problem stringlengths 10 5.15k | answer stringlengths 0 1.23k |
|---|---|
Each letter represents a non-zero digit. What is the value of $t?$ \begin{align*}
c + o &= u \\
u + n &= t \\
t + c &= s \\
o + n + s &= 12
\end{align*} | 6 |
What is the value of $x^2+y^2-z^2+2xy$ when $x=-3$, $y=5$, and $z=-4$? | -12 |
Around a circular table, there are 18 girls seated, 11 dressed in blue and 7 dressed in red. Each girl is asked if the girl to her right is dressed in blue, and each one responds with either yes or no. It is known that a girl tells the truth only when both of her neighbors, the one to her right and the one to her left,... | 11 |
Given a rectangle with dimensions \(100 \times 101\), divided by grid lines into unit squares. Find the number of segments into which the grid lines divide its diagonal. | 200 |
Simplify first, then evaluate: \\((x+2)^{2}-4x(x+1)\\), where \\(x= \sqrt {2}\\). | -2 |
In the plane Cartesian coordinate system \( xO y \), the circle \( \Omega \) and the parabola \( \Gamma: y^{2} = 4x \) share exactly one common point, and the circle \( \Omega \) is tangent to the x-axis at the focus \( F \) of \( \Gamma \). Find the radius of the circle \( \Omega \). | \frac{4 \sqrt{3}}{9} |
Find the smallest positive integer $k$ such that $
z^{10} + z^9 + z^6+z^5+z^4+z+1
$ divides $z^k-1$. | 84 |
Calculate the value of the following expression and find angle $\theta$ if the number can be expressed as $r e^{i \theta}$, where $0 \le \theta < 2\pi$:
\[ e^{11\pi i/60} + e^{21\pi i/60} + e^{31 \pi i/60} + e^{41\pi i /60} + e^{51 \pi i /60} \] | \frac{31\pi}{60} |
A cookfire burns three logs every hour. It was built with six logs to start. If it gets two more logs added to it at the end of every hour, how many logs will be left after 3 hours? | After 3 hours, it will have 3 * 2 = <<3*2=6>>6 logs added to it.
After 3 hours, it will have burned 3 * 3 = <<3*3=9>>9 logs.
After 3 hours, the cookfire will have 6 + 6 - 9 = <<6+6-9=3>>3 logs left.
#### 3 |
Troy had 300 straws. He fed 3/5 of the straws to the adult pigs and an equal number of straws to the piglets. If there were 20 piglets, how many straws did each piglet eat? | The adult pigs ate 3/5*300 = <<3/5*300=180>>180 straws.
The total number of straws that the piglets shared is 300-180 = <<300-180=120>>120
Since there were 20 piglets, each piglet ate 120/20 = <<120/20=6>>6 straws
#### 6 |
Suppose the graph of $y=f(x)$ includes the points $(1,5),$ $(2,3),$ and $(3,1)$.
Based only on this information, there are two points that must be on the graph of $y=f(f(x))$. If we call those points $(a,b)$ and $(c,d),$ what is $ab+cd$? | 17 |
Into how many parts can a plane be divided by four lines? Consider all possible cases and make a drawing for each case. | 11 |
Given the ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \left( a > b > 0 \right)$ has an eccentricity of $\frac{\sqrt{2}}{2}$, and the distance from one endpoint of the minor axis to the right focus is $\sqrt{2}$. The line $y = x + m$ intersects the ellipse $C$ at points $A$ and $B$.
$(1)$ Find the equation of the... | \frac{\sqrt{2}}{2} |
Simplify the product \[\frac{9}{3}\cdot\frac{15}{9}\cdot\frac{21}{15}\dotsm\frac{3n+6}{3n}\dotsm\frac{3003}{2997}.\] | 1001 |
\begin{align*}
4a + 2b + 5c + 8d &= 67 \\
4(d+c) &= b \\
2b + 3c &= a \\
c + 1 &= d \\
\end{align*}
Given the above system of equations, find \(a \cdot b \cdot c \cdot d\). | \frac{1201 \times 572 \times 19 \times 124}{105^4} |
There are $10$ horses, named Horse $1$, Horse $2$, . . . , Horse $10$. They get their names from how many minutes it takes them to run one lap around a circular race track: Horse $k$ runs one lap in exactly $k$ minutes. At time $0$ all the horses are together at the starting point on the track. The horses start running... | 6 |
Given the arithmetic sequence $\{a_n\}$, it is given that $a_2+a_8-a_{12}=0$ and $a_{14}-a_4=2$. Let $s_n=a_1+a_2+\ldots+a_n$, then determine the value of $s_{15}$. | 30 |
How many ways are there to put 5 balls in 3 boxes if the balls are distinguishable but the boxes are not? | 41 |
Given $(x^2+1)(x-2)^8 = a + a_1(x-1) + a_2(x-1)^2 + \ldots + a_{10}(x-1)^{10}$, find the value of $a_1 + a_2 + \ldots + a_{10}$. | -2 |
Given the polynomial $f(x) = 12 + 35x - 8x^2 + 79x^3 + 6x^4 + 5x^5 + 3x^6$, use Horner's Rule to calculate the value of $v_4$ when $x = -4$. | 220 |
Find the constant term in the expansion of \\((x+ \frac {2}{x}+1)^{6}\\) (Answer with a numerical value) | 581 |
Let $a, b, c$, and $d$ be positive real numbers such that
\[
\begin{array}{c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c@{\hspace{3pt}}c}
a^2+b^2 &=& c^2+d^2 &=& 2016, \\
ac &=& bd &=& 1024.
\end{array}
\]
If $S = a + b + c + d$, compute the value of $\lfloor S \rfloor$. | 127 |
Compute the number of ways to color the vertices of a regular heptagon red, green, or blue (with rotations and reflections distinct) such that no isosceles triangle whose vertices are vertices of the heptagon has all three vertices the same color. | 294 |
A standard deck of 52 cards is divided into 4 suits, with each suit containing 13 cards. Two of these suits are red, and the other two are black. The deck is shuffled, placing the cards in random order. What is the probability that the first three cards drawn from the deck are all the same color? | \frac{40}{85} |
Let $P$ be a point outside a circle $\Gamma$ centered at point $O$ , and let $PA$ and $PB$ be tangent lines to circle $\Gamma$ . Let segment $PO$ intersect circle $\Gamma$ at $C$ . A tangent to circle $\Gamma$ through $C$ intersects $PA$ and $PB$ at points $E$ and $F$ , respectively. Given tha... | 12 \sqrt{3} |
At a pool party, there are 4 pizzas cut into 12 slices each. If the guests eat 39 slices, how many slices are left? | There’s a total of 4 x 12 = <<4*12=48>>48 slices.
After the guests eat, there are 48 - 39 = <<48-39=9>>9 pieces.
#### 9 |
If $\sqrt2 \sin 10^\circ$ can be written as $\cos \theta - \sin\theta$ for some acute angle $\theta,$ what is $\theta?$ (Give your answer in degrees, not radians.) | 35^\circ |
What is the remainder when $2^{19}$ is divided by $7$? | 2 |
What is the remainder when 1,234,567,890 is divided by 99? | 72 |
A $16$-quart radiator is filled with water. Four quarts are removed and replaced with pure antifreeze liquid. Then four quarts of the mixture are removed and replaced with pure antifreeze. This is done a third and a fourth time. The fractional part of the final mixture that is water is: | \frac{81}{256} |
There are 100 points marked on a circle, painted either red or blue. Some points are connected by segments, with each segment having one blue end and one red end. It is known that no two red points are connected to the same number of segments. What is the maximum possible number of red points? | 50 |
Given four points \( K, L, M, N \) that are not coplanar. A sphere touches the planes \( K L M \) and \( K L N \) at points \( M \) and \( N \) respectively. Find the surface area of the sphere, knowing that \( M L = 1 \), \( K M = 2 \), \( \angle M N L = 60^\circ \), and \( \angle K M L = 90^\circ \). | \frac{64\pi}{11} |
If \( x_{i}=\frac{i}{101} \), then the value of \( S=\sum_{i=0}^{101} \frac{x_{i}^{3}}{3 x_{i}^{2}-3 x_{i}+1} \) is | 51 |
Given that \( t = \frac{1}{1 - \sqrt[4]{2}} \), simplify the expression for \( t \). | -(1 + \sqrt[4]{2})(1 + \sqrt{2}) |
In three-dimensional space, the volume of the geometric body formed by points whose distance to line segment $A B$ is no greater than three units is $216 \pi$. What is the length of the line segment $A B$? | 20 |
In triangle $ABC$, $AB = 3$, $AC = 5$, and $BC = 4$. The medians $AD$, $BE$, and $CF$ of triangle $ABC$ intersect at the centroid $G$. Let the projections of $G$ onto $BC$, $AC$, and $AB$ be $P$, $Q$, and $R$, respectively. Find $GP + GQ + GR$.
[asy]
import geometry;
unitsize(1 cm);
pair A, B, C, D, E, F, G, P, Q... | \frac{47}{15} |
Compute $\cos 210^\circ$. | -\frac{\sqrt{3}}{2} |
Let $\theta$ be the smallest acute angle for which $\sin \theta,$ $\sin 2 \theta,$ $\sin 3 \theta$ form an arithmetic progression, in some order. Find $\cos \theta.$ | \frac{3}{4} |
Find the leading coefficient in the polynomial $-3(x^4 - x^3 + x) + 7(x^4 + 2) - 4(2x^4 + 2x^2 + 1)$ after it is simplified. | -4 |
Matt can paint a house in 12 hours. Patty can paint the same house in one third the time. Rachel can paint the same house in 5 more than double the amount of hours as Patty. How long will it take Rachel to paint the house? | Patty: 12/3=<<12/3=4>>4 hours
Rachel: 5+2(4)=13 hours
#### 13 |
Find the minimum value of
\[f(x) = x + \frac{1}{x} + \frac{1}{x + \frac{1}{x}}\]for $x > 0.$ | \frac{5}{2} |
Distribute 10 volunteer positions among 4 schools, with the requirement that each school receives at least one position. How many different ways can the positions be distributed? (Answer with a number.) | 84 |
What is the positive difference between the two largest prime factors of $159137$? | 14 |
In product inspection, the method of sampling inspection is often used. Now, 4 products are randomly selected from 100 products (among which there are 3 defective products) for inspection. The number of ways to exactly select 2 defective products is ____. (Answer with a number) | 13968 |
Let $m$ be the smallest integer whose cube root is of the form $n+s$, where $n$ is a positive integer and $s$ is a positive real number less than $1/2000$. Find $n$. | 26 |
A nine-digit integer is formed by repeating a positive three-digit integer three times. For example, 123,123,123 or 307,307,307 are integers of this form. What is the greatest common divisor of all nine-digit integers of this form? | 1001001 |
For positive integers $n,$ let $\tau (n)$ denote the number of positive integer divisors of $n,$ including 1 and $n.$ For example, $\tau (1)=1$ and $\tau(6) =4.$ Define $S(n)$ by $S(n)=\tau(1)+ \tau(2) + \cdots + \tau(n).$ Let $a$ denote the number of positive integers $n \leq 2005$ with $S(n)$ odd, and let $b$ denote ... | 25 |
Debora has 12 more dresses than Melissa. Melissa has half the number of dresses Emily has. If Emily has 16 dresses, how many dresses do the three of them have in total? | Melissa has 16 / 2 = <<16/2=8>>8 dresses.
Debora has 8 + 12 = <<8+12=20>>20 dresses.
In total, they have 16 + 8 + 20 = <<16+8+20=44>>44 dresses.
#### 44 |
How many integer values of $x$ satisfy $|x|<3\pi$? | 19 |
In which acute-angled triangle is the value of the product \(\operatorname{tg} \alpha \cdot \operatorname{tg} \beta \cdot \operatorname{tg} \gamma\) minimized? | \sqrt{27} |
Numbers between $200$ and $500$ that are divisible by $5$ contain the digit $3$. How many such whole numbers exist? | 24 |
Six ants simultaneously stand on the six vertices of a regular octahedron, with each ant at a different vertex. Simultaneously and independently, each ant moves from its vertex to one of the four adjacent vertices, each with equal probability. What is the probability that no two ants arrive at the same vertex? | \frac{5}{256} |
Given the equation in terms of \( x \)
$$
x^{4}-16 x^{3}+(81-2a) x^{2}+(16a-142) x + a^{2} - 21a + 68 = 0
$$
where all roots are integers, find the value of \( a \) and solve the equation. | -4 |
Let $\triangle ABC$ be a triangle in the plane, and let $D$ be a point outside the plane of $\triangle ABC$, so that $DABC$ is a pyramid whose faces are all triangles.
Suppose that every edge of $DABC$ has length $20$ or $45$, but no face of $DABC$ is equilateral. Then what is the surface area of $DABC$? | 40 \sqrt{1925} |
The sum of the first thirteen terms of an arithmetic progression is $50\%$ of the sum of the last thirteen terms of this progression. The sum of all terms of this progression, excluding the first three terms, is to the sum of all terms excluding the last three terms in the ratio $5:4$. Find the number of terms in this ... | 22 |
Shuxin begins with 10 red candies, 7 yellow candies, and 3 blue candies. After eating some of the candies, there are equal numbers of red, yellow, and blue candies remaining. What is the smallest possible number of candies that Shuxin ate? | 11 |
A positive integer is called primer if it has a prime number of distinct prime factors. A positive integer is called primest if it has a primer number of distinct primer factors. A positive integer is called prime-minister if it has a primest number of distinct primest factors. Let $N$ be the smallest prime-minister nu... | 378000 |
The function \[f(x) = \left\{ \begin{aligned} x-3 & \quad \text{ if } x < 5 \\ \sqrt{x} & \quad \text{ if } x \ge 5 \end{aligned} \right.\] has an inverse $f^{-1}.$ Find the value of $f^{-1}(0) + f^{-1}(1) + \dots + f^{-1}(9).$ | 291 |
Simplify $(3p^3 - 5p + 6) + (4 - 6p^2 + 2p)$. Express your answer in the form $Ap^3 + Bp^2 + Cp +D$, where $A$, $B$, $C$, and $D$ are numbers (possibly negative). | 3p^3 - 6p^2 - 3p + 10 |
The area of the closed region formed by the line $y = nx$ and the curve $y = x^2$ is \_\_\_\_\_\_ when the binomial coefficients of the third and fourth terms in the expansion of $(x - \frac{2}{x})^n$ are equal. | \frac{125}{6} |
How many even natural-number factors does $n = 2^2 \cdot 3^1 \cdot 7^2$ have? | 12 |
The equation $x^2-kx-12=0$ has only integer solutions for certain positive integers $k$. What is the sum of all such values of $k$? | 16 |
The number of solutions to \{1,~2\} \subseteq~X~\subseteq~\{1,~2,~3,~4,~5\}, where $X$ is a subset of \{1,~2,~3,~4,~5\} is | 6 |
$2014$ points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of the $2014$ points, the sum of the numbers written on their sides is less or equal than $1$. Find the maximum possible va... | 507024.5 |
Ana and Bonita were born on the same date in different years, $n$ years apart. Last year Ana was $5$ times as old as Bonita. This year Ana's age is the square of Bonita's age. What is $n?$ | 12 |
Given the function $f(x)=\cos (\omega x+\varphi)$ ($\omega > 0$, $|\varphi| \leqslant \frac {\pi}{2}$), when $x=- \frac {\pi}{4}$, the function $f(x)$ can achieve its minimum value, and when $x= \frac {\pi}{4}$, the function $y=f(x)$ can achieve its maximum value. Moreover, $f(x)$ is monotonic in the interval $( \frac ... | - \frac {\pi}{2} |
Let \( m \) be the largest positive integer such that for every positive integer \( n \leqslant m \), the following inequalities hold:
\[
\frac{2n + 1}{3n + 8} < \frac{\sqrt{5} - 1}{2} < \frac{n + 7}{2n + 1}
\]
What is the value of the positive integer \( m \)? | 27 |
Given vectors $\overrightarrow{a}=(2\sin x,-\cos x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x,2\cos x)$, and function $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}+1$.
(I) Find the smallest positive period of function $f(x)$, and find the range of $f(x)$ when $x\in\left[\dfrac{\pi}{12},\dfrac{2\pi}{3}\right]$;
(II) ... | \sqrt{3} |
What is the smallest positive integer with exactly 12 positive integer divisors? | 288 |
Recently, many cities in China have been intensifying efforts to develop the "night economy" to meet the diverse consumption needs of different groups and to boost employment, drive entrepreneurship, and enhance regional economic development vitality. A handicraft seller at a night market found through a survey of dail... | 441 |
Each row of a seating arrangement seats either 9 or 10 people. A total of 100 people are to be seated. How many rows seat exactly 10 people if every seat is occupied? | 10 |
Square $BCFE$ is inscribed in right triangle $AGD$, as shown in the problem above. If $AB = 34$ units and $CD = 66$ units, what is the area of square $BCFE$? | 2244 |
Given points $A(\sin\theta, 1)$, $B(\cos\theta, 0)$, $C(-\sin\theta, 2)$, and $\overset{→}{AB}=\overset{→}{BP}$.
(I) Consider the function $f\left(\theta\right)=\overset{→}{BP}\cdot\overset{→}{CA}$, $\theta\in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, discuss the monotonicity of the function and find its range.
(II) ... | \frac{\sqrt{74}}{5} |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $4\sin A\sin B-4\cos ^{2} \frac {A-B}{2}= \sqrt {2}-2$.
$(1)$ Find the magnitude of angle $C$;
$(2)$ Given $\frac {a\sin B}{\sin A}=4$ and the area of $\triangle ABC$ is $8$, find the value of side lengt... | c=4 |
When a certain biased coin is flipped five times, the probability of getting heads exactly once is not equal to $0$ and is the same as that of getting heads exactly twice. Let $\frac ij$, in lowest terms, be the probability that the coin comes up heads in exactly $3$ out of $5$ flips. Find $i+j$.
| 283 |
In right triangle $ABC$ with $\angle A = 90^\circ$, $AC = 3$, $AB = 4$, and $BC = 5$, point $D$ is on side $BC$. If the perimeters of $\triangle ACD$ and $\triangle ABD$ are equal, then what is the area of $\triangle ABD$? | $\frac{12}{5}$ |
Alli rolls a standard $8$-sided die twice. What is the probability of rolling integers that differ by $3$ on her first two rolls? Express your answer as a common fraction. | \dfrac{7}{64} |
For lines $l_1: x + ay + 3 = 0$ and $l_2: (a-2)x + 3y + a = 0$ to be parallel, determine the values of $a$. | -1 |
A certain product was bought in the fall, and 825 rubles were paid for it. In the fall, the price per kilogram of this product was 1 ruble cheaper than in the spring. Therefore, for the same amount in the spring, 220 kg less was bought. How much does 1 kg of the product cost in the spring, and how much was bought in th... | 550 |
Find a number \( N \) with five digits, all different and none zero, which equals the sum of all distinct three-digit numbers whose digits are all different and are all digits of \( N \). | 35964 |
A baker has 10 cheesecakes on the display while 15 more are still in the fridge. If the baker has sold 7 cheesecakes from the display, how many more cheesecakes are left to be sold? | A baker has a total of 10 + 15 = <<10+15=25>>25 cheesecakes both from the display and the fridge.
Therefore, there are 25 - 7 = <<25-7=18>>18 more cheesecakes to be sold.
#### 18 |
The arithmetic mean of these six expressions is 30. What is the value of $y$? $$y + 10 \hspace{.5cm} 20 \hspace{.5cm} 3y \hspace{.5cm} 18 \hspace{.5cm} 3y + 6 \hspace{.5cm} 12$$ | \frac{114}{7} |
When Bendegúz boarded the 78-seat train car with his valid seat reservation, he was shocked to find that all seats were already taken. What had happened was that Dömötör boarded without a seat reservation. The other 77 passengers, including Elek, had purchased a seat reservation, but did not necessarily sit in their as... | 1/2 |
Let $A$ and $B$ be points in space for which $A B=1$. Let $\mathcal{R}$ be the region of points $P$ for which $A P \leq 1$ and $B P \leq 1$. Compute the largest possible side length of a cube contained within $\mathcal{R}$. | \frac{\sqrt{10}-1}{3} |
There are 2016 points arranged on a circle. We are allowed to jump 2 or 3 points clockwise at will.
How many jumps must we make at least to reach all the points and return to the starting point again? | 2017 |
If $a$, $b$, $c$, and $d$ are four positive numbers whose product is 1, calculate the minimum value of the algebraic expression $a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+cd$. | 10 |
Are there integers $m$ and $n$ such that
\[5m^2 - 6mn + 7n^2 = 1985 \ ?\] | \text{No} |
Determine the distance that the origin $O(0,0)$ moves under the dilation transformation that sends the circle of radius $4$ centered at $B(3,1)$ to the circle of radius $6$ centered at $B'(7,9)$. | 0.5\sqrt{10} |
The random variable $X$ follows a normal distribution $N(1,4)$. Given that $P(X \geqslant 2) = 0.2$, calculate the probability that $0 \leqslant X \leqslant 1$. | 0.3 |
Given that $x \in (1,5)$, find the minimum value of the function $y= \frac{2}{x-1}+ \frac{1}{5-x}$. | \frac{3+2 \sqrt{2}}{4} |
Going into the final game, Duke is so close to breaking the school's record for most points scored in a basketball season. He only needs 17 more points to tie the record. By the end of the game, Duke breaks the record by 5 points. The old record was 257 points. In the final game Duke made 5 free throws (worth one point... | Duke scored 22 points in the final game because 17 + 5 = <<17+5=22>>22
He scored 5 points of free-throws, because 5 x 1 = <<5*1=5>>5
He scored 8 points of regular baskets because 4 x 2 = <<4*2=8>>8
He made 9 points of three-pointers because 22 - 5 - 8 = <<22-5-8=9>>9
He scored 3 three-pointers in the final game because... |
In the plane Cartesian coordinate system, the area of the region corresponding to the set of points $\{(x, y) \mid(|x|+|3 y|-6)(|3 x|+|y|-6) \leq 0\}$ is ________. | 24 |
A yogurt shop sells four flavors of yogurt and has six different toppings. How many combinations of one flavor and two different toppings are available? | 60 |
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 |
How many prime numbers are between 20 and 30? | 2 |
Given that $α$ and $β ∈ ( \frac{π}{2},π)$, and $sinα + cosα = a$, $cos(β - α) = \frac{3}{5}$.
(1) If $a = \frac{1}{3}$, find the value of $sinαcosα + tanα - \frac{1}{3cosα}$;
(2) If $a = \frac{7}{13}$, find the value of $sinβ$. | \frac{16}{65} |
Consider all 4-digit palindromes that can be written as $\overline{abba}$, where $a$ is non-zero and $b$ ranges from 1 to 9. Calculate the sum of the digits of the sum of all such palindromes. | 36 |
Given that $α$ is an obtuse angle, if $\sin (α- \frac{3π}{4})= \frac{3}{5}$, find $\cos (α+ \frac{π}{4})=\_\_\_\_\_\_\_\_\_\_\_\_\_.$ | - \frac{4}{5} |
Find $\frac{1}{3}+\frac{2}{7}$. | \frac{13}{21} |
What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$? | 8 |
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