question stringlengths 16 2.12k | final_answer stringlengths 1 30 |
|---|---|
If $\tan(\theta) + \cot(\theta) = 2$, find $\tan^7(\theta) + \cot^7(\theta)$. | 2 |
In triangle \(ABC\), if \(\sin A : \sin B : \sin C = 4 : 5 : 6\), find the value of \(\frac{(\sin B)(\sin C)}{\sin A}\). | \(\frac{15\sqrt{7}}{32}\) |
In triangle ABC, angle C is a right angle. AD is the bisector of angle A such that CD = 4 and BD = 5. Find the area of triangle ABC. | 54 |
The positive real numbers $a, b, c, d$ satisfy the equality
$$a + bc + cd + db + \frac{1}{ab^2c^2d^2} = 18.$$
Find the maximum possible value of $a$. | 16 |
Find the sum of the cubes of the real roots of the equation \(x^3 - 2x^2 - x + 1 = 0\). | 11 |
Given a rhombus \(ABCD\), a point \(M\) is chosen on its side \(BC\). The lines passing through \(M\) and perpendicular to \(BD\) and \(AC\) meet line \(AD\) at points \(P\) and \(Q\) respectively. Suppose that the lines \(PB\), \(QC\), and \(AM\) have a common point. Find all possible values of the ratio \(\frac{BM}{M... | \(\frac{1}{2}\) |
Find the coefficient of $x^4$ in the expansion of $\left(1-\frac{3}{2}x-x^2\right)^5$. | \(-\frac{515}{16}\) |
Let \( P(x) \) be a polynomial with degree 3. Consider the polynomial \( Q(x) = (x^3 - 2x + 1 - P(x))(2x^3 - 5x^2 + 4 - P(x)) \). Assume that \( Q(x) \le 0 \) for all \( x \) and \( P(0) = 3 \). Calculate \( Q(-1) \). | \(-\frac{50}{9}\) |
Evaluate the integral \(\displaystyle\int_2^4\frac{\sqrt{\ln(9-x)}}{\sqrt{\ln(9-x)}-\sqrt{\ln(3+x)}}dx\). | 1 |
Let \( z \in \mathbb{C} \) be such that the points \( z^3 \), \( 2z^3 + z^2 \), \( 3z^3 + 3z^2 + z \), and \( 4z^3 + 6z^2 + 4z + 1 \) are the vertices of an inscribed quadrilateral in the complex plane. Find \( \text{Re}(z) \). | \(-\frac{1}{2}\) |
In how many ways is it possible to select six letters, including at least one vowel, from the letters of FLABELLIFORM? (Note: F is repeated twice, L is repeated three times; the word has 12 letters, 4 vowels, and 8 consonants.) | 296 |
Find the integers \( x \) and \( y \) such that \( x^2 + (x+1)^2 = y^4 \). | \( x = 119, y = 13 \) |
If \( n \) is a perfect square, solve the equation \((n+2)(n-3)(n+6) \equiv 0 \pmod{1989}\). | 1069 |
Find $\text{gcd}\left(\binom{2n}{1}, \binom{2n}{3}, \ldots, \binom{2n}{2n-1}\right)$. | $2^{v_2(2n)}$ |
Find the smallest positive integer $n$ with the property that the polynomial $x^4 - nx + 63$ can be written as a product of two nonconstant polynomials with integer coefficients. | 008 |
Two noncongruent triangles have the same area but different perimeters. Triangle A has side lengths of 8 cm, 8 cm, and 6 cm. Triangle B has two sides with lengths of 8 cm each. Find the length of the third side of Triangle B. Round your answer to the nearest hundredth of a cm. | 14.83 |
The average of $x+6$, $6x+2$, and $2x+7$ is $4x-7$. What is $x$? | 12 |
Define $A\star B$ as $A\star B = \frac{(A+B)}{3}$. What is the value of $(2\star 10) \star 5$? | 3 |
Determine the maximum value of \( n \) such that every subset of 10 elements from the set \( S = \{7, 8, 9, \ldots, n\} \) can form the side lengths of a triangle. | 418 |
In an equilateral triangle \(ABC\) with side length 2, let \(M\) and \(N\) be the midpoints of \(AB\) and \(AC\), respectively. The triangle is inscribed in a circle, and the line segment \(MN\) is extended to meet the circle at \(P\). Determine the length of the line segment \(NP\). | \(\frac{\sqrt{5} - 1}{2}\) |
Evaluate the limit \(\mathop {\lim }\limits_{n \to \infty } \frac{{a^{n + 1} + b^{n + 1}}}{{a^n + b^n}}\) where \(a, b > 0\). | \(\max \{a, b\}\) |
Compute $\sqrt{(31)(30)(29)(28)+1}.$ | 869 |
Given a line segment $AB=7$, with point $C$ on $AB$ such that $AC=5$. Two equilateral triangles are constructed on the same side of $AB$ with $AC$ and $BC$ as sides. Find the length of the segment connecting the circumcenters of these two equilateral triangles. | $\sqrt{13}$ |
Determine the smallest constant $\alpha > 1$ ($\alpha \in \mathbb{R}$) such that
\[
\frac{\alpha + \sin x}{\alpha + \sin y} \le e^{y - x}
\]
for all $x \le y$. | \(\sqrt{2}\) |
Find \( n \in \mathbb{N}, n \geq 2 \) such that the integer part of \( (n^2 - n)(\sqrt[n]{e} - 1) \) is equal to 2018. | 2019 |
Evaluate the integral $\int_0^4 x d([x] - x)$, where $[x]$ denotes the greatest integer not exceeding $x$. | 2 |
Let \( x, y, z \) be three different real numbers not equal to 0 that satisfy the equations \( x^2 - xy = y^2 - yz = z^2 - zx \). Find all the values of \( \frac{x}{z} + \frac{y}{x} + \frac{z}{y} \). | -3 |
Find all polynomials \( P(x) \) (in either \(\mathbb{C}[X]\) or \(\mathbb{R}[X]\)) such that the derivative \( P'(x) \) divides \( P(x) \). | \( P(x) = k(x-u)^n \) |
In ∆ABC, AB = AC = 115, AD = 38, and CF = 77 where D lies on AB and F lies on AC produced. DF intersects BC at E. Compute $\frac{[CEF]}{[DBE]}.$ | \(\frac{19}{96}\) |
Regular hexagon $ABCDEF$ is inscribed in a circle. $X$ is the midpoint of arc $\widehat{DC}$. Determine the exact value of $\frac{AX}{DX}$. | \(2+\sqrt{3}\) |
A car has a defected odometer that skips the digit $4$. For example, it goes from $39$ to $50$. If the odometer reads $2005$, what is the actual distance traveled by the car? | 1462 |
Compute the integral \(\int_{0}^{1} f(t) \, dt\) where \(f(t) = |x_1(t)| + |x_2(t)|\) and \(x_1(t)\) and \(x_2(t)\) are the roots of the quadratic equation \(x^2 + x + t = 0\). | \(\frac{17}{12}\) |
In a triangle \(ABC\), given that \(\frac{\cos A}{1 + \sin A} = \frac{\sin 2B}{1 + \cos 2B}\), find the minimum value of \(\frac{a^2 + b^2}{c^2}\). | \(4\sqrt{2} - 5\) |
A sequence $a_1,$ $a_2,$ $a_3,$ $\dots,$ is defined recursively by $a_1 = 1,$ $a_2 = 1,$ and for $k \ge 3,$
\[a_k = \frac{1}{3} a_{k - 1} + \frac{1}{4} a_{k - 2}.\]Evaluate $a_1 + a_2 + a_3 + \dotsb.$ | 4 |
Let \((a_1, a_2, \ldots, a_6)\) be a permutation of \((1, 2, 3, 4, 5, 6)\) such that the minimum number of transpositions needed to transform \((a_1, a_2, \ldots, a_6)\) into \((1, 2, 3, 4, 5, 6)\) is four. Find the number of such permutations. | 274 |
Given that 10 is the arithmetic mean of the set $\{6, 13, 18, 4, x\}$, what is the value of $x$? | 9 |
For \( a > 0 \), the cubic equation \( \frac{1}{2}(x^3 + 3x) = a \) has only one real solution \( x(a) > 0 \). For positive number \( R \), let \( a \) vary in the range \( 0 < a \leq R \). Denote \( N(R) \) as the number of \( a \) such that the corresponding real solution \( x(a) \) is an integer. Find the range of \... | \( 2 \leq R < 7 \) |
Two positive integers differ by $60$. The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? | 156 |
For any positive integer \( n \), let \( f(n) \) be the closest integer to \( \sqrt{n} \). Find the sum
\[
\sum_{n=1}^{\infty} \frac{2^{f(n)} + 2^{-f(n)}}{2^n}
\] | 3 |
Evaluate $tan\left(\frac{\pi}{11}\right)tan\left(\frac{2\pi}{11}\right)tan\left(\frac{3\pi}{11}\right)tan\left(\frac{4\pi}{11}\right)tan\left(\frac{5\pi}{11}\right)$. | \(\sqrt{11}\) |
Participation in the local soccer league this year is $10\%$ higher than last year. The number of males increased by $5\%$ and the number of females increased by $20\%$. What fraction of the soccer league is now female? | \(\frac{4}{11}\) |
Find the flux of the vector field \(\mathbf{V}(x,y,z) = 3xy^2 \mathbf{i} + 3x^2y \mathbf{j} + z^3 \mathbf{k}\) out of the unit sphere. | \(\frac{12\pi}{5}\) |
In a rectangle with dimensions $2$ and $50$, what is the maximal number of disjoint circles with diameter $1$ that can fit inside the rectangle? | 100 |
Consider the following 50-term sums:
\[ S = \frac{1}{1 \cdot 2} + \frac{1}{3 \cdot 4} + \cdots + \frac{1}{99 \cdot 100} \]
\[ T = \frac{1}{51 \cdot 100} + \frac{1}{52 \cdot 99} + \cdots + \frac{1}{99 \cdot 52} + \frac{1}{100 \cdot 51} \]
Express \( \frac{S}{T} \) as an irreducible fraction. | \(\frac{151}{2}\) |
Given \(0 < z \leq y \leq x \leq 3\) and the conditions:
\[ \frac{3}{xy} + \frac{2}{y^2} \geq 1 \]
\[ \frac{18}{x^2y} + \frac{4}{y^2z} + \frac{3}{z^2x} \geq 3 \]
find the maximum value of the expression:
\[ A = \frac{xyz}{2} + \frac{80x^3}{27} + \frac{18y^3}{8} \] | 101 |
Evaluate the double integral \(\iint_{R} [x + y] \, dx \, dy\) over the rectangle \(R = [0,1] \times [0,2]\), where \([ \cdot ]\) denotes the greatest integer function. | 2 |
Find the value of \( x + y \), given that
\[
\begin{cases}
x\sqrt{y} + y\sqrt{x} &= 182 \\
x\sqrt{x} + y\sqrt{y} &= 183
\end{cases}
\] | \(\frac{365}{9}\) |
In how many ways can the sum of the numbers on a 6-faced die, an 8-faced die, and a 10-faced die be a multiple of 3 when all three dice are rolled together? | 160 |
For how many values of $n$ is it possible to insert $+$ signs into a string of $n$ 7's so that the resulting expression has a value of 7000? | 108 |
The edges of $K_{2017}$ are each labeled with $1, 2,$ or $3$ such that any triangle has a sum of labels at least $5$. Determine the minimum possible average of all $\binom{2017}{2}$ labels. (Here $K_{2017}$ is the complete graph on 2017 vertices, with an edge between every pair of vertices.) | $2 - \frac{1}{2017}$ |
Determine the number of solutions to the equation $g^n = e$ in a cyclic group $G$ whose order is divisible by $n$. | \( n \) |
Let \( p(X) \in \mathbb{Z}_{p}[X] \) be an odd polynomial, where \( p \) is a prime and \( p \equiv 3 \pmod{4} \). How many solutions \((X, Y)\) does the equation \( Y^{2} = p(X) \) have in \( \mathbb{Z}_{p}^{2} \)? | \( p \) |
There are 10 horses, named Horse 1, Horse 2, ..., Horse 10, where Horse \( k \) runs one lap in exactly \( k \) minutes. At time 0, all horses are together at the starting point on a circular track. They start running in the same direction and keep running at their constant speeds. Let \( T > 0 \) be the least time, in... | 3 |
In a regular 18-sided polygon with center \( O \), let \( A, B, C, D \) be four consecutive vertices. Let \( P \) be the midpoint of \( AC \) and \( Q \) be the midpoint of \( DO \). Find the measure of \( \angle OPQ \) in degrees. | 30 |
Find the number of positive integers less than 10000 such that the sum of its digits is 25. | 348 |
Evaluate \( \csc^4\left(\frac{\pi}{9}\right) + \csc^4\left(\frac{4\pi}{9}\right) + \csc^4\left(\frac{7\pi}{9}\right) \). | 80 |
Given the Fibonacci sequence defined by \( F_0 = F_1 = 1 \) and \( F_{n+2} = F_{n+1} + F_n \) for \( n \geq 0 \), find the value of the sum \( S = \sum_{n=0}^{2019} \left\lfloor \frac{F_{n+3}}{F_n} \right\rfloor \). | 8080 |
Find all natural numbers \(a\) and \(b\) such that \(a^n + b^n = c^{n+1}\), where \(c\) and \(n\) are natural numbers. | \(a = 2, b = 2\) |
Given the sequence defined by \( a_1 = 1 \) and \( a_{n+1} = 2a_n + \sqrt{a_n^2 + 10^n} \), find the limit of \( \frac{a_{n+1}}{a_n} \). | \(\sqrt{10}\) |
Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$, where $n$ is a positive integer. | \( P(x) = 0 \) |
Find the limit \(\lim_{\varepsilon \rightarrow 0} \sum_{n=0}^{+\infty} \frac{(-1)^n}{1+n\epsilon}\). | \(\frac{1}{2}\) |
If \(A\), \(B\), and \(C\) are acute angles of triangle \(ABC\), determine the quadrant in which the point \((\cos B - \sin A, \sin B - \cos A)\) lies. | Second quadrant |
Maya lists all the positive divisors of \(2010^2\). She then randomly selects two distinct divisors from this list. Let \( p \) be the probability that exactly one of the selected divisors is a perfect square. The probability \( p \) can be expressed in the form \( \frac{m}{n} \), where \( m \) and \( n \) are relative... | 107 |
Let \( x, y, z \in [1,3] \) such that \( x^2 + y^2 + z^2 = 14 \). Find the minimum value of \( P = \left(1 - \frac{y}{x}\right)\left(\frac{z}{x} + 2\right) \). | -8 |
Each of five, standard, six-sided dice is rolled once. What is the probability that there is at least one pair but not a three-of-a-kind (that is, there are two dice showing the same value, but no three dice show the same value)? | \frac{25}{36} |
I'm thinking of a 6-digit number. The sum of the digits is 43. Only two of the following three statements about the number are true: (1) it's a square number, (2) it's a cube number, and (3) the number is under 500000. Identify the number. | 499849 |
Let \( a = 2^{2^{35}} + 1 \) and \( b = 2^{2^{21}} + 1 \). Find the greatest common divisor of \( a \) and \( b \), i.e., \( \gcd(a, b) \). | 1 |
Let \( AH \) be an altitude of \( \triangle ABC \), \( DH \) be an altitude of \( \triangle ABH \), \( EH \) be an altitude of \( \triangle AHC \), \( DF \) be an altitude of \( \triangle BDH \), and \( EG \) be an altitude of \( \triangle HEC \). If \( BF = 1 \), \( FH = 4 \), and \( HG = 5 \), find the area of \( \tr... | 75 |
Given three positive numbers \(a, b, c\) such that \(a + b + c = \sqrt{6051}\), find the maximum value of \( \frac{a}{\sqrt{a^2 + 2017}} + \frac{b}{\sqrt{b^2 + 2017}} + \frac{c}{\sqrt{c^2 + 2017}} \). | \(\frac{3}{2}\) |
Brent rolls a fair dodecahedral die with numbers $1,2,3,...,12$ on its faces. What is the expected number of digits in the number he obtains? Express your answer in a decimal. | 1.25 |
Given the polynomial \( p(x) = x^3 + ax^2 + bx + c \) with roots \( 2016 + 2017i\pi \), \( 2016 \), and \( 2017 \), find the number of non-real roots of the polynomial \( x^{12} + ax^8 + bx^4 + c \). | 8 |
Solve the system of equations:
\[x > y > z\]
\[x + y + z = 1\]
\[x^2 + y^2 + z^2 = 69\]
\[x^3 + y^3 + z^3 = 271\] | \(x=7, y=-2, z=-4\) |
Find all natural numbers \( n \) such that \( 5^{2n+1} - 5^n + 1 \) is a perfect square. | \( n = 1 \) |
In parallelogram $ABCD$ with $\angle B < 90^\circ$ and $AB < BC$, points $E$ and $F$ are on the circumference of the circle $\omega$ inscribing triangle $ABC$. The tangents to $\omega$ at points $E$ and $F$ pass through $D$, and $\angle EDA = \angle FDC$. Find $\angle ABC$. | 60^\circ |
Determine for which values of \( x \ge 1 \) the inequality \( f(x) > \log_e x \) holds, where \( f(x) = \int_{1}^{x} \frac{e^t}{t} \, dt \). | for all \( x > 1 \) |
Let \( f \) be a function defined for the non-negative integers such that:
a) \( f(n) = 0 \) if \( n = 2^j - 1 \) for some \( j \geq 0 \).
b) \( f(n+1) = f(n) - 1 \) otherwise.
Find \( f(2^{1990}) \). | -1 |
John’s birthday cake is a scrumptious cylinder of radius 6 inches and height 3 inches. If his friends cut the cake into 8 equal sectors, what is the total surface area of a piece of birthday cake? | \(\frac{27\pi}{2} + 36\) |
Azar, Carl, Jon, and Sergey are the four players left in a singles tennis tournament. They are randomly assigned opponents in the semifinal matches, and the winners of those matches play each other in the final match to determine the winner of the tournament. When Azar plays Carl, Azar will win the match with probabili... | 125 |
Solve the equation $\sin (9x)\sin (5x) = \sin (16x)\sin (4x)$ for $0 < x < \frac{\pi}{25}$. | \( x = \frac{\pi}{30} \) |
How many polynomials \( p(x) \) exist such that the coefficients of \( p(x) \) are a rearrangement of \(\{0, 1, 2, \ldots, \deg(p)\}\) and all of the roots of \( p(x) \) are rational? (Note that the leading coefficient of \( p(x) \) cannot be zero.) | 5 |
Find all positive integers \( x \geq 1 \) such that \( 1 + 2^{n+1} + 4^{n+1} \) is divisible by \( 1 + 2^n + 4^n \). | \( n = 1 \) |
If \( n \) is an odd positive integer and \( m \) and \( n \) are positive integers, find the number of solutions to the equation \(\frac{1}{m} + \frac{4}{n} = \frac{1}{12}\). | 3 |
Let \( a, b, c, d, e \), and \( f \) be 6 variables such that:
- \( ab = 2 \)
- \( bc = 3 \)
- \( cd = 4 \)
- \( de = 5 \)
- \( ef = 6 \)
Compute all possible values of \( fa \). | \(\frac{16}{5}\) |
Given \(a, b, c, d \ge 0\) and \(2c+3d, 2d+3a, 2a+3b, 2b+3c > 0\), find the minimum value of the expression
\[
\frac{a+2b}{2c+3d} + \frac{b+2c}{2d+3a} + \frac{c+2d}{2a+3b} + \frac{d+2a}{2b+3c}.
\] | \(\frac{7}{3}\) |
Suppose a certain brand of washing machine has an average usage of 11.5 years with a standard deviation of 2.4 years. For a sample of \( n = 45 \) machines, what is the margin of error, \( E \), for a 90% confidence interval? | \( E \approx 0.589 \) |
Given the equation \( x^{(x-1)^2} = 2x + 1 \), find the value of \( \frac{x-1}{x} \). | \( 2 - \sqrt{2} \) |
Evaluate the integral \[ \int_{0}^{1} \dfrac{\log (1+x)}{1+x^{2}} \, dx \] | \(\frac{\pi}{8} \log 2\) |
Denote $D=\{(x,y,z)\in\mathbb{R}^{3}|x>0,y>0,z>0;xyz=1\}$. Let the function $f: D\to\mathbb{R}$ be defined by $f(x,y,z)=\frac{xy+yz+zx}{x+y+z}$. Find the set $\{f(x,y,z) \mid (x,y,z)\in D\}$. | $\mathbb{R}_{+}$ |
Find the last three digits of $1998^{1999^{2000}}$. | 248 |
Determine the integral values of \( a \) for which the quadratic expression \( ax^2 + (a-2)x - 2 < 0 \) holds for exactly two integral values of \( x \). | \( 1 \) |
How many ways are there to color the vertices of a square with six colors if rotations are considered the same? | 336 |
Given a 4-digit number with distinct digits, if you take the largest possible four-digit number that can be made by rearranging the four digits and subtract the smallest possible four-digit number that can be made by rearranging the four digits, you get the number with its digits in reverse order. Find the number. | 4716 |
A standard deck of playing cards with 26 red and 26 black cards is split into two non-empty piles. In pile A, there are four times as many black cards as red cards. In pile B, the number of red cards is an integer multiple of the number of black cards. How many red cards are in pile B? | 20 |
In how many ways can one choose 6 candies from 8 available brands? | 1716 |
If the equation $x^4 - 2ax^2 - x + a^2 - a = 0$ has exactly two distinct roots, find the range of $a$. | \((-0.25, 0.75)\) |
Find the maximum value of $k$ such that
\[
\sum_{cyc}a^2 + k\sum_{cyc}ab \geq (1+k)\sqrt{3\sum_{cyc}a^2b^2}
\]
for all $a, b, c > 0$. | \(\frac{\sqrt{3}-1}{2}\) |
Given the conditions for positive integers \(a\), \(b\), and \(c\):
- \(b^2 \geq 4ac + 1\)
- \(b \geq 2a + 1\)
- \(b \leq 4a - 1\)
- \(b \leq a + c - 1\)
- \(2b \leq 4a + c - 1\)
Find the smallest possible value of \(a\). | 5 |
How many ten-digit numbers composed only of the digits 1, 2, or 3 exist, in which any two neighboring digits differ by 1? | 64 |
Find the number of $4 \times 4$ arrays whose entries are from the set $\{0, 1, 2, 3\}$ and such that the sum of the numbers in each of the four rows and in each of the four columns is divisible by $4$. | \(4^9\) |
In Mr. Abraham's class, $10$ of the $15$ students received an $A$ on the latest exam. If the same ratio of students received an $A$ on Mrs. Berkeley's latest exam, and if Mrs. Berkeley has $24$ students total, how many students in Mrs. Berkeley's class received an $A$? | 16 |
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