prompt stringlengths 19 3.19k | ground_truth stringlengths 1 23 |
|---|---|
Let $g(x) = 1 + 2x + 3x^2 + 4x^3 + \ldots$. Find the coefficient of $x^{2015}$ in $f(x) = \frac{g(x)}{1-x}$. | 2033136 |
Define a sequence $\{a_n\}_{n \geq 1}$ recursively by $a_1=1$, $a_2=2$, and for all integers $n \geq 2$, $a_{n+1}=(n+1)^{a_n}$. Determine the number of integers $k$ between $2$ and $2020$, inclusive, such that $k+1$ divides $a_k - 1$. | 1009 |
How many ordered triples $(a, b, c)$, where $a$, $b$, and $c$ are from the set $\{ 1, 2, 3, \dots, 17 \}$, satisfy the equation: \[ a^3 + b^3 + c^3 + 2abc = a^2b + a^2c + b^2c + ab^2 + ac^2 + bc^2 \] | 408 |
In the triangle $ABC$, we have $\angle ABC = 100^\circ$, $\angle ACB = 65^\circ$, $M \in AB$, $N \in AC$, and $\angle MCB = 55^\circ$, $\angle NBC = 80^\circ$. Find $\angle NMC$. | 25 |
In quadrilateral $ABCD$, $AB \parallel CD$ and $BC \perp AB$. Lines $AC$ and $BD$ intersect at $E$. If $AB = 20$, $BC = 2016$, and $CD = 16$, find the area of $\triangle BCE$. | 8960 |
Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class. Each year at the college, one-third of the freshman class drops out of school, 40 students in the sophomore class drop out of school, and ... | 1185 |
Given that \[\sum_{x=1}^{70} \sum_{y=1}^{70} \frac{x^{y}}{y} = \frac{m}{67!}\] for some positive integer $m$, find $m \pmod{71}$. | 12 |
On a $3 \times 3$ board, the numbers from $1$ to $9$ are written in some order without repeating. We say that the arrangement obtained is Isthmian if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements. Note: Two arrangements are considered equal if one... | 720 |
39 students participated in a math competition. The exam consisted of 6 problems, and each problem was worth 1 point for a correct solution and 0 points for an incorrect solution. For any 3 students, there is at most 1 problem that was not solved by any of the three. Let $B$ be the sum of all of the scores of the 39 st... | 165 |
Suppose $N$ is a $6$-digit number with base-$10$ representation $\underline{a}\,\underline{b}\,\underline{c}\,\underline{d}\,\underline{e}\,\underline{f}$. If $N$ is $\frac{6}{7}$ of the number with base-$10$ representation $\underline{d}\,\underline{e}\,\underline{f}\,\underline{a}\,\underline{b}\,\underline{c}$, find... | 461538 |
Let $a, b$ be integers such that all the roots of the equation $(x^2+ax+20)(x^2+17x+b) = 0$ are negative integers. What is the smallest possible value of $a + b$? | 25 |
Let $S$ be a subset of the natural numbers such that $0 \in S$, and for all $n \in \mathbb{N}$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$? | 47 |
Call an integer $k$ debatable if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.) | 17 |
The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer. | 15 |
For a permutation $\pi$ of the integers from 1 to 10, define \[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \] where $\pi (i)$ denotes the $i$th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. ... | 40320 |
One hundred billion light years from Earth is planet Glorp. The inhabitants of Glorp are intelligent, uniform, amorphous beings with constant density which can modify their shape in any way, and reproduce by splitting. Suppose a Glorpian has somehow formed itself into a spinning cylinder in a frictionless environment. ... | 1024 |
There is a rectangle $ABCD$ such that $AB = 12$ and $BC = 7$. Points $E$ and $F$ lie on sides $AB$ and $CD$, respectively, such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Let $X$ be the intersection of lines $AF$ and $DE$. What is the area of pentagon $BCFXE$? | 47 |
How many ways are there to choose (not necessarily distinct) integers $a, b, c$ from the set $\{1, 2, 3, 4\}$ such that $a^{(b^c)}$ is divisible by $4$? | 28 |
Find $a + b + c$, where $a$, $b$, and $c$ are the hundreds, tens, and units digits of the six-digit number $123abc$, which is a multiple of $990$. | 12 |
David, when submitting a problem for CMIMC, wrote his answer as $100\frac{x}{y}$, where $x$ and $y$ are two positive integers with $x < y$. Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double ... | 299 |
The sweet shop named "Olympiad" sells boxes containing either 6, 9, or 20 chocolates. Groups of students from a nearby school collect money to buy chocolates so that each student receives one. They buy a box and distribute the chocolates accordingly. For example, students can form groups of 15 (using one box of 6 and o... | 43 |
Let $a > b$ be positive integers. Compute the smallest possible integer value of \( \frac{a! + 1}{b! + 1} \). | 103 |
Let $n$ be a positive integer with $k$ digits. A number $m$ is called an \( alero \) of \( n \) if there exist distinct digits \( a_1, a_2, \dotsb, a_k \), all different from each other and from zero, such that \( m \) is obtained by adding the digit \( a_i \) to the \( i \)-th digit of \( n \), and no sum exceeds 9. F... | 10120 |
Consider a right-angled triangle $ABC$ with $\angle C = 90^\circ$. Suppose that the hypotenuse $AB$ is divided into four equal parts by the points $D, E, F$, such that $AD = DE = EF = FB$. If $CD^2 + CE^2 + CF^2 = 350$, find the length of $AB$. | 20 |
In a tournament, each player played exactly one game against each of the other players. In each game, the winner was awarded 1 point, the loser got 0 points, and each player earned \(\frac{1}{2}\) point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by... | 25 |
This sequence lists the perfect squares in increasing order: $0, 1, 4, 9, 16, \cdots, a, 10^8, b, \cdots$. Determine the value of $b-a$. | 40000 |
In a mathematical competition, $n = 10,000$ contestants participate. During the final party, the process of distributing a cake is as follows: 1. The first contestant takes $\frac{1}{n}$ of the cake. 2. The second contestant takes $\frac{2}{n}$ of the remaining cake. 3. The third contestant takes $\frac{3}{n}$ of what ... | 100 |
An ant starts at the origin of the Cartesian coordinate plane. Each minute, it moves randomly one unit in one of the directions: up, down, left, or right, with all four directions being equally likely. Its direction each minute is independent of its direction in any previous minutes. It stops when it reaches a point $(... | 3907 |
A 10-digit arrangement $0,1,2,3,4,5,6,7,8,9$ is called beautiful if: 1. When read from left to right, $0,1,2,3,4$ form an increasing sequence. 2. $5,6,7,8,9$ form a decreasing sequence. 3. $0$ is not the leftmost digit. For example, $9807123654$ is a beautiful arrangement. Determine the number of beautiful arrangements... | 126 |
How many $6$-tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a > b > c > d > e > f$ and $a + f = b + e = c + d = 30$ are simultaneously true? | 364 |
$A_1, A_2, \cdots, A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side. | 9 |
In a country consisting of $2015$ cities, there is exactly one direct round flight between any two cities, operated by some air company. Find the minimal possible number of air companies required if direct flights between any three cities are operated by three different air companies. | 2015 |
A positive integer $n > 1$ is called beautiful if $n$ can be written in one and only one way as $n = a_1 + a_2 + \cdots + a_k = a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k > 1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. For example, 6 is beautiful since $6 = 3 \cdot 2 \cdot 1 = ... | 95 |
In a drawer, there are at most $2009$ balls. Some of these balls are white, and the rest are blue, randomly distributed. If two balls are taken at the same time, the probability that both balls are either blue or both white is $\frac{1}{2}$. Determine the maximum number of white balls in the drawer such that the probab... | 990 |
$f(x)$ is a quartic polynomial with a leading coefficient $1$ where $f(2)=4$, $f(3)=9$, $f(4)=16$, and $f(5)=25$. Compute $f(8)$. | 424 |
In a country, there are $28$ cities, and between some cities, there are two-way flights. Each city has exactly one airport, which can be either small, medium, or big. For every route that: 1. Contains more than two cities, 2. Does not repeat any city, 3. Ends where it begins, It must include all types of airports. What... | 53 |
Alex the Kat has written $61$ problems for a math contest, and there are a total of $187$ problems submitted. How many more problems does he need to write and submit before he has written half of the total problems? | 65 |
Find the greatest possible sum of integers $a$ and $b$ such that $\frac{2021!}{20^a \cdot 21^b}$ is a positive integer. | 837 |
Compute the number of ordered triples \((x, y, z)\) of integers satisfying \[x^2 + y^2 + z^2 = 9.\] | 30 |
Evaluate $\lim_{n \to \infty} n^2 \int_{-\frac{1}{n}}^{\frac{1}{n}} (2005\sin x + 2006\cos x)|x| \, dx.$ | 2006 |
How many non-congruent isosceles triangles, including equilateral triangles, have positive integer side lengths and a perimeter less than 20? | 41 |
Show that there exists an infinite arithmetic progression of natural numbers such that the first term is $16$ and the number of positive divisors of each term is divisible by $5$. Of all such sequences, find the one with the smallest possible common difference. | 32 |
In triangle $ABC$, $AB = 52$, $BC = 34$, and $CA = 50$. The side $BC$ is divided into $n$ equal segments by placing $n-1$ new points. Among these points are the feet of the altitude, median, and angle bisector from vertex $A$. What is the smallest possible value of $n$? | 102 |
Find the numerator of \(\frac{1010\overbrace{11 \ldots 11}^{2011 \text{ ones}}0101}{1100\underbrace{11 \ldots 11}_{2011\text{ ones}}0011}\) when reduced. | 9091 |
Let $ABCD$ be an isosceles trapezoid with $AD \parallel BC$ and $BC > AD$ such that the distance between the incenters of $\triangle ABC$ and $\triangle DBC$ is $16$. If the perimeters of $ABCD$ and $ABC$ are $120$ and $114$ respectively, then the area of $ABCD$ can be written as $m\sqrt{n}$, where $m$ and $n$ are posi... | 7114 |
Anders is solving a math problem, and he encounters the expression $\sqrt{15!}$. He attempts to simplify this radical as $a\sqrt{b}$ where $a$ and $b$ are positive integers. The sum of all possible values of $ab$ can be expressed in the form $q \cdot 15!$ for some rational number $q$. Find $q$. | 4 |
Two quadratic polynomials $A(x)$ and $B(x)$ have a leading term of $x^2$. For some real numbers $a$ and $b$, the roots of $A(x)$ are $1$ and $a$, and the roots of $B(x)$ are $6$ and $b$. If the roots of $A(x) + B(x)$ are $a + 3$ and $b + \frac{1}{2}$, then compute $a^2 + b^2$. | 3 |
Ted loves creating haikus (Japanese three-line poems with $5$, $7$, $5$ syllables each), but his vocabulary is rather limited. For integers $1 \le n \le 7$, he knows $2n$ words with $n$ syllables. Words cannot cross between lines but may be repeated. Compute the number of distinct haikus Ted can make, and find the rema... | 28 |
Which is the smallest natural number $n$ such that $n!$ has at least 2013 divisors? | 14 |
Amandine and Brennon play a turn-based game, with Amandine starting. On their turn, a player must select a positive integer which cannot be represented as a sum of multiples of any of the previously selected numbers. For example, if $3, 5$ have been selected so far, only $1, 2, 4, 7$ are available to be picked; if only... | 192 |
There is a sequence of numbers $+1$ and $-1$ of length $n$. It is known that the sum of every $10$ neighboring numbers in the sequence is $0$, and that the sum of every $12$ neighboring numbers in the sequence is not zero. What is the maximal value of $n$? | 15 |
A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime, while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers? | 60 |
Let $ABCD$ be a square with $AB=6$. A point $P$ in the interior is $2$ units away from side $BC$ and $3$ units away from side $CD$. What is the distance from $P$ to $A$? | 5 |
Let $\alpha ,\ \beta$ be the distinct positive roots of the equation $2x=\tan x$. Evaluate the following definite integral: \[ \int_{0}^{1}\sin(\alpha x)\sin(\beta x)\, dx \] | 0 |
The sequence $\{a_{n}\}_{n \geqslant 0}$ is defined by the initial conditions $a_{0}=1$ and $a_{1}=-4$, with the recurrence relation $a_{n+2}=-4a_{n+1}-7a_{n}$ for $n \geqslant 0$. Find the number of positive integer divisors of $a^2_{50} - a_{49}a_{51}$. | 51 |
Andrew writes down all of the prime numbers less than $50$. How many times does he write the digit $2$? | 3 |
A knight is placed at the origin of the Cartesian plane. Each turn, the knight moves in an L-shape (2 units parallel to one axis and 1 unit parallel to the other) to one of eight possible locations, chosen at random. After 2016 such turns, what is the expected value of the square of the distance of the knight from the ... | 10080 |
Given a rectangle $ABCD$, where the segment $MA$ is perpendicular to the plane $ABC$. The lengths are as follows: $MB = 15$, $MC = 24$, and $MD = 20$. Determine the length of $MA$. | 7 |
For $a > 0$, find the minimum value of $\int_{0}^{\frac{1}{a}}(a^{3}+4x-a^{5}x^{2})e^{ax}\ dx.$ | 4 |
Let $(x, y, z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \[ \begin{align*} x + y^2 + z^4 &= 0, \\ y + z^2 + x^4 &= 0, \\ z + x^2 + y^4 &= 0. \end{align*} \] If $m$ is the minimum possible value of $\lfloor x^3 + y^3 + z^3 \rfloor$, find the modulo $2007$ residue of $m$. | 2006 |
Define a sequence $a_i$ as follows: \( a_1 = 181 \) and for \( i \ge 2 \), \( a_i = a_{i-1}^2 - 1 \) if \( a_{i-1} \) is odd and \( a_i = \frac{a_{i-1}}{2} \) if \( a_{i-1} \) is even. Find the least \( i \) such that \( a_i = 0 \). | 105 |
A $20 \times 20 \times 20$ block is cut up into 8000 non-overlapping unit cubes, and a number is assigned to each cube. It is known that in each column of 20 cubes parallel to any edge of the block, the sum of their numbers is equal to 1. The number assigned to one of the unit cubes is 10. Three $1 \times 20 \times 20$... | 333 |
For what digit $d$ is the base $9$ numeral $7d35_9$ divisible by $8$? | 1 |
A real number $x$ satisfies $2 + \log_{25} x + \log_8 5 = 0$. Find $\log_2 x - (\log_8 5)^3 - (\log_{25} x)^3$. | 8 |
Let $\sigma_1 : \mathbb{N} \to \mathbb{N}$ be a function that takes a natural number $n$, and returns the sum of the positive integer divisors of $n$. For example, $\sigma_1(6) = 1 + 2 + 3 + 6 = 12$. What is the largest number $n$ such that $\sigma_1(n) = 1854$? | 1234 |
Suppose $a$, $b$, $c$, and $d$ are non-negative integers such that \[(a+b+c+d)(a^2+b^2+c^2+d^2)^2=2023.\] Find $a^3+b^3+c^3+d^3$. | 43 |
A cup with a volume of $8$ fluid ounces is filled at a rate of $0.5$ ounces per second. However, a hole at the bottom of the cup also drains it at a rate of $0.3$ ounces per second. Once the cup is full, how many ounces of water will have drained out of the cup? | 12 |
Evaluate the expression: \[ \frac{(2 + 2)^2}{2^2} \cdot \frac{(3 + 3 + 3 + 3)^3}{(3 + 3 + 3)^3} \cdot \frac{(6 + 6 + 6 + 6 + 6 + 6)^6}{(6 + 6 + 6 + 6)^6} \] | 108 |
Given a table with numbers $1, 2, \ldots, 18$, determine the minimal number of numbers that must be erased so that the sum of any two remaining numbers is not a perfect square of a positive integer. | 9 |
Let $\triangle ABC$ be an equilateral triangle with side length $s$ and $P$ a point in the interior of this triangle. Suppose that $PA$, $PB$, and $PC$ are the roots of the polynomial $t^3-18t^2+91t-89$. Then $s^2$ can be written in the form $m+\sqrt{n}$ where $m$ and $n$ are positive integers. Find $100m+n$. | 7208 |
Let $\{a_n\}_{n\geq 1}$ be an arithmetic sequence and $\{g_n\}_{n\geq 1}$ be a geometric sequence such that the first four terms of $\{a_n+g_n\}$ are $0$, $0$, $1$, and $0$, in that order. What is the $10$th term of $\{a_n+g_n\}$? | -54 |
Each of the squares in a $2 \times 2018$ grid is to be colored black or white such that in any $2 \times 2$ block, at least one of the four squares is white. Let $P$ be the number of ways of coloring the grid. Find the largest $k$ such that $3^k$ divides $P$. | 1009 |
A quadrilateral is given, where the lengths of two sides are $1$ and $4$. Additionally, a diagonal of length $2$ divides the quadrilateral into two isosceles triangles. Determine the perimeter of this quadrilateral. | 11 |
What is the maximum possible value for the sum of the squares of the roots of \(x^4 + ax^3 + bx^2 + cx + d\) where \(a\), \(b\), \(c\), and \(d\) are \(2\), \(0\), \(1\), and \(7\) in some order? | 49 |
Let $Q$ be a set of permutations of $1, 2, \ldots, 100$ such that for all $1 \leq a, b \leq 100$, $a$ can be found to the left of $b$ and adjacent to $b$ in at most one permutation in $Q$. Find the largest possible number of elements in $Q$. | 100 |
The non-zero natural number $n$ is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223 - \frac{3}{2} \cdot n$. Find the quotient of the division. | 14 |
Let $a = 3^{\frac{1}{223}} + 1$ and for all $n \geq 3$, let: \[ f(n) = \binom{n}{0} a^{n-1} - \binom{n}{1} a^{n-2} + \binom{n}{2} a^{n-3} - \ldots + (-1)^{n-1} \binom{n}{n-1} a^0. \] Find $f(2007) + f(2008).$ | 19683 |
Suppose that $f(x) = \frac{x}{x^2 - 2x + 2}$ and $g(x_1, x_2, \ldots , x_7) = f(x_1) + f(x_2) + \ldots + f(x_7)$. If $x_1, x_2, \ldots, x_7$ are non-negative real numbers with sum $5$, determine for how many tuples $(x_1, x_2, \ldots , x_7)$ does $g(x_1, x_2, \ldots , x_7)$ obtain its maximal value. | 1 |
How many positive integers $N$ less than $10^{1000}$ are such that $N$ has $x$ digits when written in base ten and $\frac{1}{N}$ has $x$ digits after the decimal point when written in base ten? For example, 20 has two digits and $\frac{1}{20} = 0.05$ has two digits after the decimal point, so $20$ is a valid $N$. | 3997 |
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences such that $a_i b_i - a_i - b_i = 0$ and $a_{i+1} = \frac{2-a_i b_i}{1-b_i}$ for all $i \ge 1$. If $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$, then what is $b_{6}$? | 257 |
Lynnelle took 10 tests in her math class at Stanford. Her score on each test was an integer from 0 through 100. She noticed that, for every four consecutive tests, her average score on those four tests was at most 47.5. What is the largest possible average score she could have on all 10 tests? | 57 |
We are given five watches which can be wound forward. What is the smallest sum of winding intervals that allows us to set them to the same time, regardless of their initial settings? | 24 |
The vertices of a regular nonagon are colored such that: 1. Adjacent vertices are different colors. 2. If 3 vertices form an equilateral triangle, they are all different colors. Let $m$ be the minimum number of colors needed for a valid coloring, and $n$ be the total number of colorings using $m$ colors. Determine $mn$... | 54 |
A subset of $\{1, 2, 3, \ldots, 2017, 2018\}$ has the property that none of its members are 5 times another. What is the maximum number of elements that such a subset could have? | 1682 |
Let $x$, $y$, and $z$ be real numbers such that: \[ 12x - 9y^2 = 7 \\ 6y - 9z^2 = -2 \\ 12z - 9x^2 = 4 \] Find the value of $6x^2 + 9y^2 + 12z^2$. | 9 |
The figure shows a convex polygon with nine vertices. The six diagonals drawn dissect the polygon into the following seven triangles: - $P_{0}P_{1}P_{3}$ - $P_{0}P_{3}P_{6}$ - $P_{0}P_{6}P_{7}$ - $P_{0}P_{7}P_{8}$ - $P_{1}P_{2}P_{3}$ - $P_{3}P_{4}P_{6}$ - $P_{4}P_{5}P_{6}$ In how many ways can these triangles be labele... | 1 |
Two circles with radius $2$ and radius $4$ have a common center at $P$. Points $A$, $B$, and $C$ on the larger circle are the vertices of an equilateral triangle. Point $D$ is the intersection of the smaller circle and the line segment $PB$. Find the square of the area of triangle $ADC$. | 192 |
Mark writes the expression $\sqrt{d}$ for each positive divisor $d$ of $8!$ on the board. Rishabh simplifies each expression to the form $a \sqrt{b}$, where $a$ and $b$ are integers such that $b$ is not divisible by the square of a prime number. For example, $\sqrt{20}$, $\sqrt{16}$, and $\sqrt{6}$ simplify to $2\sqrt{... | 3480 |
Karl likes the number $17$. His favorite polynomials are monic quadratics with integer coefficients such that $17$ is a root of the quadratic and the roots differ by no more than $17$. Compute the sum of the coefficients of all of Karl's favorite polynomials. (A monic quadratic is a quadratic polynomial whose $x^2$ ter... | 8960 |
Adi the Baller is shooting hoops, and makes a shot with probability $p$. He keeps shooting hoops until he misses. The value of $p$ that maximizes the chance that he makes between 35 and 69 (inclusive) buckets can be expressed as $\frac{1}{\sqrt[b]{a}}$ for a prime $a$ and positive integer $b$. Find $a+b$. | 37 |
How many pairs $(m, n)$ of positive integers with $m < n$ fulfill the equation $\frac{3}{2008} = \frac{1}{m} + \frac{1}{n}$? | 5 |
What is the number of ordered pairs $(A,B)$ where $A$ and $B$ are subsets of $\{1,2,..., 5\}$ such that neither $A \subseteq B$ nor $B \subseteq A$? | 570 |
Compute the base 10 value of $14641_{99}$. | 100000000 |
When two distinct digits are randomly chosen in $N=123456789$ and their places are swapped, one gets a new number $N'$ (for example, if 2 and 4 are swapped, then $N'=143256789$). The expected value of $N'$ is equal to $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute the remainder when $m... | 962963 |
Find the largest value of $a^b$ such that the positive integers $a,b > 1$ satisfy: \[ a^b b^a + a^b + b^a = 5329 \] | 81 |
Find the smallest positive integer $n$ for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit $0$ when written in base ten. | 4 |
There are real numbers $a, b, c, d$ such that for all $(x, y)$ satisfying $6y^2 = 2x^3 + 3x^2 + x$, if $x_1 = ax + b$ and $y_1 = cy + d$, then $y_1^2 = x_1^3 - 36x_1$. What is $a + b + c + d$? | 90 |
Find the greatest integer less than the number $$1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{1000000}}$$ | 1998 |
Find the smallest positive $\alpha$ (in degrees) for which all the numbers $\cos{\alpha}, \cos{2\alpha}, \ldots, \cos{2^n\alpha}, \ldots$ are negative. | 120 |
**Problem Statement:** One hundred friends, including Alice and Bob, live in several cities. Alice has determined the distance from her city to the city of each of the other 99 friends and totaled these 99 numbers. Alice’s total is \(1000\) km. Bob similarly totaled his distances to everyone else. What is the largest t... | 99000 |
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