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Define a sequence $\{u_n\}$ by $u_1=7$ and the recurrence relation $u_{n+1}-u_n=5+2(n-1)$ for $n=1,2,3,\cdots$. If $u_n$ is expressed as a polynomial in $n$, find the algebraic sum of its coefficients. | 7 | 0.833333 |
In a regular hexagon with side length 4, there are six alternating semicircles along its sides such that the diameters of three of them equal the side of the hexagon, and the other three have diamimeters equal to half the side length. Calculate the area of the region inside the hexagon but outside all the semicircles. | 24\sqrt{3} - \frac{15\pi}{2} | 0.916667 |
Given the sequence $1, 4, 9, -16, 25, -36,\ldots$, whose $n$th term is $(-1)^{n+1}\cdot n^2$, find the average of the first $100$ terms of the sequence. | -50.5 | 0.916667 |
Evaluate $x^{x^x}$ at $x = 3$. | 3^{27} | 0.833333 |
A rectangular area is to be fenced off on three sides using part of a 150 meter rock wall as the fourth side, with fence posts placed every 10 meters along the fence, including the two posts where the fence meets the rock wall. Given the area dimensions are 40 m by 100 m, determine the fewest number of posts required to fence this area. | 19 | 0.333333 |
How many 4-digit numbers greater than 1000 can be formed using the digits of 2025? | 9 | 0.666667 |
The length of the paper used to make a $3$ cm diameter cylindrical cardboard tube into an $11$ cm diameter cylindrical cardboard tube by wrapping it $400$ times is what length in meters? | 28\pi | 0.166667 |
Given a right circular cone with a base that is a circle with radius twice that of a given sphere, and the volume of the cone is one-third that of the sphere, determine the ratio of the altitude of the cone to the radius of its base. | \frac{1}{6} | 0.833333 |
Given that soda is sold in packs of 6, 12, 24, and 48 cans, find the minimum number of packs needed to buy exactly 126 cans of soda. | 4 | 0.583333 |
Given $A$ can finish a task in $10$ days, and $B$ is $75\%$ more efficient than $A$, calculate the time it takes for $B$ to complete the same task. | \frac{40}{7} | 0.916667 |
Given Alice took 80% of a whole pie. Bob took one fourth of the remainder. Cindy then took half of what remained after Alice and Bob. What portion of the whole pie was left? | 7.5\% | 0.25 |
Given the equation $\sin(3x) = \sin(x)$, determine the number of solutions on the interval $[0,2\pi]$. | 7 | 0.833333 |
A regular decagon is given. A triangle is formed by connecting three randomly chosen vertices of the decagon. Calculate the probability that none of the sides of the triangle is a side of the decagon. | \frac{5}{12} | 0.333333 |
If the perimeter of rectangle ABCD is 30 inches, determine the least value of diagonal AC in inches. | 7.5\sqrt{2} | 0.416667 |
Points $M$ and $N$ are on a circle of radius $10$ and $MN = 12$. Point $P$ is the midpoint of the minor arc $MN$. What is the length of the line segment $MP$? | 2\sqrt{10} | 0.25 |
Given that Bob spends a total of $36.00 for lunch, including a 12% sales tax and an 18% tip on the pre-tax amount, calculate the cost of his lunch without tax or tip in dollars. | 27.69 | 0.916667 |
What is $12\cdot\left(\tfrac{1}{3}+\tfrac{1}{4}+\tfrac{1}{6}+\tfrac{1}{12}\right)^{-1}$? | \frac{72}{5} | 0.833333 |
Given a minor league soccer team with $25$ players, where each player must be paid at least $20,000$ dollars, and the total of all players' salaries cannot exceed $900,000$ dollars, determine the maximum possible salary for a single player. | 420,000 | 0.916667 |
Given that a grocer stacks oranges in a pyramid-like stack whose rectangular base is $6$ oranges by $7$ oranges, and each orange above the first level rests in a pocket formed by four oranges below, determine the total number of oranges in the stack. | 112 | 0.75 |
Suppose $m$ and $n$ are positive integers such that $108m = n^3$. Find the minimum possible value of $m + n$. | 8 | 0.75 |
Given the original price of a jacket is $120.00, an 8% sales tax is applied, and then a 25% discount is applied, find the difference between the total bill calculated by Bob and the total bill calculated by Alice. | 0 | 0.666667 |
The median of the numbers 3, 7, x, 14, 20 is equal to the mean of those five numbers. Calculate the sum of all real numbers \( x \) for which this is true. | 28 | 0.416667 |
Macy wants to buy the minimum number of standard jars necessary to fill at least a large pack that contains 2000 grams of jam, without exceeding it by more than the weight of one standard jar that can hold 140 grams of jam. What is the number of jars she must buy? | 15 | 0.916667 |
Chips are drawn randomly, one at a time without replacement, from a magician's hat containing 4 red chips and 3 green chips until all 4 red chips are drawn or until at least 2 green chips are drawn. Calculate the probability that all 4 red chips are drawn. | \frac{1}{7} | 0.25 |
A blue ball and a yellow ball are randomly and independently tossed into bins numbered with the non-negative integers so that for each ball, the probability that it is tossed into bin $0$ is $\frac{1}{4}$ and into bin $k$ (where $k \geq 1$) is $\left(\frac{1}{2}\right)^k \cdot \frac{3}{4}$. Find the probability that the blue ball is tossed into a higher-numbered bin than the yellow ball. | \frac{3}{8} | 0.416667 |
The complex number $z$ satisfies $z + |z| = 5 - 3i$. What is $|z|^{2}$. | 11.56 | 0.083333 |
Given the quadratic equation $x^2 + px + q = 0$, where $p$ and $q$ are positive numbers, and the roots of this equation differ by $2$, find the value of $p$. | 2\sqrt{q+1} | 0.25 |
Given that a recipe to make $8$ servings of a smoothie requires $3$ bananas, $2$ cups of strawberries, $1$ cup of yogurt, and $4$ cups of milk, and if Sarah has $10$ bananas, $5$ cups of strawberries, $3$ cups of yogurt, and $10$ cups of milk, determine the greatest number of servings of smoothie she can make maintaining the same ratio of ingredients. | 20 | 0.916667 |
Two tangents are drawn to a circle from an exterior point A; they touch the circle at points B and C respectively. A third tangent intersects segment AB in P and AC in R, and touches the circle at Q. If AB = 24, and the lengths BP = PQ = x and QR = CR = y with x + y = 12, find the perimeter of triangle APR. | 48 | 0.916667 |
Given Professor Miller has nine different language books on a bookshelf: two Arabic, three German, and four Spanish, calculate the number of ways to arrange the nine books on the shelf keeping all German books together and all Spanish books also together. | 4! \times 3! \times 4! = 24 \times 6 \times 24 = 3456 | 0.083333 |
Given the modified sequence where the first number is $1$, and for all $n\ge 2$, the product of the first $n$ numbers in the sequence is $n^3$, find the sum of the third and the fifth numbers in this sequence. | \frac{341}{64} | 0.916667 |
Given that $2^{10} \cdot 3^6 \cdot 5^4 = d^e$, find the smallest possible value for $d+e$. | 21602 | 0.833333 |
What is the sum of the numbers 12345, 23451, 34512, 45123, and 51234? | 166665 | 0.583333 |
What is the maximum number of possible points of intersection between a circle and a rectangle? | 8 | 0.916667 |
Sara lists the whole numbers from 1 to 50. Lucas copies Sara's numbers, replacing each occurrence of the digit '3' with the digit '2'. Calculate the difference between Sara's sum and Lucas's sum. | 105 | 0.25 |
Consider those functions $f$ that satisfy $f(x+6) + f(x-6) = f(x)$ for all real $x$. Find the least common positive period $p$ for all such functions. | 36 | 0.416667 |
Given that $3x^3 - tx + q$ has $x - 3$ and $x + 1$ as its factors, determine the value of $|3t - 2q|$. | 99 | 0.75 |
Two poles, one 30 feet high and the other 90 feet high, are 150 feet apart. Find the height at which the lines joining the top of each pole to the foot of the opposite pole intersect. | 22.5 | 0.25 |
Given a triangle with side lengths $15, 20,$ and $x$, determine the number of integers $x$ for which the triangle has all its angles acute. | 11 | 0.833333 |
Given that Mary thought of a positive two-digit number and performed the following operations: multiplied it by $4$ and subtracted $7$, then switched the digits of the result, and obtained a number between $91$ and $95$, inclusive, determine the original two-digit number. | 14 | 0.333333 |
Given Alice's weekly allowance was $B$ dollars, the cost of the book was $25\%$ of the difference between $B$ and the magazine cost, and the cost of the magazine was $10\%$ of the difference between $B$ and the book's cost, calculate the fraction of $B$ that Alice spent on the book and magazine combined. | \frac{4}{13} | 0.833333 |
Josanna's current test average is $\dfrac{85 + 75 + 80 + 70 + 90}{5}$. If she wants to raise her test average by $5$ points, then her new overall test average will be $\dfrac{85 + 75 + 80 + 70 + 90}{5} + 5$. Find the minimum test score Josanna would need to achieve this new average. | 110 | 0.916667 |
Suppose $d$ is a digit. For how many values of $d$ is $3.1d3 > 3.123$? | 7 | 0.666667 |
Given a circle with center O and radius r, alongside two horizontal parallel tangents to the circle that are each at a distance of d from the center where d > r, calculate the number of points equidistant from the circle and both tangents. | 2 | 0.583333 |
If the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{a}{b} = 18$, calculate the sum of $a$ and $b$. | 107 | 0.666667 |
Given the radii ratio of two concentric circles is $1:4$, if $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB=8$, determine the radius of the larger circle. | 16 | 0.583333 |
For how many positive integers $n$ is $\frac{n}{40-n}$ also a positive integer? | 7 | 0.75 |
Consider a map with points labeled as P, Q, R, S, and T. The allowed paths are P to Q, Q to R, Q to S, S to T, Q directly to T, and R to T. Determine how many different routes are there from P to T. | 3 | 0.583333 |
Evaluate the expression $\sqrt{7+4\sqrt{3}} - \sqrt{7-4\sqrt{3}}$. | 2\sqrt{3} | 0.916667 |
Given the water tower in real life is 60 meters high and the spherical top holds 200,000 liters of water, and Logan wants his model's water tower to hold 0.2 liters of water, determine the height of Logan's model tower. | 0.6 | 0.583333 |
Given a class of $60$ students with $20$ blue-eyed blondes, $36$ brunettes, and $23$ brown-eyed students, calculate the number of blue-eyed brunettes in the class. | 17 | 0.083333 |
Given the product \(6^3 \cdot 15^4\), calculate the number of digits in the result. | 8 | 0.833333 |
A shopkeeper purchases 2000 pencils at $0.20 each. If he plans to sell them for $0.50 each, determine the number of pencils he must sell to make a profit of exactly $200. | 1200 | 0.916667 |
Let $ABCD$ be a square. Let $E, F, G, H$ be the centers, respectively, of isosceles right-angled triangles with bases $\overline{AB}, \overline{BC}, \overline{CD},$ and $\overline{DA},$ each exterior to the square. Find the ratio of the area of square $EFGH$ to the area of square $ABCD$. | 2 | 0.416667 |
Given that the fourth degree polynomial function y=p(x) and the fifth degree polynomial function y=q(x) have leading coefficients of 1, determine the maximum number of points of intersection of their graphs. | 5 | 0.916667 |
Given the expression $G = \frac{5x^2 + 20x + 4n}{5}$, which is the square of an expression linear in $x$, determine the range between which $n$ must fall. | 5 | 0.916667 |
The sums of three whole numbers taken in pair are 18, 23, and 27. Express the middle number in terms of the other two numbers and solve for its value. | 11 | 0.333333 |
A telephone number has the form $\text{ABC-DEF-GHIJ}$, where each letter represents a different digit. In this number, $A > B > C$, $D > E > F$, and $G > H > I > J$. Moreover, $D$, $E$, and $F$ are consecutive odd digits; $G$, $H$, $I$, and $J$ are consecutive even digits. Additionally, $A + B + C = 12$. Determine the value of $A$. | 8 | 0.083333 |
Given that Alice visits every 4th day, Bianca visits every 6th day, and Carmen visits every 8th day, all three friends visited Daphne yesterday, and the next 365-day period is considered, calculate the number of days when exactly two friends will visit. | 45 | 0.666667 |
What is the greatest possible sum of the digits in the base-eight representation of a positive integer less than $1728$? | 23 | 0.083333 |
In a trapezoid, the line segment joining the midpoints of the diagonals has length 5. The height of the trapezoid is 10 and the longer base is 63. Find the area of the trapezoid. | 580 | 0.666667 |
A trapezoidal field with one pair of parallel sides of lengths 15 meters and 9 meters respectively is part of a rectangular plot. The non-parallel sides of the trapezoid are equal and perpendicular to the parallel sides, and the trapezoid fills one end of the rectangle. Given that the longer parallel side of the trapezoid is also the length of the short side of the rectangle, and the total perimeter of the rectangle is 52 meters, calculate the perimeter of the trapezoidal field. | 46 | 0.25 |
Each of 8 balls is randomly and independently painted either black or white with equal probability. Calculate the probability that every ball is different in color from at least half of the other 7 balls. | \frac{35}{128} | 0.5 |
Suppose that $x$ and $y$ are nonzero real numbers such that $\frac{4x+2y}{2x-4y} = 3$. Find the value of $\frac{2x+4y}{4x-2y}$. | \frac{9}{13} | 0.916667 |
The difference between the largest and smallest roots of the cubic equation \(x^3 - px^2 + \frac{p^2 - 1}{4}x = 0\), calculate the difference. | 1 | 0.75 |
The population of Cyros doubles every 30 years and is currently 500, with each inhabitant requiring 2 acres. Given the island can support 32000 acres, calculate the number of years from 2020 until the population meets or exceeds the island's capacity. | 150 | 0.916667 |
Three different numbers are randomly selected from the set $\{ -3, -2, -1, 0, 1, 3, 4\}$ and multiplied together. What is the probability that the product is positive? | \frac{2}{7} | 0.416667 |
Given a triangle with vertices $A=(2,2)$, $B=(6,2)$, and $C=(5,5)$ plotted on an $8\times6$ grid, and a smaller rectangle within the grid defined by its diagonal with endpoints $(1,1)$ and $(6,5)$. Determine the fraction of the grid area covered by the portion of the triangle that lies within the smaller rectangle. | \frac{1}{8} | 0.083333 |
Given a positive integer $N$, with four digits in its base four representation, is chosen at random with each such number having an equal chance of being chosen, calculate the probability that $\log_2 N$ is an integer and $N$ is even. | \frac{1}{96} | 0.833333 |
Determine the twenty-fifth number in a sequence obtained by counting in base 5. | 100_5 | 0.166667 |
A woman has part of $6000 invested at 3% and the rest at 5%. If her annual return on each investment is the same, calculate the average rate of interest she realizes on the $6000. | 3.75\% | 0.916667 |
Given that the faces of each of $9$ standard dice are labeled with the integers from $1$ to $6$, let \( p \) be the probability that when all $9$ dice are rolled, the sum of the numbers on the top faces is $15$. What other sum occurs with the same probability as \( p \)? | 48 | 0.5 |
Let \( S_n = 2-4+6-8+\cdots+(-1)^{n-1}(2n) \), where \( n = 1, 2, \cdots \). Compute \( S_{18} + S_{34} + S_{51} \). | 0 | 0.25 |
Given the enclosure dimensions are 15 feet long, 8 feet wide, and 7 feet tall, with each wall and floor being 1 foot thick, determine the total number of one-foot cubical blocks used to create the enclosure. | 372 | 0.333333 |
If $1010+1012+1014+1016+1018 = 5100 - P$, calculate the value of $P$. | 30 | 0.916667 |
How many positive even multiples of $3$ less than $3000$ are perfect squares? | 9 | 0.833333 |
Given that the scores of the test are 65, 70, 75, 85, and 95, and the class average is always an integer after each score is entered, determine the last score Mrs. Johnson entered. | 70 | 0.083333 |
Fifteen points are selected on the positive $x$-axis, \( X^+ \), and six points are selected on the positive $y$-axis, \( Y^+ \). All possible segments connecting points from \( X^+ \) to \( Y^+ \) are drawn. What is the maximum possible number of points of intersection of these segments that could lie in the interior of the first quadrant if none of the points on \( Y^+ \) have an integer coordinate? | 1575 | 0.833333 |
Given the function $g(x) = 2x^3 + 5x^2 - 2x - 1$, find $g(x+h) - g(x)$. | h(6x^2 + 6xh + 2h^2 + 10x + 5h - 2) | 0.25 |
How many four-digit numbers are divisible by 17? | 530 | 0.833333 |
Given three different numbers are randomly selected from the set $\{-2, -1, 0, 0, 3, 4, 5\}$ and multiplied together, calculate the probability that the product is $0$. | \frac{5}{7} | 0.333333 |
A box contains 30 red balls, 25 green balls, 23 yellow balls, 14 blue balls, 13 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 15 balls of a single color will be drawn. | 80 | 0.25 |
Given that a product is discounted by 60% and the buyer also has a coupon for an additional 30% off the sale price, determine the percentage off the original price that the buyer effectively gets. | 72\% | 0.916667 |
Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187} - \cdots$. Determine the sum $S$ of this series. | \frac{15}{26} | 0.166667 |
Five friends did yardwork for their neighbors, and their earnings were \$18, \$22, \$30, \$36, and \$50, respectively. Calculate the amount the friend who earned \$50 will give to the others. | 18.80 | 0.666667 |
Given a list of 21 names with altered lengths recorded as follows: 6 names of 4 letters each, 4 names of 5 letters each, 1 name of 6 letters, 5 names of 7 letters, and 5 names of 8 letters, calculate the median length of these names. | 6 | 0.833333 |
Given that point P is the center of the regular hexagon ABCDEF, and Z is the midpoint of the side AB, determine the fraction of the area of the hexagon that is in the smaller region created by segment ZP. | \frac{1}{12} | 0.083333 |
Five friends earn $18, $22, $30, $35, and $45 respectively. Determine the amount the friend who earned $45 needs to give to the others. | 15 | 0.833333 |
Given Oscar buys $15$ pencils and $5$ erasers for $1.25$, and a pencil costs more than an eraser, and both items cost a whole number of cents, determine the total cost, in cents, for one pencil and one eraser. | 11 | 0.083333 |
If Michael was 15 years old when he participated in the 10th IMO, determine the year Michael was born. | 1953 | 0.416667 |
A circle is inscribed in a triangle with side lengths $9, 12$, and $15$. Let the segments of the side of length $9$, made by a point of tangency, be $r$ and $s$, with $r<s$. Find the ratio $r:s$. | 1:2 | 0.333333 |
A class contains \( k > 20 \) students and their quiz scores have a mean of \( 10 \). A group of \( 20 \) students from the class has a mean score of \( 16 \). Determine the mean of the quiz scores of the remaining students in terms of \( k \). | \frac{10k-320}{k-20} | 0.416667 |
Circles of diameter $2$ inches and $6$ inches have the same center. Find the ratio of the blue-painted area to the red-painted area. | 8 | 0.333333 |
When the radius of a circle is incremented by \( k \) units, find the ratio of the new circumference to the increase in area compared to the original. | \frac{2(r+k)}{2rk + k^2} | 0.083333 |
Given Tamara has $3030$ coins, consisting of pennies ($1$-cent coins) and dimes ($10$-cent coins), and she has at least one penny and one dime, calculate the difference in cents between the greatest possible and least amounts of money Tamara can have. | 27252 | 0.75 |
Given the expression $\dfrac{12-11+10-9+8-7+6-5+4-3+2-1}{2-3+4-5+6-7+8-9+10-11+12}$, evaluate the given expression. | \dfrac{6}{7} | 0.083333 |
Given Linda drove the first quarter of her trip on a gravel road, the next 30 miles on pavement, and the remaining one-sixth on a dirt road, determine the total length of Linda's trip in miles. | \frac{360}{7} | 0.5 |
A novel is recorded onto compact discs, taking a total of 505 minutes to read aloud. Each disc can hold up to 53 minutes of reading. Assuming the smallest possible number of discs is used and each disc contains the same length of reading, calculate the number of minutes of reading each disc will contain. | 50.5 | 0.916667 |
Given that you and five friends and four of their family members are raising $1800 in donations for a charity event, determine the amount each person will need to raise for equal distribution. | 180 | 0.75 |
As the number of sides of a polygon increases from $3$ to $n$, determine how the sum of the interior angles of the polygon changes. | (n-2) \times 180^\circ | 0.083333 |
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