problem stringlengths 10 5.15k | answer stringlengths 0 1.22k | solution stringlengths 0 11.1k | reward float64 0 1 | length float64 172 8.19k | correct_length float64 -1 8.19k | incorrect_length float64 -1 8.19k |
|---|---|---|---|---|---|---|
For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"? | 2016 | 0.0625 | 8,192 | 8,192 | 8,192 | |
What is the first digit (from left to right) of the base $8$ representation of $473_{10}$? | 7 | 1 | 2,300.3125 | 2,300.3125 | -1 | |
The center of the circle with equation $x^2+y^2=4x+12y-39$ is the point $(h,k)$. What is $h+k$? | 8 | 1 | 2,372.875 | 2,372.875 | -1 | |
Cheburashka spent his money to buy as many mirrors from Galya's store as Gena bought from Shapoklyak's store. If Gena were buying from Galya, he would have 27 mirrors, and if Cheburashka were buying from Shapoklyak, he would have 3 mirrors. How many mirrors would Gena and Cheburashka have bought together if Galya and S... | 18 | 0.125 | 7,075.375 | 5,411 | 7,313.142857 | |
Given two arithmetic sequences $\{a_n\}$ and $\{b_n\}$, the sums of the first $n$ terms are $S_n$ and $T_n$, respectively. For any positive integer $n$, it holds that $$\frac {S_{n}}{T_{n}} = \frac {3n+5}{2n+3}$$, then $$\frac {a_{7}}{b_{7}} = \_\_\_\_\_\_ ．$$ | \frac {44}{29} | 0.8125 | 5,505.375 | 4,885.384615 | 8,192 | |
In $\triangle ABC$, $AB = 6$, $BC = 10$, $CA = 8$, and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Calculate the length of $PC$. | 40 | 0 | 7,813.25 | -1 | 7,813.25 | |
Given the function $f(x)=2\sin (2x+ \frac {\pi}{4})$, let $f_1(x)$ denote the function after translating and transforming $f(x)$ to the right by $φ$ units and compressing every point's abscissa to half its original length, then determine the minimum value of $φ$ for which $f_1(x)$ is symmetric about the line $x= \frac ... | \frac{3\pi}{8} | 0.5 | 6,035.4375 | 4,981.75 | 7,089.125 | |
A certain collection of numbered indexed cards includes one card with a 1 written on it, two cards with a 2, and so forth up to $n$ cards showing an $n,$ for some positive integer $n$. Determine $n,$ if the average value of a card in this collection is 2017. | 3025 | 1 | 1,749.375 | 1,749.375 | -1 | |
Find the positive integer $n$ such that
\[\arctan\frac {1}{3} + \arctan\frac {1}{4} + \arctan\frac {1}{5} + \arctan\frac {1}{n} = \frac {\pi}{4}.\] | 47 | Adding a series of angles is the same as multiplying the complex numbers whose arguments they are. In general, $\arctan\frac{1}{n}$, is the argument of $n+i$. The sum of these angles is then just the argument of the product
\[(3+i)(4+i)(5+i)(n+i)\]
and expansion give us $(48n-46)+(48+46n)i$. Since the argument of this... | 1 | 3,519.25 | 3,519.25 | -1 |
Let \( x \) and \( y \) be positive real numbers, and \( x + y = 1 \). Find the minimum value of \( \frac{x^2}{x+2} + \frac{y^2}{y+1} \). | 1/4 | 0.5 | 7,110.5625 | 6,113.375 | 8,107.75 | |
Positive integers less than 900 that can be written as a product of two or more consecutive prime numbers. Find their count. | 14 | 0.4375 | 7,391 | 6,550.571429 | 8,044.666667 | |
The values of $f$, $g$, $h$ and $j$ are 5, 6, 7 and 8, but not necessarily in that order. What is the largest possible value of the sum of the four products $fg$, $gh$, $hj$ and $fj$? | 169 | 0.9375 | 3,406.5 | 3,425.933333 | 3,115 | |
A mother gives pocket money to her children sequentially: 1 ruble to Anya, 2 rubles to Borya, 3 rubles to Vitya, then 4 rubles to Anya, 5 rubles to Borya, and so on until Anya receives 202 rubles, and Borya receives 203 rubles. How many more rubles will Anya receive compared to Vitya? | 68 | 0.25 | 7,084.875 | 7,201.5 | 7,046 | |
What is the smallest positive integer $x$ for which $x^{2}+x+41$ is not a prime? | 40 | 40. | 0.8125 | 5,117.375 | 4,407.846154 | 8,192 |
Fifty numbers have an average of 76. Forty of these numbers have an average of 80. What is the average of the other ten numbers? | 60 | If 50 numbers have an average of 76, then the sum of these 50 numbers is $50(76)=3800$. If 40 numbers have an average of 80, then the sum of these 40 numbers is $40(80)=3200$. Therefore, the sum of the 10 remaining numbers is $3800-3200=600$, and so the average of the 10 remaining numbers is $ rac{600}{10}=60$. | 1 | 1,652.6875 | 1,652.6875 | -1 |
Given that there is 1 path from point A to the first red arrow, 2 paths from the first red arrow to the second red arrow, 3 paths from the first red arrow to each of the first two blue arrows, 4 paths from the second red arrow to each of the first two blue arrows, 5 paths from each of the first two blue arrows to each ... | 4312 | 0 | 7,639.875 | -1 | 7,639.875 | |
Six men and some number of women stand in a line in random order. Let $p$ be the probability that a group of at least four men stand together in the line, given that every man stands next to at least one other man. Find the least number of women in the line such that $p$ does not exceed 1 percent. | 594 | Let $n$ be the number of women present, and let _ be some positive number of women between groups of men. Since the problem states that every man stands next to another man, there cannot be isolated men. Thus, there are five cases to consider, where $(k)$ refers to a consecutive group of $k$ men:
_(2)_(2)_(2)_
_(3)_(3)... | 0 | 8,185.875 | -1 | 8,185.875 |
Compute $\dbinom{133}{133}$. | 1 | 1 | 1,390.3125 | 1,390.3125 | -1 | |
Given three points in space: A(0,1,5), B(1,5,0), and C(5,0,1), if the vector $\vec{a}=(x,y,z)$ is perpendicular to both $\overrightarrow{AB}$ and $\overrightarrow{AC}$, and the magnitude of vector $\vec{a}$ is $\sqrt{15}$, then find the value of $x^2y^2z^2$. | 125 | 1 | 3,703 | 3,703 | -1 | |
Find the rightmost non-zero digit of the expansion of (20)(13!). | 6 | We can rewrite this as $(10 \times 2)(13 \times 12 \times 11 \times 10 \times 9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1)=\left(10^{3}\right)(2 \times 13 \times 12 \times 11 \times 9 \times 8 \times 7 \times 6 \times 4 \times 3)$; multiplying together the units digits for the terms not eq... | 0.5 | 7,281.375 | 6,370.75 | 8,192 |
What is the value of the product
\[\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)?\] | 7 | 1. **Identify the Expression**: We start by simplifying each term in the product:
\[
\left(1+\frac{1}{1}\right)\cdot\left(1+\frac{1}{2}\right)\cdot\left(1+\frac{1}{3}\right)\cdot\left(1+\frac{1}{4}\right)\cdot\left(1+\frac{1}{5}\right)\cdot\left(1+\frac{1}{6}\right)
\]
Simplifying each term, we get:
\[
... | 1 | 2,088.3125 | 2,088.3125 | -1 |
A rectangle PQRS has a perimeter of 24 meters and side PQ is fixed at 7 meters. Find the minimum diagonal PR of the rectangle. | \sqrt{74} | 1 | 4,205.375 | 4,205.375 | -1 | |
The graph relates the distance traveled [in miles] to the time elapsed [in hours] on a trip taken by an experimental airplane. During which hour was the average speed of this airplane the largest? | second (1-2) | To determine during which hour the average speed of the airplane was the largest, we need to analyze the slope of the graph of distance versus time. The average speed for any given hour is calculated as the change in distance divided by the change in time (which is 1 hour in this case). Mathematically, this is represen... | 0 | 8,176.75 | -1 | 8,176.75 |
The product $(8)(888\dots8)$, where the second factor has $k$ digits, is an integer whose digits have a sum of $1000$. What is $k$? | 991 |
1. **Understanding the Problem:**
We need to find the value of $k$ such that the product of $8$ and a number consisting of $k$ eights, i.e., $(8)(888\ldots8)$, results in a number whose digits sum to $1000$.
2. **Exploring the Pattern:**
Let's examine the pattern formed by multiplying $8$ with numbers consistin... | 0.5 | 6,805 | 5,418 | 8,192 |
A random number selector can only select one of the nine integers 1, 2, ..., 9, and it makes these selections with equal probability. Determine the probability that after $n$ selections ( $n>1$ ), the product of the $n$ numbers selected will be divisible by 10. | \[ 1 - \left( \frac{8}{9} \right)^n - \left( \frac{5}{9} \right)^n + \left( \frac{4}{9} \right)^n \] | For the product to be divisible by 10, there must be a factor of 2 and a factor of 5 in there.
The probability that there is no 5 is $\left( \frac{8}{9}\right)^n$ .
The probability that there is no 2 is $\left( \frac{5}{9}\right)^n$ .
The probability that there is neither a 2 nor 5 is $\left( \frac{4}{9}\right)^n$ , wh... | 0 | 5,249.375 | -1 | 5,249.375 |
In right triangle $A B C$, a point $D$ is on hypotenuse $A C$ such that $B D \perp A C$. Let $\omega$ be a circle with center $O$, passing through $C$ and $D$ and tangent to line $A B$ at a point other than $B$. Point $X$ is chosen on $B C$ such that $A X \perp B O$. If $A B=2$ and $B C=5$, then $B X$ can be expressed ... | 8041 | Note that since $A D \cdot A C=A B^{2}$, we have the tangency point of $\omega$ and $A B$ is $B^{\prime}$, the reflection of $B$ across $A$. Let $Y$ be the second intersection of $\omega$ and $B C$. Note that by power of point, we have $B Y \cdot B C=B B^{\prime 2}=4 A B^{2} \Longrightarrow B Y=\frac{4 A B^{2}}{B C}$. ... | 0.4375 | 7,050.9375 | 5,723.714286 | 8,083.222222 |
A thousand points form the vertices of a convex polygon with 1000 sides. Inside this polygon, there are another 500 points placed such that no three of these 500 points are collinear. The polygon is triangulated in such a way that all of these 1500 points are vertices of the triangles, and none of the triangles have a... | 1998 | 0.9375 | 5,000.9375 | 4,788.2 | 8,192 | |
If the fractional equation in terms of $x$, $\frac{x-2}{x-3}=\frac{n+1}{3-x}$ has a positive root, then $n=\_\_\_\_\_\_.$ | -2 | 0 | 8,192 | -1 | 8,192 | |
Alice is sitting in a teacup ride with infinitely many layers of spinning disks. The largest disk has radius 5. Each succeeding disk has its center attached to a point on the circumference of the previous disk and has a radius equal to $2 / 3$ of the previous disk. Each disk spins around its center (relative to the dis... | 18 \pi | Suppose the center of the largest teacup is at the origin in the complex plane, and let $z=\frac{2}{3} e^{\pi i t / 6}$. The center of the second disk is at $5 e^{\pi i t / 6}$ at time $t$; that is, \frac{15}{2} z$. Then the center of the third disk relative to the center of the second disk is at \frac{15}{2} z^{2}$, a... | 0.0625 | 7,384.375 | 6,209 | 7,462.733333 |
\(x, y\) are real numbers, \(z_{1}=x+\sqrt{11}+yi\), \(z_{6}=x-\sqrt{11}+yi\) (where \(i\) is the imaginary unit). Find \(|z_{1}| + |z_{6}|\). | 30(\sqrt{2} + 1) | 0 | 7,956.1875 | -1 | 7,956.1875 | |
Given that $F_1$ and $F_2$ are the two foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a>0$, $b>0$), an isosceles right triangle $MF_1F_2$ is constructed with $F_1$ as the right-angle vertex. If the midpoint of the side $MF_1$ lies on the hyperbola, calculate the eccentricity of the hyperbola. | \frac{\sqrt{5} + 1}{2} | 0 | 4,381.5 | -1 | 4,381.5 | |
Circles of radius 4 and 5 are externally tangent and are circumscribed by a third circle. Calculate the area of the region outside the smaller circles but inside the larger circle. | 40\pi | 0.625 | 6,411.75 | 5,343.6 | 8,192 | |
How many positive 3-digit numbers are multiples of 25, but not of 60? | 33 | 0.875 | 5,595.5 | 5,569.142857 | 5,780 | |
In the plane rectangular coordinate system $xOy$, the parametric equations of the line $l_{1}$ are $\left\{\begin{array}{l}{x=t}\\{y=kt}\end{array}\right.$ (where $t$ is the parameter), and the parametric equations of the line $l_{2}$ are $\left\{\begin{array}{l}{x=-km+2}\\{y=m}\end{array}\right.$ (where $m$ is the par... | 1+\frac{5\sqrt{2}}{2} | 0 | 5,556.9375 | -1 | 5,556.9375 | |
Amy and Ben need to eat 1000 total carrots and 1000 total muffins. The muffins can not be eaten until all the carrots are eaten. Furthermore, Amy can not eat a muffin within 5 minutes of eating a carrot and neither can Ben. If Amy eats 40 carrots per minute and 70 muffins per minute and Ben eats 60 carrots per minute a... | 23.5 | Amy and Ben will continuously eat carrots, then stop (not necessarily at the same time), and continuously eat muffins until no food is left. Suppose that Amy and Ben finish eating the carrots in $T_{1}$ minutes and the muffins $T_{2}$ minutes later; we wish to find the minimum value of $T_{1}+T_{2}$. Furthermore, suppo... | 0 | 6,351.25 | -1 | 6,351.25 |
A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure 1. The cube is then cut in the same manner along the dashed lines shown in Figure 2. This creates nine pieces. What is the volume of the piece that contains vertex $W$?
[asy]
path a=(0,0)--(10,0)--(10,10)--(0,10)--... | \frac{1}{12} | 0 | 8,099.8125 | -1 | 8,099.8125 | |
$\triangle PQR$ is similar to $\triangle STU$. The length of $\overline{PQ}$ is 10 cm, $\overline{QR}$ is 12 cm, and the length of $\overline{ST}$ is 5 cm. Determine the length of $\overline{TU}$ and the perimeter of $\triangle STU$. Express your answer as a decimal. | 17 | 0.125 | 7,638.625 | 8,171.5 | 7,562.5 | |
A point $(x, y)$ is randomly selected from inside the rectangle with vertices $(0, 0)$, $(4, 0)$, $(4, 3)$, and $(0, 3)$. What is the probability that both $x < y$ and $x + y < 5$? | \frac{3}{8} | 0.0625 | 7,947.4375 | 8,192 | 7,931.133333 | |
In triangle \( \triangle ABC \), \(E\) and \(F\) are the midpoints of \(AC\) and \(AB\) respectively, and \( AB = \frac{2}{3} AC \). If \( \frac{BE}{CF} < t \) always holds, then the minimum value of \( t \) is ______. | \frac{7}{8} | 0.5 | 7,046.125 | 5,900.25 | 8,192 | |
Each of the first $150$ positive integers is painted on a different marble, and the $150$ marbles are placed in a bag. If $n$ marbles are chosen (without replacement) from the bag, what is the smallest value of $n$ such that we are guaranteed to choose three marbles with consecutive numbers? | 101 | 0.4375 | 7,109.1875 | 5,833.571429 | 8,101.333333 | |
Calculate the value of $n$ such that
\[(1 + \tan 1^\circ)(1 + \tan 2^\circ)(1 + \tan 3^\circ) \dotsm (1 + \tan 30^\circ) = 2^n.\] | 15 | 0.0625 | 8,094.1875 | 6,627 | 8,192 | |
Let $Z$ denote the set of points in $\mathbb{R}^n$ whose coordinates are 0 or 1. (Thus $Z$ has $2^n$ elements, which are the vertices of a unit hypercube in $\mathbb{R}^n$.) Given a vector subspace $V$ of $\mathbb{R}^n$, let $Z(V)$ denote the number of members of $Z$ that lie in $V$. Let $k$ be given, $0 \leq k \leq n$... | 2^k | The maximum is $2^k$, achieved for instance by the subspace \[\{(x_1, \dots, x_n) \in \mathbb{R}^n: x_1 = \cdots = x_{n-k} = 0\}.\]
\textbf{First solution:} More generally, we show that any affine $k$-dimensional plane in $\mathbb{R}^n$ can contain at most $2^k$ points in $Z$. The proof is by induction on $k+n$; the c... | 0.0625 | 8,176.375 | 7,942 | 8,192 |
Calculate the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin ^{2} x \, dx}{(1+\cos x+\sin x)^{2}}
$$ | \frac{1}{2} - \frac{1}{2} \ln 2 | 0.0625 | 7,609.375 | 8,192 | 7,570.533333 | |
A permutation $(a_1,a_2,a_3,a_4,a_5)$ of $(1,2,3,4,5)$ is heavy-tailed if $a_1 + a_2 < a_4 + a_5$. What is the number of heavy-tailed permutations? | 48 | We analyze the problem by considering the position of $a_3$ and how it affects the possible values of $a_1 + a_2$ and $a_4 + a_5$. We need to ensure $a_1 + a_2 < a_4 + a_5$ for the permutation to be heavy-tailed.
1. **Case 1: $a_3 = 1$.**
- The remaining numbers are $2, 3, 4, 5$. We need $a_1 + a_2 < a_4 + a_5$.
... | 0 | 7,992.0625 | -1 | 7,992.0625 |
The solution to the inequality
$$
(x-1)^{[\sqrt{1}]}(x-2)^{[\sqrt{2}]} \ldots(x-k)^{[\sqrt{k}]} \ldots(x-150)^{[\sqrt{150}]}<0
$$
is a union of several non-overlapping intervals. Find the sum of their lengths. If necessary, round the answer to the nearest 0.01.
Recall that $[x]$ denotes the greatest integer less tha... | 78.00 | 0 | 8,142.8125 | -1 | 8,142.8125 | |
Given the parabola $y^2=2px$ ($p>0$) with focus $F(1,0)$, and the line $l: y=x+m$ intersects the parabola at two distinct points $A$ and $B$. If $0\leq m<1$, determine the maximum area of $\triangle FAB$. | \frac{8\sqrt{6}}{9} | 0 | 6,380.375 | -1 | 6,380.375 | |
Alice and Bob each arrive at a meeting at a random time between 8:00 and 9:00 AM. If Alice arrives after Bob, what is the probability that Bob arrived before 8:45 AM? | \frac{9}{16} | 0 | 6,543.625 | -1 | 6,543.625 | |
Determine the value of $x$ that satisfies $\sqrt[5]{x\sqrt{x^3}}=3$. | 9 | 1 | 1,825.375 | 1,825.375 | -1 | |
Given that $\binom{23}{5}=33649$, $\binom{23}{6}=42504$, and $\binom{23}{7}=33649$, find $\binom{25}{7}$. | 152306 | 0.0625 | 7,793.1875 | 7,693 | 7,799.866667 | |
Given that $\binom{24}{5}=42504$, and $\binom{24}{6}=134596$, find $\binom{26}{6}$. | 230230 | 0.625 | 5,877.9375 | 4,925.5 | 7,465.333333 | |
I planned to work 25 hours a week for 15 weeks to earn $3750$ for a vacation. However, due to a family emergency, I couldn't work for the first three weeks. How many hours per week must I work for the remaining weeks to still afford the vacation? | 31.25 | 0.125 | 621.25 | 480.5 | 641.357143 | |
Find the minimum value of $m$ such that any $m$ -element subset of the set of integers $\{1,2,...,2016\}$ contains at least two distinct numbers $a$ and $b$ which satisfy $|a - b|\le 3$ . | 505 | 0.375 | 7,502.375 | 6,353 | 8,192 | |
What is the smallest positive value of $m$ such that the equation $10x^2 - mx + 660 = 0$ has integral solutions? | 170 | 0 | 8,192 | -1 | 8,192 | |
If the integers \( a, b, \) and \( c \) satisfy:
\[
a + b + c = 3, \quad a^3 + b^3 + c^3 = 3,
\]
then what is the maximum value of \( a^2 + b^2 + c^2 \)? | 57 | 0.1875 | 8,100.8125 | 7,705.666667 | 8,192 | |
If $f(1) = 3$, $f(2)= 12$, and $f(x) = ax^2 + bx + c$, what is the value of $f(3)$? | 21 | 0.125 | 6,169.5 | 1,650 | 6,815.142857 | |
There are integers $a, b,$ and $c,$ each greater than $1,$ such that
\[\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = \sqrt[36]{N^{25}}\]
for all $N \neq 1$. What is $b$? | 3 | 1. **Expression Simplification**:
Start by simplifying the left-hand side of the equation:
\[
\sqrt[a]{N\sqrt[b]{N\sqrt[c]{N}}} = N^{\frac{1}{a} + \frac{1}{ab} + \frac{1}{abc}}
\]
This simplification comes from the properties of exponents, where $\sqrt[k]{x} = x^{\frac{1}{k}}$ and the chain rule for expo... | 0.4375 | 7,145.375 | 5,799.714286 | 8,192 |
Let $D$ be the circle with equation $x^2 - 4y - 4 = -y^2 + 6x + 16$. Find the center $(c,d)$ and the radius $s$ of $D$, and compute $c+d+s$. | 5 + \sqrt{33} | 1 | 4,083.625 | 4,083.625 | -1 | |
Given an arithmetic sequence $\{a_n\}$ with a common difference $d \neq 0$ and the first term $a_1 = d$, the sum of the first $n$ terms of the sequence $\{a_n^2\}$ is $S_n$. A geometric sequence $\{b_n\}$ has a common ratio $q$ less than 1 and consists of rational sine values, with the first term $b_1 = d^2$, and the s... | \frac{1}{2} | 0.0625 | 7,875.0625 | 4,446 | 8,103.666667 | |
Given triangle PQR with PQ = 60 and PR = 20, the area is 240. Let M be the midpoint of PQ and N be the midpoint of PR. An altitude from P to side QR intersects MN and QR at X and Y, respectively. Find the area of quadrilateral XYMR. | 80 | 0 | 8,192 | -1 | 8,192 | |
V is the pyramidal region defined by the inequalities \( x, y, z \geq 0 \) and \( x + y + z \leq 1 \). Evaluate the integral:
\[ \int_V x y^9 z^8 (1 - x - y - z)^4 \, dx \, dy \, dz. \ | \frac{9! 8! 4!}{25!} | 0 | 8,070 | -1 | 8,070 | |
A metal bar with a temperature of $20{ }^{\circ} \mathrm{C}$ is placed into water that is initially at $80{ }^{\circ} \mathrm{C}$. After thermal equilibrium is reached, the temperature is $60{ }^{\circ} \mathrm{C}$. Without removing the first bar from the water, another metal bar with a temperature of $20{ }^{\circ} \m... | 50 | 0 | 6,736.625 | -1 | 6,736.625 | |
There are seven cards in a hat, and on the card $k$ there is a number $2^{k-1}$ , $k=1,2,...,7$ . Solarin picks the cards up at random from the hat, one card at a time, until the sum of the numbers on cards in his hand exceeds $124$ . What is the most probable sum he can get? | 127 | 0 | 8,192 | -1 | 8,192 | |
A triline is a line with the property that three times its slope is equal to the sum of its \(x\)-intercept and its \(y\)-intercept. For how many integers \(q\) with \(1 \leq q \leq 10000\) is there at least one positive integer \(p\) so that there is exactly one triline through \((p, q)\)? | 57 | 0 | 7,807.1875 | -1 | 7,807.1875 | |
Suppose $105 \cdot 77 \cdot 132 \equiv m \pmod{25}$, where $0 \le m < 25$. | 20 | 0.9375 | 4,832.5625 | 4,608.6 | 8,192 | |
In recent years, the awareness of traffic safety among citizens has gradually increased, leading to a greater demand for helmets. A certain store purchased two types of helmets, type A and type B. It is known that they bought 20 type A helmets and 30 type B helmets, spending a total of 2920 yuan. The unit price of type... | 1976 | 0.4375 | 5,089.25 | 3,878.571429 | 6,030.888889 | |
How many positive factors does 30 have? | 8 | 1 | 1,532.3125 | 1,532.3125 | -1 | |
There are seven students taking a graduation photo in a row. Student A must stand in the middle, and students B and C must stand together. How many different arrangements are possible? | 192 | 0.3125 | 6,343.3125 | 7,072.8 | 6,011.727273 | |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that $b\sin(C+\frac{π}{3})-c\sin B=0$.
$(1)$ Find the value of angle $C$.
$(2)$ If the area of $\triangle ABC$ is $10\sqrt{3}$ and $D$ is the midpoint of $AC$, find the minimum value of $BD$. | 2\sqrt{5} | 0.625 | 6,177.625 | 5,183.8 | 7,834 | |
An ice cream cone has radius 1 inch and height 4 inches, What is the number of inches in the radius of a sphere of ice cream which has the same volume as the cone? | 1 | 1 | 1,662.6875 | 1,662.6875 | -1 | |
A decagon is inscribed in a rectangle such that the vertices of the decagon divide each side of the rectangle into five equal segments. The perimeter of the rectangle is 160 centimeters, and the ratio of the length to the width of the rectangle is 3:2. What is the number of square centimeters in the area of the decagon... | 1413.12 | 0 | 8,192 | -1 | 8,192 | |
The side of a triangle are 2, 2, and $\sqrt{6} - \sqrt{2}.$ Enter the angles of the triangle in degrees, separated by commas. | 75^\circ, 75^\circ | 0 | 3,348.375 | -1 | 3,348.375 | |
The diagram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in the diagram. The ratio of the longer dimension of the rectangle to the shorter dimension can be written as $\dfrac{1}{2}(\sqrt{p}-q)$ where ... | 154 | Let the radius of the circles be $r$. The longer dimension of the rectangle can be written as $14r$, and by the Pythagorean Theorem, we find that the shorter dimension is $2r\left(\sqrt{3}+1\right)$.
Therefore, $\frac{14r}{2r\left(\sqrt{3}+1\right)}= \frac{7}{\sqrt{3} + 1} \cdot \left[\frac{\sqrt{3}-1}{\sqrt{3}-1}\rig... | 0.125 | 7,595.0625 | 6,656 | 7,729.214286 |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is planning to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | 12\% | 0.125 | 3,644.75 | 2,029.5 | 3,875.5 | |
A train has five carriages, each containing at least one passenger. Two passengers are said to be 'neighbours' if either they are in the same carriage or they are in adjacent carriages. Each passenger has exactly five or exactly ten neighbours. How many passengers are there on the train? | 17 | 0.125 | 7,049.5625 | 5,913.5 | 7,211.857143 | |
Let $T$ be the set of ordered triples $(x,y,z)$ of real numbers where
\[\log_{10}(2x+2y) = z \text{ and } \log_{10}(x^{2}+2y^{2}) = z+2.\]
Find constants $c$ and $d$ such that for all $(x,y,z) \in T$, the expression $x^{3} + y^{3}$ equals $c \cdot 10^{3z} + d \cdot 10^{z}.$ What is the value of $c+d$?
A) $\frac{1}{16}$... | \frac{5}{16} | 0 | 8,192 | -1 | 8,192 | |
Two externally tangent circles $\omega_1$ and $\omega_2$ have centers $O_1$ and $O_2$, respectively. A third circle $\Omega$ passing through $O_1$ and $O_2$ intersects $\omega_1$ at $B$ and $C$ and $\omega_2$ at $A$ and $D$, as shown. Suppose that $AB = 2$, $O_1O_2 = 15$, $CD = 16$, and $ABO_1CDO_2$ is a convex hexagon... | 140 | Let points $A'$ and $B'$ be the reflections of $A$ and $B,$ respectively, about the perpendicular bisector of $O_1 O_2.$ \[B'O_2 = BO_1 = O_1 P = O_1 C,\] \[A'O_1 = AO_2 = O_2 P = O_2 D.\] We establish the equality of the arcs and conclude that the corresponding chords are equal \[\overset{\Large\frown} {CO_1} + \overs... | 0 | 8,192 | -1 | 8,192 |
As shown in the diagram, a cube with a side length of 12 cm is cut once. The cut is made along \( IJ \) and exits through \( LK \), such that \( AI = DL = 4 \) cm, \( JF = KG = 3 \) cm, and the section \( IJKL \) is a rectangle. The total surface area of the two resulting parts of the cube after the cut is \( \quad \) ... | 1176 | 0 | 8,192 | -1 | 8,192 | |
Find the largest prime divisor of \( 16^2 + 81^2 \). | 53 | 0 | 2,287.4375 | -1 | 2,287.4375 | |
Given the mean of the data $x_1, x_2, \ldots, x_n$ is 2, and the variance is 3, calculate the mean and variance of the data $3x_1+5, 3x_2+5, \ldots, 3x_n+5$. | 27 | 1 | 1,872.375 | 1,872.375 | -1 | |
A circular cylindrical post has a circumference of 6 feet and a height of 18 feet. A string is wrapped around the post which spirals evenly from the bottom to the top, looping around the post exactly six times. What is the length of the string, in feet? | 18\sqrt{5} | 0.8125 | 3,124.125 | 1,954.615385 | 8,192 | |
A rectangular photograph is placed in a frame that forms a border two inches wide on all sides of the photograph. The photograph measures $8$ inches high and $10$ inches wide. What is the area of the border, in square inches? | 88 | 1. **Calculate the area of the photograph**:
The photograph measures $8$ inches in height and $10$ inches in width. Therefore, the area of the photograph is calculated as:
\[
\text{Area of photograph} = 8 \times 10 = 80 \text{ square inches}
\]
2. **Determine the dimensions of the entire framed area**:
... | 0.9375 | 4,763.125 | 4,534.533333 | 8,192 |
In the Cartesian coordinate system $xOy$, the curve $C$ is given by $\frac{x^2}{4} + \frac{y^2}{3} = 1$. Taking the origin $O$ of the Cartesian coordinate system $xOy$ as the pole and the positive half-axis of $x$ as the polar axis, and using the same unit length, a polar coordinate system is established. It is known t... | 2\sqrt{5} | 0.375 | 5,971.1875 | 5,020 | 6,541.9 | |
Calculate the volume of the solid bounded by the surfaces \(x + z = 6\), \(y = \sqrt{x}\), \(y = 2\sqrt{x}\), and \(z = 0\) using a triple integral. | \frac{48}{5} \sqrt{6} | 0.0625 | 4,949.375 | 3,401 | 5,052.6 | |
Given that tetrahedron P-ABC is a 'Bie'zhi' with PA⊥ plane ABC, PA=AB=2, and AC=4, and all four vertices of the tetrahedron P-ABC lie on the surface of sphere O, calculate the surface area of the sphere O. | 20\pi | 0 | 6,017.3125 | -1 | 6,017.3125 | |
What is the remainder of $5^{2010}$ when it is divided by 7? | 1 | 1 | 2,066.875 | 2,066.875 | -1 | |
What is the result when we compute $$1^3 + 2^3 + 3^3 + 4^3 + \dots + 99^3 + 100^3 $$and $$(-1)^3 + (-2)^3 + (-3)^3 + (-4)^3 + \dots + (-99)^3 + (-100)^3,$$and then add the two results? | 0 | 1 | 1,893.5 | 1,893.5 | -1 | |
A regular hexahedron with an edge length of $1$ is cut by planes passing through the common vertex of three edges and their respective midpoints. After removing the $8$ triangular pyramids, the volume of the remaining convex polyhedron is $\_\_\_\_\_\_$. | \frac{5}{6} | 0.5625 | 7,274.1875 | 6,560.333333 | 8,192 | |
On the eve of the 2010 Guangzhou Asian Games, a 12-person tour group took a commemorative photo near a venue of the Asian Games. They initially stood in a formation with 4 people in the front row and 8 people in the back row. Now, the photographer plans to keep the order of the front row unchanged, and move 2 people fr... | 560 | 0.3125 | 7,348.5625 | 6,233 | 7,855.636364 | |
Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? | 75 | 1. **Identify the areas of sub-triangles**:
- The area of $\triangle ABC$ is given as $120$.
- Since $D$ and $E$ are midpoints of $AB$ and $AC$ respectively, $AD = DB = 25$ and $AE = EC = 5$.
- The area of $\triangle ADE$ can be calculated using the formula for the area of a triangle, $\frac{1}{2} \cdot \text... | 0.25 | 7,899 | 7,346.25 | 8,083.25 |
What is the largest integer less than $\log_2 \frac{2}{1} + \log_2 \frac{3}{2} + \cdots + \log_2 \frac{2009}{2008} + \log_2 \frac{2010}{2009}$? | 10 | 0.875 | 5,519 | 5,137.142857 | 8,192 | |
For positive integers $a$, $b$, and $c$ with $a < b < c$, consider collections of postage stamps in denominations $a$, $b$, and $c$ cents that contain at least one stamp of each denomination. If there exists such a collection that contains sub-collections worth every whole number of cents up to $1000$ cents, let $f(a, ... | 188 | Notice that we must have $a = 1$, otherwise $1$ cent stamp cannot be represented. At least $b-1$ numbers of $1$ cent stamps are needed to represent the values less than $b$. Using at most $c-1$ stamps of value $1$ and $b$, it can have all the values from $1$ to $c-1$ cents. Plus $\lfloor \frac{999}{c} \rfloor$ stamps o... | 0 | 8,192 | -1 | 8,192 |
Given the function $f(x)=\sin (x+ \frac{7\pi}{4})+\cos (x- \frac{3\pi}{4})$, where $x\in R$.
(1) Find the smallest positive period and the minimum value of $f(x)$;
(2) Given that $f(\alpha)= \frac{6}{5}$, where $0 < \alpha < \frac{3\pi}{4}$, find the value of $f(2\alpha)$. | \frac{31\sqrt{2}}{25} | 0 | 6,523.6875 | -1 | 6,523.6875 | |
Find the minimum value of $\sin^4 x + \cos^4 x.$ | \frac{1}{2} | 1 | 3,026.625 | 3,026.625 | -1 | |
Jindra collects dice, all of the same size. Yesterday he found a box in which he started stacking the dice. He managed to fully cover the square base with one layer of dice. He similarly stacked five more layers, but he ran out of dice halfway through the next layer. Today, Jindra received 18 more dice from his grandmo... | 234 | 0.0625 | 3,380.4375 | 3,179 | 3,393.866667 | |
If $\det \mathbf{M} = -2,$ then find $ \det (\mathbf{M}^4).$ | 16 | 1 | 1,561.0625 | 1,561.0625 | -1 | |
I won a trip for four to the Super Bowl. I can bring three of my friends. I have 8 friends. In how many ways can I form my Super Bowl party? | 56 | 1 | 1,734.0625 | 1,734.0625 | -1 | |
The diagram shows a shaded semicircle of diameter 4, from which a smaller semicircle has been removed. The two semicircles touch at exactly three points. What fraction of the larger semicircle is shaded? | $\frac{1}{2}$ | 0 | 8,192 | -1 | 8,192 | |
Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([4, 6]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{9 - \sqrt{19}}{2}\right)) \ldots) \). If necessary, round the answer to two decimal places. | 6.68 | 0.0625 | 8,045.8125 | 7,244 | 8,099.266667 | |
Petya considered moments in his life to be happy when his digital clock showed the number of hours to be six times the number of minutes, or vice versa. Petya fell asleep and woke up at a happy moment in his life, without missing any such moment during his sleep. What is the maximum whole number of minutes Petya's slee... | 361 | 0 | 7,977.8125 | -1 | 7,977.8125 | |
Find the solutions to
\[\frac{13x - x^2}{x + 1} \left( x + \frac{13 - x}{x + 1} \right) = 42.\]Enter all the solutions, separated by commas. | 1, 6, 3 + \sqrt{2}, 3 - \sqrt{2} | 0 | 4,685.5625 | -1 | 4,685.5625 |
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