn.
|
85
| 0.5 |
Given that teams $X$ and $Y$ are participating in a hockey league, and team $X$ has won $\frac{3}{4}$ of its games and team $Y$ has won $\frac{2}{3}$ of its games, while team $Y$ has won $4$ more games, lost $5$ more games, and drawn $3$ more games than team $X$, determine the number of games played by team $X$.
|
48
| 0.416667 |
$\frac{3+6+9}{2+5+8} - \frac{4 \cdot (1+2+3)}{5+10+15}$
|
\frac{2}{5}
| 0.916667 |
Given that the product of the digits of a 3-digit positive integer equals 36, calculate the number of such integers.
|
21
| 0.333333 |
In how many ways can $10003$ be written as the sum of two primes?
|
0
| 0.083333 |
Gabriela has a younger brother and an older sister. The product of their three ages is 72. Find the sum of their three ages.
|
13
| 0.916667 |
Ana's monthly salary was $2500 in May. In June, she received a 15% raise and an additional $300 bonus. In July, she received a 25% pay cut. Calculate Ana's monthly salary after these adjustments in June and July.
|
2381.25
| 0.916667 |
Given that the distance light travels in one year is approximately $6,000,000,000,000$ miles, calculate the distance light travels in $50$ years.
|
3 \times 10^{14} \text{ miles}
| 0.75 |
Calculate the smallest product obtainable by multiplying any two numbers from the set $\{-9, -5, -1, 1, 4\}$.
|
-36
| 0.916667 |
In a trapezoid, the line segment joining the midpoints of the diagonals has length $5$, and the longer base has a length of $115$. Calculate the shorter base.
|
105
| 0.916667 |
If Yen has a 5 × 7 index card and reduces the length of the shorter side by 1 inch, the area becomes 24 square inches. Determine the area of the card if instead she reduces the length of the longer side by 2 inches.
|
25
| 0.833333 |
Given that Chelsea is leading by 60 points halfway through a 120-shot archery competition, each shot can score 10, 7, 3, or 0 points, and Chelsea always scores at least 3 points. If Chelsea's next \(n\) shots are all for 10 points, she will secure her victory regardless of her opponent's scoring in the remaining shots. Find the minimum value for \(n\).
|
52
| 0.083333 |
If $a > 2$, then find the sum of the real solutions of the equation:
\[ \sqrt{a - \sqrt{a - x}} = x \]
A) $\frac{\sqrt{4a} - 1}{2}$
B) $\frac{\sqrt{4a-1} - 1}{2}$
C) $\frac{\sqrt{4a-2} - 1}{2}$
D) $\frac{\sqrt{4a-3} - 2}{2}$
E) $\frac{\sqrt{4a-3} - 1}{2}$
|
\frac{\sqrt{4a-3} - 1}{2}
| 0.75 |
Given the product $\frac{5}{4}\cdot \frac{6}{5}\cdot \frac{7}{6}\cdot \ldots \cdot \frac{a}{b} = 42$, calculate the sum of $a$ and $b$.
|
335
| 0.5 |
$\triangle ABC$ has a right angle at $C$ and $\angle A = 30^{\circ}$. If $BD$ ($D$ in $\overline{AC}$) is the bisector of $\angle ABC$, then find the measure of $\angle BDC$.
|
60^\circ
| 0.916667 |
Let $M$ be the greatest five-digit number whose digits have a product of $72$. Calculate the sum of the digits of $M$.
|
20
| 0.666667 |
For how many positive integers $x$ is $\log_{10}(x-50) + \log_{10}(70-x) < 2$?
|
18
| 0.833333 |
Evaluate the expression $(2(2(2(2(2(2(3+2)+2)+2)+2)+2)+2)+2)$.
|
446
| 0.083333 |
Given $g(x) = 2 - 3x^2$ and $f(g(x)) = \frac{2-3x^2}{2x^2}$ when $x \neq 0$, determine $f\left(\frac{2}{3}\right)$.
|
\frac{3}{4}
| 0.75 |
Evaluate the product: $\left(\frac{2\cdot4}{3\cdot3}\right)\left(\frac{3\cdot5}{4\cdot4}\right)\cdots\left(\frac{98\cdot100}{99\cdot99}\right)\left(\frac{99\cdot101}{100\cdot100}\right)$.
|
\frac{101}{150}
| 0.25 |
For all real numbers $x$, calculate the value of the expression $x[x\{x(x-3)-5\}+11]+2$.
|
x^4 - 3x^3 - 5x^2 + 11x + 2
| 0.666667 |
If Billy Bob counts a total of 29 wheels and observes 10 children riding past his house, using bicycles, tricycles, and scooters, determine the number of tricycles among the children.
|
9
| 0.333333 |
Given that Connor takes four tests, each worth a maximum of 100 points, and his scores on the first two tests are 82 and 75, calculate the lowest score he could earn on one of the remaining two tests in order to have a mean score of 85 for all four tests.
|
83
| 0.833333 |
If the product $\dfrac{5}{3}\cdot \dfrac{6}{5}\cdot \dfrac{7}{6}\cdot \dfrac{8}{7}\cdot \ldots\cdot \dfrac{a}{b} = 16$, calculate the sum of $a$ and $b$.
|
95
| 0.583333 |
Given a basketball player made 7 shots during a game, each shot was worth either 1, 2, or 3 points, determine how many different numbers could represent the total points scored by the player.
|
15
| 0.833333 |
If Shauna takes five tests, each worth a maximum of 100 points, and her scores on the first three tests are 76, 94, and 87, while aiming for an average of 85 across all five tests, given that she scores 92 on the fourth test, determine the lowest score she could earn on the fifth test to meet her goal.
|
76
| 0.833333 |
Let $p$ and $q$ be the roots of the quadratic equation $x^2 - 2px + q = 0$, where $p \neq 0$ and $q \neq 0$. Calculate the sum of the roots $p$ and $q$.
|
2
| 0.833333 |
Given David's previous math contest scores of 88, 92, 75, 83, and 90, what is the minimum score he must achieve on his next contest to increase his average score by at least 4 points?
|
110
| 0.916667 |
The region consisting of all points in three-dimensional space within $4$ units of line segment $\overline{CD}$ has volume $544\pi$. Calculate the length of $CD$.
|
\frac{86}{3}
| 0.75 |
For how many three-digit whole numbers does the sum of the digits equal 27?
|
1
| 0.916667 |
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