What is the difference between the sum of the first 1000 even counting numbers including 0, and the sum of the first 1000 odd counting numbers?


A picture 5 feet across is hung on a wall that is 25 feet wide with a 2-foot offset from being centered toward one end of the wall. Determine the distance from the end of the wall to the nearest edge of the picture.


A point $P(a, b)$ in the $xy$-plane is first rotated counterclockwise by $180^\circ$ around the point $(2, 3)$ and then reflected about the line $y = x$. The final position of $P$ after these transformations is $(5, -4)$. Determine the value of $a - b$.


Given that $\angle AOE = 40^\circ$, $\angle FOB = 60^\circ$, and points $F$ and $E$ are on the same side of diameter $\overline{AB}$, calculate the ratio of the area of the smaller sector $EOF$ to the area of the whole circle.


A jar contains 6 different colors of gumdrops: 25% blue, 15% brown, 15% red, 10% yellow, 20% orange, and the rest are green. If there are 36 green gumdrops and one-third of the blue gumdrops are replaced with brown gumdrops, calculate the number of gumdrops that will be brown.


Sara had walked one-third of the way from home to the library in 9 minutes, and she ran 5 times as fast as she walked. Determine the total time it took Sara to get from home to the library.


Consider all triangles $ABC$ where $AB = AC$, $D$ is a point on $\overline{AC}$ such that $\overline{BD} \perp \overline{AC}$. Assume $AC$ and $CD$ are odd integers, and $BD^{2} = 65$. Find the smallest possible value of $AC$.


Determine the measure of the angle $\angle BAC$ in $\triangle ABC$, which is scalene with $AB = AC$, given that point $P$ is on $BC$ such that $AP = PC$ and $BP = 2PC$.


A 3x3 magic square with the same row sums and column sums is given by:
\[
\left[
\begin{matrix}
2 & 7 & 6\\
9 & 5 & 1\\
4 & 3 & 8
\end{matrix}
\right]
\]
Compute the minimum number of entries that must be changed to make all three row sums and three column sums different from one another.


Chloe chooses a real number uniformly at random from the interval $[0, 3000]$ and Laurent chooses a real number uniformly at random from the interval $[0, 4000]$. Determine the probability that Laurent's number is greater than Chloe's number.


Let \(r\) be the number that results when both the base and the exponent of \(a^b\) are quadrupled, and \(r\) equals the product of \(a^b\) and \(x^{2b}\). Find the value of \(x\).

Solution: \boxed{16 a^{3/2}}


Casper has exactly enough money to buy either 10 pieces of red candy, 18 pieces of green candy, 20 pieces of blue candy, or n pieces of purple candy, where a piece of purple candy costs 24 cents. What is the smallest possible value of n?


A rectangular floor is 15 feet wide and 35 feet long, covered entirely by one-foot square tiles. A bug walks from one corner to the opposite corner in a straight line. Including the first and last tiles, calculate the total number of tiles the bug visits.


Given that there are $121$ grid points in a square, including the points at the edges, and point $P$ is in the center of the square, determine the probability that the line $PQ$ is a line of symmetry for the square when a point $Q$ is randomly chosen among the other $120$ points.


Given that Casper can buy either 10 pieces of orange candy, 16 pieces of yellow candy, 18 pieces of maroon candy, or n pieces of violet candy, where the price of each piece of violet candy is 18 cents. Find the smallest possible value of n.


A point is chosen at random from within a circle of radius 4. Calculate the probability that the point is closer to the center of the circle than it is to the boundary of the circle.


Given the expression \(150(150-4) \div (150\cdot150 \cdot 2 -4)\), calculate the result.


Given that there are 128 players in a chess tournament, with the strongest 32 players receiving a bye in the first round and the remaining players competing in the initial round, determine the total number of matches played until only one champion is crowned.


At a local shop last month, packets of seeds were sold at 6 packets for $8. This month, they are on sale at 8 packets for $6. Determine the percent decrease in the price per packet during the sale.


Given that Crystal modifies her running course by running due north for 2 miles, then east for 3 miles, and finally southwest until she returns to her starting point, determine the length of the last portion of her run.


An equilateral triangle $ABC$ with side length $4$ has a smaller equilateral triangle $DBE$ with side length $2$ cut from it, where $D,E$ lie on sides $AB$ and $BC$ respectively. Calculate the perimeter of the remaining quadrilateral $ACED$.


Given the expression $3^{\left(1^{\left(2^8\right)}\right)} + \left(\left(3^1\right)^2\right)^8$, evaluate its value.


A circle of radius 3 is centered at A. An equilateral triangle with side 4 has a vertex at A. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle.

Solution: \boxed{9\pi - 4\sqrt{3}}


Given a bag of popcorn kernels containing $\frac{3}{4}$ white kernels and $\frac{1}{4}$ yellow kernels, with $\frac{1}{4}$ of the kernels damaged and $\frac{3}{5}$ of the white kernels and $\frac{4}{5}$ of the yellow kernels good and capable of popping, calculate the probability that a kernel selected from the bag and which pops was white.


Given that the angles of a triangle that is not right are $a^{\circ}$, $b^{\circ}$, and $c^{\circ}$, where $a$, $b$, and $c$ are prime numbers and $a > b > c$, determine the least possible value of $b$.


In a classroom, $65$ students are wearing green shirts, and another $79$ students are wearing red shirts. The $144$ students are grouped into $72$ pairs. In exactly $27$ of these pairs, both students are wearing green shirts. Determine the number of pairs that consist of both students wearing red shirts.


What is the remainder when $5^1 + 5^2 + 5^3 + \cdots + 5^{4020}$ is divided by 10?


Find the number of pairs (m, n) of integers such that the equation $m^3 + 3m^2 + 2m = 8n^3 + 12n^2 + 6n + 1$ holds.


How many subsets containing three different numbers can be selected from the set $\{ 90, 94, 102, 135, 165, 174 \}$ so that the sum of the three numbers is divisible by 5?


If 60 students wear red shirts and 90 students wear green shirts, making a total of 150 students, and the students are grouped into 75 pairs, and 28 of these pairs contain students both wearing red shirts, determine the number of pairs that have students both wearing green shirts.


Jessica wants to walk from her apartment to her friend Anna's apartment, which is located four blocks east and three blocks north of Jessica's apartment. Jessica needs to avoid a risky construction site, which is located two blocks east and one block north of her apartment. Calculate the number of ways Jessica can walk to Anna's apartment by walking a total of seven blocks.


The radius of Earth at the equator is approximately 4200 miles. Suppose a jet flies once around this new Earth's equator at a speed of 525 miles per hour relative to Earth. If the flight path is negligibly above the equator, find the approximate number of hours of flight.


A number is formed by repeating a two-digit number three times. Determine the smallest prime number by which any number of this form is always exactly divisible.


An uncrossed belt is fitted without slack around two circular pulleys with radii of $10$ inches and $6$ inches. If the distance between the points of contact of the belt with the pulleys is $30$ inches, then calculate the distance between the centers of the pulleys in inches.


Jacob wants to tile the floor of a 14-foot by 20-foot hallway. He plans to use 2-foot by 2-foot square tiles for a border along the edges of the hallway and fill in the rest of the floor with three-foot by three-foot square tiles. Calculate the total number of tiles he will use.


Given an ellipse with semi-major axis 2 and semi-minor axis 1, find the number of points interior to the region such that the sum of the squares of the distances from these points to the foci of the ellipse is 5.


Given the state income tax rate is $q\%$ for the first $\$30000$ of yearly income plus $(q + 1)\%$ for any amount above $\$30000$, and Samantha's state income tax amounts to $(q + 0.5)\%$ of her total annual income, determine Samantha's annual income.


A half-sector of a circle with a radius of $6$ 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. Calculate the volume of the cone in cubic inches.

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


Given the quadratic polynomial $ax^2 + bx + c$ and a sequence of increasing, equally spaced $x$ values: $441, 484, 529, 576, 621, 676, 729, 784$, determine which function value is incorrect.


Given the sequence [..., p, q, r, s, 5, 8, 13, 21, 34, 55, 89...], where each term is the sum of the two terms to its left, find p.


What is the smallest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is not a divisor of the square of the product of the first $n$ positive integers?


The ratio of the length to the width of a rectangle is $5$ : $2$. If the rectangle has a diagonal of length $13$, find the area expressed as $kd^2$ for some constant $k$.

Solution: \boxed{\frac{10}{29} d^2}


Given a two-digit positive integer $N$, determine how many have the property that the sum of $N$ and the number obtained by reversing the order of the digits of $N$ is a prime number.


One fifth of Sarah's marbles are blue, one sixth of them are red, and one tenth of them are green. What is the smallest number of yellow marbles Sarah could have?


A shopkeeper buys $2000$ pens at $0.15$ each. If he sells them at $0.30$ each, determine the number of pens he must sell to make a profit of exactly $150.00$.


Five blue beads, three green beads, and one red bead are placed in line in random order. Calculate the probability that no two blue beads are immediately adjacent to each other.


Given that it is currently between 4:00 and 5:00 o'clock, and eight minutes from now, the minute hand of a clock will be exactly opposite to the position where the hour hand was six minutes ago, determine the exact time now.


When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 2001 is greater than zero and is equal to the probability of obtaining a sum of $S$. Determine the smallest possible value of $S$.


Let the bisectors of the interior angles at $B$ and $C$ of triangle $ABC$ meet at point $E$. Express the measure of angle $BEC$ in terms of angle $A$.

Solution: \boxed{90^\circ + \frac{A}{2}}


A flower arrangement consists of yellow tulips, white tulips, yellow daisies, and white daisies. Given that half of the yellow flowers are tulips, two-thirds of the white flowers are daisies, and seven-tenths of the flowers are yellow, calculate the percentage of the flowers that are daisies.


Determine the difference between the two smallest integers greater than $1$ that, when divided by any integer $k$ such that $2 \le k \le 13$, leave a remainder of $1$. Then, subtract $2^3\cdot 3^2\cdot 5\cdot 7\cdot 11\cdot 13$ from this difference and find the result.


Given the expressions $(2401^{\log_7 3456})^{\frac{1}{2}}$, calculate its value.


Find the sum of all numbers of the form $3k + 2$, where $k$ takes on integral values from $0$ to $n-1$.

Solution: \boxed{\frac{3n^2 + n}{2}}


Let \(p\), \(q\), and \(r\) be the distinct roots of the polynomial \(x^3 - 27x^2 + 98x - 72\). It is given that there exist real numbers \(A\), \(B\), and \(C\) such that
\[
\frac{1}{s^3 - 27s^2 + 98s - 72} = \frac{A}{s-p} + \frac{B}{s-q} + \frac{C}{s-r}
\]
for all \(s \not\in \{p, q, r\}\). What is \(\tfrac{1}{A} + \tfrac{1}{B} + \tfrac{1}{C}\)?
A) 255
B) 256
C) 257
D) 258


Determine the value of $A + B + C$, where $A$, $B$, and $C$ are the dimensions of a three-dimensional rectangular box with faces having areas $40$, $40$, $90$, $90$, $100$, and $100$ square units.


Given the polynomial function $f$ of degree exactly 3, determine the number of possible values of $f(x)$ that satisfy $f(x^2) = [f(x)]^2 = f(f(x))$.


Keiko walks once around a track at exactly the same constant speed every day. The track has straight sides and the ends are semi-circular. The track has a width of $6$ meters, and it takes her $48$ seconds longer to walk around the outer edge of the track than around the inner edge. Calculate Keiko's speed in meters per second.


Given that $x \neq 1$, $y \neq 2$, and $z \neq 3$, compute the value in simplest form of the expression $\frac{x-1}{3-z} \cdot \frac{y-2}{1-x} \cdot \frac{z-3}{2-y}$.


A solid cube of side length $4$ has a smaller cube of side length $1.5$ removed from each corner. Calculate the total number of edges of the remaining solid.


Given parallelogram EFGH has side lengths EF = 40 cm, FG = 30 cm, and the longer diagonal EH = 50 cm, calculate the area of parallelogram EFGH.


Given that two angles of an isosceles triangle measure $60^\circ$ and $x^\circ$, find the sum of the three possible values of $x$.


A rectangle has a ratio of length to width of 5:2. If the diagonal of the rectangle is 13 units, determine the value of $k$ such that the area of the rectangle may be expressed as $kd^2$, where $d$ is the diagonal.


An item's price is reduced by 15%. What percentage increase is required on this new price to bring it back to its original value?


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


Johann has 64 fair coins. He flips all the coins, and any coin that lands on tails is tossed again, with the process repeating up to three more times for each coin, for a total of four tosses per coin. Calculate the expected number of coins that show heads after these tosses.


Given parallelogram $ABCD$ with $E$ the midpoint of diagonal $AC$. Point $E$ is connected to a point $F$ in $DA$ so that $DF=\frac{2}{5}DA$. Determine the ratio of the area of $\triangle DFE$ to the area of $\triangle ABE$.


Given $6 \cdot 3^x = 7^{y + 3}$, find the value of $x$ when $y = -3$.


Given that N is a two-digit positive integer, find the number of N such that the sum of N and the number obtained by reversing the order of the digits of N is a perfect cube.


In the list where each integer $n$ appears $n$ times for $1 \leq n \leq 300$, find the median of the numbers.


Let $M = 36 \cdot 36 \cdot 85 \cdot 128$. Calculate the ratio of the sum of the odd divisors of $M$ to the sum of the even divisors of $M$.


A bug starts crawling from position $-3$ on a number line. It first moves to $-7$, then reverses direction to head towards $8$, and finally turns around again and stops at $2$. Calculate the total distance the bug has crawled.


Given hexadecimal numbers use the digits $0$ to $9$ and the letters $A$ to $F$ for values $10$ to $15$, determine how many of the first $512$ decimal numbers only contain numeric digits in their hexadecimal representation, and find the sum of the digits of this quantity.


For how many positive integer values of \(n\) are both \(\frac{n}{5}\) and \(5n\) four-digit whole numbers?


Find the smallest integer n such that (x+y+z)^2 ≤ n(x^2+y^2+z^2) for all real numbers x, y, and z.


Starting with the display "1," calculate the fewest number of keystrokes needed to reach "240" using only the keys [+1] and [x2].


Consider the first four terms of an arithmetic sequence given by $a, x, b, 3x$, and determine the ratio of $a$ to $b$.


Given $M = 57^{5} + 5\cdot 57^{4} + 10\cdot 57^{3} + 10\cdot 57^{2} + 5\cdot 57 + 1$, calculate the number of positive integers that are factors of $M$.


Farmer Pythagoras has now expanded his field, which remains a right triangle. The lengths of the legs of this field are $5$ units and $12$ units, respectively. He leaves an unplanted rectangular area $R$ in the corner where the two legs meet at a right angle. This rectangle has dimensions such that its shorter side runs along the leg of length $5$ units. The shortest distance from the rectangle $R$ to the hypotenuse is $3$ units. Find the fraction of the field that is planted.
A) $\frac{151}{200}$
B) $\frac{148}{200}$
C) $\frac{155}{200}$
D) $\frac{160}{200}$


Shelby drives her scooter at a speed of 40 miles per hour if it is sunny, and 25 miles per hour if it is raining. One particular day, she drove in the sun for the first part of the journey and in the rain for the last part, covering a total of 25 miles in 45 minutes. Calculate the time, in minutes, she drove in the rain.


Given the function g defined on the set of positive rational numbers by g(x \cdot y) = g(x) + g(y) for all positive rational numbers x and y, and g(n) = n^2 for every prime number n, calculate g(x) for x = \frac{25}{21}.


Given that $65^7$ has positive integer divisors that are perfect squares or perfect cubes (or both), calculate the number of such divisors.


Determine the number of pairs of positive integers $(x,y)$ which satisfy the system of equations $x^2 + y^2 = x^4$ and $x > y$.


A rectangular piece of metal with dimensions $2a$ by $b$ (where $b \leq 2a$) is used to cut out a circular piece of maximum size, and then a square piece of maximum size is cut out of the circular piece. Find the total amount of metal wasted.

Solution: \boxed{2ab - \frac{b^2}{2}}


After the year 2002, which is a palindrome, identify the next year where the sum of the product of its digits is greater than 15. Find the sum of the product of the digits of that year.


Given Paige calculates the sum of the interior angles of a convex polygon, mistakenly omitting two angles this time, and arrives at a sum of $3240^\circ$, calculate the combined degree measure of the two forgotten angles.


The price of an item is decreased by 20%. To bring it back to its original value and then increase it by an additional 10%, the price after restoration must be increased by what percentage.


Given a list of $3000$ positive integers with a unique mode occurring exactly $12$ times, calculate the least number of distinct values that can occur in the list.


Julia drives from her home to the theatre to watch a show. She drives 40 miles in the first hour, but realizes that she will be 1.5 hours late if she continues at this speed. She increases her speed by 20 miles per hour for the rest of the way to the theatre and arrives 1 hour early. Let $d$ be the distance from her home to the theatre; calculate $d$.


A committee has $60 to buy flowers for a conference room. Orchids cost $4 each, and lilies cost $3 each. Find the number of different combinations of orchids ($o$) and lilies ($l$) that can be purchased for exactly $60, given that the committee must buy at least 10 flowers in total.


Find the solution of the equation $\sqrt{5x^3-1} + \sqrt{x^3-1} = 4$.


If $\theta$ is an acute angle and $\cos 2\theta = b$, find the value of $\sin\theta + \cos\theta$.

Solution: \boxed{\sqrt{1 + \sqrt{1 - b^2}}}


Given that three generations of the Patel family are attending a theater play, with three members from each generation, and where the members of the youngest generation enjoy a $40\%$ discount as students, the members of the middle generation pay the full price, and the members of the oldest generation receive a $20\%$ discount as senior citizens, and a discounted ticket for Mr. Patel costs $10.00, determine the total amount Mr. Patel must pay in order to cover all the ticket costs.


Consider the infinite series $1 - \frac{1}{3} - \frac{1}{9} + \frac{1}{27} - \frac{1}{81} - \frac{1}{243} + \frac{1}{729} - \frac{1}{2187} - \cdots$. Evaluate the sum of this series.


Let $x$ be a real number selected uniformly at random between 100 and 300. If $\lfloor {\sqrt{x}} \rfloor = 15$, find the probability that $\lfloor {\sqrt{100x}} \rfloor = 150$.


Calculate the total amount of metal wasted by cutting a circular piece of maximum size from a rectangle with sides $a$ and $b$ (where $a < b$), and then a square piece of maximum size from this circular piece.

Solution: \boxed{ab - \frac{a^2}{2}}


In quadrilateral $EFGH$, the internal angles form an arithmetic sequence. Furthermore, triangles $EFG$ and $HGF$ are similar with $\angle EFG = \angle HGF$ and $\angle EGF = \angle HFG$. Each of these triangles' angles also forms an arithmetic sequence. In degrees, what is the largest possible sum of the largest and smallest angles of $EFGH$?


The maximum value of $\frac{(3^t - 4t)t}{9^t}$ for real values of $t$.


Marvin had a birthday on Friday, May 27 in the non-leap year 2007. In what year will his birthday next fall on a Thursday?


A sequence contains one red ball, one blue ball, and $N$ green balls where $N$ is a positive multiple of 7. Calculate the probability $P(N)$ such that at least $\frac{2}{3}$ of the green balls are on the same side as the red ball but on the opposite side of the blue ball. Determine the least value of $N$ for which $P(N) < \frac{7}{10}$, and find the sum of its digits.


Given the 9 data values are $70, 110, x, 50, x, 210, 100, 85, 40$, and both the mean, median, and mode of these values are all equal to $x$. What is the value of $x$?


What is the largest integer n for which 7^n is a factor of the sum 47! + 48! + 49!?


Given that 20% of the participants scored 60 points, 25% scored 75 points, 15% scored 85 points, 30% scored 90 points, and the rest scored 95 points, calculate the difference between the mean and median score of the participants' scores on this competition.


Let $S_n$ and $T_n$ be the respective sums of the first $n$ terms of two arithmetic series. If $S_n:T_n=(7n^2+1):(4n^2+27)$ for all $n$, find the ratio of the fifteenth term of the first series to the fifteenth term of the second series.


A particle moves such that its time for the third and subsequent miles varies directly as the square of the integral number of miles already traveled. For each subsequent mile the time is constant. If the third mile is traversed in $8$ hours, determine the expression for the time, in hours, needed to traverse the $n$th mile.


Let \( p \) be a positive integer. Carlos and Alice play a game on a whiteboard where they interact with numbers derived from powers of integers. Carlos starts by writing the smallest power of 2 (exponentiation, not square) with \( p+1 \) digits. Every time Carlos writes a number, Alice erases the last \( p \) digits of it. Carlos then writes the next higher power of 2, Alice erases the last \( p \) digits, and this process continues until the last two numbers that remain on the board differ by at least 3. Let \( h(p) \) be the smallest positive integer not written on the board. Determine \( h(1) \).


Given that in triangle ABC with ∠C = 120°, M is the midpoint of AB, D is a point on BC such that BD:DC = 3:2, and E is the midpoint of AC, determine the area of triangle BME if the area of triangle ABC is 36 square units.


Calculate the number of diagonals that can be drawn in a regular polygon with 150 sides, where no diagonal that can divide the polygon into two polygons with equal number of sides is considered.


Let $x$ and $y$ be two-digit positive integers with mean $70$. Find the maximum value of the ratio $\frac{x}{y}$.


Given that three primes $p, q,$ and $r$ satisfy $p+q=r$, $1 < p < q$, and $r > 10$, find the smallest value of $p$.


Given a circle of radius 3 centered at B(3,3) is dilated to a circle of radius 5, and the new center of the circle is at B'(7,10), determine the distance the origin O(0,0) moves under this transformation.


Find the smallest positive integer greater than 3000 that is neither prime nor square and has no prime factor less than 60.


Given the graph showing Suzanna's variable rate of 12 mph, with each subsequent 5-minute interval resulting in a 1 mph decrease, calculate the total distance she would ride in half an hour.


Calculate the area enclosed by a pentagon on a geoboard with vertices at coordinates (4,1), (2,6), (6,7), (9,5), and (7,2).


A rectangular table with dimensions $9'\times 12'$ needs to be moved from one corner to another inside a rectangular room. The shorter wall of the room is denoted by $W$ feet and the longer wall by $L$ feet. What is the smallest integer value for $W$, assuming $L \geq W$, such that the table can be relocated within the room without tilting or disassembling?


Given that \( n! \) is evenly divisible by \( 1 + 2 + \cdots + n \), find the number of positive integers \( n \) less than or equal to 50.


Given the expression $(xy-z)^2 + (x+y+z)^2$, find the least possible value for this expression for real numbers $x$, $y$, and $z$.


Given the equations \(3x - 4y - 2z = 0\) and \(x - 2y + 5z = 0\) with \(z \neq 0\), find the numerical value of \(\frac{2x^2 - xy}{y^2 + 4z^2}\).


Find the number of 3-digit positive integers whose digits multiply to 30.


Given the definition of a rising number as a positive integer each digit of which is larger than each of the digits to its left, find the smallest digit in the $103^{\text{rd}}$ five-digit rising number.


How many four-digit whole numbers are there such that the leftmost digit is a prime number, the second digit is even, and all four digits are different?


Square $IJKL$ has one vertex on each side of square $WXYZ$. Point $I$ is on $WZ$ with $WI = 3 \cdot IZ$. What is the ratio of the area of $IJKL$ to the area of $WXYZ$?
A) $\frac{1}{8}$
B) $\frac{1}{16}$
C) $\frac{1}{32}$
D) $\frac{1}{64}$
E) $\frac{1}{128}$


Set A has 30 elements, and set B has 25 elements. Set C has 10 elements. Calculate the smallest possible number of elements in the union A ∪ B ∪ C.


Given $P(x) =(x-1^2)(x-2^2)\cdots(x-50^2)$, determine the number of integers $n$ such that $P(n)\leq 0$.


Given the circle with a circumference of 18 units, a tangent drawn from an external point P, and a secant that divides the circle into arcs of lengths m and n, where m = 2n, and the tangent's length, t, is the geometric mean between m and n, find the number of integer values that t can take.


A set of 36 square blocks is arranged into a 6 × 6 square. How many different combinations of 4 blocks can be selected from that set so that no two blocks are in the same row or column?


Given a two-digit integer, determine how many two-digit positive integers equal the sum of their nonzero tens digit and the cube of their units digit.


Given a pizza shared with $12$ equally-sized slices, where Mark ate all the pepperoni slices and two plain slices. Mark preferred pepperoni on one-third of the pizza. Mark paid an additional $3 for the pepperoni topping. The cost of a plain pizza was $12. Find how much more Mark paid than Anne.


The number $15!$ expressed in the base $18$ system, determine the value of $k$ such that the result ends with exactly $k$ zeros.


Calculate the height of Camille’s model water tower, given that the real water tower is 50 meters high and can contain 200,000 liters of water, while her model's version contains only 0.05 liters of water.


Given that $r$, $s$, and $t$ are the roots of the cubic equation $ax^3 + bx^2 + cx + d = 0$, find the value of $\frac{1}{r^2} + \frac{1}{s^2} + \frac{1}{t^2}$.

Solution: \boxed{\frac{c^2 - 2bd}{d^2}}


Starting with the number 1 on the display of a calculator with only the [+1] and [x3] keys, calculate the fewest number of keystrokes needed to reach 243.


An uncrossed belt is fitted without slack around two circular pulleys with radii of 10 inches and 6 inches respectively. If the distance between the points of contact of the belt with the pulleys is 30 inches, determine the distance between the centers of the pulleys in inches.


In triangle $\triangle ABC$, $\angle C = 90^\circ$ and $\angle A = 30^\circ$. If $BD$ (with $D$ on $\overline{AC}$) is the angle bisector of $\angle ABC$, find $\angle BDC$.


Given positive integers $A$, $B$, and $C$, with no common factor greater than 1, such that $ A \log_{100} 5 + B \log_{100} 2 = C$, calculate the value of $A + B + C$.


Given six different awards are to be given to four students, with each student receiving at least one award. Calculate the number of different ways the awards can be distributed.


Given the set of all fractions $\frac{x}{y}$, where $x$ and $y$ are relatively prime positive integers, determine the number of fractions with the property that if both numerator and denominator are increased by 1, the value of the fraction is increased by $20\%$.


A triangle has a base of 18 inches. Two lines are drawn parallel to the base, terminating in the other two sides, and dividing the triangle into three areas with a ratio of 1:2:1, from the top to the bottom respectively. Find the length of the middle segment parallel to the base.

Solution: \boxed{9\sqrt{3} \text{ inches}}


Given the dimensions of a metallic rectangular sheet are 6 inches by 4 inches, and each dimension should be at least $x - 1.0$ inch and at most $x + 1.0$ inch from the reported measurement, and the final processed sheet dimensions shrink to 90% of its original size, calculate the minimum possible area of the rectangular sheet after processing.


Calculate the sum of all numbers of the form $3k + 2$ plus the sum of all numbers of the form $3k$, where $k$ takes integral values from $0$ to $n$.


Three standard 6-sided dice are rolled once. The sum of the numbers rolled determines the radius of a circle. Calculate the probability that the numerical value of the circumference of the circle is greater than the numerical value of twice the area of the circle.


Given that a basketball player made 7 baskets during a game, and each basket was worth either 1, 2, or 3 points, and the player made no more than 4 three-point baskets, determine the total number of different numbers that could represent the total points scored by the player.


If two poles are 30'' and 50'' high and are 150'' apart, find the height of the intersection of the lines joining the top of each pole to the foot of the opposite pole.


Given that Josie jogs parallel to a canal along which a boat is moving at a constant speed in the same direction and counts 130 steps to reach the front of the boat from behind it, and 70 steps from the front to the back, find the length of the boat in terms of Josie's steps.


Let $p$, $q$, $r$, $s$, and $t$ be distinct integers such that $(8-p)(8-q)(8-r)(8-s)(8-t) = -120$. Calculate the sum $p+q+r+s+t$.


Given that the product of the ages of Kiana and her two younger twin sisters is $72$, find the sum of their three ages.


Given that there are 25 students participating in an after-school program, 12 taking yoga, 15 taking bridge, 11 taking painting, 10 taking at least two classes, and 7 taking exactly two classes, determine the number of students taking all three classes.


Two spinners are spun once. Spinner 1 has sectors numbered: 1, 2, 4, while Spinner 2 has sectors numbered: 1, 3, 5, 7. Calculate the probability that the sum of the numbers on which the spinners land is both odd and prime.


Let $z = a + bi$ be a complex number such that $z + |z| = 1 + 12i$. Calculate $|z|^2$.


A rectangular floor measures $a$ by $b$ feet, where $a$ and $b$ are positive integers with $b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $2$ feet around the painted rectangle and occupies half of the area of the entire floor. Find the number of possible ordered pairs $(a,b)$.


Given that five students from Maplewood school worked for 6 days, six students from Oakdale school worked for 4 days, and eight students from Pinecrest school worked for 7 days, and the total amount paid for the students' work was 1240 dollars, determine the total amount earned by the students from Oakdale school, ignoring additional fees.


A point is chosen at random within the rectangle in the coordinate plane whose vertices are $(0, 0), (1000, 0), (1000, 2000),$ and $(0, 2000)$. The probability that the point lies within $d$ units of a lattice point is $\frac{3}{4}$. Calculate the value of $d$ to the nearest tenth.


If $x = \frac{a}{b}$ with $a \neq b$ and $b \neq 0$, and $y = \frac{b}{a}$, calculate $\frac{a + 2b}{a - 2b}$.


What is the sum of the digits of the square of $\text 999999999$?


Determine the number of significant digits in the measurement of the side of a square whose computed area is $3.2400$ square inches to the nearest ten-thousandth of a square inch.


Given a triangle with a base of 24 inches, a line is drawn parallel to the base, terminating in the other two sides and dividing the triangle into two equal areas, calculate the length of this parallel line.

Solution: \boxed{12\sqrt{2}\text{ inches}}


Given that a 4-digit positive integer has only even digits (0, 2, 4, 6, 8) and is divisible by 4, calculate the number of such integers.


When $10^{93} - 95$ is expressed as a single whole number, calculate the sum of its digits.


Let $(1-x+x^2)^n = b_0 + b_1x + b_2x^2 + \cdots + b_{2n}x^{2n}$ be an identity in $x$. Find the sum $t = b_0 + b_2 + b_4 + \cdots + b_{2n}$.

Solution: \boxed{\frac{1 + 3^n}{2}}


The common ratio of the geometric progression is $\frac{\sqrt[6]{2}}{\sqrt[4]{2}}=\frac{\sqrt[12]{2}}{\sqrt[6]{2}}$, so a general term of the sequence is $(\sqrt[4]{2})\left(\frac{\sqrt[6]{2}}{\sqrt[4]{2}}\right)^{n-1}=(\sqrt[4]{2})\left(\sqrt[12]{2}\right)^{2n-3}=\sqrt[4]{2}\cdot 2^{(2n-3)/12}=\sqrt[4]{2}\cdot 2^{n-3/4}$, so find the fourth term


For how many integer values of $n$ is the value of $8000 \cdot \left(\frac{3}{4}\right)^n$ an integer?


Given a square initially painted red, with each middle ninth of the square turning white every time the process is executed, calculate the fractional part of the original area of the square that stays red after four changes.


A top hat contains 4 red chips and 2 green chips. Chips are drawn randomly, one at a time without replacement, until all 4 reds are drawn or until both green chips are drawn. Calculate the probability that all 4 red chips are drawn.


Oliver has a training loop for his weekend cycling. He starts by cycling due north for 3 miles. He then cycles northeast, making a 30° angle with the north for 2 miles, followed by cycling southeast, making a 60° angle with the south for 2 miles. He completes his loop by cycling directly back to the starting point. Find the distance of this final segment of his ride.

Solution: \boxed{\sqrt{11 + 6\sqrt{3}}}


If Jo and Blair take turns counting, starting from $1$ with each number being two more than the last number, but repeating the number every 5th turn, find the 34th number said.


The ratio of the interior angles of two regular polygons with sides of unit length is $4:3$, and one of them has fewer than 15 sides. Determine the number of such pairs of polygons.


A lemming is placed at a corner of a square with a side length of 8 meters. It runs 6.8 meters diagonally towards the opposite corner. After reaching this point, it makes a 90-degree right turn and runs another 2.5 meters. Calculate the average of the shortest distances from the lemming's final position to each side of the square.


Determine the values of $p$ and $q$ for which the equation $\log p + \log q^2 = \log (p + q^2)$ holds.

Solution: \boxed{p = \frac{q^2}{q^2 - 1}}


If $x$ men working $x$ hours a day for $x$ days produce $x^2$ articles and each man's efficiency decreases by $\frac{1}{2}$ for every additional 10 men beyond 10 men in the workforce, calculate the number of articles produced by $y$ men working $y$ hours a day for $y$ days if $y = 20$.


Determine the upcoming year in which Andrea's birthday next falls on a Tuesday, given that Andrea's birthday in 2015 fell on a Friday, December 18.


For how many positive integers $m$ is $\frac{2310}{m^2 - 2}$ a positive integer?


Given that the squares of the positive integers $m$ and $n$ differ by 144, determine the number of ordered pairs $(m,n)$ such that $m \ge n$.


A square with a side length of $1$ is divided into one triangle and three trapezoids by joining the center of the square to points on each side. These points divide each side into segments such that the length from a vertex to the point is $\frac{1}{4}$ and from the point to the center of the side is $\frac{3}{4}$. If each section (triangle and trapezoids) has an equal area, find the length of the longer parallel side of the trapezoids.


Given that Jo and Blair take turns counting from 1, incrementing the previous number by 2 during each turn, find the $30^{\text{th}}$ number said.


Let $N$ be a two-digit number. We subtract the number with the digits reversed from $N$ and find that the result is a positive perfect fourth power. What is the digit $N$ can end?


Find the $1234$th digit to the right of the decimal point in $x = .123456789101112...498499500$.


A cube with side length $2$ is sliced by a plane that passes through a vertex $A$, the midpoint $M$ of an adjacent edge, and the midpoint $P$ of the face diagonal of the top face, not containing vertex $A$. Find the area of the triangle $AMP$.


Find the number of distinct points in the xy-plane common to the graphs of $(x+y-7)(3x-2y+6)=0$ and $(x-y-2)(4x+y-10)=0$.


Two circles, one with radius 2 and the other with radius 3, are to be placed as follows. The center of the circle with radius 2 is to be chosen uniformly and at random from the line segment joining (0,0) to (5,0). The center of the other circle is placed at (3,2). Determine the probability that the two circles intersect.


Rectangle ABCD with AB = 7 and AD = 10 is joined with right triangle DCE so that DC is common to both shapes. The areas of ABCD and DCE are equal. If both ABCD and DCE form a pentagon, find the length of DE.


Sandhya must save 35 files onto disks, each with 1.44 MB space. 5 of the files take up 0.6 MB, 18 of the files take up 0.5 MB, and the rest take up 0.3 MB. Files cannot be split across disks. Calculate the smallest number of disks needed to store all 35 files.


Given Gilda starts with a bag of marbles, she gives $30\%$ of them to Pedro, $10\%$ of the remaining to Ebony, $15\%$ of the remaining to Jimmy, and $5\%$ of the remaining to Maria. Calculate the percentage of her original bag of marbles that Gilda has left.


Given the first term of a geometric progression be $s$ and the common ratio be $(r+1)$, where $-1 < r < 0$, find the limit of the sum of the cubes of the terms in this progression as the number of terms approaches infinity.

Solution: \boxed{\frac{s^3}{1 - (r+1)^3}}


Two boys start at the same point A on an elliptical track and move in opposite directions. The speeds of the boys are 7 ft. per second and 11 ft. per second, respectively. The elliptical track has a major axis of 100 ft and a minor axis of 60 ft. Calculate the number of times they meet, excluding the start and finish, assuming they start at the same time and finish when they first meet at point A again.


The number $17!$ has a certain number of positive integer divisors. What is the probability that one of them is odd?


A lemming sits at a corner of a rectangle with side lengths 12 meters and 8 meters. The lemming runs 7.5 meters along a diagonal toward the opposite corner. It then makes a $90^{\circ}$ right turn and runs 3 meters, followed by another $90^{\circ}$ right turn running an additional 2 meters. Calculate the average of the shortest distances from the lemming's final position to each side of the rectangle.


Given a two-digit positive integer, determine how many integers satisfy the condition of being equal to the sum of its nonzero tens digit and the cube of its units digit.


Given an equilateral triangle $\triangle B_1B_2B_3$, for each positive integer $k$, point $B_{k+3}$ is the reflection of $B_k$ across the line $B_{k+1}B_{k+2}$, determine the measure of $\measuredangle B_{47}B_{45}B_{46}$.


Find the smallest positive integer n for which (n-17)/(7n+8) is a non-zero reducible fraction.


Given the digits $A$, $B$, $C$, and $D$, in the equations:
$$
\begin{array}{cc}
 & A\ B \\
+ & C\ D \\
\hline
 & A\ E \\
\end{array}
$$
and
$$
\begin{array}{cc}
 & A\ B \\
- & D\ C \\
\hline
 & A\ F \\
\end{array}
$$
Find the value of the digit $E$.


Given that $\underbrace{9999\cdots 99}_{80\text{ nines}}$ is multiplied by $\underbrace{7777\cdots 77}_{80\text{ sevens}}$, calculate the sum of the digits in the resulting product.


Three fair coins are to be tossed once. For each head that results, one fair die is to be rolled. If all three coins show heads, roll an additional fourth die. Determine the probability that the sum of the die rolls is odd.


Starting from "1," using a calculator with only the [ +1 ] and [ x2 ] keys, find the fewest number of keystrokes needed to reach "400," ensuring the display shows "50" at least once during the sequence.


Given that each of 24 balls is tossed independently and at random into one of the 4 bins, calculate $\frac{p'}{q'}$, where $p'$ is the probability that one bin ends up with 6 balls, and the other three bins end up with 6 balls each, and $q'$ is the probability that each bin ends up with exactly 6 balls.


Given a $60$-question multiple-choice math exam where students are scored $5$ points for a correct answer, $0$ points for an unpicked answer, and $-2$ points for a wrong answer, determine the maximum number of questions that Maria could have answered correctly, given that her total score on the exam was $150$.


For every $n$, the sum of n terms of an arithmetic progression is $5n + 4n^3$. Find the $r$th term of the progression.

Solution: \boxed{12r^2 - 12r + 9}


The circle having (2,2) and (10,8) as the endpoints of a diameter intersects the x-axis at a second point. Find the x-coordinate of this point.


With all three valves open, the tank fills in 1.2 hours, with only valves X and Z open it takes 2 hours, and with only valves Y and Z open it takes 3 hours, calculate the number of hours required with only valves X and Y open.


Given that Four times Dick's age plus twice Tom's age equals three times Harry's age, and Three times the square of Harry's age is equal to twice the square of Dick's age added to four times the square of Tom's age, and their respective ages are relatively prime to each other, find the sum of the cubes of their ages.


Given that the dimensions of a box are $4\text{-in} \times 3\text{-in}\times2\text{-in}$ and the dimensions of a solid block are $3\text{-in} \times 1\text{-in}\times1\text{-in}$, calculate the maximum number of blocks that can fit in the box.


Given the expression \( \left(1-\frac{1}{2^{2}}\right)\left(1-\frac{1}{3^{2}}\right)\ldots\left(1-\frac{1}{12^{2}}\right) \), compute its value.


Two lines with slopes $\dfrac{1}{3}$ and $3$ intersect at $(1,1)$. What is the area of the triangle enclosed by these two lines and the line $x + y = 8$.


Given a set $T$ of $8$ integers taken from $\{1,2,\dots,20\}$ with the property that if $c$ and $d$ are elements of $T$ with $c<d$, then $d$ is not a multiple of $c$, find the greatest possible value of an element in $T$.


Distinct points A, B, C, and D lie on a line, with AB=BC=CD=1. Points E, F, and G lie on a second line, parallel to the first, with EF=FG=1. Determine the number of possible values for the area of a triangle with positive area that has three of the six points as its vertices.


Given the base $9$ representation of a perfect square that is also a multiple of $3$ is $ab4c$, where $a \ne 0$, find the value of $c$.


For all real values of $x$, simplify the expression $\sqrt{x^6 + x^4 + 1}$.

Solution: \boxed{\sqrt{x^6 + x^4 + 1}}


Determine the maximum number of integers among $a, b, c, d$ that can be negative if $2^a 3^b + 5^c 7^d = 6^e 10^f+4$.


A magician's hat contains 4 red chips and 3 green chips. Chips are drawn randomly, one at a time without replacement, until all of the green chips are drawn or all of the red chips are drawn. Determine the probability that all the red chips are drawn before all the green chips.


Evaluate the expression \(3 + \sqrt{3} + \frac{1}{3 + \sqrt{3}} + \frac{1}{\sqrt{3} - 3}\).

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


Given that Raashan, Sylvia, and Ted each start with $1, a bell rings every 15 seconds, and each player who has money gives $1 to another player chosen at random, what is the probability that after the bell has rung 2019 times, each player will have $1?


Find the number of pairs (m, n) of integers that satisfy the equation m^3 + 4m^2 + 3m = 8n^3 + 12n^2 + 6n + 1.


A traveler starts from the origin, walks 4 miles due west, then turns $120^\circ$ to his left and walks another 5 miles in the new direction. After that, he turns $60^\circ$ to his left again and walks another 2 miles. Find the distance between his starting point and the endpoint.


Given a two-digit positive integer, define it as spiky if it equals the sum of its tens digit and the cube of its units digit subtracted by twice the tens digit. How many two-digit positive integers are spiky?


Find the smaller root of the equation $\left(x-\frac{1}{3}\right)^2 + \left(x-\frac{1}{3}\right)\left(x+\frac{1}{6}\right) = 0$.


The line $10x + 8y = 80$ forms a triangle with the coordinate axes. Calculate the sum of the lengths of the altitudes of this triangle.

Solution: \boxed{18 + \frac{40\sqrt{41}}{41}}


Determine the number of distinct pairs of integers $(x, y)$ such that $0 < x < y$ and $\sqrt{2500} = \sqrt{x} + 2\sqrt{y}$.
A) $0$
B) $1$
C) $2$
D) $3$
E) $4$


The dimensions of a rectangular box in inches are positive integers with one dimension being an even number. The volume of the box is $1806$ in$^3$. Find the minimum possible sum of the three dimensions.


Lisa celebrated her 20th birthday on Monday, August 15, 2010, which was not a leap year. Find the next year when her birthday will fall on a Thursday.


A circle of radius 3 is centered at a point O. An isosceles right triangle with hypotenuse 6 has its right-angled vertex at O. Calculate the difference between the area of the region that lies inside the circle but outside the triangle and the area of the region that lies inside the triangle but outside the circle.

Solution: \boxed{9(\pi - 1)}


What is the greatest power of $2$ that is a factor of $10^{1003} - 4^{502}$?


Determine the number of points common to the graphs of $(x-2y+3)(4x+y-5)=0$ and $(x+2y-3)(3x-4y+6)=0$.


Evaluate the tens digit of \(4032^{4033} - 4036\).


During a journey from the United States to France, Lucas took $d$ U.S. dollars. At the airport, he exchanged all his money to Euros at a rate where $5$ U.S. dollars yielded $8$ Euros. After spending $80$ Euros on souvenirs, Lucas found that he had exactly $d$ Euros left. Find the sum of the digits of $d$.


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


Ann now wants to extend her toothpick staircase to a 6-step version. She already has a 4-step staircase which used 28 toothpicks. Determine how many additional toothpicks are needed to complete the 6-step staircase if the pattern observed in increases continues.


In a circle, diameter $\overline{EB}$ is parallel to chord $\overline{DC}$. If the angles $AEB$ and $ABE$ are in the ratio of $2 : 3$, determine the degree measure of angle $BCD$.


Given two integers have a sum of 29. When two more integers are added to the first two, the sum becomes 47. Finally, when three more integers are added to the sum of the previous four integers, the sum becomes 66. Determine the minimum number of even integers among the 7 integers.


Two circles have centers at $(1,3)$ and $(10,6)$ with radii $3$ and $6$, respectively. Determine the y-intercept of a common external tangent to the circles if the tangent line can be expressed in the form $y=mx+b$ with $m>0$.
A) $\frac{5}{4}$
B) $\frac{6}{4}$
C) $\frac{7}{4}$
D) $\frac{8}{4}$
E) $\frac{9}{4}$


Given Mr. and Mrs. Alpha want to name their baby Alpha so that its monogram (first, middle, and last initials) will be in alphabetical order with no letter repeated, and the first initial must be 'A', find the number of such monograms possible.


Cagney can frost a cupcake in 15 seconds and Lacey can frost a cupcake in 25 seconds. After a 1-minute warm-up, their frosting speeds increase by 20%. Calculate the total number of cupcakes they can frost together in 10 minutes.


A right triangle has sides 5, 12, and 13, with 13 being the hypotenuse. A square with side length $x$ is inscribed in this triangle so that one side of the square lies along one of the legs of the triangle (length 12). Another square with side length $y$ is inscribed in the same triangle so that one side of the square lies along the hypotenuse. What is the ratio $\frac{x}{y}$?
A) $\frac{5}{13}$
B) $\frac{17}{18}$
C) $\frac{11}{12}$
D) $\frac{12}{13}$
E) $\frac{13}{12}$


Given that the larger rectangular fort is $15$ feet long, $12$ feet wide, and $6$ feet high, with the floor, ceillings, and all four walls made with blocks that are one foot thick, determine the total number of blocks used to build the entire fort.


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


Given two positive numbers that can be inserted between 4 and 16, such that the sequence forms two different arithmetic progressions, the first three numbers form an arithmetic progression, and the last three numbers form a different arithmetic progression, find the sum of these two numbers.


A coin has a probability of $3/4$ of turning up heads. Given that this coin is tossed 60 times, calculate the probability that the number of heads obtained is divisible by 3.


Given the dimensions of a $2\text{-in} \times 1\text{-in} \times 2\text{-in}$ solid block and a $3\text{-in} \times 4\text{-in} \times 2\text{-in}$ box, calculate the largest number of blocks that can fit in the box.


Given that right triangle $ACD$ with right angle at $C$ is constructed outwards on the hypotenuse $\overline{AC}$ of isosceles right triangle $ABC$ with leg length $2$, and $\angle CAD = 30^{\circ}$, find $\sin(2\angle BAD)$.


Given a square in the coordinate plane with vertices at $(0, 0), (3030, 0), (3030, 3030),$ and $(0, 3030)$, a point is chosen at random within the square. The probability that the point is within $d$ units of a lattice point is $\tfrac{1}{3}$. Determine $d$ to the nearest tenth.


Points $M$ and $N$ are the midpoints of sides $PA$ and $PB$ of $\triangle PAB$. As $P$ moves along a line such that the length of segment $PA$ remains constant, determine how many of the following quantities change: the length of the segment $MN$, the perimeter of $\triangle PAB$, the area of $\triangle PAB$, the area of trapezoid $ABNM$.


Consider a square grid composed of $121$ points arranged in an $11 \times 11$ formation, including all the points on the edges. Point $P$ is situated at the center of this square. Calculate the probability that a line drawn from $P$ to any other randomly chosen point $Q$ from the remaining $120$ points acts as a line of symmetry for the square.


Choose three different numbers from the set $\{2, 5, 8, 11, 14\}$. Add two of these numbers and multiply their sum by the third number. What is the smallest result that can be obtained through this process?


Given a two-digit positive integer, define it as $\emph{charming}$ if it is equal to the sum of its nonzero tens digit and the cube of its units digit. Determine how many two-digit positive integers are charming.


Given that there are $6$-pound sapphires worth $\$$15 each, $3$-pound rubies worth $\$$9 each, and $2$-pound diamonds worth $\$$5 each, and there are at least $10$ of each type available, determine the maximum value of the gemstones Ellie can carry, given a weight limit of $24$ pounds.


The Fibonacci sequence starts with two 1s, and each term afterwards is the sum of its two predecessors. Determine the last digit to appear in the units position of a number in the Fibonacci sequence when considered modulo 12.


How many different real numbers x satisfy the equation (2x^2 - 7)^2 - 8x = 48?


Given $\omega = -\frac{1}{2} + \frac{1}{2}i\sqrt{3}$, define $T$ as the set of all points in the complex plane of the form $a + b\omega + c\omega^2$, where $0 \leq a, b, c \leq 2$. Find the area of $T$.


The greatest possible sum of the digits in the base-eight representation of a positive integer less than $5000$.


What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{30} \cdot 5^3}$ as a decimal?


Given two circles are internally tangent at a point, with circles centered at points $A$ and $B$ having radii $7$ and $4$ respectively, find the distance from point $B$ to the point where an internally tangent line intersects ray $AB$ at point $C$.


Given a lawn measuring 100 feet by 140 feet, a mower swath of 30 inches wide with a 6-inch overlap, and a walking speed of 4500 feet per hour, determine the closest estimate for the number of hours it will take to mow the entire lawn.


The numbers $-3, 1, 5, 8$, and $10$ are rearranged according to these rules:
        1. The largest isn't first, but it is in one of the first three places.
        2. The smallest isn't last, but it is in one of the last three places.
        3. The median isn't in the second or fourth position.
        4. The sum of the first and last numbers is greater than 12. Find the average of the first and last numbers.


Given the population of Brazil in 2010 as approximately 195,000,000 and the area of Brazil as 3,288,000 square miles, determine the average number of square feet per person in Brazil.


Given the school store sells 9 pencils and 10 notebooks for $\mathdollar 5.06$ and 6 pencils and 4 notebooks for $\mathdollar 2.42$, determine the cost of 20 pencils and 14 notebooks.


Given a three-digit positive integer and a two-digit positive integer with all five digits in these numbers being different, the sum of the two numbers is a three-digit number. Find the smallest possible value for the sum of the digits of this three-digit sum.


A wooden block with dimensions $12 \times 10 \times 9$ is formed by gluing together unit cubes. Determine the greatest number of unit cubes that can be seen from a single point.


Given the digits 1, 2, 3, 7, 8, 9, find the smallest sum of two 3-digit numbers that can be obtained by placing each of these digits in one of the six boxes in the given addition problem, with the condition that each number must contain one digit from 1, 2, 3 and one digit from 7, 8, 9.


It is now between 9:00 and 10:00 o'clock, and nine minutes from now, the minute hand of a watch will be exactly opposite the place where the hour hand was six minutes ago. Determine the exact time now.


Chantal starts hiking from a trailhead to a fire tower 6 miles away, walking initially at 5 miles per hour on a flat portion for 3 miles. She then hikes uphill at 3 miles per hour for 3 miles, and descends at 4 miles per hour. If Chantal meets Jean exactly at the 3-mile point as Chantal descends, calculate Jean's average speed in miles per hour.


Each vertex of a convex hexagon $ABCDEF$ is to be assigned a color. There are $7$ colors to choose from, and both adjacent vertices and the ends of each diagonal must have different colors. Calculate the total number of different colorings possible.


Let $n$ be the number of pairs of values of $a$ and $d$ such that the equations $ax+3y+d=0$ and $4x-ay+8=0$ represent the same line. Find $n$.


Starting with the display "1," calculate the fewest number of keystrokes required to reach "300" by only allowing up to three consecutive [+1] operations before a [x2] must be pressed.


How many unique pairs of parallel edges does a regular octahedron have?


What is the largest number of solid $2\text{-in} \times 3\text{-in} \times 1\text{-in}$ blocks that can fit in a $4\text{-in} \times 3\text{-in}\times 3\text{-in}$ box?


The number $2024$ needs to be expressed in the form $\frac{a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$, where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_m + b_n$ is as small as possible. Find the value of $|a_m - b_n|$.


A $4 \times 4$ square is partitioned into $16$ smaller squares. Each square is painted either red or blue randomly and independently with each color being equally likely. The square is rotated $90^\circ$ counterclockwise. After the rotation, any red square that has moved into a position that was occupied by a blue square is repainted blue. What is the probability that the entire grid is blue after this operation?
A) $\frac{1}{256}$
B) $\frac{1}{4096}$
C) $\frac{1}{65536}$
D) $\frac{1}{16384}$
E) $\frac{1}{1024}$


The product of the two 99-digit numbers 707,070,707,...,070,707 and 909,090,909,...,090,909 has a thousands digit A and a units digit B. Calculate the sum of A and B.


Let $A,B,C$ and $D$ be four different digits selected from the set $\{1,2,3,4,5,6,7,8 \}$. Find the minimum value of the expression $\dfrac{A}{B} + \dfrac{C}{D}$.


Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3, \dots, 1004\}$. What is the probability that $abc + a + b + c$ is divisible by $4$?
A) $\frac{9}{64}$
B) $\frac{1}{8}$
C) $\frac{1}{64}$
D) $\frac{1}{16}$
E) $\frac{1}{4}$


Given numbers $5, 6, 7, 8, 9, 10, 11, 12, 13$ are written in a $3\times3$ array, with the condition that two consecutive numbers must share an edge. If the sum of the numbers in the four corners is $32$, calculate the number in the center of the array.


Given $\log 216$, evaluate this expression.

Solution: \boxed{3(\log 2 + \log 3)}


Ridley designs nonstandard checkerboards with 32 squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. Calculate the total number of black squares on such a checkerboard.


A square in the coordinate plane has vertices at $(0, 0), (20, 0), (20, 20),$ and $(0, 20)$. Find the radius $d$ such that the probability the point is within $d$ units of a lattice point is $\tfrac{3}{4}$, and determine $d$ to the nearest tenth.


Let $C$ and $D$ be consecutive positive integers with $C < D$, and $C$, $D$, and $C+D$ represent number bases in the equation $231_C + 56_D = 105_{C+D}$. Determine the value of $C+D$.


Given a two-digit positive integer is called $\emph{entangled}$ if it is equal to twice the sum of its nonzero tens digit and the cube of its units digit, how many two-digit positive integers are entangled?


Two tangents are drawn to a circle from an exterior point $A$; they touch the circle at points $B$ and $C$ respectively. A third tangent intersects segment $AB$ in $P$—a point halfway along segment $AB$—and $AC$ in $R$—a point halfway along segment $AC$, and touches the circle at $Q$. Given that $AB=24$, calculate the perimeter of $\triangle APR$.


In their base $10$ representations, the integer $a$ consists of a sequence of $1000$ sixes and the integer $b$ consists of a sequence of $1000$ sevens. What is the sum of the digits of the base $10$ representation of $9ab$?
A) 18876
B) 19986
C) 20000
D) 21000


Determine the value of \(\text{rem} \left(\frac{5}{7}, -\frac{3}{4}\right)\).


What is the minimum number of digits to the right of the decimal point needed to express the fraction $\frac{987654321}{2^{28}\cdot 5^3}$ as a decimal?


For how many integers $n$ is $(n+2i)^5$ a real number?


Given a $3 \times 3$ square is placed inside a larger diagram, with circles centered at each corner with radius equal to the distance from the corner to the center of the square, construct a new square $EFGH$ such that each of its sides is exactly tangent to one of these circles, with sides parallel to the original square. Determine the area of square $EFGH$.

Solution: \boxed{27 + 18\sqrt{2}}


The sum of the first $2k + 3$ terms of the arithmetic series with first term $k^2 + k + 1$ is computed.

Solution: \boxed{2k^3 + 7k^2 + 10k + 6}


A rectangle $ABCD$ has its diagonal $DB$ divided into three segments by two lines $L$ and $L'$, which are parallel to each other and perpendicular to $DB$. Line $L$ passes through $A$, and $L'$ passes through $C$. The segments along $DB$ created by $L$ and $L'$ have lengths $DE = 2$, $EF = 1$, and $FB = 1$. Find the area of rectangle $ABCD$, rounded to one decimal place.
A) $6.8$
B) $6.9$
C) $7.0$
D) $7.1$


Given a triangle with vertices at $(0,0)$, $(2,2)$, and $(8m,0)$, find the sum of all possible values of $m$ such that a line defined by $y = mx$ divides the triangle into two triangles of equal area.


Jenny and Jack run on a circular track. Jenny runs counterclockwise and completes a lap every 75 seconds, while Jack runs clockwise and completes a lap every 70 seconds. They start at the same place and at the same time. Between 15 minutes and 16 minutes from the start, a photographer standing outside the track takes a picture that shows one-third of the track, centered on the starting line. What is the probability that both Jenny and Jack are in the picture?
A) $\frac{23}{60}$
B) $\frac{12}{60}$
C) $\frac{13}{60}$
D) $\frac{46}{60}$
E) $\frac{120}{60}$


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


Using only pennies, nickels, dimes, quarters, and half-dollars, determine the smallest number of coins needed to pay any amount of money less than a dollar and a half.


Given that in triangle $\triangle ABC$, point D is on line segment AC such that $\angle ABD$ measures $70^\circ$ and the total sum of angles around point B is $200^\circ$, and $\angle CBD = 60^\circ$, find the measure of $\angle ABC$.


Given that vertex $E$ of right $\triangle ABE$ (with $\angle ABE = 90^\circ$) is inside square $ABCD$, and $F$ is the point of intersection of diagonal $BD$ and hypotenuse $AE$, find the area of $\triangle ABF$ where the length of $AB$ is $\sqrt{2}$.


A school club buys 1200 candy bars at a price of four for $3 dollars, and sells all the candy bars at a price of three for $2 dollars, or five for $3 dollars if more than 50 are bought at once. Calculate their total profit in dollars.


Let there be $k$ red balls and $N$ green balls arranged in a line, where $k$ is a positive integer and $N$ is a positive multiple of 10. Determine the least value of $N$ such that the probability that at least $\frac{3}{5}$ of the green balls are on the same side of all the red balls combined is less than $\frac{8}{10}$ when there are $2$ red balls, and calculate the sum of its digits.


Given $12345 \times 5 + 23451 \times 4 + 34512 \times 3 + 45123 \times 2 + 51234 \times 1$, evaluate the expression.


In a round-robin tournament with 8 teams, where each team plays one game against each other team, resulting in a win or loss, determine the maximum number of teams that could be tied for the most wins at the end of the tournament.


A clock is set correctly at 1 P.M. on March 15 and subsequently loses exactly 3 minutes per day. By what amount of time, in minutes, must the clock be adjusted on the morning of March 22 at 9 A.M. to reflect the correct time?
A) $17$ minutes
B) $18$ minutes
C) $19$ minutes
D) $20$ minutes
E) $21$ minutes


Given the quadratic equation \( ax^2 + bx + c \) and the table of values \( 6300, 6481, 6664, 6851, 7040, 7231, 7424, 7619, 7816 \) for a sequence of equally spaced increasing values of \( x \), determine the function value that does not belong to the table.


Let $d(m)$ denote the number of positive integers that divide $m$, including $1$ and $m$. Define the function $g(m)=\frac{d(m)}{\sqrt[4]{m}}$. Find the unique positive integer $M$ such that $g(M) < g(m)$ for all positive integers $m \ne M$, and calculate the product of the digits of $M$.


A circular disc with a diameter of $2D$ is placed on a $6 \times 6$ checkerboard, aiming for the centers to coincide. Calculate the number of checkerboard squares which are completely covered by the disc.


In their base $10$ representations, the integer $a$ consists of a sequence of $2000$ eights and the integer $b$ consists of a sequence of $2000$ fives. What is the sum of the digits of the base $10$ representation of $9ab$?
A) 17765
B) 17965
C) 18005
D) 18100
E) 18200


Define a positive real number as "modified special" if it has a decimal representation consisting entirely of digits $0$ and $3$. Find the smallest positive integer $n$ such that $1$ can be written as a sum of $n$ modified special numbers.


Determine the units digit of the decimal expansion of (17 + √224)^21 + (17 + √224)^85.


A collection of circles in the upper half-plane, all tangent to the $x$-axis, is constructed in layers as follows. Layer $L_0$ consists of two circles of radii $50^2$ and $55^2$ that are externally tangent. For $k \ge 1$, the circles in $\bigcup_{j=0}^{k-1}L_j$ are ordered according to their points of tangency with the $x$-axis. For every pair of consecutive circles in this order, a new circle is constructed externally tangent to each of the two circles in the pair. Layer $L_k$ consists of $2^{k-1}$ circles constructed in this way. Let $S=\bigcup_{j=0}^{7}L_j$, and for every circle $C$ denote by $r(C)$ its radius. Calculate:
\[
\sum_{C\in S} \frac{1}{\sqrt{r(C)}}
\]
A) $\frac{105}{2750}$
B) $\frac{21}{550}$
C) $\frac{2688}{550}$
D) $\frac{1344}{275}$


Let $ABCD$ be a square with side length 3. Points $E$ and $F$ are on sides $AB$ and $AD$ respectively such that $AE=2AF$. Determine the maximum area of quadrilateral $CDFE$.
A) $\frac{1}{2}$
B) $\frac{3}{2}$
C) $\frac{5}{2}$
D) $\frac{7}{2}$
E) $\frac{9}{2}$


In a badminton tournament, $2n$ women and $3n$ men participate, and each player competes exactly once against every other player. If there are no ties and the ratio of the number of matches won by women to the number of matches won by men is $\frac{4}{3}$, find the value of $n$.


Let $ABCD$ be a regular tetrahedron, and let $E$ be a point lying exactly in the middle of edge $AB$. Denote $s$ as the sum of the distances from $E$ to the planes $DAC$, $DBC$, and the plane containing the face $BCD$, and $S$ as the sum of the distances from $E$ to the remaining edges $AC$, $BC$, $CD$, $DA$, and $DB$. Calculate $\frac{s}{S}$.
A) $\frac{9}{10}$
B) $\frac{3}{2}$
C) $1$
D) $\frac{1}{3}$
E) $\frac{\sqrt{2}}{2}$


The number $395$ is expressed in the form $395 = \frac{a_1!a_2!}{b_1!b_2!b_3!}$, where $a_1 \ge a_2$ and $b_1 \ge b_2 \ge b_3$ are positive integers, and $a_1 + b_1$ is as small as possible. Calculate the value of $|a_1 - b_1|$.


Let $b_1, b_2, \dots, b_{2018}$ be a strictly increasing sequence of positive integers such that
\[ b_1 + b_2 + \cdots + b_{2018} = 2018^3.\]
Determine the remnants of $b_1^3 + b_2^3 +\cdots + b_{2018}^3$ and $b_1^4 + b_2^4 + \cdots + b_{2018}^4$ when divided by 5.
A) Cube remainder 3, Fourth Power remainder 3
B) Cube remainder 3, Fourth Power remainder 1
C) Cube remainder 1, Fourth Power remainder 3
D) Cube remainder 2, Fourth Power remainder 1

Solution: \boxed{B) Cube remainder 3, Fourth Power remainder 1}


In a modified game, each of $5$ players rolls a standard $6$-sided die. The winner is identified as the player who rolls the highest number. In case of a tie for the highest roll, the tied players roll again, repeating this process until a single player wins. Hugo is among these players. What is the probability that Hugo's first roll was a $6$, given that he won the game?
A) $\frac{1024}{7776}$
B) $\frac{3125}{7776}$
C) $\frac{2000}{7776}$
D) $\frac{5000}{7776}$
E) $\frac{625}{7776}$


Consider an expanded diamond-shaped grid where you need to spell the word "DIAMOND" by moving rightward, downward, or diagonally downward to the right between adjacent letters in the given diagram. For how many paths, consisting of a sequence of horizontal, vertical, or diagonal line segments, with each segment connecting a pair of adjacent letters, is the word DIAMOND spelled out as the path is traversed from beginning to end?


Let $S=\{(x,y) : x\in \{0,1,2,3,4,5\}, y\in \{0,1,2,3,4,5,6\}\}$. Find the product of the values of $\tan(\angle{CBA})$ for all right triangles $t=\triangle{ABC}$ with vertices $A$, $B$, and $C$ in counter-clockwise order and right angle at $A$.


In $\triangle DEF$, $\angle E = 90^\circ$ and $DE = 10$. An equilateral triangle $DEF_{1}$ and a regular pentagon $EF_{2}F_{3}F_{4}F_{5}$ are constructed on the sides $DE$ and $EF$ outside the triangle, respectively. The points $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$ and $F_{5}$ lie on a circle. Compute the perimeter of triangle $DEF$.


My friend reads four times as fast as I do. If it takes me 3 hours to read a novel, and immediately after reading, I take another hour to write a summary, how many minutes will it take my friend to read the same novel and write a summary assuming writing speed is the same for both of us?

Solution: \boxed{105 \text{ min}}


Terrell usually lifts two 25-pound weights 10 times. If he decides to lift one 20-pound weight instead, how many times must Terrell lift it in order to lift the same total weight?


The cost of 1 piece of gum is 2 cents. What is the cost of 500 pieces of gum, in cents and in dollars?


I have 7 books, among which three are identical copies of a physics book, and the remaining books are all different from each other. In how many ways can I arrange these books on a shelf?


A 16-slice pizza is made with only pepperoni and mushroom toppings, and every slice has at least one topping. Only nine slices have pepperoni, and exactly twelve slices have mushrooms. How many slices have only mushrooms?


The arithmetic mean of 15 scores is 90. When the highest and lowest scores are removed, the new mean becomes 92. If the highest of the 15 scores is 110, what is the lowest score?


Parrots absorb 40% of the seeds they consume. Today, a parrot absorbed 8 ounces of seeds. What were the total ounces of seeds consumed by the parrot today? Additionally, how many ounces would be twice the total amount of seeds consumed?


What is the remainder when $1632 \cdot 2024$ is divided by $400$?


A student needs to provide his waist size in centimeters for a customized lab coat, based on the measurements in inches. If there are $10$ inches in a foot and $25$ centimeters in a foot, then what size should the student specify in centimeters if his waist size is $40$ inches?


The sum of two positive integers is 40 and their difference is 8. What is the product of the two integers and the value of the positive difference of the squares of these integers?


Given a modified operation $\nabla$ such that for $a>0$ and $b>0$, $$a \nabla b = \frac{a^2 + b^2}{1 + a^2b^2},$$ calculate $3 \nabla 4$.


The sides of a triangle have lengths of $8$, $15$, and $17$. Find the length of the shortest altitude.


To calculate $49^2$, Sam mentally computes the value of $50^2$ and subtracts some quantity. Additionally, Sam adds 1. What number does Sam subtract before adding 1 to compute $49^2$ correctly?


What is the least positive multiple of 25 that is greater than 475?


In triangle $ABC$, $\angle C = 90^\circ$, $AC = 4$, and $AB = \sqrt{41}$. What is $\tan B$?


What is the least integer value of \( x \) such that \( |3x + 10| \leq 25 \)?


What is the least positive integer that can be added to 525 to yield a multiple of 5?


Liam rolls a fair icosahedral die with numbers $1,2,3,...,20$ on its faces. What is the expected number of digits in the number Liam obtains? Express your answer as a decimal.


Bertha has 5 daughters and no sons. Some of her daughters have 5 daughters, and the rest have none. Bertha has a total of 25 daughters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and granddaughters have no daughters?


In a rectangular coordinate system, what is the number of units in the distance from the origin to the point $(20, 21)$?


A sphere is inscribed in a cube. The edge of the cube measures 8 feet. Find the volume of the sphere in cubic feet, and express your answer in terms of \(\pi\).


In right triangle $ABC$, it is given that $\sin A = \frac{3}{5}$ and $\sin B = 1$. Find $\sin C$.


On a high-performance racing bicycle, the front wheel has a diameter of 28 inches, and the back wheel has a diameter of 20 inches. If the bicycle does not slip, how many revolutions will the back wheel make while the front wheel makes 50 revolutions?


If the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is $60^\circ$, what is the angle between the vectors $-2\mathbf{a}$ and $\mathbf{b}$?


What is the value of $c$ if the lines with equations $y = \frac{5}{2}x + 5$ and $y = (3c)x + 3$ are parallel?


Points $A$, $B$, $C$, and $T$ are in space such that each of $\overline{TA}$, $\overline{TB}$, and $\overline{TC}$ is perpendicular to the other two. If $TA = 8$, $TB = 15$, and $TC = 12$, what is the volume of pyramid $TABC$?


Sarah constructs a conical sand sculpture for a beach competition. The sculpture has a height of 15 inches and a circular base with a radius of 8 inches. Sarah needs to find the smallest cube-shaped box to transport her sculpture upright without tilting. What is the volume of this cube-shaped box, in cubic inches?


Define a new operation $a \text{ Z } b = a^3 - 3a^2b + 3ab^2 - b^3$. What is the value of $4 \text{ Z } 3$?


Four concentric circles are drawn with radii of 2, 4, 6, and 8. The innermost circle is painted black, the next ring is painted white, the ring after that is black, and the outermost ring is white. What is the ratio of the black area to the white area? Express your answer as a common fraction.


How many five-digit positive integers are there with thousands digit $3$ and ten-thousands digit $1$?


$\textbf{Elisa's Stamp Album}$

Elisa sorts the stamps in her collection by country and decade of issuance. She paid the following prices: China and Japan, $7$ cents each, Italy $5$ cents each, and Germany $6$ cents each. (China and Japan are Asian countries; Italy and Germany are European countries.) The numbers of stamps per decade and country are given in the table below:

\[
\begin{array}{|c|c|c|c|c|}
\hline
\text{Country} & \text{'50s} & \text{'60s} & \text{'70s} & \text{'90s} \\
\hline
China & 5 & 9 & 11 & 8 \\
Japan & 10 & 6 & 9 & 12 \\
Italy & 7 & 8 & 5 & 11 \\
Germany & 8 & 7 & 10 & 9 \\
\hline
\end{array}
\]

Calculate the total cost of her Asian stamps from before the '80s and the total number of European stamps from the '90s.


A regular polygon has a perimeter of 180 cm and each side has a length of 15 cm. How many sides does this polygon have and what is its area if the apothem is 12 cm?

Solution: \boxed{1080 \text{ cm}^2}


Makoto drove from his office to a client's location and back. The odometer read 82,435 when he left his office, and it read 82,475 when he returned. Makoto's car consumes fuel at a rate of 25 miles per gallon and the price of one gallon of gas is \$3.75. What was the cost of the gas used for Makoto's round trip? (Express your answer in dollars and round to the nearest cent.)


Determine the area of the region bounded by the equation $x^2 + y^2 + 6x - 8y - 5 = 0$.


Chris, David, Eva, and Fiona weigh themselves in pairs. Together, Chris and David weigh 330 pounds, David and Eva weigh 290 pounds, and Eva and Fiona weigh 310 pounds. How many pounds do Chris and Fiona weigh together?


Let $p$ and $q$ be angles such that $\cos p + \cos q = \frac{1}{3}$ and $\sin p + \sin q = \frac{5}{13}.$ Find
\[\tan \left( \frac{p + q}{2} \right).\]


When a polynomial is divided by \(3x^3 - 4x^2 + 5x - 6\), what are the possible degrees of the remainder? Enter all the possible values, separated by commas.

Solution: \boxed{0, 1, 2}


Find the units digit of the following within the indicated number base: $63_9 + 74_9$


A number is chosen at random from the set of consecutive natural numbers $\{1, 2, 3, \ldots, 120\}$. What is the probability that the number chosen is a factor of $5!$? Express your answer as a common fraction.


Find the distance between the points (0,12) and (9,0).


The sum of 81 consecutive integers is $9^5$. What is their median?


Marty wants to paint a box. He can choose to use either blue, green, yellow, black, or red paint. However, if he chooses to use a sponge, he can only do so with green or yellow paints. Marty can also style the paint by painting with a brush, a roller, or a sponge. How many different combinations of color and painting method can Marty choose?


In a Biology class, Jake wants to prepare a nutrient solution by mixing 0.05 liters of nutrient concentrate with 0.03 liters of water to obtain 0.08 liters of solution. He needs to make a total of 0.64 liters of this solution for an experiment. How many liters of water will he need to use?


A flagpole is supported by a wire which extends from the top of the pole to a point on the ground 5 meters from its base. When Ben walks 3 meters from the base of the pole toward the point where the wire is attached to the ground, his head just touches the wire. Ben is 1.8 meters tall. How many meters tall is the flagpole?

Solution: \boxed{4.5 \text{ meters}}


We know the following to be true:
- $Z$ and $K$ are integers where $1000 < Z < 2000$ and $K > 0$.
- $Z = K \times K^3$.

What is the value of $K$ for which $Z$ is a perfect fourth power?


In MODIFIED SHORT BINGO, a $5\times5$ card is filled by marking the middle square as WILD. Modify the assignment of numbers such that the first column consists of 5 distinct numbers from the set $10-25$. The rest of the squares are filled with numbers as follows: 5 distinct numbers from $26-35$ in the second column, 4 distinct numbers $36-45$ in the third column (skipping the WILD square), 5 distinct numbers from $46-55$ in the fourth column, and 5 distinct numbers from $56-65$ in the fifth column.

How many distinct possibilities are there for the values in the first column of a MODIFIED SHORT BINGO card, where the order of the numbers matters?


What is the length of the diagonal of a square with side length $100\sqrt{3}$ cm? Express your answer in simplest form.

Solution: \boxed{100\sqrt{6} \text{ cm}}


Define the operation $X \diamond Y$ such that $X \diamond Y = 4X - 3Y + 2$. Calculate $X$ if $X \diamond 6 = 35$.


In the prime factorization of $30!$, what is the exponent of $2$ and the exponent of $5$?


Below is the graph of $y = a \tan bx$ for some positive constants $a$ and $b.$ The graph has a period of $\frac{2\pi}{5}$ and passes through the point $\left(\frac{\pi}{10}, 1\right)$. Find $ab.$


What is the sum and the count of all positive integer divisors of 91?


The Great Eighteen Soccer League has three divisions, with six teams in each division. Each team plays each of the other teams in its own division three times and every team in the other divisions twice. How many league games are scheduled?


Suppose the graph of a function $y = g(x)$ has the property that if it is shifted $30$ units to the right, then the resulting graph is identical to the original graph of $y = g(x)$.

What is the smallest positive $b$ such that if the graph of $y = g\left(\frac{x}{4}\right)$ is shifted $b$ units to the right, then the resulting graph is identical to the original graph of $y = g\left(\frac{x}{4}\right)$?


Calculate the sum of the coefficients in the polynomial $4(2x^{6} + 9x^3 - 6) + 8(x^4 - 6x^2 + 3)$ when it is fully simplified.


If $4$ lunks can be traded for $2$ kunks, and $2$ kunks will buy $6$ apples, how many lunks are needed to purchase two dozen apples?


I have 7 marbles numbered 1 through 7 in a bag. Suppose I take out three different marbles at random. What is the expected value of the sum of the numbers on the marbles?


John now has six children. What is the probability that exactly three of them are girls? (Assume a boy is equally likely to be born as a girl.)


In triangle $ABC$, altitudes $AD$, $BE$, and $CF$ intersect at the orthocenter $H$. If $\angle ABC = 58^\circ$ and $\angle ACB = 23^\circ$, then find the measure of $\angle BHC$, in degrees.


What is the smallest number with three different prime factors, none of which can be less than 10?


Solution: \boxed{6x^3 - x^2 + 23x - 3}


Two fair, six-sided dice are rolled. What is the probability that the sum of the two numbers showing is less than 12?


Given $\begin{vmatrix} x & y & z \\ a & b & c \\ p & q & r \end{vmatrix} = 2$, find $\begin{vmatrix} 3x & 3y & 3z \\ 3a & 3b & 3c \\ 3p & 3q & 3r \end{vmatrix}.$


What is the measure of the smaller angle between the hands of a 12-hour clock at 3:45 pm, in degrees? Express your answer as a decimal to the nearest tenth.


Determine the value of the positive number $b$ such that the terms $25, b, \frac{1}{4}$ are the first, second, and third terms, respectively, of a geometric sequence.


Factor the following expression: $75x^2 + 50x$.

Solution: \boxed{25x(3x + 2)}


A circular plaza includes a three-layer arrangement: a central water feature, surrounded by a decorative tile area, then bordered by a circular walkway. The water feature has a diameter of 16 feet. The decorative tile area is 12 feet wide uniformly around the water feature, and the walkway is 10 feet wide. What is the diameter of the circle that forms the outer boundary of the walkway?

Solution: \boxed{60 \text{ feet}}


I have 6 books, three of which are identical copies of the same math book, and the rest are different. In how many ways can I arrange them on a shelf?


Find the least positive integer $x$ that satisfies $x + 3567 \equiv 1543 \pmod{14}$.


In a configuration, two concentric circles form two regions: an outer gray region and an inner white region. The area of the gray region within the larger circle is equal to four times the area of the white circular region. What is the ratio of the radius of the small circle to the radius of the large circle?


$\textbf{Mia's Classic Postcard Collection}$

Mia organizes her postcard collection by country and by the decade in which they were printed. The prices she paid for them at an antique shop were: Italy and Germany, $8$ cents each, Canada $5$ cents each, and Japan $7$ cents each. (Italy and Germany are European countries, Canada and Japan are in North America and Asia respectively.)

In her collection:
- 1950s: Italy 5, Germany 9, Canada 3, Japan 6
- 1960s: Italy 12, Germany 5, Canada 7, Japan 8
- 1970s: Italy 11, Germany 13, Canada 6, Japan 9
- 1980s: Italy 10, Germany 15, Canada 10, Japan 5
- 1990s: Italy 6,  Germany 7,  Canada 11, Japan 9

In dollars and cents, how much did her Asian and North American postcards issued in the '60s, '70s, and '80s cost her?


How many possible distinct arrangements are there of the letters in the word "BALLOON"?


Three years ago, there were 30 trailer homes on Oak Street with an average age of 15 years. In that year, several brand new trailer homes were added to Oak Street. Today, the average age of all the trailer homes on Oak Street is 12 years. How many new trailer homes were added three years ago?


A sphere-shaped balloon with a radius of 2 feet is filled with helium. This helium is then evenly transferred into 64 smaller sphere-shaped balloons. What is the radius of each of the smaller balloons, in feet?


Fully simplify the following expression: $[(1+2+3+4+5+6)\div 3] + [(3\cdot5 + 12) \div 4]$.


Rebecca purchases hot dogs that come in packages of seven, and hot dog buns that come in packages of nine. What is the smallest number of hot dog packages she must buy to have an equal number of hot dogs and hot dog buns?


Find \( n \) such that \( 2^7 \cdot 3^4 \cdot n = 9! \).


How many one-quarters are there in one-eighth?


Nora has three stamp albums. Each page she fills contains an equal number of stamps. If she fills the first album with 945 stamps, the second with 1260 stamps, and the third with 1575 stamps, what is the maximum number of stamps she could be putting on each page?


Simplify this expression to a common fraction:  \(\frac{1}{\frac{1}{(\frac{1}{3})^1}+\frac{1}{(\frac{1}{3})^2}+\frac{1}{(\frac{1}{3})^3}}\)


The hypotenuse of a 30-60-90 triangle is $6\sqrt{2}$ units. How many square units are in the area of this triangle?


Let $a\star b = a^2b + 2b - a$. If $7\star x = 85$, find the value of $x$.


Four concentric circles are drawn with radii of 2, 4, 6, and 8. The innermost circle is painted black, the ring around it is white, the next ring beyond it is black, and the outermost ring is white. What is the ratio of the black area to the white area?


Allie and Betty play a game where they alternately roll a standard die. The scoring function $g(n)$ awards points as per the following criteria:
\[g(n) = \left\{
\begin{array}{cl}
6 & \text{if } n \text{ is a multiple of 2 and 3}, \\
3 & \text{if } n \text{ is a multiple of 3 but not 2}, \\
2 & \text{if } n \text{ is a multiple of 2 but not 3}, \\
0 & \text{otherwise}.
\end{array}
\right.\]
Allie rolls the die four times, receiving the outcomes 6, 3, 2, and 4. Betty rolls the die four times as well, getting 5, 2, 3, and 6. Calculate the product of Allie's total points and Betty's total points.


Find the monic quadratic polynomial, in $x,$ with real coefficients, which has $-3 - i \sqrt{8}$ as a root.

Solution: \boxed{x^2 + 6x + 17}


Andy and Alexa baked a total of 30 cookies. This time, Alexa ends up eating three times the number of cookies eaten by Andy. Determine the maximum number of cookies Andy could have eaten.


In right triangle $XYZ$, $\angle X = \angle Z$ and $XY = 10$. What is the area of $\triangle XYZ$?


Determine the ratio of $a$ to $b$ if: $\frac{9a-4b}{12a-3b} = \frac{4}{7}$.


What is the smallest odd number with five different prime factors?


Eight people can paint two identical houses in four hours. If their efficiency decreases by 20% when they are tired, how long would it take five people to paint the same two houses, assuming they are tired?

Solution: \boxed{8 \text{ hours}}


In the land of Eldoria, a valid license plate consists of two letters followed by three digits. However, the first letter of these license plates must be a vowel (A, E, I, O, U). How many valid license plates are possible in Eldoria?


In a right triangle $ABC$, $\tan A = \frac{3}{4}$, $AC = 6$, and $\angle C = 90^\circ$. Find the length of side $AB$.


What is the smallest integer \( k \) such that \( k > 1 \) and \( k \) has remainder \( 1 \) when divided by any of \( 23, \) \( 7, \) \( and \) \( 3? \)


Find the smallest composite number that has no prime factors less than 20.


Coach Grunt is preparing the 6-person starting lineup for his basketball team, the Grunters. There are 15 players on the team. Two of them, Ace and Zeppo, are league All-Stars and will definitely be in the starting lineup. However, one player, Xander, is recovering from an injury and cannot be in the lineup. How many different starting lineups are possible?


A domino set is modified such that each integer from 0 to 12 is paired with every other integer from 0 to 12 exactly once. A $\textit{double}$ is defined as a domino with the same number on both squares. Calculate the probability that a randomly selected domino from this set will be a $\textit{double}$. Express the answer as a common fraction.


The arithmetic mean of 15 scores is 85. When the highest and lowest scores are removed, the new mean becomes 90. If the highest of the 15 scores is 105, what is the lowest score?


A square is divided into four congruent rectangles, each rectangle is as shown. If the perimeter of each of these four rectangles is 40 inches, what is the perimeter of the square, in inches?

[asy]
draw((0,0)--(0,4)--(4,4)--(4,0)--cycle);
draw((1,0)--(1,4));
draw((2,0)--(2,4));
draw((3,0)--(3,4));
[/asy]

Solution: \boxed{64 \text{ inches}}


April has five different basil plants and four different tomato plants. In how many ways can she arrange the plants in a row if she puts all the tomato plants next to each other?


How many distinct sequences of four letters can be made from the letters in EQUALS if each sequence must begin with L and must include the letter 'S' somewhere in the sequence, while no letter can appear in the sequence more than once?


A valid license plate in New Xanadu consists of two letters followed by three digits and then one letter again. How many valid license plates are possible?


Find the units digit of the following within the indicated number base: $65_8 + 75_8$.


An infinite geometric series has a first term of $512$ and a sum of $3072$. What is its common ratio?


Allie and Betty play a game where they take turns rolling a standard die. If a player rolls $n$, she is awarded $f(n)$ points, where \[f(n) = \left\{
\begin{array}{cl}
9 & \text{ if } n \text{ is a multiple of 3}, \\
3 & \text{ if } n \text{ is only a multiple of 2}, \\
1 & \text{ otherwise.}
\end{array}
\right.\] This time, Allie rolls the die three times, getting a 6, 3, and 4. Betty rolls the die four times, getting 1, 2, 5, and 6. Calculate the product of Allie's total points and Betty's total points.


You recently purchased over 150 eggs. The eggs are arranged in containers, each ideally holding 12 eggs. This time, three containers each contain only 11 eggs, while all other containers are full with 12 eggs each. What is the smallest number of eggs you could have now?


Contractor Maria agreed to complete a job in 40 days. After 10 days, she found that the 10 people assigned to the work had already completed $\frac{2}{5}$ of the job. Assuming all workers maintain the same rate of productivity, what is the minimum number of workers she must retain to ensure the job is completed on time?


Yan is somewhere between his home and the school. He can either walk directly to school or walk back home and then ride his bicycle to school. He rides 5 times as fast as he walks, and both choices require the same amount of time. What is the ratio of Yan's distance from his home to his distance from the school?


If $x$ is a positive number such that \[\sqrt{12x}\cdot\sqrt{18x}\cdot\sqrt{6x}\cdot\sqrt{9x}=27,\] find all possible values for $x$.


As $n$ ranges over the positive integers, what is the maximum possible value for the greatest common divisor of $13n + 4$ and $7n + 3$?


The hypotenuse of a right triangle measures 12 inches and one angle is $30^{\circ}$. What is the area of the triangle in square inches?


The noon temperatures for ten consecutive days were recorded as $75^\circ$, $74^\circ$, $76^\circ$, $77^\circ$, $80^\circ$, $81^\circ$, $83^\circ$, $85^\circ$, $83^\circ$, and $85^\circ$ Fahrenheit. Calculate the mean noon temperature, in degrees Fahrenheit, for these ten days.


Define $X\star Y$ as $X\star Y = \frac{(X+Y)}{4}$. What is the value of $(3\star 11) \star 7$?


What is the area, in square units, of a triangle with vertices at $A(2, 3), B(9, 3), C(4, 12)$?

Solution: \boxed{31.5 \text{ square units}}


How many positive integers less than $201$ are multiples of either $5$ or $11$, but not both at once?


Express $3x^2 + 9x + 20$ in the form $a(x - h)^2 + k$. What is the value of $h$?


How many positive three-digit integers are there in which each of the first two digits is prime and the last digit is a non-prime odd number?


Compute: $\left(\frac{3}{5}\right)^4 \cdot \left(\frac{2}{9}\right)^{1/2}$.


I have six apples and eight oranges. If a fruit basket must contain at least one piece of fruit, how many different kinds of fruit baskets can I create? (The apples are identical, and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and the arrangement of the fruit in the basket does not matter.)


Compute $4 \begin{pmatrix} 3 \\ -9 \end{pmatrix} - 3 \begin{pmatrix} 2 \\ -8 \end{pmatrix} + 2 \begin{pmatrix} 1 \\ -6 \end{pmatrix}$.

Solution: \boxed{\begin{pmatrix} 8 \\ -24 \end{pmatrix}}


The diameter, in inches, of a sphere with three times the volume of a sphere of radius 6 inches can be expressed in the form $a\sqrt[3]{b}$ where $a$ and $b$ are positive integers and $b$ contains no perfect cube factors. Compute $a+b$.


What is the sum of all of the positive even factors of $720$?


The mean of the set of numbers $\{90,88,81,84,87,x\}$ is 85. Calculate the median of this set of six numbers.


Multiply the base-10 numbers 312 and 57, then write the product in base-7. What is the units digit of the base-7 representation?


Simplify $\sin (2x - y) \cos (3y) + \cos (2x - y) \sin (3y)$.

Solution: \boxed{\sin (2x + 2y)}


For how many different digits $n$ is the three-digit number $23n$ divisible by $n$?

Note: $23n$ refers to a three-digit number with the unit digit of $n,$ not the product of $23$ and $n.$


A relatively prime date is a date where the numbers of the month and the day are relatively prime. For example, February 17 is a relatively prime date because the greatest common factor of 2 and 17 is 1. Determine how many relatively prime dates are in February during a leap year.


How many even natural-number factors divisible by 5 does $n = 2^3 \cdot 5^2 \cdot 11^1$ have?


Rationalize the denominator of $\frac{4}{3\sqrt[4]{7}}$. The answer can be written in the form of $\frac{A\sqrt[4]{B}}{C}$, where $A$, $B$, and $C$ are integers, $C$ is positive, and $B$ is not divisible by the fourth power of any prime. Find $A+B+C$.


Consider a generalized domino set where each domino contains two squares, and each square can contain an integer from 0 to 11. Each integer from this range is paired with every integer (including itself) exactly once to form a complete set of dominos. What is the probability that a domino randomly selected from this set will be a $\textit{double}$ (containing the same integer on both squares)?


Compute $\cos 225^\circ$ and $\sin 225^\circ$.


Calculate the residue of $-963 + 100 \pmod{35}$. The answer should be an integer in the range $0,1,2,\ldots,33,34$.


A 6-foot by 12-foot floor is tiled with square tiles of size 1 foot by 1 foot. Each tile has a pattern consisting of four white quarter circles of radius 1/3 foot centered at each corner of the tile. The remaining portion of the tile is shaded. Calculate the total square feet of the floor that are shaded.

Solution: \boxed{72 - 8\pi \text{ square feet}}


An infinite geometric series has a first term of $400$ and sums up to $2500$. What is the common ratio?


If \( a, b, c, \) and \( d \) are real numbers satisfying:
\begin{align*}
a+b+c &= 5, \\
a+b+d &= 3, \\
a+c+d &= 8, \text{ and} \\
b+c+d &= 17,
\end{align*}
find \( ab + cd \).


A square is divided into four congruent rectangles, each rectangle having the longer side parallel to the sides of the square. If the perimeter of each rectangle is 40 inches, what is the perimeter of the square?

Solution: \boxed{64 \text{ inches}}


How many license plates consist of 3 letters followed by 3 digits, where the sequence is letter-digit-letter-digit-letter-digit, and each digit alternates between odd and even starting with an odd number?


Find the sum of the coefficients in the polynomial $3(x^8 - 2x^5 + x^3 - 6) - 5(2x^4 + 3x^2) + 2(x^6 - 5)$.


What is the constant term of the expansion of \( \left(5x + \frac{2}{5x}\right)^8 \)?


Find the ordered pair $(x,y)$ if
\begin{align*}
x + y &= (7-x) + (7-y),\\
x - y &= (x-2) + (y-2).
\end{align*}


Find the largest value of $c$ such that $4$ is in the range of $f(x) = x^2 + 5x + c$.


If the two roots of the quadratic $9x^2 + 5x + m$ are $\frac{-5 \pm i\sqrt{371}}{18}$, what is $m$?


A convex polyhedron $Q$ has $30$ vertices, $72$ edges, and $44$ faces consisting of $30$ triangular and $14$ quadrilateral faces. Calculate the number of space diagonals in $Q$.


Chandra has four bowls of different colors (red, blue, yellow, green) and five glasses, with an extra glass having no matching bowl color (purple). If she randomly chooses a bowl and a glass, how many different color pairings are possible considering that one glass color has no pairing dish color?


If $x$, $y$, and $z$ are positive with $xy=30$, $xz=60$, and $yz=90$, what is the value of $x+y+z$?

Solution: \boxed{x + y + z = 11\sqrt{5}}


Convert $\rm{B}2\rm{F}_{16}$ to a base 10 integer.


How many positive integers less than 300 are divisible by 4, 5, and 6?


What is the smallest positive four-digit integer equivalent to 3 mod 8?


Each box in the new grid diagram forms part of a larger layout composed of squares. Determine the total number of different-sized squares that can be traced using the lines in this expanded grid:

[asy]
unitsize(0.2 inch);
for (int i=0; i <= 6; ++i) {
  draw((i,0)--(i,6));
}
for (int j=0; j <= 6; ++j) {
  draw((0,j)--(6,j));
}
[/asy]


A line is parameterized by a parameter $t$, so that the vector on the line at $t = 1$ is $\begin{pmatrix} 2 \\ 7 \end{pmatrix},$ and the vector on the line at $t = 4$ is $\begin{pmatrix} 8 \\ -5 \end{pmatrix}.$ Find the vector on the line at $t = 5.$

Solution: \boxed{\begin{pmatrix} 10 \\ -9 \end{pmatrix}}


Jayden borrowed money from his friend to buy a bicycle. His friend agreed to let him repay the debt by gardening, with the payment structure as follows: first hour of gardening earns $\$2$, second hour $\$3$, third hour $\$4$, fourth hour $\$5$, fifth hour $\$6$, sixth hour $\$7$, and then it resets. If Jayden repays his debt by gardening for 47 hours, how much money did he borrow?


Three of the following test scores are Alex's and the other three are Jamie's: 80, 85, 90, 95, 100, 105. Alex's mean score is 85. What is Jamie's mean score?


Convert the point \((\rho,\theta,\phi) = \left( 15, \frac{5 \pi}{4}, \frac{\pi}{4} \right)\) in spherical coordinates to rectangular coordinates.

Solution: \boxed{\left(-\frac{15}{2}, -\frac{15}{2}, \frac{15\sqrt{2}}{2}\right)}


Olivia's Omelette Oasis offers a range of omelettes that include various fillings: cheese, ham, mushrooms, peppers, onions, tomatoes, spinach, and olives. A customer can choose omelette egg base ranging from one to four eggs, and any combination of fillings. How many different kinds of omelettes can be ordered?


The graphs of the function $f(x) = 2x + b$ and its inverse function $f^{-1}(x)$ intersect at the point $(-4, a)$. Given that $b$ and $a$ are both integers, what is the value of $a$?


If $\log (xy^2) = 2$ and $\log (x^3y) = 3$, what is $\log (xy)$?


Determine the range of the function
\[ g(x) = \frac{3(x + 5)(x - 4)}{x + 5}. \]

Solution: \boxed{(-\infty, -27) \cup (-27, \infty)}


The probability it will rain on Friday is $30\%$, the probability it will rain on Saturday is $50\%$, and the probability it will rain on Sunday is $40\%$. Assuming the probability that it rains on any day is independent of the other days except that if it rains on Saturday, the probability it will rain on Sunday increases to $70\%$. What is the probability it will rain on all three days, expressed as a percent?


Find all solutions to the inequality \[\frac{x^2}{(x+2)^2} \ge 0.\]

Solution: \boxed{(-\infty, -2) \cup (-2, \infty)}


In a sports conference, ten sports star players are seated. The players are from four teams: Cubs (3 players), Red Sox (3 players), Yankees (2 players), and Dodgers (2 players). If teammates insist on sitting together, how many ways can the ten sports stars be seated in a row?


If a polynomial of degree $15$ is divided by a polynomial $d(x)$, resulting in a quotient of degree $8$ and a remainder of $2x^3 + 5x^2 + x + 7$, what is $\deg d(x)$?