Given a rectangular grid sized at $5 \times 8$, consisting of alternating light and dark squares, calculate the difference between the number of dark squares and the number of light squares.


A standard deck now includes 4 red cards, 4 green cards, and 4 blue cards, each labeled A, B, C, D. Two cards are dealt from this deck. A winning pair is defined as two cards either of the same color or sharing the same label. Calculate the probability of drawing a winning pair.


Given the base exchange rate of 1 EUR = 0.85 GBP and a 5% fee on the total exchanged amount, calculate the amount of GBP the student will receive for exchanging 100 EUR.


A chef bakes a $24$-inch by $15$-inch pan of brownies. Each brownie is cut into pieces that measure $3$ inches by $2$ inches. Calculate the number of pieces of brownie the pan contains.


How many distinct integers can be expressed as the sum of three distinct members of the set $\{2, 5, 8, 11, 14, 17, 20\}$?


A 4x4x4 cube is made of $64$ normal dice. Each die's opposite sides sum to $7$. Calculate the smallest possible sum of all of the values visible on the $6$ faces of the large cube.


Given a box containing $30$ red balls, $24$ green balls, $16$ yellow balls, $14$ blue balls, $12$ white balls, and $4$ purple balls, determine the minimum number of balls that must be drawn from the box without replacement to guarantee that at least $12$ balls of a single color will be drawn.


Real numbers $x$ and $y$ are chosen independently and uniformly at random from the interval $(0,\pi/2)$. Calculate the probability that $\lfloor \sin x \rfloor = \lfloor \sin y \rfloor$.


Simplify the expression $(-\frac{1}{343})^{-2/3}$.


Suppose $d$ is a digit. For how many values of $d$ is $20.d05 > 20.05$?


The area of the ring between two concentric circles is $16\pi$ square inches. Determine the length of a chord of the larger circle that is tangent to the smaller circle.


The volume of the top portion of the water tower is equal to the volume of a sphere, which can be calculated using the formula $V = \frac{4}{3}\pi r^3$. The top portion of the real tower has a volume of 50,000 liters, so we can solve for the radius:

$\frac{4}{3}\pi r^3 = 50,000$

Simplifying, we get $r^3 = \frac{50,000 \times 3}{4\pi}$. We can now calculate the radius on the right-hand side and then take the cube root to find $r$.

Now, we want to find the volume of the top portion of Logan’s model. It is given that this sphere should have a volume of 0.2 liters. Using the same formula, we can solve for the radius:

$\frac{4}{3}\pi r^3 = 0.2$

Simplifying, we get $r^3 = \frac{0.2 \times 3}{4\pi}$. We can now calculate the radius on the right-hand side and then take the cube root to find $r$.

Now, since the sphere is a model of the top portion of the water tower, the radius of the model is proportional to the radius of the real tower. Therefore, the ratio of the radius of the model to the radius of the real tower is equal to the ratio of the volume of the model to the volume of the real tower.

$\frac{r_{model}}{r_{real}} = \frac{0.2}{50000}$

We can now equate the two expressions for the radius and solve for the height of the model:

 $\frac{r_{model}}{60} = \frac{0.2}{50000}$


In an obtuse isosceles triangle, the product of the lengths of the two congruent sides is twice the product of the base length and the triangle's height to the base. What is the measure, in degrees, of the vertex angle of this triangle?


Alex, Mel, and Chelsea play a game that involves 8 rounds. In each round, only one of them wins, and the outcomes of the rounds are independent. Each round, the probability of Alex winning is $\frac{1}{2}$. Mel and Chelsea have equal probabilities of winning any round. Calculate the probability that Alex wins four rounds, Mel wins three rounds, and Chelsea wins one round.


A choir consisting of boys and girls is planning a fundraiser. Initially, $60\%$ of the choir are girls. Shortly thereafter, four girls leave and six boys join while two boys leave, and then $50\%$ of the choir are girls. Calculate the initial number of girls in the choir.


The Hawks scored a certain number of points, and the total points scored by both teams together is 82. If the difference between the points scored by the Eagles and the Hawks is 18, and the spectator claimed that the Hawks scored 40 points, then determine the actual number of points the Hawks scored.


Nine friends dined at a restaurant and agreed to split the bill equally. Unfortunately, Judi forgot her wallet and could only pay $5, and another friend, Tom, could only pay half of his share. The remaining friends covered these shortages by each paying an extra $3. Calculate the total bill.


Given the quadratic equation $x^2 - 5x + k = 0$ where $r$ and $s$ are the roots, create a new quadratic equation $x^2 + ax + b = 0$ such that the roots of this new equation are $r^2 + 1$ and $s^2 + 1$, find the value of $a$ given that $k = 6$.


Given that Ron mistakenly reversed the digits of the two-digit number $a$, and the product of $a$ and $b$ was mistakenly calculated as $221$, determine the correct product of $a$ and $b$.


Mina drives 20 miles at an average speed of 40 miles per hour. Calculate the number of additional miles she must drive at 60 miles per hour to average 55 miles per hour for her entire trip.


For how many integers $x$ does a triangle with side lengths $15, 20$ and $x$ have all its angles acute?


Given the equations $ab = 2c$, $bc = 2a$, and $ca = 2b$, determine the number of ordered triples $(a, b, c)$ of non-zero real numbers that satisfy these equations.


Two numbers have their difference, their sum, and their product related to one another as $1:5:15$. Find the product of these two numbers.


Given the product of the digits of a 3-digit positive integer equals 30, find the number of such integers.


Let $r$ be the number that results when the base of $a^b$ is quadrupled and the exponent is doubled, where $a,b>0$. If $r$ equals the square of the product of $a^b$ and $x^b$ where $x>0$, find $x$.


The sum of the dimensions of a rectangular prism is the sum of the number of edges, corners, and faces, where the dimensions are 2 units by 3 units by 4 units. Calculate the resulting sum.


Find the value of \( d \) given that runner \( A \) can beat \( B \) by \( 15 \) meters, \( B \) can beat \( C \) by \( 30 \) meters, and \( A \) can beat \( C \) by \( 40 \) meters.


Given the product sequence $\frac{5}{3} \cdot \frac{6}{5} \cdot \frac{7}{6} \cdot \ldots \cdot \frac{a}{b} = 12$, determine the sum of $a$ and $b$.


Given that the tallest building in the campus is 120 meters high, and the top portion is a cone that holds 30,000 liters of water, and Ella's model of this building has a top portion that holds 0.03 liters of water, determine the height of the tallest building in Ella's model.


Let $c_1, c_2, \ldots$ and $d_1, d_2, \ldots$ be arithmetic progressions such that $c_1 = 10, d_1 = 90$, and $c_{50} + d_{50} = 500$. Find the sum of the first fifty terms of the progression $c_1 + d_1, c_2 + d_2, \ldots$


Solution: \boxed{1 + x^3 - 2x^4 - 2x^7 + x^8 + x^{11}}


Consider the set of all four-digit rising numbers using the digits 1 through 7. Find the digit that the 35th number in the list from smallest to largest does not contain.


Find the sum of all possible values of $s$ between $0$ and $360$ such that the triangle in the coordinate plane whose vertices are \[(\cos 30^\circ, \sin 30^\circ), (\cos 45^\circ, \sin 45^\circ), \text{ and } (\cos s^\circ, \sin s^\circ)\] is isosceles and its area is greater than $0.1$.
A) 15
B) 30
C) 45
D) 60


A palindrome between $1000$ and $10,000$ is chosen at random. Find the probability that it is divisible by $11$.


If $a=\log_5 243$ and $b=\log_3 27$, find the relationship between $a$ and $b$.

Solution: \boxed{a = \frac{5}{3}b}


In an isosceles triangle, one of the angles measures $60^\circ$. Determine the sum of the three possible values of another angle $y^\circ$ in the triangle.


Determine the number of integer solutions (x, y) to the equation xy - x - y = 3.


A box contains $30$ red balls, $25$ green balls, $18$ yellow balls, $15$ blue balls, $12$ purple balls, $10$ white balls, and $7$ black balls. What is 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?


Let $a + 2 = b + 4 = c + 6 = d + 8 = a + b + c + d + 10$. Solve for the value of $a + b + c + d$.


Erin the ant starts at a given vertex of a tetrahedron and crawls along exactly three edges in such a way that she visits every vertex exactly once. Calculate the number of paths meeting these conditions.


Claire's car's odometer reading was 12321, a palindrome. She drove to a location where her car's odometer showed another palindrome. If Claire drove for 4 hours and her speed never exceeded 65 miles per hour, calculate her greatest possible average speed.


The mean of the 9 data values $70, 110, x, 55, 45, 220, 85, 65, x$ is $x$. Calculate the value of $x$ given that the sum of the 9 data values is $9x$.


Given that $5.17171717\ldots$ can be written as a fraction, find the sum of the numerator and the denominator of this fraction when reduced to lowest terms.


Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate that is faster than the ship. She counts 300 equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts 60 steps of the same size from the front of the ship to the back. Determine the length of the ship in terms of Emily's equal steps.


A 2x2x2 cube is made of $8$ normal dice. Each die's opposite faces sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the small cube?


Given a particle projected with an initial velocity of $180$ feet per second from the ground at an angle of $45^\circ$ to the horizontal, derive the expressions for the horizontal range $R$ and the maximum height $H$ by using the given expression $s = 180t \sin(45^\circ) - 16t^2$ where $s$ is the vertical displacement in feet as a function of time $t$ in seconds.


Determine the number of even positive $3$-digit integers that are divisible by $5$ but do not contain the digit $5$.


In an International track meet, 256 sprinters participate in a 100-meter dash competition. If the track has 8 lanes, and only the winner of each race advances to the next round while the others are eliminated, how many total races are needed to determine the champion sprinter?


For how many integers $n$ is $(n+i)^6$ an integer?


A triangle has an area of $24$, one side of length $8$, and the median to that side of length $7$. Let $\theta$ be the acute angle formed by that side and the median. Find $\cos{\theta}$.


Change the digit of $0.123456$ to an $8$ to result in the smallest number.


Consider two events A and B where the probability that event A occurs is $\frac{5}{6}$, and the probability that event B occurs is $\frac{3}{4}$. Determine the smallest interval that necessarily contains the probability that both A and B occur.

Solution: \boxed{\left[\frac{7}{12}, \frac{3}{4}\right]}


The miniature tower holds 0.2 liters of water and is a scaled version of the original tower that holds 200,000 liters of water. Calculate the height of the miniature tower given that the original tower is 60 meters high.


Determine the area of the polygon with vertices at $(0,1)$, $(3,4)$, $(7,1)$, and $(3,7)$.


The highest power of 3 that is a factor of the integer $N=181920\cdots9293$, formed by writing the 2-digit integers from 18 to 93 consecutively, must be calculated.


A cylinder has a radius of 5 cm and a height of 10 cm. Find the height of another cylinder that has twice the volume of this original cylinder, assuming the radius is doubled.


A right rectangular prism has edge lengths \(\log_{3}x, \log_{5}x,\) and \(\log_{6}x,\) and its surface area and volume are numerically equal. Find the value of \(x\).


In a coordinate plane, points $A$ and $B$ are $12$ units apart. Determine the number of points $C$ such that for $\triangle ABC$, the perimeter is $60$ units, and the area is $240$ square units.


It takes Mina 90 seconds to walk down an escalator when it is not operating, and 30 seconds to walk down when it is operating. Additionally, it takes her 40 seconds to walk up another escalator when it is not operating, and only 15 seconds to walk up when it is operating. Calculate the time it takes Mina to ride down the first operating escalator and then ride up the second operating escalator when she just stands on them.


Given the expression $[(a+2b)^3(a-2b)^3]^3$, determine the number of terms in the simplified expansion of this expression.


At Archimedes Academy, there are three teachers, Mrs. Algebra, Mr. Calculus, and Ms. Statistics, teaching students preparing for the AMC 8 contest. Mrs. Algebra has 13 students, Mr. Calculus has 10 students, and Ms. Statistics has 12 students. If 3 students are enrolled in both Mrs. Algebra's and Ms. Statistics' classes, and no other overlaps exist, how many distinct students are preparing for the AMC 8 at Archimedes Academy?


Given a cashier mistakenly counts $y$ half-dollar coins as $1$ dollar bills and $y$ $5$ dollar bills as $10$ dollar bills, calculate how much the cashier should adjust the total cash amount to account for these mistakes.


A rectangle has length $AC = 48$ and width $AE = 30$. Points $B$ and $F$ divide lines $\overline{AC}$ and $\overline{AE}$ in the ratio $1:3$ and $2:3$, respectively. Find the area of quadrilateral $ABDF$.


Given the operation \( x \clubsuit y = (x^2 + y^2)(x - y) \), calculate \( 2 \clubsuit (3 \clubsuit 4) \).


If $a @ b = \frac{a \times b}{a + b + 2}$, calculate the result of $7 @ 21$ after adding 3 to the result.


In $\triangle ABC$, point $F$ divides side $AC$ in the ratio $1:3$. Let $E$ be the point of intersection of side $BC$ and $AG$ where $G$ is the midpoint of $BF$. Determine the ratio in which the point $E$ divides side $BC$.


Consider a modified finite sequence of four-digit integers where the tens, hundreds, and units digits of each term are, respectively, the thousands, hundreds, and tens digits of the next term, and the tens, hundreds, and units digits of the last term are, respectively, the thousands, hundreds, and tens digits of the first term. Let S be the sum of all terms in the sequence. Determine the largest prime factor that always divides S.


Given a box containing $30$ red balls, $22$ green balls, $18$ yellow balls, $15$ blue balls, and $10$ black balls, determine the minimum number of balls that must be drawn from the box to guarantee that at least $12$ balls of a single color will be drawn.


Given a circle of radius $7$ inscribed in a rectangle, where the ratio of the length of the rectangle to its width is $3:1$, calculate the area of the rectangle.


Given trapezoid $ABCD$, $\overline{AD}$ is perpendicular to $\overline{DC}$, $AD = AB = 5$, and $DC = 10$. In addition, $E$ is on $\overline{DC}$ such that $DE = 4$. If $\overline{BE}$ is parallel to $\overline{AD}$, find the area of $\triangle ADE$.


The mean of three numbers is 20 more than the least of the numbers and 18 less than the greatest. The median of the three numbers is 9. Determine their sum.


Given spinners $P$, $Q$, and $R$ with values 1, 2, 3, 4; 2, 4, 6; and 1, 3, 5 respectively, what is the probability that the sum of the resulting numbers from spinning each is an odd number?


For how many integers $n$ is $(n+i)^6$ an integer?


A point is chosen at random within the square in the coordinate plane whose vertices are $(0, 0), (1000, 0), (1000, 1000),$ and $(0, 1000)$. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{4}$. Find the value of $d$ to the nearest tenth.


Find the number of ordered pairs $(a, b)$ of positive integers that satisfy the equation: $a\cdot b + 100 = 25\cdot \text{lcm}(a, b) + 15\cdot\text{gcd}(a,b)$.


Given that the car ran exclusively on battery for the first $75$ miles, then ran exclusively on gasoline for the rest of the journey, consuming gasoline at a rate of $0.05$ gallons per mile, and the car averaged $50$ miles per gallon for the entire trip, determine the total length of the trip in miles.


Evaluate the expression: $[x + (y-z)] - [(x+z) - y]$.

Solution: \boxed{2y - 2z}


For how many integers \( n \) in the set \(\{1, 2, 3, \ldots, 200\}\) is the units digit of \( n^3 \) greater than 5?


Given a frog starting at point (0,0) and making jumps of length 4, determine the smallest possible number of jumps the frog must make to reach the point (6,2).


Given \( x = 1 + 3^p \) and \( y = 1 + 3^{-p} \), express y in terms of x.


Susie buys $5$ muffins and $2$ bananas. Calvin spends three times as much as Susie on $3$ muffins and $12$ bananas. Determine how many times a muffin is as expensive as a banana.


Given g(x) = ax^2 - \sqrt{3} for some positive constant a, find the value of a if g(g(\sqrt{3})) = -\sqrt{3}.


Jonas sets his watch correctly at 8:00 AM and notices that his watch reads 9:48 AM at the actual time of 10:00 AM. Assuming his watch loses time at a constant rate, calculate the actual time when his watch will first read 5:00 PM.

Solution: \boxed{6:00 PM}


If James has taken four tests, with scores of 82, 70, and 88, and wants to average 85 for a total of six tests, what is the lowest score he could earn on one of the remaining three tests?


How many pairs $(m,n)$ of integers satisfy the equation $m+n=mn-1$?


Evaluate the expression \(\left(\frac{x^2+2}{x}\right)\left(\frac{y^2+2}{y}\right)+\left(\frac{x^2-2}{y}\right)\left(\frac{y^2-2}{x}\right)\), given \(xy \neq 0\).

Solution: \boxed{2xy + \frac{8}{xy}}


Liam read for 4 days at an average of 42 pages per day, and for 2 days at an average of 50 pages per day, then read 30 pages on the last day. What is the total number of pages in the book?


Given $x=\frac{-1+i\sqrt{3}}{2}$ and $y=\frac{-1-i\sqrt{3}}{2}$, where $i^2=-1$, calculate the value of $x^{15} + y^{15}$.


Given the equation $x^4y^2 - 10x^2y + 9=0$, determine the number of distinct ordered pairs $(x,y)$ where $x$ and $y$ have positive integral values.


A lemming starts at a corner of a square with side length 12 meters. It runs 7.8 meters along a diagonal towards the opposite corner, then turns 60 degrees to the right relative to its original direction and runs 3 meters. Find the average of the shortest distances from the lemming to each side of the square.


Given that the median of the numbers $3, 5, 7, 23,$ and $x$ is equal to the mean of those five numbers, calculate the sum of all real numbers $x$.


Given that four $\Delta$'s and two $\diamondsuit$'s balance twelve $\bullet$'s, and two $\Delta$'s balance a $\diamondsuit$ and three $\bullet$'s, calculate the number of $\bullet$'s that balance three $\diamondsuit$'s in this balance.


Determine the number of sets of two or more consecutive positive integers whose sum is 120.


It takes Clea 75 seconds to walk down an escalator when it is not moving, and 30 seconds when it is moving. Determine the time it would take Clea to ride the escalator down when she is not walking.


Given Hammie and his triplet siblings' weights are 120, 4, 7, and 10 pounds respectively, calculate the difference between the mean and the median of their weights.


If it costs two cents for each plastic digit used to number each locker and it costs $294.94 to label all lockers up to a certain number, calculate the highest locker number labeled.


The product of two positive numbers is 16. The reciprocal of one of these numbers is 3 times the reciprocal of the other number. Find the sum of the two numbers.


Using a calculator with only [+1] and [x2] keys and starting with the display "1", calculate the fewest number of keystrokes needed to reach "500".


Determine the percentage by which the price was increased and then decreased, given that the resulting price is 75% of the original price.


Moe, Loki, Nick, and Ott are friends. Thor initially had no money, while the other friends did have money. Moe gave Thor one-sixth of his money, Loki gave Thor one-fifth of his money, Nick gave Thor one-fourth of his money, and Ott gave Thor one-third of his money. Each friend gave Thor $2. Determine the fractional part of the group's total money that Thor now has.


A bag contains four pieces of paper, each labeled with one of the prime digits 2, 3, 5, or 7. Three of these pieces are drawn, one at a time without replacement, to construct a three-digit number. Calculate the probability that this three-digit number is a multiple of 3.


Let $x$ and $y$ be three-digit positive integers with a mean of $505$. Find the maximum value of the difference $x - y$.


Given the fraction $\frac{987654321}{2^{30}\cdot 5^6}$, calculate the minimum number of digits to the right of the decimal point needed to express it as a decimal.


Set $A$ has $30$ elements, and set $B$ has $25$ elements. Set $C$ has $10$ elements and is a subset of both $A$ and $B$. Determine the smallest possible number of elements in $A \cup B$.


The centers of two circles are $50$ inches apart. One circle has a radius of $7$ inches and the other has a radius of $10$ inches. Determine the length of the common internal tangent.

Solution: \boxed{\sqrt{2211}\text{ inches}}


A sphere with center \(O\) has a radius of \(8\). An isosceles triangle with sides \(17, 17,\) and \(16\) is situated in space such that each of its sides is tangent to the sphere. Determine the distance between \(O\) and the plane determined by the triangle.


Given that Chelsea is ahead by 60 points halfway through a 120-shot archery contest, with each shot scoring 10, 8, 5, 3, or 0 points and Chelsea scoring at least 5 points on every shot, determine the smallest number of bullseyes (10 points) Chelsea needs to shoot in her next n attempts to ensure victory, assuming her opponent can score a maximum of 10 points on each remaining shot.


In a right triangle, the sides adjacent to the right angle are $a$ and $b$, with the hypotenuse being $c$. A perpendicular from the right angle vertex divides the hypotenuse $c$ into two segments $r$ and $s$, where $r$ is adjacent to $a$ and $s$ is adjacent to $b$. Given that the ratio of $a$ to $b$ is $2 : 5$, find the ratio of $r$ to $s$.


A woman buys a house for $20,000 and rents it out. She sets aside $15%$ of each month's rent for maintenance, pays $400 a year in taxes, and wants to realize a $6%$ return on her investment. Calculate the monthly rent she needs to charge.


A triangle has vertices at $(0,0)$, $(3,3)$, and $(8m,0)$, where $m$ is a nonzero constant. The line $y = mx$ divides the triangle into two regions of equal area. Determine the sum of all possible values of $m$.


Given that the ceiling is 3.0 meters above the ground, Bob is 1.8 meters tall and can reach 50 centimeters above the top of his head, and the light fixture is 15 centimeters below the ceiling, calculate the height of the box in centimeters.


Given two integers, their sum is 30. After adding two more integers to the first two, the sum is 47. Finally, after adding two more integers to the sum of the previous four, the sum is 65. Determine the minimum number of odd integers among the six integers.


Find the difference of the roots of the quadratic equation $3x^2 + 4x - 15 = 0$.


Determine the number of positive integer divisors of $255^8$ that are either perfect squares or perfect cubes (or both).


If $M$ is $40 \%$ of $Q$, $Q$ is $30 \%$ of $P$, and $N$ is $60 \%$ of $2P$, calculate the value of $\frac {M}{N}$.


Given Orvin went to the store with just enough money to buy 40 balloons at the regular price, and he noticed a promotion: buy 1 balloon at the regular price and get the next one at 1/2 off. What is the greatest number of balloons he could purchase?


Given a skewed six-sided die is structured so that rolling an odd number is twice as likely as rolling an even number, calculate the probability that, after rolling the die twice, the sum of the numbers rolled is odd.


Given the equation $x^{4} + x + y^2 = 2y + 3$, determine the number of ordered pairs of integers $(x, y)$ that satisfy the equation.


For values of \( x \) between $0$ and $5$, evaluate the maximum and minimum values of the expression
$$\frac{x^2 - 4x + 5}{2x - 4}.$$
A) Minimum value of -1 and maximum value of 1
B) Minimum value of 1 and maximum value of -1
C) Maximum value only of 0
D) No extremum values within the domain

Solution: \boxed{A) Minimum value of -1 and maximum value of 1}


Given that eight teams play each other three times, with a team earning $3$ points for a win and $2$ points for a draw, determine the maximum possible number of total points that each of the four teams could have earned.


Given the equation $x^{2024} + y^2 = 2y + 1$, calculate the number of ordered pairs of integers $(x, y)$.


For how many integers \( x \) does a triangle with side lengths \( 13, 15 \) and \( x \) have all its angles acute?


Given Chloe chooses a real number uniformly at random from the interval $[0, 100]$ and Laurent chooses a real number uniformly at random from the interval $[0, 200]$, find the probability that Laurent's number is greater than Chloe's number.


Calculate the area of a quadrilateral with vertices at \((2,1)\), \((4,3)\), \((7,1)\), and \((4,6)\).


A mixture of $50$ liters of paint is composed of $20\%$ red tint, $40\%$ yellow tint, and $40\%$ water. Six liters of red tint are added to the original mixture. What is the percent of red tint in the new mixture?


Given that 40% of the birds are sparrows, 20% are pigeons, 15% are parrots, and 25% are crows in Oakwood Park, calculate the percentage of the birds that are not pigeons which are sparrows.


Shelby drives her car at a speed of $40$ miles per hour if it is not raining, and $25$ miles per hour if it is raining. Today she drove in the sun in the morning and in the rain in the evening, for a total of $20$ miles in $36$ minutes. Determine the number of minutes she drove in the rain.


The probability that when all 8 dice are rolled, the sum of the numbers on the top faces is 12. What other sum occurs with the same probability as this?


A half-sector of a circle of radius 6 inches together with its interior can be rolled up to form the lateral surface area of a right circular cone by taping along the two radii shown. Calculate the volume of the cone.

Solution: \boxed{9\pi \sqrt{3}}


Given that Maria's pedometer flips 50 times from $99999$ to $00000$ and reads $25000$ on December 31, and she covers $1500$ steps per mile, calculate the total distance Maria walked during the year.


A company has 28 employees, among which 16 have brand A computers and 12 have brand B computers. The technician can install cables between brand A and brand B computers, ensuring no redundant connections between the same pairs. Determine the maximum possible number of cables the technician could use.


Given that $x$ is a positive real number, find an equivalent expression for $\sqrt[4]{x\sqrt[3]{x}}$.


Segments $AD=15$, $BE=9$, $CF=30$ are drawn from vertices $A$, $B$, and $C$ of triangle $ABC$, each perpendicular to a straight line $RS$, which does not intersect the triangle. Points $D$, $E$, and $F$ are the intersection points of $RS$ with the perpendiculars. If $x$ is the length of the perpendicular segment $GH$ drawn to $RS$ from the intersection point $G$ of the medians of the triangle, find the length of $x$.


The price of a ticket to a concert is \( x \) dollars, where \( x \) is an even whole number. A group of students from one class buys tickets costing a total of \( \$72 \), and another group from a different class buys tickets costing a total of \( \$108 \). Determine the number of possible values for \( x \).


During a mathematics competition preparation period, John aims to score a perfect "100" on at least $85\%$ of his $40$ practice tests. He has scored a "100" on $30$ of the first $36$ tests. Calculate the maximum number of the remaining tests on which he can score less than "100".


Suppose $3 + \frac{1}{2 + \frac{1}{3 + \frac{3}{4+x}}} = \frac{225}{68}$. Determine the value of $x$.


Given that $M = 58^3 + 3 \cdot 58^2 + 3 \cdot 58 + 1$, determine the number of positive integers that are factors of $M$.


A list of $3042$ positive integers has a unique mode, which occurs exactly $15$ times. Calculate the least number of distinct values that can occur in the list.


Calculate the probability that a randomly chosen divisor of $24!$ is odd.


Given that spinner S has numbers 1, 4, 3, spinner T has numbers 2, 4, 6, and spinner U has numbers 2, 3, 5, determine the probability that the sum of the numbers obtained from spinning each of the three spinners is an even number.


Professors Alpha, Beta, Gamma, and Delta choose their chairs so that each professor will be between two students. Given that there are 13 chairs in total, determine the number of ways these four professors can occupy their chairs.


Alice has $30$ apples. In how many ways can she share them with Becky and Chris so that each of the three people has at least four apples?


Given that a recipe that makes $4$ servings requires $3$ squares of chocolate, $1/2$ cup of sugar, $2$ cups of water, and $3$ cups of milk, if Emily maintains the same ratio of ingredients, calculate the greatest number of servings of the dessert she can make with $9$ squares of chocolate, $3$ cups of sugar, plenty of water, and $10$ cups of milk.


The relationship between the arithmetic mean and the geometric mean of the segments $a$, $b$, and $c$ can be accurately represented.

Solution: \boxed{\frac{a+b+c}{3} \geq \sqrt[3]{abc}}


When $\left(1 - \frac{1}{a}\right)^8$ is expanded, calculate the sum of the last three coefficients.


Given that \( x \) and \( y \) are nonzero real numbers and \(\frac{4x+y}{x-4y} = 3\), calculate the value of \(\frac{x+4y}{4x-y}\).


Jean makes a mixed drink using $150$ grams of cranberry juice, $50$ grams of honey, and $300$ grams of water. There are $30$ calories in $100$ grams of cranberry juice, and $304$ calories in $100$ grams of honey. Water contains no calories. Calculate the total calories in $250$ grams of Jean's mixed drink.


The two concentric circles have the same center $O$. A chord $\overline{EF}$ of the larger circle is tangent to the smaller circle at point $G$. The distance from $O$ to $F$ is $12$ units, and the length of chord $\overline{EF}$ is $20$ units. Find the area between the two circles.


Given that the length of AB is 3 times the length of BD and the length of AC is 5 times the length of CD on line segment AD, express the length of BC as a fraction of the length of AD.


Given the sequence $x, 3x+3, 6x+6, \dots$ are in geometric progression, find the fourth term.


What is the maximum number of solid $3\text{-in} \times 1\text{-in} \times 1\text{-in}$ blocks that can be placed inside a $3\text{-in} \times 4\text{-in} \times 3\text{-in}$ box?


A group of friends collects $120 to buy flowers for their teacher. Each rose costs $4, and each daisy costs $3. They want to buy at least 5 roses. Find the number of different bouquets they could purchase for exactly $120.


Starting with a display of "1," calculate the fewest number of keystrokes needed to reach "1458" using the keys [+1], [x2], and [x3].


How many four-digit whole numbers are there such that the leftmost digit is an odd prime, the second digit is a multiple of 3, and all four digits are different?


Each of 8 balls is randomly and independently painted either black or white with equal probability. Find the probability that every ball is different in color from at least four of the other 7 balls.


Two pitchers, each with a capacity of 800 mL, are partially filled with beverages. The first pitcher is $\frac{1}{4}$ full of orange juice, and the second pitcher is $\frac{3}{8}$ full of apple juice. Both pitchers are then filled to capacity with water and the contents are poured into a larger container. What fraction of the mixture in the large container is orange juice?


Determine how many positive integers $n$ satisfy the condition that $\frac{n}{40-n}$ is also a positive integer.


Let \( m \) be a positive integer and let the lines \( 17x+7y=1000 \) and \( y=mx+2 \) intersect in a point whose coordinates are integers. Find the possible value of \( m \).


Given a right rectangular prism with edge lengths $\log_{5}y, \log_{6}y,$ and $\log_{10}y$, find the value of $y$ such that the numerical value of its surface area equals its volume.


Given a pair of $8$-sided dice each with numbers from $1$ to $8$ is rolled once, the sum of the numbers rolled determines the diameter of a circle. Determine the probability that the numerical value of the area of the circle is less than the numerical value of the circumference.


Given that a contest began at noon one day and ended $1500$ minutes later, determine the time at which the contest ended.

Solution: \boxed{1:00 \text{ p.m.}}


Given a square is displayed with 121 grid points, uniformly spaced, including those on the edges, and point $P$ is precisely at the center of the square, determine the probability that the line $PQ$ is a line of symmetry for the square if point $Q$ is selected at random from the remaining 120 points.


A triangle has one of its sides divided by the point of tangency of the inscribed circle into segments of $9$ and $5$ units. If the radius of the inscribed circle is $3$ units, calculate the length of the longest side of the triangle.


Given Ahn chooses a three-digit integer, subtracts it from $500$, and triples the result, find the largest number Ahn can get.


Circle A has a radius of 150. Circle B has an integer radius r < 150 and remains internally tangent to circle A as it rolls once around the circumference of circle A. Determine the number of possible values for r.


What was the regular price of one tire, given that Sam paid 250 dollars for a set of four tires at the sale, with the fourth tire being offered for 10 dollars when three tires are bought at the regular price?


In a small town, Sarah usually bikes to her bookstore in 30 minutes. However, one windy day she decides to ride faster, biking 12 miles per hour faster than usual, getting her to the bookstore in 18 minutes. Determine the distance to the bookstore in miles.


If 1 yen is worth 0.0075 dollars, what percent is the value of Etienne's 5000 yen greater than or less than the value of Diana's 500 dollars?


The point (c, d) in the xy-plane is first rotated clockwise by 180° around the point (2, -3) and then reflected about the line y = x. The image of (c, d) after these two transformations is at (5, -4). Find the value of d - c.


What is the tens digit of $2035^{2037} - 2039$?


The even number 134 can be represented as the sum of two distinct prime numbers. Find the largest possible difference between these two prime numbers.


Given that $n$ is a positive integer, the sum of the positive factors of $n$ is represented as $\boxed{n}$, where $\boxed{5}$ equals $1 + 5 = 6$. Determine the value of $\boxed{\boxed{7}}$.


The number of positive integers less than $500$ divisible by neither $5$ nor $7$.


A ferry boat begins shuttling tourists to an island every hour starting at 10 AM, with its last trip starting at 4 PM. On the 10 AM trip, there were 100 tourists on the ferry, and on each successive trip, the number of tourists decreased by 2 from the previous trip. Calculate the total number of tourists transported to the island that day.


Given the number $9,680$, find the number of its positive factors.


Anna selects a real number uniformly at random from the interval $[0, 3000]$, and Bob chooses a real number uniformly at random from the interval $[0, 6000]$. Determine the probability that Bob's number is greater than Anna's number.


What is the hundreds digit of $(18! + 14!)$?


Find the number of distinct points common to the curves $x^2 + 4y^2 = 4$, $4x^2 + y^2 = 4$, and $x^2 + y^2 = 1$.


The perimeter of a rectangular garden with dimensions 32 meters by 72 meters must be encircled by a fence, with fence posts placed every 8 meters, including the posts at each corner. Determine the fewest number of posts required.


Given that Marie has 2500 coins consisting of pennies (1-cent coins), nickels (5-cent coins), and dimes (10-cent coins) with at least one of each type of coin, calculate the difference in cents between the greatest possible and least amounts of money that Marie can have.


Point $O$ is the center of the regular octagon $ABCDEFGH$, and $Y$ is the midpoint of side $\overline{CD}$. Calculate the fraction of the area of the octagon that is shaded, where the shaded region includes triangles $\triangle DEO$, $\triangle EFO$, $\triangle FGO$, and half of $\triangle DCO$.


Points E and F lie on segment GH. The length of segment GE is 3 times the length of segment EH, and the length of segment GF is 4 times the length of segment FH. Express the length of segment EF as a fraction of the length of segment GH.


Given that liquid $Y$ forms a circular film $0.2$ cm thick when poured onto a water surface, find the radius, in centimeters, of the resulting circular film created when a rectangular box with dimensions $10$ cm by $5$ cm by $20$ cm is filled with liquid $Y$ and its contents are poured onto a large body of water.


Maria drives 25 miles at an average speed of 40 miles per hour. Calculate the distance Maria needs to drive at 75 miles per hour to average 60 miles per hour for the entire trip.


The difference between the sum of the first 1000 even numbers, starting from 0, and the sum of the first 1000 odd counting numbers.


For what value of $x$ does $2^{3x} \cdot 8^{x-1} = 32^{2x}$


If $\frac{b}{a} = 3$, $\frac{c}{b} = 2$, and $\frac{d}{c} = 4$, calculate the ratio of $a + c$ to $b + d$.


A driver travels for 3 hours at a speed of 50 miles per hour. The car has a fuel efficiency of 25 miles per gallon, earns $0.60 per mile, and gasoline costs $2.50 per gallon. Calculate her net rate of pay per hour after considering gasoline expenses.


Cagney can frost a cupcake every 25 seconds and Lacey can frost a cupcake every 20 seconds. If they both take a 30-second break after every 3 minutes of work, how many cupcakes can they frost together in 10 minutes?


If a total of 28 conference games were played during the 2017 season, determine the number of teams that were members of the BIG N conference.


With all three valves open, the tank fills in 0.5 hours, with only valves A and C open it takes 1 hour, and with only valves B and C open it takes 2 hours. Determine the number of hours required with only valves A and B open.


Ted's grandfather used his treadmill on 4 days this week with distances of 3 miles on Monday, Wednesday, and Friday, and 4 miles on Sunday. The times taken to travel these distances were 0.5 hours, 0.75 hours, and 0.6 hours on Monday, Wednesday, and Friday, respectively, and 4/3 hours on Sunday. If Grandfather had always walked at 5 miles per hour, calculate the time saved in minutes.


Let $x$ be the smallest positive real number such that $\cos(x) = \cos(3x)$, where $x$ is measured in radians. Express $x$ in radians, rounded to two decimal places.


If two factors of $3x^3 - mx + n$ are $x-3$ and $x+4$, determine the value of $|3m-2n|$.


Given a real number $x$, simplify the expression $\sqrt{x^4 - x^2}$.


Sam drove $150$ miles in $135$ minutes. His average speed during the first $45$ minutes was $50$ mph, and his average speed during the second $45$ minutes was $60$ mph. Calculate the average speed, in mph, during the last $45$ minutes.


A $3\times 3$ block of calendar dates is shown. First, the order of the numbers in the first and the third rows are reversed. Then, the numbers on each diagonal are added. What will be the positive difference between the two diagonal sums?


Given that $\frac{x}{3} = y^2$ and $\frac{x}{6} = 3y$, solve for the value of $x$.


Given the quadratic polynomial $ax^2+bx+c$, identify the value that is incorrect among the outcomes $2107, 2250, 2402, 2574, 2738, 2920, 3094, 3286$.


Given that Mrs. Wanda Wake drives to work at 7:00 AM, if she drives at an average speed of 30 miles per hour, she will be late by 2 minutes, and if she drives at an average speed of 50 miles per hour, she will be early by 2 minutes. Find the required average speed for Mrs. Wake to get to work exactly on time.


Given that there are $400$ adults in total in City Z, with $370$ adults owning bikes and $75$ adults owning scooters, determine the number of bike owners who do not own a scooter.


In a round-robin tournament with 8 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. At the end of the tournament, the teams are ranked by the number of games won, calculate the maximum number of teams that could be tied for the most wins.


Given that John begins the first task at 8:00 AM and finishes the third task at 11:30 AM, determine the time when he finishes the fourth task.

Solution: \boxed{12:40 \text{ PM}}


A merchant buys goods at a 30% discount off the list price and wants to mark up the goods. He plans to offer a 20% discount on his marked price but still aims to achieve a 30% profit on the selling price. What percentage of the list price should he set as the marked price?


The fifth term in the expansion of $(\frac{b}{x}-x^2b)^7$ when simplified is $-35b^7x^5$.


Alice is baking two batches of cookies, one batch requiring $3 \frac{1}{2}$ cups of sugar. Her measuring cup can hold $\frac{1}{3}$ cup of sugar. Determine the total number of times she must fill her cup to measure the sugar needed for both batches.


Given that a semipro baseball league has 25 players per team and each player's minimum salary is $15,000, while the total salary cap for each team is $875,000, calculate the maximum possible salary for a single player.


Given that in a bag of marbles, $\frac{2}{3}$ of the marbles are blue and the rest are red, find the new fraction of the marbles that will be red if the number of red marbles is tripled while the number of blue marbles stays the same.


Given two circles with centers 50 inches apart, where the radius of the smaller circle is 7 inches and the radius of the larger circle is 10 inches, calculate the length of the common internal tangent between these two circles.

Solution: \boxed{\sqrt{2211} \text{ inches}}


A number, $n$, is added to the set $\{4, 8, 12, 15\}$ to make the mean of the set of five numbers equal to its median. Find the smallest possible value of $n$.


Jenny and Joe shared a pizza that was cut into 12 slices. Jenny prefers a plain pizza, but Joe wanted mushrooms on four of the slices. A whole plain pizza costs $12. Each slice with mushrooms adds an additional $0.50 to the cost of that slice. Joe ate all the mushroom slices and three of the plain slices. Jenny ate the remaining plain slices. Each paid for what they had eaten. What is the difference, in dollars, that Joe paid compared to Jenny?


In parallelogram $\text{EFGH}$, point $\text{J}$ is on $\text{EH}$ such that $\text{EJ} + \text{JH} = \text{EH} = 12$ and $\text{JH} = 8$. If the height of parallelogram $\text{EFGH}$ from $\text{FG}$ to $\text{EH}$ is 10, find the area of the shaded region $\text{FJGH}$.


The sum of five integers is $3$. Calculate the maximum number of these integers that can be larger than $26$.


Given real numbers $x$ and $y$, find the least possible value of $(x^2y-1)^2 + (x^2+y)^2$.


Given two pipes, one with an inside diameter of $2$ inches and a length of $2$ meters, and the other with an inside diameter of $8$ inches and a length of $1$ meter, determine the number of circular pipes with an inside diameter of $2$ inches and a length of $2$ meters that will carry the same amount of water as one pipe with an inside diameter of $8$ inches and a length of $1$ meter.


Given that $f(x+6) + f(x-6) = f(x)$ for all real $x$, determine the least positive period $p$ for these functions.


Consider a line segment whose endpoints are (5, 10) and (68, 178), determine the number of lattice points on this line segment, including both endpoints.


Given that Eva is using 24 fence posts to fence her rectangular vegetable patch, with each post spaced 6 yards apart, including the corners, and the longer side of the patch has three times as many posts as the shorter side, calculate the area of Eva’s vegetable patch.


Evaluate the expression $\frac{1}{1+\frac{1}{2+\frac{1}{1+\frac{1}{4}}}}$.


Find the area of quadrilateral $EFGH$, given that $m\angle F = m \angle G = 135^{\circ}$, $EF=4$, $FG=6$, and $GH=8$.


Given the annual incomes of $500$ families ranging from $12,000$ dollars to $150,000$ dollars, where the largest income was incorrectly entered as $1,500,000$ dollars, calculate the difference between the mean of the incorrect data and the mean of the actual data.


Lucas, Emma, and Noah collected shells at the beach. Lucas found four times as many shells as Emma, and Emma found twice as many shells as Noah. Lucas decides to share some of his shells with Emma and Noah so that all three will have the same number of shells. What fraction of his shells should Lucas give to Emma?


Square $ABCD$ has its vertex $A$ at the origin of the coordinate system, and side length $AB = 2$. Vertex $E$ of isosceles triangle $\triangle ABE$, where $AE=BE$, is inside the square. A circle is inscribed in $\triangle ABE$ tangent to side $AB$ at point $G$, exactly at the midpoint of $AB$. Find the area of $\triangle ABF$, where $F$ is the point of intersection of diagonal $BD$ of the square and line segment $AE$.


Calculate the number of integer values of $n$ such that $3200 \cdot \left(\frac{4}{5}\right)^n$ remains an integer.


Given seven positive consecutive integers starting with $c$, find the average of seven consecutive integers starting with $d$, where $d$ is the average of the first set of integers.

Solution: \boxed{c + 6}


Given a positive number $x$ that satisfies the inequality $\sqrt{x} > 3x$, determine the inequality that must be true.

Solution: \boxed{0 < x < \frac{1}{9}}


What is the expression $\frac{x-2}{4-z} \cdot \frac{y-3}{2-x} \cdot \frac{z-4}{3-y}$ in its simplest form?


At $5:50$ o'clock, calculate the angle between the hour and minute hands of a clock.


Given that circle $O$ contains points $C$ and $D$ on the same side of diameter $\overline{AB}$, $\angle AOC = 40^{\circ}$, and $\angle DOB = 60^{\circ}$, calculate the ratio of the area of the smaller sector $COD$ to the area of the circle.


The largest number by which the expression $n^4 - n^2$ is divisible for all possible integral values of $n$.


Jeremy wakes up at 6:00 a.m., catches the school bus at 7:00 a.m., has 7 classes that last 45 minutes each, enjoys 45 minutes for lunch, and spends an additional 2.25 hours (which includes 15 minutes for miscellaneous activities) at school. He takes the bus home and arrives at 5:00 p.m. Calculate the total number of minutes he spends on the bus.


Given $a,b>0$ and the triangle in the first quadrant bounded by the coordinate axes and the graph of $ax+by=8$ has area 8, determine the value of $ab$.


Jack drove 150 miles in 2.5 hours. His average speed during the first hour was 50 mph. After a 15-minute stop, he resumed travel for another hour at an average speed of 55 mph. Calculate his average speed, in mph, during the last 30 minutes.


The interior of a quadrilateral is bounded by the graphs of $(x+by)^2 = 9b^2$ and $(bx-y)^2 = 4b^2$, where $b$ is a positive real number. Determine the area of this region in terms of $b$, valid for all $b > 0$.

Solution: \boxed{\frac{24b^2}{1 + b^2}}


Let $s$ be the result of tripling both the base and exponent of $c^d$, where $d$ is a non-zero integer. If $s$ equals the product of $c^d$ by $y^d$, determine the value of $y$.


Given that John divides a circle into 15 sectors with central angles that form an arithmetic sequence, calculate the degree measure of the smallest possible sector angle.


Let the roots of $px^2 + qx + r = 0$ be $u$ and $v$. Determine the quadratic equation with roots $p^2u - q$ and $p^2v - q$.

Solution: \boxed{x^2 + (pq + 2q)x + (p^3r + pq^2 + q^2) = 0}


Suppose that $\frac{2}{3}$ of $15$ bananas are worth as much as $12$ oranges, calculate the number of oranges that are worth as much as $\frac{1}{4}$ of $20$ bananas.


Find how many 3-digit whole numbers have a digit-sum of 27 and are even.


Given a bag of popping corn that contains $\frac{3}{4}$ white kernels and $\frac{1}{4}$ yellow kernels, only $\frac{1}{3}$ of the white kernels will pop, while $\frac{3}{4}$ of the yellow ones will pop, calculate the probability that the kernel selected was white.


Determine how many integers $n$ exist such that $(n+i)^6$ is an integer.


The sum other than $11$ which occurs with the same probability when all 8 dice are rolled is equal to what value.


Four runners start running simultaneously from the same point on a 600-meter circular track. They each run clockwise around the course maintaining constant speeds of 5.0, 5.5, 6.0, and 6.5 meters per second. Calculate the total time until the runners are together again somewhere on the circular course.


Calculate the simplified value of the expression:
\[
\frac{4^2 \times 7}{8 \times 9^2} \times \frac{8 \times 9 \times 11^2}{4 \times 7 \times 11}
\]


Given that an ATM password is composed of four digits from $0$ to $9$, and the fourth digit cannot be the same as the third digit, calculate the number of valid passwords possible.


Given a rectangular floor that is $11$ feet wide and $19$ feet long, and tiled with $209$ one-foot square tiles, calculate the number of tiles that a bug visits when it walks from one corner to the opposite corner in a straight line, including the first and the last tile.


A cylindrical tank with radius 6 feet and height 7 feet is lying on its side. The tank is filled with water to a depth of 3 feet. Find the volume of water in the tank, in cubic feet.

Solution: \boxed{84\pi - 63\sqrt{3}}


Given $g(x) = 3x^2 + 4x + 5$, calculate $g(x + h) - g(x)$.

Solution: \boxed{h(6x + 3h + 4)}


Given that a 3-digit whole number's digit-sum is $27$, how many of these numbers are even?


Given that Connie divided a number by $3$ and got $27$, determine the correct number that she should have multiplied by $3$ to get the correct answer.


Michael jogs daily around a track consisting of long straight lengths connected by a full circle at each end. The track has a width of 4 meters, and the length of one straight portion is 100 meters. The inner radius of each circle is 20 meters. It takes Michael 48 seconds longer to jog around the outer edge of the track than around the inner edge. Calculate Michael's speed in meters per second.


Determine $r_2$ if $q_1(x)$ and $r_1$ are the quotient and remainder, respectively, when the polynomial $x^6$ is divided by $x - \tfrac{1}{3}$, and $q_2(x)$ and $r_2$ are the quotient and remainder, respectively, when $q_1(x)$ is divided by $x - \tfrac{1}{3}$.


If $\frac{x}{3} = y^2$ and $\frac{x}{5} = 5y$, find the value of $x$.


In a class test, $15\%$ of the students scored $60$ points, $20\%$ scored $75$ points, another $40\%$ scored $85$ points, $20\%$ scored $95$ points, and the remainder scored $100$ points, calculate the difference between the mean and the median score.


The numbers from -5 to 10 are arranged in a 4-by-4 square such that the sum of the numbers in each row, each column, and each of the two main diagonals are all the same, find the value of this common sum.


If 300 students voted on two proposals with 230 students voting in favor of the first proposal and 190 students voting in favor of the second proposal, and 40 students voted against both proposals, determine the number of students who voted in favor of both proposals.


Given a real and positive variable $x$ approaching infinity, evaluate the limit of the expression $\log_4{(8x-3)} - \log_4{(3x+4)$.

Solution: \boxed{\log_4 \left(\frac{8}{3}\right)}


Given $G=\frac{9x^2+21x+4m}{9}$ be the square of an expression linear in $x$, and valid when $x = p - 1$ where $p \neq 0$, determine the values of $m$ that satisfy these conditions.


There are births in West Northland every 6 hours, deaths every 2 days, and a net immigration every 3 days. Calculate the approximate annual increase in population.


Given the expression $(30! - 25!)$, calculate the thousands digit of the result.


The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(2,6)$ and then reflected about the line $y = x$. The image of $P$ after these two transformations is at $(-7,4)$. Determine the value of $b - a$.


The numeral 36 in base a represents the same number as 63 in base b. Assuming that both bases are positive integers, find the least possible value of a+b.


Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be arithmetic progressions with $a_1 = 30, b_1 = 70$. If $a_{50} + b_{50} = 200$, calculate the sum of the first fifty terms of the progression $a_1 + b_1, a_2 + b_2, \ldots$.


The function $g(x)$ satisfies $g(3+x) = g(3-x)$ for all real numbers $x$. If $g(x) = 0$ has exactly three distinct real roots, find the sum of these roots.


Given that points $A$ and $B$ are 12 units apart in a plane, determine how many points $C$ are there such that the perimeter of $\triangle ABC$ is 48 units and the area of $\triangle ABC$ is 72 square units.


What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30} \cdot 5^6}$ as a decimal?


Given the equations $x^2+kx+10=0$ and $x^2-kx+10=0$. If, when the roots of the equations are suitably listed, each root of the second equation is $3$ more than the corresponding root of the first equation, then determine the value of $k$.


At noon on a certain day, Denver is \( N \) degrees warmer than Boulder. By 6:00 PM, the temperature in Denver falls by 8 degrees while the temperature in Boulder rises by 2 degrees. At that time, the temperature difference between the two cities is 3 degrees. Find the product of all possible values of \( N \).


Solve the equation $x + \sqrt{3x-2} = 6$.

Solution: \boxed{\frac{15 - \sqrt{73}}{2}}


Let $x = -2500$. Evaluate the expression $\bigg| \big| |x|-x \big| - |x| \bigg| + x$.


Given a bag of popping corn contains $\frac{3}{4}$ white kernels and $\frac{1}{4}$ neon yellow kernels, where only $\frac{2}{3}$ of the white kernels and $\frac{3}{4}$ of the neon yellow ones will pop, and given that $\frac{1}{4}$ of the popped kernels will also fizz automatically regardless of color, determine the probability that a kernel selected, which pops and then fizzes, was white.


Given that the fraction \(\frac{3}{5}\) is modified by subtracting \(2x\) from the numerator and adding \(x\) to the denominator, and that the resulting fraction is equal to \(\frac{1}{2}\), determine the value of \(x\).


What is the greatest number of consecutive integers whose sum is $36$?


A two-digit number is reduced by the sum of its digits and the units digit of the result is $7$. How many two-digit numbers satisfy this condition?


If \( 4^{x+2} = 68 + 4^x \), solve for the value of \( x \).

Solution: \boxed{\log_4\left(\frac{68}{15}\right)}


Given rectangle $ABCD$ with $E$ the midpoint of diagonal $BD$. Point $E$ is connected to a point $F$ on $DA$ so that $DF=\frac{1}{4}DA$. Find the ratio of the area of $\triangle DFE$ to the area of quadrilateral $ABEF$.


If $x$ varies as the inverse square root of $y$, and $y$ varies as the fourth power of $z$, then determine the value of $n$ for which $x$ varies as the nth power of $z$.


Given the four-digit number 3003, calculate the total number of different numbers that can be formed by rearranging the four digits.


Given that five circles are tangent to each other and to two parallel lines, and the radius of the largest circle is $20$ cm while that of the smallest circle is $6$ cm, with the radii forming an arithmetic sequence, find the radius of the middle circle.


If Ella completed 9, 13, 8, 14, 12, 10, and 11 hours in the first 7 weeks, calculate how many hours she needs to work in the eighth week to average 12 hours per week over an 8-week period.


The sum of $n$ terms of an arithmetic progression with a common difference of $3$ is $180$, and the first term is the square of an integer. Given $n>1$, find the number of possible values for $n$.


Samantha took three exams with a varying number of questions. On the first exam, which had 30 questions, she scored 75%. On the second exam with 50 questions, she secured 80%. The third exam had 20 questions, with each question being double weighted and she answered 65% correctly. What percentage of the overall weighted questions did Samantha answer correctly?


The sums of three whole numbers taken in pairs are 20, 25, and 29. Find the middle number.


Given a large 15 by 21 rectangular region, calculate the fraction of the region that is shaded if the shaded region constitutes one-fourth of one-half of the rectangle.


How many digits are in the product $3^8 \cdot 6^4$?


Let $n=x-y^{x-y}$. If $x > y$, also add $xy$ to $n$. Find $n$ when $x=3$ and $y=1$.


A poster 4 feet wide is centrally placed on a wall that is 25 feet wide. Calculate the distance from the end of the wall to the nearest edge of the poster.


Given that a spaceship orbits around a fictional planet with a radius of 3500 miles at a speed of 550 miles per hour relative to the planet, determine the number of hours required for one complete orbit.


Three fair coins are tossed once. For each head that results, two fair dice are rolled. What is the probability that the total sum of all dice rolls is odd?


Given a large semicircle with radius R and N congruent smaller semicircles inside, each lying on the diameter of the large semicircle and having their diameters completely covering this diameter without gaps or overlaps, find the value of N if the ratio of the total area of the smaller semicircles to the area of the region inside the large semicircle but outside all the smaller semicircles is 1:36.


A circle of radius 6 is inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Find the area of the rectangle.


Given that $f(x+5) + f(x-5) = f(x)$ for all real $x$, determine the least common positive period $p$ of all such functions $f$.


Given Professor Lewis has twelve different language books lined up on a bookshelf: three Arabic, four German, three Spanish, and two French, calculate the number of ways to arrange the twelve books on the shelf keeping the Arabic books together, the Spanish books together, and the French books together.


Determine the value of $n$, the number of integer values of $x$ for which $Q = x^4 + 4x^3 + 9x^2 + 2x + 17$ is a prime number.


Anna and her brother observed a freight train as it started crossing a tunnel. Anna counted 8 cars in the first 15 seconds. It took the train 3 minutes to completely pass through the tunnel at a constant speed. Determine the most likely number of cars in the train.


Given a pair of standard $6$-sided dice is rolled once, the product of the numbers rolled determines the diameter of a circle. Find the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference.


Two externally tangent circles have radii of lengths 7 and 3, respectively, and their centers at points $A$ and $B$. A line externally tangent to both circles intersects ray $AB$ at point $C$. Calculate the length of $BC$.


Given $3^{7} + 1$ and $3^{15} + 1$ inclusive, how many perfect cubes lie between these two values?


Calculate $(2.1)(50.5 + 0.15)$ after increasing $50.5$ by $5\%$. What is the product closest to?


Given that $5y$ varies inversely as the square of $x$ and $3z$ varies directly as $x$, find the value of $y$ when $x = 4$ and $z = 6$ given that when $x = 2$ and $y = 25$, the relationship holds true.


Given Jones traveled 100 miles on his first trip and 500 miles on a subsequent trip at a speed four times as fast, compare his new time to the old time.


Given Mindy made four purchases for $2.96, 6.57, 8.49, and 12.38. Each amount needs to be rounded up to the nearest dollar except the amount closest to a whole number, which should be rounded down. Calculate the total rounded amount.


A regular 17-gon has L lines of symmetry, and the smallest positive angle for which it has rotational symmetry is R degrees. Calculate the value of L + R.


Given soda is sold in packs of 8, 14, and 28 cans, determine the minimum number of packs needed to buy exactly 100 cans of soda.


A box contains $23$ red balls, $24$ green balls, $12$ white balls, and $21$ blue balls. Determine the minimum number of balls that must be drawn from the box without replacement to ensure that at least $20$ balls of a single color are drawn.


Given that Elmer's new van has 40% better fuel efficiency than his old truck, and the fuel for the new van costs 30% more per liter than the fuel for the old truck, determine the percent by which Elmer will save or spend more if he uses his new van instead of his old truck for a trip covering 300 kilometers.


What is the largest quotient that can be formed using two numbers chosen from the set $\{-30, -6, 0, 3, 5, 15\}$?


A cell phone plan costs $25$ each month, plus $8$ cents per text message sent, plus $15$ cents for each minute used over $50$ hours. Determine the total cost of the plan for John for the month, given that he sent $250$ text messages and talked for $50.75$ hours.


Determine the value of $N$ such that when $(a+b+c+d+e+1)^N$ is expanded and like terms are combined, the expression contains exactly $252$ terms that include all five variables $a, b, c, d,$ and $e$, each to some positive power.


Given integers between 2000 and 6999 have four distinct digits, calculate the number of such integers.


Two poles are 30 feet and 90 feet high and 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.


In $\triangle PQR$, $S$ is the midpoint of side $QR$ and $T$ is on side $PR$. If the length of $PR$ is $12$ units and $\angle QPR = 45^\circ, \angle PQR = 90^\circ, \angle PRQ = 45^\circ$ and $\angle RTS = 45^\circ$, find the area of $\triangle PQR$ plus twice the area of $\triangle RST$.


A list of $2023$ positive integers has a unique mode, which occurs exactly $11$ times. What is the least number of distinct values that can occur in the list?


Given that vertices P, Q, R, and S of a quadrilateral have coordinates (a, a), (a, -a), (-a, -a), and (-a, a), and the area of the quadrilateral PQRS is 36, calculate the value of a + b.


Given that Liam has written down one integer three times and another integer four times, and the sum of the seven numbers is 161, and one of the numbers is 17, find the value of the other number.


Circle C has a radius of $120$. Circle D, with an integer radius $s$, rolls externally around circle C and returns to its original position after one revolution while remaining externally tangent throughout. Determine the number of possible values for $s$ that are less than $120$.


Cagney can frost a cupcake every 15 seconds, while Lacey can frost every 40 seconds. They take a 10-second break after every 10 cupcakes. Calculate the number of cupcakes that they can frost together in 10 minutes.


Suppose $a$, $b$, and $c$ are positive integers such that $a+b+c=2010$ and $b = 2a$. If $a!b!c! = m \cdot 10^n$ where $m$ and $n$ are integers and $m$ is not divisible by 10, what is the smallest possible value of $n$?
A) 586
B) 589
C) 592
D) 595
E) 598


Consider the graphs of $y=3\log{x}$ and $y=\log{(x+4)}$. Determine the number of intersection points of the two graphs.


A large container can hold 420 grams of flour, whereas a smaller container can hold 28 grams of flour. A baker wants to transfer flour from a sack into the smaller containers and then use exactly 3 of these filled smaller containers each to fill medium containers. Calculate the number of medium containers the baker can fill using flour from one large container.


Let's consider the sum of five consecutive odd numbers starting from 997 up to 1005. If the sum $997 + 999 + 1001 + 1003 + 1005 = 5100 - M$, find $M$.


Let \( t_n = \frac{n(n+1)}{2} \) be the \( n \)th triangular number. Find the sum of the reciprocals of the first 2003 triangular numbers, from \( t_1 \) to \( t_{2003} \).


Calculate the sum of the digits when $10^{95}-95$ is expressed as a single whole number.


A woman purchases a property for $12,000 and decides to rent it. She saves $15\%$ of each month's rent for maintenance; pays $400 a year in taxes, and targets a $6\%$ return on her investment. Calculate the monthly rent.


A solid sphere is enclosed within a right circular cylinder. The volume of the cylinder is three times that of the sphere. If the height of the cylinder equals the diameter of the sphere, find the ratio of the height of the cylinder to the radius of its base.


Consider all 4-digit palindromes that can be written as $\overline{abba}$, where $a$ is non-zero and $b$ ranges from 1 to 9. Calculate the sum of the digits of the sum of all such palindromes.


Calculate the value of $\left(9^{-1} - 5^{-1}\right)^{-1}$.


Let $s$ be the result of tripling both the base and exponent of $c^d$, where $d > 0$. Express the result of this operation in terms of $c$ and $d$, and set it equal to the product of $c^d$ by $y^d$. Solve for $y$.


A circle is inscribed in a triangle with side lengths $10, 15$, and $19$. Let the segments of the side of length $10$, made by point of tangency, be $r$ and $s$, with $r < s$. Determine the ratio $r:s$.


Given that Anne cycles 3 miles on Monday, 4 miles on Wednesday, and 5 miles on Friday, and her cycling speeds on these days are 6 miles per hour, 4 miles per hour, and 5 miles per hour respectively, calculate the time difference in minutes if she cycled every day at 5 miles per hour.


How many primes less than $200$ have $3$ as the ones digit?


Given a line $x=m$ intersects the graph of $y=\log_2 x$ and the graph of $y=\log_2 (x + 6)$, find the value of $c+d$ where $m = c + \sqrt{d}$ and $c$ and $d$ are integers.


Given an unfair coin with a $3/4$ probability of turning up heads, find the probability that the total number of heads is odd when the coin is tossed $60$ times.

Solution: \boxed{\frac{1}{2}(1 - \frac{1}{2^{60}})}


Given that Professor Lee has ten different language books lined up on a bookshelf: three German, four Spanish, and three French, find the number of ways to arrange the ten books on the shelf keeping both the Spanish and French books together as separate units.


Determine the sum of all values of $z$ for which $f(4z) = 9$, given that $f\left(\dfrac{x}{4}\right) = x^2 - 3x + 2$.


A circle has a diameter of $D$ divided into $2n$ equal parts. On each of these parts, a quarter-circle is constructed standing vertically from each point. Calculate the limiting value of the sum of the lengths of the arcs of these quarter-circles as $n$ becomes very large.

Solution: \boxed{\frac{\pi D}{4}}


Liam has written down one integer three times and another integer twice. The sum of these five numbers is 130, and one of the numbers is 35. What is the other number?


The least common multiple of $a$ and $b$ is $20$, and the least common multiple of $b$ and $c$ is $21$. Find the least possible value of the least common multiple of $a$ and $c$.


A wooden cube, $n$ units on each side, is painted red on four of its six faces (the top and bottom faces are left unpainted), and then cut into $n^3$ smaller unit cubes. Exactly one-third of the total number of faces of these unit cubes are red. What is the value of $n$, and what is the volume of the original cube in cubic units?


Given \(5^a + 5^b = 2^c + 2^d + 17\), determine the number of integers \(a, b, c, d\) which can possibly be negative.


In a round-robin tournament with 8 teams, each team plays one game against each other team, and each game results in one team winning and one team losing. Find the maximum number of teams that could be tied for the most wins at the end of the tournament.


Jane, Bob, and Alice are playing a game in which they take turns throwing a ball at a bottle, with Jane throwing first, then Bob, then Alice, and then back to Jane, in a cyclic order. The probability that Jane or Alice knocks the bottle off on their turn is $\frac{1}{3}$, while the probability that Bob knocks it off is $\frac{2}{3}$. Calculate the probability that Jane wins the game.


Given Emily converts her $e$ U.S. dollars into Euros at an exchange bureau that offers $5$ Euros for every $4$ U.S. dollars, and after spending $75$ Euros she has exactly half of her original amount of U.S. dollars converted into Euros, calculate the value of $e$.


What is the value of $\dfrac{13! - 12!}{11!}$?


A set $T$ of points in the $xy$-plane is symmetric about the origin, both coordinate axes, the line $y=x$, and the line $y=-x$. If $(3,4)$ is in $T$, determine the smallest number of points in $T$.


For each positive integer $n > 1$, let $P(n)$ denote the greatest prime factor of $n$. Find the number of positive integers $n$ for which $P(n) = \sqrt{n}$ and $P(n+60) = \sqrt{n+60}$.


If ten friends split a bill equally, but nine friends paid an additional $3.00 each to cover the share of one friend, calculate the total bill.


For how many values of $d$ is $2.00d5 > 2.007$?


Bella begins to walk from her house toward her friend Ella's house which is 3 miles away. At the same time, Ella begins to ride her bicycle toward Bella's house. Ella rides at a speed that is 4 times as fast as Bella walks. If Bella covers 3 feet with each step, how many steps will Bella take by the time she meets Ella?


Given the repeating decimal $3.71717171\ldots$, express this number as a fraction in lowest terms and find the sum of the numerator and denominator.


A cube has faces whose surface areas are $36$, $36$, $36$, $36$, $36$, and $36$ square units. Each side length is reduced uniformly by $1$ unit. Find the sum of the new dimensions of the cube.


How many primes less than $50$ have $3$ as the ones digit?


Given $M = 35^5 + 5\cdot35^4 + 10\cdot35^3 + 10\cdot35^2 + 5\cdot35 + 1$, calculate how many positive integers are factors of $M$.


Given the expression $3000(3000^{2500}) \cdot 2$, evaluate the expression.

Solution: \boxed{2 \cdot 3000^{2501}}


Suppose that 1 euro is now worth 1.5 dollars. Diana has 600 dollars and Etienne has 350 euros. Additionally, there is a transaction fee of 2% when converting euros to dollars. Calculate the percent by which the value of Etienne's money is greater than or less than the value of Diana's money after accounting for the conversion fee.


In a new arrangement of letters and numerals, one wants to spell "AMC10". Starting from an 'A' in the center, move only to adjacent letters (above, below, left, or right, but not diagonal) to spell out the sequence. Given that there are 4 'M’s around the central 'A', 4 'C’s next to each 'M', and now 5 '10’s reachable from each 'C', find the total number of different paths one can spell "AMC10".


A rectangular floor is covered with congruent square tiles. If the total number of tiles that lie on the two diagonals is 57, how many tiles cover the floor.


Let \( x = -2023 \). Evaluate the expression \( \Bigg\vert 2 \times \Big\vert |x|-x\Big\vert - |x| \Bigg\vert - x \).


Given the product of two positive integers $a$ and $b$ is $143$, where Alice mistakenly reversed the digits of the two-digit number $a$ to obtain this value, calculate the correct value of the product of $a$ and $b$.


Given the equations $x^2 + y^2 = m^2$ and $x + y = m$, determine the value of $m$ such that the graph of $x^2 + y^2 = m^2$ is tangent to the graph of $x + y = m$.


Given that $20\%$ of the students scored $60$ points, $25\%$ scored $75$ points, $25\%$ scored $85$ points, and the remaining students scored $95$ points, calculate the difference between the mean and median score of the students' scores on this test.


Consider a parabola with its vertex $V$ and focus $F$. Let there be a point $B$ on this parabola such that the distance $BF$ is 25 and $BV$ is 26. Determine the sum of all possible values for the length $FV$.


The largest number by which the expression $n^4 - n^2$ is divisible for all possible integral values of $n$.


Given that the city's water tower stands 60 meters high and holds 150,000 liters of water, and Logan's miniature water tower holds 0.15 liters, determine the height, in meters, that Logan should make his tower.


Given that the floor of the hall is tiled in a repeated pattern of an 8x8 layout, with the configuration in a 4x4 section including 10 dark tiles, calculate the fraction of the hall's floor that is composed of darker tiles.


What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $15!$?


A collector offers to buy state quarters for 1500% of their face value. If Bryden has 10 state quarters, calculate the total amount he will receive.

Solution: \boxed{37.5\text{ dollars}}


Chubby designs another nonstandard checkerboard, but this time it has 33 squares on each side. The board still follows the pattern of having a black square in every corner and alternates red and black squares along every row and column. Determine how many black squares are there on this checkerboard.


Given that $528$ be written as the sum of an increasing sequence of two or more consecutive positive integers, starting with an even number, determine the total number of such ways.


If one side of a triangle is $16$ inches and the opposite angle is $30^{\circ}$, calculate the diameter of the circumscribed circle.

Solution: \boxed{32\text{ inches}}


Given that the Highest Common Divisor (HCD) of $12348$ and $2448$ is multiplied by $3$ and then diminished by $14$, determine the result.


Alice, Bob, Carol, and Dave repeatedly take turns tossing a die, with Alice beginning, followed by Bob, then Carol, and finally Dave before returning to Alice. Find the probability that Dave will be the first one to toss a six.


A shopkeeper purchases 2000 pens at a cost of $0.15 each. If the shopkeeper wants to sell them for $0.30 each, calculate the number of pens that need to be sold to make a profit of exactly $120.00.


Alice and Bob play a game around a circle divided into 15 equally spaced points, numbered 1 through 15. Alice moves 7 points clockwise per turn, and Bob moves 4 points counterclockwise per turn. Determine how many turns will be required for Alice and Bob to land on the same point for the first time.


Jo and Blair take turns counting. Jo starts by saying 1, and each time Blair follows by saying the last number Jo said plus 2, while Jo says the last number Blair said plus 3. Determine the 20th number said.


Given the dimensions of a rectangular lawn $120$ feet by $180$ feet, and a mower swath of $30$ inches with a $6$-inch overlap, calculate the total time it will take Moe to mow the entire lawn while walking at a rate of $4000$ feet per hour.


How many primes less than $50$ have $3$ as the ones digit?


In our number system, the base is ten. If the base were changed to seven, count the twenty-fifth number in the new base.


Determine the number of distinct points common to the curves $x^2 + y^2 = 4$ and $x^2 + 2y^2 = 2$.


Let \( x = -2023 \). Determine the value of \(\Bigg\vert\Big\vert |x|-x\Big\vert-|x|+5\Bigg\vert-x+3\).


Simplify the expression $(-\frac{1}{343})^{-2/3}$.


A cell phone plan charges $20$ dollars each month, plus $0.03$ dollars per text message sent, plus $0.10$ dollars for each minute used over $30$ hours, plus $0.15$ dollars for each megabyte of data used over $500$ MB. In January, Michelle sent $200$ text messages, talked for $31$ hours, and used $550$ MB of data. Calculate the total amount she has to pay.


Given that the stationary shop owner purchased 2000 pencils at $0.15 each and sells them for $0.30 each, determine how many pencils she must sell to achieve a profit of exactly $180.00.


Sharon usually drives from her house to her office and it takes her 180 minutes at a constant speed. On a particular day, after she has covered half the distance, she runs into a heavy snowstorm and has to reduce her speed by 30 miles per hour for the remainder of the trip. Due to the weather conditions, her journey takes a total of 300 minutes today. Find the total distance of Sharon's journey from her house to her office.


Given a cylindrical cheese log with a height of 8 cm and a radius of 5 cm, determine the volume of a wedge representing one-third of the cylinder's total volume.


A large equilateral triangle with a side length of 20 cm is to be completely covered by non-overlapping equilateral triangles of side length 2 cm. Determine the number of smaller triangles needed if each row of small triangles is rotated by 180 degrees relative to the row immediately below it.


The positive integers $x$ and $y$ are the two smallest positive integers such that the product of $540$ and $x$ is a square and the product of $540$ and $y$ is a cube, determine the sum of $x$ and $y$.


A fair coin is tossed 4 times. What is the probability of at least three consecutive heads?


Let $Q$ equal the product of 1,000,000,001 and 10,000,000,007. Calculate the number of digits in $Q$.


Let $A$ be the set of the $1500$ smallest positive multiples of $7$, and let $B$ be the set of the $1500$ smallest positive multiples of $9$. What is the number of elements common to $A$ and $B$?


Given two right isosceles triangles are placed inside a square of side length $2$ such that their hypotenuses are the opposite sides of the square. Their intersection forms a rhombus. Calculate the area of this rhombus.


Given that Carlos took 60% of a whole pie and Maria took half of the remainder, find the portion of the whole pie that was left.


Points P and Q are on a circle of radius 7 and the chord PQ=8. Point R is the midpoint of the minor arc PQ. Calculate the length of the line segment PR.

Solution: \boxed{\sqrt{98 - 14\sqrt{33}}}


Given that Paula initially had enough paint for 40 identically sized rooms and had reduced capacity to just enough paint for 30 rooms after losing 5 cans of paint, calculate the original number of cans of paint she had.


If $y$ cows produce $y+2$ cans of milk in $y+3$ days, determine the number of days it will take $y+4$ cows to produce $y+7$ cans of milk.


Calculate the number of circular pipes with an inside diameter of 2 inches required to carry twice the amount of water as one pipe with an inside diameter of 8 inches.


Given the function \(\frac{(4^t - 2t)t}{16^t}\), find the maximum value for real values of \(t\).


Given two poles 30'' and 60'' high are 120'' apart, calculate the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole.


Given real numbers $m$ and $n$, define $m \star n = (3m - 2n)^2$. Evaluate $(3x-2y)^2 \star (2y-3x)^2$.


Let $a_1,a_2,\dots,a_{2021}$ be a strictly increasing sequence of positive integers such that \[a_1+a_2+\cdots+a_{2021}=2021^{2021}.\] Find the remainder when $a_1^3+a_2^3+\cdots+a_{2021}^3$ is divided by $6$.


Given the expression $3 + \sqrt{3} + \frac{1}{3 + \sqrt{3}} + \frac{1}{\sqrt{3} - 3}$, evaluate the expression.

Solution: \boxed{3 + \frac{2\sqrt{3}}{3}}


Find the unique positive integer $n$ such that $\log_3{(\log_{27}{n})} = \log_9{(\log_3{n}).


Given a round-robin tournament with 8 teams, each team plays one game against each other team, and each game results in one team winning and one team losing, determine the maximum number of teams that could be tied for the most wins at the end of the tournament.


The perimeter of an equilateral triangle exceeds the perimeter of a square by $108 \ \text{cm}$. The length of each side of the triangle exceeds the length of each side of the square by $d+2 \ \text{cm}$. The square has a perimeter greater than 0. Calculate the number of positive integers that are NOT possible values for $d$.


The sum of the squares of the roots of the equation \(x^2 - 4hx = 8\) is 20. Find the value of \(h\).

Solution: \boxed{\pm \frac{1}{2}}


Given that the sum of any two numbers equals the third number, find the number of ordered triples $(a, b, c)$ of non-zero real numbers.


Thirty percent of the objects in the urn are beads and rings, and beads make up half of that volume. Additionally, thirty-five percent of the coins are silver. Determine the percentage of objects in the urn that are gold coins.


Given that $3/8$ of the people in a room are wearing gloves and $5/6$ of the people are wearing hats, find the minimum number of people in the room wearing both a hat and a glove.


Given that Al, Bert, and Carl are to divide a pile of stickers in a ratio of $4:3:2$, calculate the fraction of stickers that go unclaimed when each of them takes his share, where Al takes the correct total amount, Bert mistakenly takes his share based on a total that is 10% less, and Carl takes his share based on what remains.


Two angles of an isosceles triangle measure $80^\circ$ and $y^\circ$. Find the sum of the three possible values of $y$.


Lily uses a mower to cut her rectangular 120-foot by 180-foot lawn with a 30-inch wide swath and 6-inch overlap. Calculate the time it will approximately take her to mow the lawn, given that she walks at a rate of 6000 feet per hour.


A circle with a radius of 7 is perfectly inscribed in a rectangle. The ratio of the length of the rectangle to its width is 3:1. Calculate the area of the rectangle.


When $\left ( 1 - \frac{1}{a} \right ) ^8$ is expanded, calculate the sum of the last three coefficients.


Determine the length of the median of the trapezoid formed by combining two equilateral triangles with side lengths of 4 units and 3 units, where the triangles form the two bases.


Given a rectangular array of numbers with 50 rows and 100 columns, find the value of A/B, where A is the average of the 100 sums calculated by Andy (each sum being the sum of a column) and B is the average of the 50 sums calculated by Bethany (each sum being the sum of a row).


What is the greatest possible sum of the digits in the base-six representation of a positive integer less than $1728$?


The number $22!$ has many positive integer divisors. If one is chosen at random, calculate the probability that it is odd.


A rectangular grazing area is to be fenced off on three sides using part of a $120$ meter rock wall as the fourth side. Fence posts are to be placed every $15$ meters along the fence, including the two posts where the fence meets the rock wall. Determine the fewest number of posts required to fence an area $45$ m by $75$ m.


Given points G, H, I, and J lie on a line, with GH = 2, HI = 3, IJ = 4, and points K, L, and M lie on another line, parallel to the first, with KL = 2 and LM = 3, determine the number of possible different values for the area of a triangle with positive area formed by three of these points.


Seven students, Abby, Bridget, Charlie, and four others are to be seated in two rows of four. If the seating positions are assigned randomly, what is the probability that Abby and Bridget are adjacent to each other in the same row or the same column, and Charlie is seated in a different row from Abby?
A) $\frac{1}{4}$
B) $\frac{5}{21}$
C) $\frac{1}{3}$
D) $\frac{2}{7}$
E) $\frac{1}{5}$


A particle moves such that its speed for each subsequent mile varies inversely as the square of the integral number of miles already traveled. If the speed for the first mile was $1$ mile per hour and the second mile was traversed in $2$ hours, determine the time in hours needed to traverse the $n$-th mile.


Determine the roots of the polynomial \( p(x) = x(x+3)^2(5-x) = 0 \).

Solution: \boxed{0, -3, 5}


Given $g(3x) = \frac{3}{3 + 2x}$ for all $x > 0$, calculate the value of $3g(x)$.

Solution: \boxed{\frac{27}{9 + 2x}}


Given all four-digit palindromes that can be represented as $\overline{abba}$, where $a, b$ are digits and $a \neq 0$, find the sum of the digits of the total of these palindromes.


What is the area of the polygon with vertices at $(2, 1)$, $(4, 3)$, $(6, 1)$, $(5, -2)$, and $(3, -2)$?


Twenty-five percent of the audience listened to the entire 90-minute talk, and fifteen percent did not pay attention at all. Of the remainder, 40% caught half of the talk, and the rest heard only one fourth of it. Calculate the average time in minutes the talk was heard by the audience members.


What is the probability that a randomly selected positive factor of $90$ is less than $8$?


Given that half of Julia's marbles are blue, one-third of them are red, and twelve of them are green, find the smallest number of yellow marbles that Julia could have.


Given $a$ and $b$ are positive numbers such that $a^b = b^a$ and $b = 4a$, solve for the value of $a$.


Let $x=2023$. Find the value of $\bigg|$ $||x|-x|-|x|$ $\bigg|$ $-x$.


A cylindrical tank with a radius of $5$ feet and a height of $10$ feet is lying on its side. The tank is half-filled with water, reaching the top of the cylinder's horizontal diameter. Calculate the volume of water in cubic feet.


The two spinners are spun once. Spinner A has sectors numbered 0, 2, 3 while spinner B has sectors numbered 1, 3, 5. Determine the probability that the sum of the numbers on which the spinners land is a prime number.


A particle moves in such a way that its speed for each mile after the first is inversely proportional to the square of the number of miles already traveled. If it takes 4 hours to travel the third mile, find the time, in hours, needed to traverse the $n$th mile.


When three different numbers from the set $\{-4, -3, -1, 5, 6\}$ are multiplied, calculate the largest possible product.


Calculate the sum $E(1)+E(2)+E(3)+\cdots+E(500)$, where $E(n)$ denotes the sum of the even digits of $n$. For example, $E(5681) = 6 + 8 = 14$.
A) 2000
B) 2200
C) 2400
D) 2500
E) 2600


What is the greatest number of consecutive integers, including both negative and non-negative integers, whose sum is $120$?


Given a magician's hat contains 4 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 4 of the reds are drawn or until both green chips are drawn, calculate the probability that all 4 red chips are drawn before both green chips are drawn.


Given the four-digit number $2055$, calculate the number of different four-digit numbers that can be formed by rearranging its digits.


Given that every 7-digit whole number can be a telephone number, except those that begin with numbers 0, 1, or 2, or end with 9, calculate the fraction of telephone numbers that begin with 7 and end with 5.


A computer can perform 15,000 additions per second, calculate the total number of additions it can perform in 2 hours and 30 minutes.


Let $A$, $B$ and $C$ be three distinct points on the graph of $y=x^2$ such that line $AB$ is parallel to the $x$-axis and $\triangle ABC$ is a right triangle with area $2016$. Find the sum of the digits of the $y$-coordinate of $C$.


The number of positive integers less than $1200$ divisible by neither $6$ nor $8$ must be calculated.


Given a fair coin is tossed 4 times, calculate the probability of getting at least two consecutive heads.


If the radius of a circle is increased by $k$ units (where $k > 0$), determine the ratio of the new circumference to the original diameter.

Solution: \boxed{\pi \left(1 + \frac{k}{r}\right)}


Given circle $O$, point $C$ is on the opposite side of diameter $\overline{AB}$ from point $A$, and point $D$ is on the same side as point $A$. Given $\angle AOC = 40^{\circ}$, and $\angle DOB = 60^{\circ}$, calculate the ratio of the area of the smaller sector $COD$ to the area of the circle.


Suppose a square piece of paper with side length 4 units is folded in half diagonally, then cut along the fold, producing two right-angled triangles. Calculate the ratio of the perimeter of one of the triangles to the perimeter of the original square.

Solution: \boxed{\frac{1}{2} + \frac{\sqrt{2}}{4}}


Given $f(x) = x^2 + 5x + 6$, consider the set $S = \{0, 1, 2, \dots, 30\}$. Find the number of integers $s$ in $S$ such that $f(s)$ is divisible by $5$.


What is the value of the expression $2 + \frac{1}{2 + \frac{1}{2 + \frac{1}{2}}}$?


Given that each vertex of a regular tetrahedron is to be labeled with an integer $1$ through $4$, with each integer being used exactly once, such that the sum of the numbers on the vertices of each triangular face is the same, determine the number of distinct arrangements possible.


Given 2 gallons of paint are required for the room's complete coverage, determine the fraction of the original amount of paint available to use on the third day, given that one quarter of the paint is used on the first day and one third of the remaining paint is used on the second day.


Given that Brianna is planning to use some of her earnings from a weekend job to purchase several music albums at equal prices, if she spends one quarter of her money to buy one quarter of the albums, determine the fraction of her money that she will have left after purchasing all the albums.


Solution: \boxed{h(6x + 3h + 4)}


Given a quadrilateral with vertices at the points (2,1), (0,7), (5,5), and (6,9), calculate the area enclosed by this quadrilateral.


Given the podcast series with a total content of 500 minutes, evenly distributed throughout the week with each day featuring up to 65 minutes of content, calculate the number of minutes of content that each day will feature and the number of days it will take to release the entire series.


What is the greatest number of consecutive integers whose sum is $36$?


Given that the new track has a width of 8 meters, and it takes Keiko 48 seconds longer to walk around the outside edge of the track than around the inside edge, determine Keiko's new speed in meters per second.


What is the area of the polygon with vertices at $(2,1)$, $(4,3)$, $(6,1)$, and $(4,6)$?


Lucas wakes up at 6:00 a.m., walks to the bus stop at 6:45 a.m., attends 7 classes each lasting 45 minutes, gets 40 minutes for lunch, and spends an additional 1.5 hours at school for extracurricular activities. He catches the bus home and arrives at 3:30 p.m. Calculate the number of minutes Lucas spends on the bus.


A triangle has a fixed base $AB$ that is $6$ inches long. The median from $A$ to side $BC$ is $4$ inches long and must be at a 60-degree angle with the base. Determine the locus of vertex $C$.
A) A circle with center A and radius 6 inches
B) A circle with center A and radius 7 inches
C) An ellipse with A as focus
D) A straight line perpendicular to AB at A

Solution: \boxed{B) A circle with center A and radius 7 inches}


Given a sequence where each term increases by 1 from the previous term and the first term is $x_1 = 3$, find the sum of the first $n$ terms of the sequence where $n$ is a positive integer such that $n \leq 10$.

Solution: \boxed{\frac{n(n + 5)}{2}}


Between $5^{5} - 1$ and $5^{10} + 1$, inclusive, calculate the number of perfect cubes.


Samia jogged to her friend's house at an average speed of 8 kilometers per hour, and then walked the remaining two-thirds of the way at 4 kilometers per hour. If the entire trip took her 105 minutes to complete, calculate the distance in kilometers that Samia walked.


Calculate the product of \(\frac{4}{3} \times \frac{5}{4} \times \frac{6}{5} \times \cdots \times \frac{2010}{2009}\).


For how many positive integer values of N is the expression $\dfrac{48}{N+3}$ an integer?


A vehicle is equipped with five tires (four standard tires and one enhanced durability spare tire) is driven for a total of 45,000 miles. The spare tire can endure 20% more mileage than the regular tires. If the rotation policy ensures each tire is used the maximum amount without exceeding its durability, how many miles was each type of tire used?
A) 30,000 miles, 36,000 miles
B) 34,615 miles, 41,538 miles
C) 37,500 miles, 45,000 miles
D) 40,000 miles, 48,000 miles

Solution: \boxed{B) 34,615 miles, 41,538 miles}


Given that the number is $180^8$, calculate the number of positive integer divisors that are perfect squares or perfect cubes (or both).


Three jars each contain the same number of marbles, and each marble is either blue or green. In Jar $1$ the ratio of blue to green marbles is $7:3$, in Jar $2$ it is $6:4$, and in Jar $3$ it is $5:5$. There are a total of $240$ green marbles across all jars. Calculate the difference in the number of blue marbles in Jar $1$ and Jar $3$.


The maximum number of points of intersection of the graphs of two different third degree polynomial functions $y=f(x)$ and $y=g(x)$, each with leading coefficient 2, must be determined.


A divisor of $10!$ that is a multiple of 12 is chosen at random. What is the probability that this divisor is also a perfect square? Express your answer as a fraction $\frac{m}{n}$ where $m$ and $n$ are relatively prime integers.


Given that Sangho's video had a score of 140, and 75% of the total votes were like votes, determine the total number of votes cast on Sangho's video.


Given that Maria's video had a score of $120$, with $75\%$ of the votes being likes, determine the total number of votes cast on her video.


Given a quadrilateral with vertices $P(k,a)$, $Q(a,k)$, $R(-k,-a)$, and $S(-a,-k)$, where $k$, $a$, and $b$ are integers with $k > a > 0$, find the value of $k+a$, given that the area of the quadrilateral $PQRS$ is $32$.


Given that $x \not= 0$ or $3$ and $y \not= 0$ or $4$, solve the equation $\frac{3}{x} + \frac{2}{y} = \frac{5}{6}$ and find $x$ in terms of $y$.

Solution: \boxed{\frac{18y}{5y - 12}}


Casper started with some candies. On the first day, he ate $\frac{1}{4}$ of his candies then gave $3$ candies to his brother. On the second day, he ate $\frac{1}{5}$ of the remaining candies and gave $5$ candies to his sister. On the third day, he gave $7$ candies to his friend and finally ate $\frac{1}{6}$ of what was left, leaving him with $10$ candies. Determine the number of candies that Casper had at the beginning.


Hadi has a $5 \times 7$ index card. If he shortens the length of one side of this card by $2$ inches, the card would have an area of $21$ square inches. Calculate the area of the card if instead he shortens the length of the other side by $2$ inches.


Given a pentagonal grid with a base of 10 toothpicks and four other sides made of 8 toothpicks each, each vertex joint consumes an additional toothpick for structural support. Calculate the total number of toothpicks needed.


Calculate the sum $\triangle$ for two triangles, where the first triangle has numbers 2, 3, and 5, and the second triangle has numbers 3, 4, and 6.


Given Ben's test scores $95, 85, 75, 65,$ and $90$, and his goal to increase his average by at least $5$ points and score higher than his lowest score of $65$ with his next test, calculate the minimum test score he would need to achieve both goals.


Evaluate the sum $2345 + 3452 + 4523 + 5234$ and then subtract $1234$ from the result.


Given points $A, B, C,$ and $D$ on a line with $AB=2$, $BC=1$, and $CD=3$, and points $E, F,$ and $G$ on a second line, parallel to the first, with $EF=1$ and $FG=2$, and the perpendicular distance between the two lines is a fixed, positive value $h$, determine the number of possible distinct areas that can be formed using any three of these seven points as vertices.


Given that Mary is 30% older than Sally, and Sally is 50% younger than Danielle, and the sum of their ages is 45 years, determine Mary's age on her next birthday.


Given the expressions $(6+16+26+36+46)$ and $(14+24+34+44+54)$, evaluate their sum.


How many primes less than $100$ have $3$ as the ones digit?


A regular hexagon has a point at each of its six vertices, and each side of the hexagon measures one unit. If two of the 6 points are chosen at random, calculate the probability that the two points are one unit apart.


If the shopper saves $\textdollar{3.75}$ when buying a coat on sale and the final price paid for the coat was $\textdollar{50}$, determine the percentage that the shopper saved.


Determine the number of digits in the expression $(8^4 \cdot 4^{12}) / 2^8$.


In a distant kingdom, four apples can be traded for three bottles of juice, and each bottle of juice can be traded for five slices of cake. Calculate the value in slices of cake that one apple is worth.


Find the sum of the squares of all real numbers satisfying the equation $x^{64} - 16^{16} = 0$.


What is the sum of the two smallest prime factors of $280$?


What is the sum of the mean, median, and mode of the numbers $1, 2, 2, 4, 5, 5, 5, 7$?


Let $M$ be the greatest four-digit number whose digits have a product of $24$. Calculate the sum of the digits of $M$.


Brianna uses one quarter of her money to buy one half of the CDs she wants. What fraction of her money will she have left after she buys all the CDs?


What is the radius of a circle inscribed in a rhombus with diagonals of length $14$ and $30$?


Shea and Ara were once the same height. Since then, Shea has grown 30%, while Ara has grown 5 inches less than Shea. Shea is now 65 inches tall. Calculate Ara's current height.


The graphs of $y = -|x-a|^2 + b$ and $y = |x-c|^2 + d$ intersect at points $(1,8)$ and $(9,4)$. Find $a+c$.


Given $x \otimes y = x^2 - 2y$, calculate the value of $k \otimes (k \otimes k)$.

Solution: \boxed{-k^2 + 4k}


The notation $N!$ represents the product of integers $1$ through $N$. Consider the expression $102! + 103! + 104! + 105!$. Determine the largest integer $n$ for which $5^n$ is a factor of this sum.


Evaluate the expression $\left(\left((5+2)^{-1} - \frac{1}{2}\right)^{-1} + 2\right)^{-1} + 2$.


In ∆ABC, AB = 6, BC = 8, AC = 10, and D is on AC with BD = 6. Find the ratio of AD:DC.


Given that one of every 400 people has a particular disease and a blood test has a 5% false positive rate, determine the probability that a person randomly selected from this population and who tests positive actually has the disease.


Given Sarah has a six-ounce cup of coffee and an identical cup containing three ounces of milk. She pours one-third of the coffee into the milk, mixes it thoroughly, and then pours half of this mixed content back into the coffee cup. What fraction of the liquid in the coffee cup is now milk.


Points $A = (2,8)$, $B = (0,2)$, $C = (6,4)$, and $D=(a,b)$ lie in the first quadrant and are the vertices of quadrilateral $ABCD$. The quadrilateral formed by joining the midpoints of $\overline{AB}$, $\overline{BC}$, $\overline{CD}$, and $\overline{DA}$ is a square. Find the sum of the coordinates of point $D$.


What is the least possible value of $(xy-2)^2 + (x-y)^2$ for real numbers $x$ and $y$?


Given that Spinner A has the numbers 1, 4, 7, 9 and Spinner B has the numbers 2, 3, 5, 8, calculate the probability that the sum of the numbers on which the spinners land is a prime number.


Consider a regular polygon with $n$ sides, $n>4$, where each side is bent inward at an angle $\theta$ such that $\theta = \frac{360^\circ}{2n}$. Calculate the total angle formed by all the bends around the entire shape.