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 |
|---|---|---|---|---|---|---|
Three distinct vertices of a cube are chosen at random. What is the probability that the plane determined by these three vertices contains points inside the cube? | \frac{4}{7} |
To solve this problem, we need to determine the probability that three randomly chosen vertices of a cube will form a plane that intersects the interior of the cube. We will use a combinatorial approach to count the favorable and total outcomes.
#### Step 1: Total number of ways to choose three vertices
A cube has 8 ... | 0.375 | 6,324.9375 | 4,094.666667 | 7,663.1 |
Let \( p \) and \( q \) be the two distinct solutions to the equation
\[ (x-6)(3x+10) = x^2 - 19x + 50. \]
What is \( (p + 2)(q + 2) \)? | 108 | 0 | 2,711.625 | -1 | 2,711.625 | |
Connecting the right-angled vertex of a right triangle and the two trisection points on the hypotenuse, the lengths of the two resulting line segments are $\sin \alpha$ and $\cos \alpha$ (where $0 < \alpha < \frac{\pi}{2}$). What is the length of the hypotenuse? | $\frac{3}{\sqrt{5}}$ | 0 | 5,226.0625 | -1 | 5,226.0625 | |
The shelf life $y$ (in hours) of a certain food product and its storage temperature $x$ (in °C) satisfy the function relationship $y=e^{kx+b}$ (where $e=2.718\ldots$ is the base of the natural logarithm, and $k$, $b$ are constants). It is known that the shelf life of this food product is 192 hours at 0°C, and 24 hours ... | 48 | 0.9375 | 4,392.3125 | 4,139 | 8,192 | |
Compute
$$\sum_{k=1}^{1000} k(\lceil \log_{\sqrt{2}}{k}\rceil- \lfloor\log_{\sqrt{2}}{k} \rfloor).$$ | 499477 | 0.8125 | 3,772.5625 | 3,516.923077 | 4,880.333333 | |
A cube 4 units on each side is composed of 64 unit cubes. Two faces of the larger cube that share an edge are painted blue, and the cube is disassembled into 64 unit cubes. Two of the unit cubes are selected uniformly at random. What is the probability that one of two selected unit cubes will have exactly two painted f... | \frac{1}{14} | 0.4375 | 5,234.75 | 4,976.285714 | 5,435.777778 | |
Find all real solutions $(x, y)$ of the system $x^{2}+y=12=y^{2}+x$. | (3,3),(-4,-4),\left(\frac{1+3 \sqrt{5}}{2}, \frac{1-3 \sqrt{5}}{2}\right),\left(\frac{1-3 \sqrt{5}}{2}, \frac{1+3 \sqrt{5}}{2}\right) | We have $x^{2}+y=y^{2}+x$ which can be written as $(x-y)(x+y-1)=0$. The case $x=y$ yields $x^{2}+x-12=0$, hence $(x, y)=(3,3)$ or $(-4,-4)$. The case $y=1-x$ yields $x^{2}+1-x-12=x^{2}-x-11=0$ which has solutions $x=\frac{1 \pm \sqrt{1+44}}{2}=\frac{1 \pm 3 \sqrt{5}}{2}$. The other two solutions follow. | 0 | 3,932.375 | -1 | 3,932.375 |
Let $a_{0}, a_{1}, a_{2}, \ldots$ be a sequence of real numbers defined by $a_{0}=21, a_{1}=35$, and $a_{n+2}=4 a_{n+1}-4 a_{n}+n^{2}$ for $n \geq 2$. Compute the remainder obtained when $a_{2006}$ is divided by 100. | 0 | No pattern is evident in the first few terms, so we look for a formula for $a_{n}$. If we write $a_{n}=A n^{2}+B n+C+b_{n}$ and put $b_{n+2}=4 b_{n+1}-4 b_{n}$. Rewriting the original recurrence, we find $$\begin{aligned} A n^{2}+(4 A+B) n+(4 A+2 B+C)+b_{n+2} & \\ =4\left(A n^{2}+(2 A+B) n+(A+B+C)\right. & \left.+b_{n+... | 0 | 7,646.875 | -1 | 7,646.875 |
Given that the side length of square $ABCD$ is 1, point $M$ is the midpoint of side $AD$, and with $M$ as the center and $AD$ as the diameter, a circle $\Gamma$ is drawn. Point $E$ is on segment $AB$, and line $CE$ is tangent to circle $\Gamma$. Find the area of $\triangle CBE$. | 1/4 | 0 | 3,433.0625 | -1 | 3,433.0625 | |
In triangle \( \triangle ABC \), given \( AB = 4 \), \( AC = 3 \), and \( P \) is a point on the perpendicular bisector of \( BC \), find \( \overrightarrow{BC} \cdot \overrightarrow{AP} \). | -\frac{7}{2} | 0.9375 | 4,882.0625 | 4,661.4 | 8,192 | |
Let $\omega = \cos\frac{2\pi}{7} + i \cdot \sin\frac{2\pi}{7},$ where $i = \sqrt{-1}.$ Find the value of the product\[\prod_{k=0}^6 \left(\omega^{3k} + \omega^k + 1\right).\] | 024 | The product can be factored into $-(r-1)(s-1)(t-1)(r-w)(s-w)(t-w)(r-w^2)(s-w^2)(t-w^2)....(r-w^6)(s-w^6)(t-w^6)$,
where $r,s,t$ are the roots of the polynomial $x^3+x+1=0$.
This is then $-(r^7-1)(s^7-1)(t^7-1)$ because $(r^7-1)$ and $(r-1)(r-w)(r-w^2)...(r-w^6)$ share the same roots.
To find $-(r^7-1)(s^7-1)(t^7-1)$... | 0 | 8,192 | -1 | 8,192 |
Two jokers are added to a 52 card deck and the entire stack of 54 cards is shuffled randomly. What is the expected number of cards that will be between the two jokers? | 52 / 3 | Each card has an equal likelihood of being either on top of the jokers, in between them, or below the jokers. Thus, on average, $1 / 3$ of them will land between the two jokers. | 0.4375 | 7,587.0625 | 6,809.285714 | 8,192 |
A trapezoid \(AEFG\) (\(EF \parallel AG\)) is positioned inside a square \(ABCD\) with a side length of 14, such that points \(E\), \(F\), and \(G\) lie on sides \(AB\), \(BC\), and \(CD\) respectively. The diagonals \(AF\) and \(EG\) are perpendicular, and \(EG = 10\sqrt{2}\). Find the perimeter of the trapezoid. | 45 | 0.5625 | 6,488.875 | 5,164.222222 | 8,192 | |
An abstract animal lives in groups of two and three.
In a forest, there is one group of two and one group of three. Each day, a new animal arrives in the forest and randomly chooses one of the inhabitants. If the chosen animal belongs to a group of three, that group splits into two groups of two; if the chosen animal ... | 4/7 | 0 | 7,996.9375 | -1 | 7,996.9375 | |
A workshop produces items of types $A$ and $B$. One item of type $A$ requires 10 kg of steel and 23 kg of non-ferrous metals, while an item of type $B$ requires 70 kg of steel and 40 kg of non-ferrous metals. The profit from selling an item of type $A$ is 80 thousand rubles, and for type $B$ it is 100 thousand rubles. ... | 2180 | 0 | 7,654.375 | -1 | 7,654.375 | |
Let \(\mathbf{a} = \begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}\), \(\mathbf{b} = \begin{pmatrix} 4 \\ 5 \\ -2 \end{pmatrix}\), and \(\mathbf{c} = \begin{pmatrix} 1 \\ 2 \\ 5 \end{pmatrix}\). Compute the following:
\[
((\mathbf{a} + \mathbf{c}) - \mathbf{b}) \cdot [(\mathbf{b} - \mathbf{c}) \times (\mathbf{c} - \mathbf{a... | 161 | 0 | 4,639 | -1 | 4,639 | |
A certain product costs $6$ per unit, sells for $x$ per unit $(x > 6)$, and has an annual sales volume of $u$ ten thousand units. It is known that $\frac{585}{8} - u$ is directly proportional to $(x - \frac{21}{4})^2$, and when the selling price is $10$ dollars, the annual sales volume is $28$ ten thousand units.
(1) ... | 135 | 0.25 | 6,582.8125 | 3,740 | 7,530.416667 | |
Three of the four endpoints of the axes of an ellipse are, in some order, \[(10, -3), \; (15, 7), \; (25, -3).\] Find the distance between the foci of the ellipse. | 11.18 | 0 | 7,551.25 | -1 | 7,551.25 | |
Let $S$ be the set of all real numbers $x$ such that $0 \le x \le 2016 \pi$ and $\sin x < 3 \sin(x/3)$ . The set $S$ is the union of a finite number of disjoint intervals. Compute the total length of all these intervals. | 1008\pi | 0.75 | 5,689.75 | 4,978.333333 | 7,824 | |
Given the sequence $\{a_n\}$ with the general term formula $a_n = -n^2 + 12n - 32$, determine the maximum value of $S_n - S_m$ for any $m, n \in \mathbb{N^*}$ and $m < n$. | 10 | 0.125 | 8,141.6875 | 7,789.5 | 8,192 | |
In $\triangle ABC$, it is known that $\cos C + (\cos A - \sqrt{3} \sin A) \cos B = 0$.
(1) Find the measure of angle $B$.
(2) If $\sin (A - \frac{\pi}{3}) = \frac{3}{5}$, find $\sin 2C$. | \frac{24 + 7\sqrt{3}}{50} | 0 | 6,471.9375 | -1 | 6,471.9375 | |
The function \( f: \mathbb{R} \rightarrow \mathbb{R} \) is continuous. For every real number \( x \), the equation \( f(x) \cdot f(f(x)) = 1 \) holds. It is known that \( f(1000) = 999 \). Find \( f(500) \). | \frac{1}{500} | 0 | 8,192 | -1 | 8,192 | |
Given a sequence $\{a_n\}$ where each term is a positive number and satisfies the relationship $a_{n+1}^2 = ta_n^2 +(t-1)a_na_{n+1}$, where $n\in \mathbb{N}^*$.
(1) If $a_2 - a_1 = 8$, $a_3 = a$, and the sequence $\{a_n\}$ is unique:
① Find the value of $a$.
② Let another sequence $\{b_n\}$ satisfy $b_n = \frac... | 32 | 0.0625 | 8,192 | 8,192 | 8,192 | |
Let $a \geq b \geq c$ be real numbers such that $$\begin{aligned} a^{2} b c+a b^{2} c+a b c^{2}+8 & =a+b+c \\ a^{2} b+a^{2} c+b^{2} c+b^{2} a+c^{2} a+c^{2} b+3 a b c & =-4 \\ a^{2} b^{2} c+a b^{2} c^{2}+a^{2} b c^{2} & =2+a b+b c+c a \end{aligned}$$ If $a+b+c>0$, then compute the integer nearest to $a^{5}$. | 1279 | We factor the first and third givens, obtaining the system $$\begin{aligned} a^{2} b c+a b^{2} c+a b c^{2}-a-b-c=(a b c-1)(a+b+c) & =-8 \\ a^{2} b+a^{2} c+b^{2} c+b^{2} a+c^{2} a+c^{2} b+3 a b c=(a b+b c+c a)(a+b+c) & =-4 \\ a^{2} b^{2} c+a b^{2} c^{2}+a^{2} b c^{2}-a b-b c-c a=(a b c-1)(a b+b c+c a) & =2 \end{aligned}... | 0 | 8,192 | -1 | 8,192 |
In $\triangle ABC$, if $\angle A=60^{\circ}$, $\angle C=45^{\circ}$, and $b=4$, then the smallest side of this triangle is $\_\_\_\_\_\_\_.$ | 4\sqrt{3}-4 | 0.625 | 5,819.9375 | 5,945.6 | 5,610.5 | |
Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$ . If $AI=3$ and the distance from $I$ to $BC$ is $2$ , what is the square of length on $BC$ ? | 80 | 0.875 | 4,860.4375 | 4,384.5 | 8,192 | |
Let $(a_1, a_2, a_3,\ldots,a_{13})$ be a permutation of $(1,2,3,\ldots,13)$ for which
$$a_1 > a_2 > a_3 > a_4 > a_5 > a_6 > a_7 \mathrm{\ and \ } a_7 < a_8 < a_9 < a_{10} < a_{11} < a_{12} < a_{13}.$$
Find the number of such permutations. | 924 | 0.5 | 5,768.1875 | 4,702 | 6,834.375 | |
Let \( N = 34 \times 34 \times 63 \times 270 \). The ratio of the sum of all odd factors of \( N \) to the sum of all even factors of \( N \) is ( ). | 1: 14 | 0 | 5,957.1875 | -1 | 5,957.1875 | |
Assume that $x$ is a positive real number. Which is equivalent to $\sqrt[3]{x\sqrt{x}}$? | $x^{\frac{1}{2}}$ | 1. **Rewrite the expression inside the cube root:**
We start by expressing the term inside the cube root in terms of powers of $x$:
\[
x\sqrt{x} = x \cdot x^{1/2} = x^{1 + 1/2} = x^{3/2}
\]
2. **Apply the cube root to the expression:**
Now, we take the cube root of $x^{3/2}$:
\[
\sqrt[3]{x^{3/... | 0 | 2,822.3125 | -1 | 2,822.3125 |
Given \( a_{n} = \mathrm{C}_{200}^{n} \cdot (\sqrt[3]{6})^{200-n} \cdot \left( \frac{1}{\sqrt{2}} \right)^{n} \) for \( n = 1, 2, \ldots, 95 \), find the number of integer terms in the sequence \(\{a_{n}\}\). | 15 | 0 | 7,132 | -1 | 7,132 | |
Given the function $f(x) = \sin(x - \varphi)$ and $|\varphi| < \frac{\pi}{2}$, and $\int_{0}^{\frac{2\pi}{3}} f(x) \, dx = 0$, find the equation of one of the axes of symmetry of the graph of function $f(x)$. | \frac{5\pi}{6} | 0.9375 | 5,508.75 | 5,329.866667 | 8,192 | |
At the moment when Pierrot left the "Commercial" bar, heading to the "Theatrical" bar, Jeannot was leaving the "Theatrical" bar on his way to the "Commercial" bar. They were walking at constant (but different) speeds. When the vagabonds met, Pierrot proudly noted that he had walked 200 meters more than Jeannot. After ... | 1000 | 0.5625 | 6,444.5 | 5,085.333333 | 8,192 | |
What is the discriminant of $3x^2 - 7x - 12$? | 193 | 1 | 1,653.3125 | 1,653.3125 | -1 | |
Let \( a, b, c, d \) be 4 distinct nonzero integers such that \( a + b + c + d = 0 \) and the number \( M = (bc - ad)(ac - bd)(ab - cd) \) lies strictly between 96100 and 98000. Determine the value of \( M \). | 97344 | 0 | 8,192 | -1 | 8,192 | |
If 500 were expressed as a sum of at least two distinct powers of 2, what would be the least possible sum of the exponents of these powers? | 32 | 0 | 8,192 | -1 | 8,192 | |
The volume of a certain rectangular solid is $8 \text{ cm}^3$, its total surface area is $32 \text{ cm}^2$, and its three dimensions are in geometric progression. The sums of the lengths in cm of all the edges of this solid is | 32 | 0.9375 | 4,234.75 | 3,970.933333 | 8,192 | |
The values of $a$ in the equation: $\log_{10}(a^2 - 15a) = 2$ are: | 20, -5 | 1. **Convert the logarithmic equation to exponential form:**
Given the equation $\log_{10}(a^2 - 15a) = 2$, we can rewrite it in exponential form:
\[
10^2 = a^2 - 15a
\]
Simplifying, we get:
\[
100 = a^2 - 15a
\]
Rearranging the terms, we obtain a quadratic equation:
\[
a^2 - 15a - 100 ... | 0 | 2,049.125 | -1 | 2,049.125 |
The graph, $G$ of $y=\log_{10}x$ is rotated $90^{\circ}$ counter-clockwise about the origin to obtain a new graph $G'$. What is the equation for $G'$? | 10^{-x} | 1. **Understanding the rotation**: Rotating a point $(x, y)$ $90^\circ$ counterclockwise about the origin results in the point $(-y, x)$. This can be verified using rotation matrices:
\[
\begin{bmatrix}
\cos(90^\circ) & -\sin(90^\circ) \\
\sin(90^\circ) & \cos(90^\circ)
\end{bmatrix}
\begin{bmatrix}
... | 0.75 | 4,454.625 | 3,716.25 | 6,669.75 |
In the third year of high school, the class organized a fun sports competition. After multiple rounds of competition, Class A and Class B entered the finals. There were three events in the finals, with the winner of each event receiving 2 points and the loser receiving -1 point. There were no draws. The class with the ... | 0.9 | 0 | 7,311.125 | -1 | 7,311.125 | |
A glass is filled to the brim with salty water. Fresh ice with mass \( m = 502 \) floats on the surface. What volume \( \Delta V \) of water will spill out of the glass by the time the ice melts? Neglect surface tension. The density of fresh ice is \( \rho_{n} = 0.92 \, \text{g/cm}^3 \), the density of salty ice is \( ... | 2.63 | 0 | 7,143.5625 | -1 | 7,143.5625 | |
Given that $\sin \alpha + \cos \alpha = -\frac{3}{\sqrt{5}}$, and $|\sin \alpha| > |\cos \alpha|$, find the value of $\tan \frac{\alpha}{2}$. | -\frac{\sqrt{5} + 1}{2} | 0 | 5,891.75 | -1 | 5,891.75 | |
Every time these two wheels are spun, two numbers are selected by the pointers. What is the probability that the sum of the two selected numbers is even? | \frac{1}{2} | 1. **Understanding the Problem**: We need to find the probability that the sum of the numbers selected by spinning two wheels is even. For the sum to be even, both numbers must either be both even or both odd.
2. **Determining the Probabilities**:
- **Probability of Even Numbers**: Let's denote the probability that... | 1 | 5,201.5625 | 5,201.5625 | -1 |
The figure is constructed from $11$ line segments, each of which has length $2$. The area of pentagon $ABCDE$ can be written as $\sqrt{m} + \sqrt{n}$, where $m$ and $n$ are positive integers. What is $m + n ?$ | 23 | 1. **Identify Key Triangles and Midpoint**: Let $M$ be the midpoint of $CD$. Since each side of the pentagon and the internal segments are of length $2$, and given the symmetry and equal lengths, we can infer that triangles $AED$ and $ABC$ are isosceles with a vertex angle of $120^\circ$ formed by extending sides of eq... | 0 | 8,192 | -1 | 8,192 |
The cards in a stack of $2n$ cards are numbered consecutively from 1 through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A.$ The remaining cards form pile $B.$ The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A,$ respectively. In this proc... | 392 | If you index the final stack $1,2,\dots,2n$, you notice that pile A resides only in the odd indices and has maintained its original order aside from flipping over. The same has happened to pile B except replace odd with even. Thus, if 131 is still at index 131, an odd number, then 131 must be from pile A. The numbers i... | 0.0625 | 8,130.875 | 7,214 | 8,192 |
Find the sum of the distinct prime factors of $7^7 - 7^4$. | 31 | 0.9375 | 2,454.125 | 2,425.8 | 2,879 | |
Determine the maximum and minimum values of the function $f(x)=x^3 - \frac{3}{2}x^2 + 5$ on the interval $[-2, 2]$. | -9 | 0.875 | 2,503.0625 | 2,570.142857 | 2,033.5 | |
If the general term of the sequence \\(\{a_n\}\) is \\(a_n = (-1)^{n+1} \cdot (3n-2)\\), then \\(a_1 + a_2 + \cdots + a_{20} = \)____. | -30 | 1 | 4,337.875 | 4,337.875 | -1 | |
A set of tiles numbered 1 through 100 is modified repeatedly by the following operation: remove all tiles numbered with a perfect square, and renumber the remaining tiles consecutively starting with 1. How many times must the operation be performed to reduce the number of tiles in the set to one? | 18 | 1. **Initial Set and Operation Definition**: We start with a set of tiles numbered from 1 to 100. The operation, denoted as $P(x)$, involves removing all tiles numbered with a perfect square and renumbering the remaining tiles consecutively starting with 1.
2. **Understanding Perfect Squares**: The perfect squares bet... | 0.6875 | 5,838.5625 | 4,768.818182 | 8,192 |
The set of points satisfying the pair of inequalities $y>2x$ and $y>4-x$ is contained entirely in quadrants: | I and II | 1. **Graph the inequalities**:
- The inequality $y > 2x$ represents the region above the line $y = 2x$. This line passes through the origin and has a positive slope, dividing the plane into two regions. The region of interest is above this line.
- The inequality $y > 4 - x$ represents the region above the line $y... | 0 | 6,462.5 | -1 | 6,462.5 |
I have five apples and ten oranges. If a fruit basket must contain at least one piece of fruit, how many kinds of fruit baskets can I make? (The apples are identical and the oranges are identical. A fruit basket consists of some number of pieces of fruit, and it doesn't matter how the fruit are arranged in the basket... | 65 | 0.75 | 4,162.25 | 3,077.166667 | 7,417.5 | |
Angelica wants to choose a three-digit code for her suitcase lock. To make it easier to remember, Angelica wants all the digits in her code to be in non-decreasing order. How many different possible codes does Angelica have to choose from? | 220 | 0.6875 | 4,654.125 | 4,154.181818 | 5,754 | |
Calculate $[(15^{15} \div 15^{13})^3 \cdot 3^2] \div 2^3$. | 3^8 \cdot 5^6 \cdot 2^{-3} | 0 | 7,593.1875 | -1 | 7,593.1875 | |
Compute $\dbinom{60}{3}$. | 57020 | 0 | 3,678 | -1 | 3,678 | |
How many 10-digit numbers exist in which at least two digits are the same? | 9 \times 10^9 - 9 \times 9! | 0 | 4,346.25 | -1 | 4,346.25 | |
Through vertex $A$ of parallelogram $ABCD$, a line is drawn that intersects diagonal $BD$, side $CD$, and line $BC$ at points $E$, $F$, and $G$, respectively. Find the ratio $BE:ED$ if $FG:FE=4$. Round your answer to the nearest hundredth if needed. | 2.24 | 0.5625 | 7,236.3125 | 6,779.555556 | 7,823.571429 | |
Given that α is in (0, π) and cosα = -$$\frac{15}{17}$$, find the value of sin($$\frac{π}{2}$$ + α) • tan(π + α). | \frac{8}{17} | 0.8125 | 4,154.9375 | 3,447.153846 | 7,222 | |
Kolya, after walking one-fourth of the way from home to school, realized that he forgot his problem book. If he does not go back for it, he will arrive at school 5 minutes before the bell rings, but if he goes back, he will be 1 minute late. How long (in minutes) does it take to get to school? | 12 | 0 | 7,905.3125 | -1 | 7,905.3125 | |
The quadratic equation $x^2+mx+n=0$ has roots that are twice those of $x^2+px+m=0,$ and none of $m,$ $n,$ and $p$ is zero. What is the value of $n/p?$ | 8 | 0.9375 | 3,171.25 | 2,836.533333 | 8,192 | |
Vasya has three cans of paint of different colors. In how many different ways can he paint a fence consisting of 10 planks so that any two adjacent planks are different colors and he uses all three colors? Provide a justification for your answer. | 1530 | 0.1875 | 7,730.6875 | 6,423.666667 | 8,032.307692 | |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$f(x^2 + y) \ge (\frac{1}{x} + 1)f(y)$$
holds for all $x \in \mathbb{R} \setminus \{0\}$ and all $y \in \mathbb{R}$. | f(x) = 0 |
To find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) that satisfy the given inequality:
\[
f(x^2 + y) \ge \left(\frac{1}{x} + 1\right)f(y)
\]
for all \( x \in \mathbb{R} \setminus \{0\} \) and \( y \in \mathbb{R} \), we'll start by analyzing and simplifying the inequality.
### Step 1: Setting \( y = 0 \... | 0 | 7,814.125 | -1 | 7,814.125 |
Given there are ten steps from the first floor to the second floor, calculate the total number of ways Xiao Ming can go from the first floor to the second floor. | 89 | 0.125 | 6,284.6875 | 4,663.5 | 6,516.285714 | |
Calculate the value of the following expressions:
1. $\sqrt[4]{(3-\pi )^{4}}+(0.008)\;^{- \frac {1}{3}}-(0.25)\;^{ \frac {1}{2}}×( \frac {1}{ \sqrt {2}})^{-4}$
2. $\log _{3} \sqrt {27}-\log _{3} \sqrt {3}-\lg 625-\lg 4+\ln (e^{2})- \frac {4}{3}\lg \sqrt {8}$ | -1 | 0.8125 | 4,185.5625 | 3,767.846154 | 5,995.666667 | |
Let $p$, $q$, $r$, and $s$ be real numbers with $|p-q|=3$, $|q-r|=5$, and $|r-s|=7$. What is the sum of all possible values of $|p-s|$? | 30 | 0.5625 | 5,529.875 | 4,610.555556 | 6,711.857143 | |
When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 8$. | \frac{1}{36} | 0 | 6,402.5 | -1 | 6,402.5 | |
$\sqrt{\frac{1}{9} + \frac{1}{16}} = $ | \frac{5}{12} | 1. **Identify the common denominator** for the fractions inside the square root:
\[
\frac{1}{9} + \frac{1}{16}
\]
The least common multiple of 9 and 16 is 144. Therefore, we rewrite the fractions with this common denominator:
\[
\frac{1}{9} = \frac{16}{144}, \quad \frac{1}{16} = \frac{9}{144}
\]
2... | 1 | 1,726.4375 | 1,726.4375 | -1 |
Given $$\frac {1}{3}$$≤a≤1, if the function f(x)=ax<sup>2</sup>-2x+1 has its maximum value M(a) and minimum value N(a) in the interval [1,3], let g(a)=M(a)-N(a).
(1) Find the expression for g(a);
(2) Describe the intervals where g(a) is increasing and decreasing (no proof required), and find the minimum value of g(a). | \frac {1}{2} | 0.5625 | 5,362.5 | 4,849 | 6,022.714286 | |
Two concentric circles have radii of 24 and 36 units, respectively. A shaded region is formed between these two circles. A new circle is to be drawn such that its diameter is equal to the area of the shaded region. What must the diameter of this new circle be? Express your answer in simplest radical form. | 720 \pi | 0.75 | 3,362.1875 | 3,453.5 | 3,088.25 | |
Find the real roots of
\[x^4 - 2x^3 - x + 2 = 0.\] | 1,2 | 0 | 2,387.625 | -1 | 2,387.625 | |
The length of the median to the hypotenuse of an isosceles, right triangle is $10$ units. What is the length of a leg of the triangle, in units? Express your answer in simplest radical form. | 10\sqrt{2} | 1 | 2,246.25 | 2,246.25 | -1 | |
Piravena must make a trip from $A$ to $B$, then from $B$ to $C$, then from $C$ to $A$. Each of these three parts of the trip is made entirely by bus or entirely by airplane. The cities form a right-angled triangle as shown, with $C$ a distance of 3000 km from $A$ and with $B$ a distance of 3250 km from $A$. To take a... | \$1012.50 | 0.3125 | 4,542.1875 | 4,581.8 | 4,524.181818 | |
Let $ABCD$ be a rectangle with $AB=10$ and $BC=26$ . Let $\omega_1$ be the circle with diameter $\overline{AB}$ and $\omega_2$ be the circle with diameter $\overline{CD}$ . Suppose $\ell$ is a common internal tangent to $\omega_1$ and $\omega_2$ and that $\ell$ intersects $AD$ and $BC$ at $E$ ... | 24 | 0 | 6,906.75 | -1 | 6,906.75 | |
A cube with a side length of 1 meter was cut into smaller cubes with a side length of 1 centimeter and arranged in a straight line. What is the length of the resulting line? | 10000 | 0.5 | 1,869.375 | 2,125.125 | 1,613.625 | |
The graph of the rational function $\frac{q(x)}{2x^5+x^4-7x^2+1}$ has a horizontal asymptote. What is the largest possible degree of $q(x)$? | 5 | 0.9375 | 2,078.9375 | 1,671.4 | 8,192 | |
Let $f(x)=3x-2$, and let $g(x)=f(f(f(f(x))))$. If the domain of $g$ is $0\leq x\leq 2$, compute the range of $g$. | -80\leq g(x)\leq 82 | 0 | 2,566 | -1 | 2,566 | |
Compute
\[\prod_{n = 1}^{20} \frac{n + 3}{n}.\] | 1771 | 0.875 | 5,075.5625 | 4,630.357143 | 8,192 | |
Given the odd function $f(x)$ is increasing on the interval $[3, 7]$ and its minimum value is 5, determine the nature of $f(x)$ and its minimum value on the interval $[-7, -3]$. | -5 | 0.1875 | 6,224.9375 | 2,281 | 7,135.076923 | |
The sum of the first four terms of an arithmetic progression, as well as the sum of the first nine terms, are natural numbers. Additionally, the first term \( b_{1} \) of this progression satisfies the inequality \( b_{1} \leq \frac{3}{4} \). What is the greatest possible value of \( b_{1} \)? | 11/15 | 0.125 | 8,051.5625 | 7,068.5 | 8,192 | |
Given that $F\_1$ and $F\_2$ are the foci of a hyperbola, a line passing through $F\_2$ perpendicular to the real axis intersects the hyperbola at points $A$ and $B$. If $BF\_1$ intersects the $y$-axis at point $C$, and $AC$ is perpendicular to $BF\_1$, determine the eccentricity of the hyperbola. | \sqrt{3} | 0.9375 | 3,956.9375 | 3,674.6 | 8,192 | |
Given the function $f(x)=\sin x+a\cos x(x∈R)$ whose one symmetric axis is $x=- \frac {π}{4}$.
(I) Find the value of $a$ and the monotonically increasing interval of the function $f(x)$;
(II) If $α$, $β∈(0, \frac {π}{2})$, and $f(α+ \frac {π}{4})= \frac { \sqrt {10}}{5}$, $f(β+ \frac {3π}{4})= \frac {3 \sqrt {5}}{5}$, f... | \frac { \sqrt {2}}{2} | 0 | 7,316.1875 | -1 | 7,316.1875 | |
The length of the chord cut by the line $y= \frac{1}{2}x+1$ on the ellipse $x^2+4y^2=16$ is ______. | \sqrt{35} | 0.75 | 6,153.0625 | 5,473.416667 | 8,192 | |
Given the sample data: $110$, $120$, $120$, $120$, $123$, $123$, $140$, $146$, $150$, $162$, $165$, $174$, $190$, $210$, $235$, $249$, $280$, $318$, $428$, $432$, find the $75$th percentile. | 242 | 0 | 7,930.5625 | -1 | 7,930.5625 | |
The maximum and minimum values of the function y=2x^3-3x^2-12x+5 on the interval [0,3] need to be determined. | -15 | 1 | 2,375.8125 | 2,375.8125 | -1 | |
The clock shows $00:00$, with both the hour and minute hands coinciding. Considering this coincidence as number 0, determine after what time interval (in minutes) they will coincide for the 21st time. If the answer is not an integer, round the result to the nearest hundredth. | 1374.55 | 0.125 | 6,164.9375 | 5,959.5 | 6,194.285714 | |
The positive integer equal to the expression
\[ \sum_{i=0}^{9} \left(i+(-9)^i\right)8^{9-i} \binom{9}{i}\]
is divisible by exactly six distinct primes. Find the sum of these six distinct prime factors.
*Team #7* | 835 | 0.375 | 6,953.375 | 4,889 | 8,192 | |
Given a rectangle $A B C D$, let $X$ and $Y$ be points on $A B$ and $B C$, respectively. Suppose the areas of the triangles $\triangle A X D$, $\triangle B X Y$, and $\triangle D Y C$ are 5, 4, and 3, respectively. Find the area of $\triangle D X Y$. | 2\sqrt{21} | 0.375 | 7,325.3125 | 6,448.333333 | 7,851.5 | |
Given two parabolas $N\_1$: $y=ax^{2}+bx+c$ and $N\_2$: $y=-ax^{2}+dx+e$ with vertices $P\_1(x\_1,y\_1)$ and $P\_2(x\_2,y\_2)$, respectively. The parabolas intersect at points $A(12,21)$ and $B(28,3)$ (both distinct from the vertices). Determine the value of $\frac{x\_1+x\_2}{y\_1+y\_2}$. | \frac{5}{3} | 0.1875 | 7,543.5625 | 4,767.333333 | 8,184.230769 | |
Given a sphere O with a radius of 2, a cone is inscribed in the sphere O. When the volume of the cone is maximized, find the radius of the sphere inscribed in the cone. | \frac{4(\sqrt{3} - 1)}{3} | 0 | 7,905.3125 | -1 | 7,905.3125 | |
On a rectangular sheet of paper, a picture in the shape of a "cross" was drawn using two rectangles $ABCD$ and $EFGH$, with their sides parallel to the edges of the sheet. It is known that $AB=9$, $BC=5$, $EF=3$, $FG=10$. Find the area of the quadrilateral $AFCH$. | 52.5 | 0 | 7,476.9375 | -1 | 7,476.9375 | |
29 boys and 15 girls came to the ball. Some boys danced with some girls (no more than once per pair). After the ball, each person told their parents how many times they danced. What is the greatest number of distinct counts that the children could report? | 29 | 0.1875 | 7,917.6875 | 6,729 | 8,192 | |
In a dark room drawer, there are 100 red socks, 80 green socks, 60 blue socks, and 40 black socks. A young person picks out one sock at a time without seeing its color. To ensure that at least 10 pairs of socks are obtained, what is the minimum number of socks they must pick out?
(Assume that two socks of the same colo... | 23 | 0.1875 | 7,704.8125 | 5,593.666667 | 8,192 | |
If
\[\begin{pmatrix} 1 & 2 & a \\ 0 & 1 & 4 \\ 0 & 0 & 1 \end{pmatrix}^n = \begin{pmatrix} 1 & 18 & 2007 \\ 0 & 1 & 36 \\ 0 & 0 & 1 \end{pmatrix},\]then find $a + n.$ | 200 | 0.4375 | 6,410.625 | 4,397.857143 | 7,976.111111 | |
Four points are randomly chosen from the vertices of a regular 12-sided polygon. Find the probability that the four chosen points form a rectangle (including square). | 1/33 | 0.125 | 7,625.0625 | 3,656.5 | 8,192 | |
Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need? | 36 | 1. **Identify the ratios**: According to the problem, the recipe requires:
- 4 times as much water as sugar.
- Twice as much sugar as lemon juice.
Let's denote the amount of lemon juice used as $L$, the amount of sugar as $S$, and the amount of water as $W$. From the problem, we have:
\[
S = 2L \quad \t... | 0 | 2,432.6875 | -1 | 2,432.6875 |
Sarah intended to multiply a two-digit number and a three-digit number, but she left out the multiplication sign and simply placed the two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah should have obtained. What is the sum of... | 126 | 0.9375 | 4,513.5625 | 4,268.333333 | 8,192 | |
In the expanded country of Mathlandia, all automobile license plates have five symbols. The first two must be vowels (A, E, I, O, U), the next two must be different non-vowels among the 21 non-vowels in the alphabet, and the fifth must be a digit (0 through 9). Determine the probability that the plate will read "AIE19" | \frac{1}{105,000} | 0 | 2,941 | -1 | 2,941 | |
In the sport of diving from a high platform, there is a functional relationship between the athlete's height above the water surface $h$ (m) and the time $t$ (s) after the jump: $h(t)=-4.9t^2+6.5t+10$. Determine the moment when the instantaneous velocity is $0 \text{ m/s}$. | \frac{65}{98} | 0.75 | 4,952.3125 | 4,226.833333 | 7,128.75 | |
19) A puck is kicked up a ramp, which makes an angle of $30^{\circ}$ with the horizontal. The graph below depicts the speed of the puck versus time. What is the coefficient of friction between the puck and the ramp?
A) 0.07
B) 0.15
C) 0.22
D) 0.29
E) 0.37 | 0.29 | 0 | 7,861.0625 | -1 | 7,861.0625 | |
Find the absolute value of the difference of the solutions of $x^2-5x+5=0$. | \sqrt{5} | 1 | 1,896.5625 | 1,896.5625 | -1 | |
Find the largest positive integer $m$ which makes it possible to color several cells of a $70\times 70$ table red such that [list] [*] There are no two red cells satisfying: the two rows in which they are have the same number of red cells, while the two columns in which they are also have the same number of red cells; ... | 32 |
To find the largest positive integer \( m \) that allows coloring several cells of a \( 70 \times 70 \) table red such that:
1. There are no two red cells satisfying: the two rows in which they are have the same number of red cells, while the two columns in which they are also have the same number of red cells.
2. The... | 0 | 8,192 | -1 | 8,192 |
Let $S$ be the set of natural numbers that cannot be written as the sum of three squares. Legendre's three-square theorem states that $S$ consists of precisely the integers of the form $4^a(8b+7)$ where $a$ and $b$ are nonnegative integers. Find the smallest $n\in\mathbb N$ such that $n$ and $n+1$ are b... | 111 | 0.3125 | 7,522.4375 | 6,639.8 | 7,923.636364 |
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