Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
11,700 | Gari is seated in a jeep, and at the moment, has one 10-peso coin, two 5-peso coins, and six 1-peso coins in his pocket. If he picks four coins at random from his pocket, what is the probability that these will be enough to pay for his jeepney fare of 8 pesos? | 37/42 | 0.78125 |
11,701 | The distance from home to work is $s = 6$ km. At the moment Ivan left work, his favorite dog dashed out of the house and ran to meet him. They met at a distance of one-third of the total route from work. The dog immediately turned back and ran home. Upon reaching home, the dog turned around instantly and ran back towar... | 12 | 40.625 |
11,702 | Simplify the expression \(1.6 \frac{\left(\frac{1}{a}+\frac{1}{b}-\frac{2 c}{a b}\right)(a+b+2 c)}{\frac{1}{a^{2}}+\frac{1}{b^{2}}+\frac{2}{a b}-\frac{4 c^{2}}{a^{2} b^{2}}}\) given that \(a = 7.4\) and \(b = \frac{5}{37}\). | 1.6 | 40.625 |
11,703 | Two cyclists started a trip at the same time from the same location. They traveled the same route and returned together. Both rested along the way. The first cyclist rode twice as long as the second cyclist rested. The second cyclist rode four times as long as the first cyclist rested. Who rides their bicycle faster an... | 1.5 | 7.03125 |
11,704 | In the numbers 1, 2, 3, ..., 399, 400, the digit 2 appears a total of times. | 180 | 75 |
11,705 | Find the smallest two-digit prime number such that reversing the digits of the number produces an even number. | 23 | 4.6875 |
11,706 |
If organisms do not die but only divide, then the population will certainly never die out.
The conditions are satisfied by the function whose graph is highlighted in the image.
$$
x(p)=\left\{\begin{array}{l}
1, \text { if } 0 \leq p \leq \frac{1}{2} \\
\frac{q}{p}, \text { if } \frac{1}{2}<p \leq 1
\end{array}\rig... | \frac{2}{3} | 78.125 |
11,707 | Find the largest integer $x$ such that the number
$$
4^{27} + 4^{1000} + 4^{x}
$$
is a perfect square. | 1972 | 10.9375 |
11,708 | Twelve candidates for mayor are participating in a televised debate. At some point, one of them says, "So far, we've lied once." A second then says, "Now it's twice." A third exclaims, "Three times now," and so on, up to the twelfth who claims that before him, they lied twelve times. The presenter then stops the discus... | 11 | 67.96875 |
11,709 | Inside a convex 13-sided polygon, there are 200 points such that no three of these 213 points (including the vertices of the polygon) lie on the same line. The polygon is divided into triangles, each vertex of which is any three of the given 213 points. What is the maximum number of triangles that could result? | 411 | 18.75 |
11,710 | Given the equation $x^2+kx+6=0$ has one root as $2$, find the other root and the value of $k$. | -5 | 25 |
11,711 | Use the digits from $0$ to $9$ to form three-digit numbers that are divisible by $5$ and do not have repeating digits. How many such numbers are there? | 136 | 92.1875 |
11,712 | An infinite geometric series has a sum of 2020. If the first term, the third term, and the fourth term form an arithmetic sequence, find the first term. | 1010(1+\sqrt{5}) | 30.46875 |
11,713 | Find a natural number \( N \) that is divisible by 5 and 49, and has exactly 10 divisors, including 1 and \( N \). | 12005 | 35.9375 |
11,714 | Given a set of data $x_1, x_2, x_3, \ldots, x_n$ with a mean of 2 and a variance of 3, calculate the mean and variance of the data set $2x_1+5, 2x_2+5, 2x_3+5, \ldots, 2x_n+5$ respectively. | 12 | 84.375 |
11,715 | Let \( n \) be the smallest positive integer such that the sum of its digits is 2011. How many digits does \( n \) have? | 224 | 99.21875 |
11,716 | The last two digits of the decimal representation of the square of a natural number are the same and are not zero. What are these digits? Find all solutions. | 44 | 42.96875 |
11,717 | Compute the number of positive integers less than or equal to $10000$ which are relatively prime to $2014$ . | 4648 | 58.59375 |
11,718 | Ten identical books cost no more than 11 rubles, whereas 11 of the same books cost more than 12 rubles. How much does one book cost? | 110 | 0 |
11,719 | The novel takes 630 minutes to read aloud. The disc can hold 80 minutes of reading with at most 4 minutes of unused space. Calculate the number of minutes of reading each disc will contain. | 70 | 23.4375 |
11,720 |
Given a rectangle \(ABCD\). On two sides of the rectangle, different points are chosen: six points on \(AB\) and seven points on \(BC\). How many different triangles can be formed with vertices at the chosen points? | 231 | 83.59375 |
11,721 | In how many ways can the number 1024 be factored into three natural factors such that the first factor is divisible by the second, and the second is divisible by the third? | 14 | 56.25 |
11,722 | Simplify \[\frac{1}{\dfrac{2}{\sqrt{5}+2} + \dfrac{3}{\sqrt{7}-2}}.\] | \frac{2\sqrt{5} + \sqrt{7} + 2}{23 + 4\sqrt{35}} | 2.34375 |
11,723 | Given \( f_{1}(x)=-\frac{2x+7}{x+3}, \) and \( f_{n+1}(x)=f_{1}(f_{n}(x)), \) for \( x \neq -2, x \neq -3 \), find the value of \( f_{2022}(2021) \). | 2021 | 60.9375 |
11,724 | What is the area of the smallest square that can completely enclose a circle of radius 5 units? | 100 | 100 |
11,725 | If a sequence of numbers \(a_{1}, a_{2}, \cdots\) satisfies, for any positive integer \(n\),
$$
a_{n}=\frac{n^{2}+n-2-\sqrt{2}}{n^{2}-2},
$$
then what is the value of \(a_{1} a_{2} \cdots a_{2016}\)? | 2016\sqrt{2} - 2015 | 19.53125 |
11,726 | Find the maximum value of the parameter \( b \) for which the inequality \( b \sqrt{b}\left(x^{2}-10 x+25\right)+\frac{\sqrt{b}}{\left(x^{2}-10 x+25\right)} \leq \frac{1}{5} \cdot \sqrt[4]{b^{3}} \cdot \left| \sin \frac{\pi x}{10} \right| \) has at least one solution. | 1/10000 | 21.09375 |
11,727 | Given $a > 0$, $b > 0$, and it satisfies the equation $3a + b = a^2 + ab$. Find the minimum value of $2a + b$. | 2\sqrt{2} + 3 | 1.5625 |
11,728 | Given that $\{a_n\}$ is an arithmetic sequence, with the first term $a_1 > 0$, $a_5+a_6 > 0$, and $a_5a_6 < 0$, calculate the maximum natural number $n$ for which the sum of the first $n$ terms $S_n > 0$. | 10 | 21.875 |
11,729 | In the number $2 * 0 * 1 * 6 * 0 *$, each of the 5 asterisks must be replaced with any of the digits $0,1,2,3,4,5,6,7,8$ (digits can be repeated) so that the resulting 10-digit number is divisible by 18. How many ways can this be done? | 3645 | 31.25 |
11,730 | Given \\(\alpha \in (0^{\circ}, 90^{\circ})\\) and \\(\sin (75^{\circ} + 2\alpha) = -\frac{3}{5}\\), calculate \\(\sin (15^{\circ} + \alpha) \cdot \sin (75^{\circ} - \alpha)\\). | \frac{\sqrt{2}}{20} | 12.5 |
11,731 | The number $947$ can be written as $23q + r$ where $q$ and $r$ are positive integers. What is the greatest possible value of $q - r$? | 37 | 88.28125 |
11,732 | Given a triangle \( \triangle ABC \) with sides \( a, b, c \) opposite to angles \( A, B, C \) respectively, and \( a^{2} + b^{2} = c^{2} + \frac{2}{3}ab \). If the circumradius of \( \triangle ABC \) is \( \frac{3\sqrt{2}}{2} \), what is the maximum possible area of \( \triangle ABC \)? | 4\sqrt{2} | 41.40625 |
11,733 | Each of the ten cards has a real number written on it. For every non-empty subset of these cards, the sum of all the numbers written on the cards in that subset is calculated. It is known that not all of the obtained sums turned out to be integers. What is the largest possible number of integer sums that could have res... | 511 | 68.75 |
11,734 | The total \( T \) is obtained as the sum of the integers from 2006 to 2036 inclusive. What is the sum of all the prime factors of \( T \)? | 121 | 41.40625 |
11,735 | Determine the total surface area of a cube if the distance between the non-intersecting diagonals of two adjacent faces of this cube is 8. If the answer is not an integer, round it to the nearest whole number. | 1152 | 9.375 |
11,736 | In a class organizing a cultural evening, they plan to select 4 programs from 8 programs, with the requirement that at least one of the programs A or B must be selected, and when both A and B are selected, their performance order cannot be adjacent. Express the number of different performance orders as a value. | 1140 | 13.28125 |
11,737 | In the rectangular prism \(ABCD-A_1B_1C_1D_1\), \(AB=2\), \(AA_1=AD=1\). Points \(E\), \(F\), and \(G\) are the midpoints of edges \(AA_1\), \(C_1D_1\), and \(BC\) respectively. What is the volume of the tetrahedron \(B_1-EFG\)? | \frac{3}{8} | 35.9375 |
11,738 | A given odd function $f(x)$, defined on $\mathbb{R}$, is symmetric about the line $x=1$, and $f(-1) = 1$. Find the value of $f(1) + f(2) + f(3) + \ldots + f(2009)$. | -1 | 60.15625 |
11,739 | Given a triangular prism \( S-ABC \) with a base that is an isosceles right triangle with \( AB \) as the hypotenuse, and \( SA = SB = SC = AB = 2 \). If the points \( S, A, B, C \) all lie on the surface of a sphere centered at \( O \), what is the surface area of this sphere? | \frac{16 \pi}{3} | 10.15625 |
11,740 | The function \[f(x) = \left\{ \begin{aligned} 2x + 1 & \quad \text{ if } x < 3 \\ x^2 & \quad \text{ if } x \ge 3 \end{aligned} \right.\] has an inverse $f^{-1}.$ Compute the value of $f^{-1}(-3) + f^{-1}(0) + \dots + f^{-1}(4) + f^{-1}(9).$ | 3.5 | 0.78125 |
11,741 | The consignment shop received for sale cameras, clocks, pens, and receivers totaling 240 rubles. The sum of the prices of the receiver and one clock is 4 rubles more than the sum of the prices of the camera and the pen, and the sum of the prices of one clock and the pen is 24 rubles less than the sum of the prices of t... | 18 | 6.25 |
11,742 | Given $a$, $b$, and $c$ are the three sides of $\triangle ABC$, and $3a^2+3b^2-3c^2+2ab=0$, then $\tan C= \_\_\_\_\_\_$. | -2 \sqrt {2} | 0 |
11,743 | Crystal decides to alter her running routine slightly. She heads due north for 2 miles, then goes northwest for 2 miles, followed by a southwest direction for 2 miles, and she finishes with a final segment directly back to her starting point. Calculate the distance of this final segment of her run. | 2\sqrt{3} | 88.28125 |
11,744 | Choose two different numbers from the set of numbers {1, 2, ..., 8, 9}, and find the probability that their product is an odd number. (Express the result as a numerical value). | \frac{5}{18} | 71.09375 |
11,745 | Determine the total surface area of a cone with a diameter of 8 cm and a height of 12 cm. Express your answer in terms of \(\pi\). | 16\pi (\sqrt{10} + 1) | 19.53125 |
11,746 | Given vectors $\overrightarrow{a} = (\cos \alpha, \sin \alpha)$ and $\overrightarrow{b} = (\cos \beta, \sin \beta)$, with $|\overrightarrow{a} - \overrightarrow{b}| = \frac{2\sqrt{5}}{5}$.
(Ⅰ) Find the value of $\cos (\alpha - \beta)$;
(Ⅱ) If $0 < \alpha < \frac{\pi}{2}$ and $-\frac{\pi}{2} < \beta < 0$, and $\sin \b... | \frac{33}{65} | 10.15625 |
11,747 | The kindergarten received flashcards for reading: some say "MA", and the others say "NYA". Each child took three cards to form words. It turned out that 20 children can form the word "MAMA" from their cards, 30 children can form the word "NYANYA", and 40 children can form the word "MANYA". How many children have all th... | 10 | 55.46875 |
11,748 | In the Cartesian coordinate system, the graphs of the functions $y=\frac{3}{x}$ and $y=x+1$ intersect at the point $\left(m,n\right)$. Evaluate the algebraic expression $\left(m-n\right)^{2}\cdot \left(\frac{1}{n}-\frac{1}{m}\right)$. | -\frac{1}{3} | 65.625 |
11,749 | How can 13 rectangles of sizes $1 \times 1, 2 \times 1, 3 \times 1, \ldots, 13 \times 1$ be combined to form a rectangle, where all sides are greater than 1? | 13 \times 7 | 7.03125 |
11,750 | Every evening, Laura gathers three socks randomly to pack for her gym session next morning. She has 12 black socks, 10 white socks, and 6 striped socks in her drawer. What is the probability that all three socks she picks are of the same type? | \frac{60}{546} | 0 |
11,751 | In the disaster relief donation, $\frac{1}{10}$ of the people in a company each donated 200 yuan, $\frac{3}{4}$ of the people each donated 100 yuan, and the remaining people each donated 50 yuan. Find the average donation per person in the company. | 102.5 | 69.53125 |
11,752 | How many factors of 2 are in the prime factorization of 1984!? | 1979 | 96.09375 |
11,753 | Given that α is an angle in the fourth quadrant, and $sin \left( \frac{\pi}{2} + \alpha \right) = \frac{4}{5}$, calculate the value of $tan(\alpha)$. | -\frac{3}{4} | 89.0625 |
11,754 | Let \( M \) and \( m \) be the maximum and minimum elements, respectively, of the set \( \left\{\left.\frac{3}{a}+b \right\rvert\, 1 \leq a \leq b \leq 2\right\} \). Find the value of \( M - m \). | 5 - 2\sqrt{3} | 0 |
11,755 | In the polar coordinate system, curve $C$: $\rho =2a\cos \theta (a > 0)$, line $l$: $\rho \cos \left( \theta -\frac{\pi }{3} \right)=\frac{3}{2}$, $C$ and $l$ have exactly one common point. $O$ is the pole, $A$ and $B$ are two points on $C$, and $\angle AOB=\frac{\pi }{3}$, then the maximum value of $|OA|+|OB|$ is ____... | 2 \sqrt{3} | 21.09375 |
11,756 | The volume of a cube in cubic meters and its surface area in square meters is numerically equal to four-thirds of the sum of the lengths of its edges in meters. What is the total volume in cubic meters of twenty-seven such cubes? | 216 | 8.59375 |
11,757 | Compute: $$\left( \frac {8}{27}\right)\,^{- \frac {2}{3}}-\log {\sqrt {2}}-\log {\sqrt {5}}=$$ \_\_\_\_\_\_ . | \frac{7}{4} | 98.4375 |
11,758 | Given that the random variable $\xi$ follows the normal distribution $N(1, 4)$, if $P(\xi > 4) = 0.1$, then $P(-2 \leq \xi \leq 4)$ equals _______. | 0.8 | 68.75 |
11,759 | Evaluate the expression $\sqrt{5+4\sqrt{3}} - \sqrt{5-4\sqrt{3}} + \sqrt{7 + 2\sqrt{10}} - \sqrt{7 - 2\sqrt{10}}$.
A) $4\sqrt{3}$
B) $2\sqrt{2}$
C) $6$
D) $4\sqrt{2}$
E) $2\sqrt{5}$ | 2\sqrt{2} | 0.78125 |
11,760 | What is the value of $\sqrt{5! \cdot (5!)^2}$ expressed as a positive integer? | 240\sqrt{30} | 0 |
11,761 | Given positive real numbers $a$ and $b$ satisfying $a+b=2$, the minimum value of $\dfrac{1}{a}+\dfrac{2}{b}$ is ______. | \dfrac{3+2 \sqrt{2}}{2} | 21.875 |
11,762 | Given \( m = \frac{\sin x}{\sin (y-z)}, n = \frac{\sin y}{\sin (z-x)}, p = \frac{\sin z}{\sin (x-y)} \), find the value of \( m n + n p + p m \). | -1 | 26.5625 |
11,763 | Jim borrows $1500$ dollars from Sarah, who charges an interest rate of $6\%$ per month (which compounds monthly). What is the least integer number of months after which Jim will owe more than twice as much as he borrowed? | 12 | 53.90625 |
11,764 | Carl decided to fence his rectangular flowerbed using 24 fence posts, including one on each corner. He placed the remaining posts spaced exactly 3 yards apart along the perimeter of the bed. The bed’s longer side has three times as many posts compared to the shorter side, including the corner posts. Calculate the area ... | 144 | 20.3125 |
11,765 | A bag contains 70 balls that differ only in color: 20 red, 20 blue, 20 yellow, and the rest are black and white.
What is the minimum number of balls that must be drawn from the bag, without looking, to ensure that among them there are at least 10 balls of a single color? | 38 | 28.90625 |
11,766 | In triangle \(ABC\), points \(P\) and \(Q\) are taken on the base \(AC\) such that \(AP < AQ\). The lines \(BP\) and \(BQ\) divide the median \(AM\) into three equal parts. It is known that \(PQ = 3\).
Find \(AC\). | 10 | 0 |
11,767 | The sequence starts with 800,000; each subsequent term is obtained by dividing the previous term by 3. What is the last integer in this sequence? | 800000 | 0.78125 |
11,768 | If \( a < b < c < d \) are distinct positive integers such that \( a+b+c+d \) is a square, what is the minimum value of \( c+d \)? | 11 | 77.34375 |
11,769 | There is a sphere with a radius of $\frac{\sqrt{3}}{2}$, on which 4 points $A, B, C, D$ form a regular tetrahedron. What is the maximum distance from the center of the sphere to the faces of the regular tetrahedron $ABCD$? | \frac{\sqrt{3}}{6} | 67.96875 |
11,770 | In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively. It is known that $c=a\cos B+b\sin A$.
(I) Find angle $A$.
(II) If $a=2$, find the maximum area of $\triangle ABC$. | \sqrt {2}+1 | 0 |
11,771 | A team of loggers was scheduled to harvest $216 \mathrm{~m}^{3}$ of wood over several days. For the first three days, the team met the daily target set by the plan. Then, they harvested an additional $8 \mathrm{~m}^{3}$ above the daily target each day. As a result, they harvested $232 \mathrm{~m}^{3}$ of wood one day a... | 24 | 53.90625 |
11,772 | Given points $A(-1,1)$, $B(1,2)$, $C(-2,-1)$, $D(2,2)$, the projection of vector $\overrightarrow{AB}$ in the direction of $\overrightarrow{CD}$ is ______. | \dfrac{11}{5} | 5.46875 |
11,773 | Find the value of the function \( f(x) \) at the point \( x_{0} = 4500 \), if \( f(0) = 1 \) and for any \( x \) the equality \( f(x + 3) = f(x) + 2x + 3 \) holds. | 6750001 | 80.46875 |
11,774 | The numbers \(1, 2, 3, \ldots, 7\) are randomly divided into two non-empty subsets. What is the probability that the sum of the numbers in the two subsets is equal? If the probability is expressed as \(\frac{p}{q}\) in its lowest terms, find \(p + q\). | 67 | 42.96875 |
11,775 | Given the function $y=\sin(\omega x)$ ($\omega>0$) is increasing in the interval $[0, \frac{\pi}{3}]$ and its graph is symmetric about the point $(3\pi, 0)$, the maximum value of $\omega$ is \_\_\_\_\_\_. | \frac{4}{3} | 13.28125 |
11,776 | How many positive integer multiples of \(3003\) can be expressed in the form \(10^j - 10^i\), where \(i\) and \(j\) are integers and \(0 \leq i < j \leq 50\)? | 192 | 4.6875 |
11,777 | Tom, Dorothy, and Sammy went on a vacation and agreed to split the costs evenly. During their trip Tom paid $105, Dorothy paid $125, and Sammy paid $175. In order to share costs equally, find the difference between the amount of money Tom gave Sammy and the amount of money Dorothy gave Sammy. | 20 | 92.1875 |
11,778 | In a certain area, there are 100,000 households, among which there are 99,000 ordinary households and 1,000 high-income households. A simple random sampling method is used to select 990 households from the ordinary households and 100 households from the high-income households for a survey. It was found that a total of ... | 4.8\% | 0 |
11,779 | The sides of rectangle $ABCD$ are $AB=3$ and $BC=2$. Point $P$ is on side $AB$ such that line $PD$ touches the circle with diameter $BC$ at point $E$. The line passing through the center of the circle and point $E$ intersects side $AB$ at point $Q$. What is the area of triangle $PQE$? | 1/24 | 3.125 |
11,780 | In a rectangular prism $A^{\prime}C$, with $AB=5$, $BC=4$, and $B^{\prime}B=6$, $E$ is the midpoint of $AA^{\prime}$. Find the distance between the skew lines $BE$ and $A^{\prime}C^{\prime}$. | \frac{60}{\sqrt{769}} | 53.125 |
11,781 | The right triangles \(MDC\) and \(ADK\) have a common right angle at \(D\). Point \(K\) lies on \(CD\) and divides it in the ratio \(2:3\) from point \(C\). Point \(M\) is the midpoint of side \(AD\). Find the sum of the degree measures of angles \(AKD\) and \(MCD\), given that \(AD : CD = 2:5\). | 45 | 35.15625 |
11,782 | Given \( A=\left\{x \mid \log _{3}\left(x^{2}-2 x\right) \leqslant 1\right\}, B=(-\infty, a] \cup(b,+\infty) \), where \( a < b \), if \( A \cup B=\mathbf{R} \), what is the minimum value of \( a - b \) ? | -1 | 8.59375 |
11,783 | A city adopts a lottery system for "price-limited housing," where winning families can randomly draw a house number from the available housing in a designated community. It is known that two friendly families, Family A and Family B, have both won the lottery and decided to go together to a certain community to draw the... | \dfrac{3}{5} | 54.6875 |
11,784 | Let \( XYZ \) be a triangle with \( \angle X = 60^\circ \) and \( \angle Y = 45^\circ \). A circle with center \( P \) passes through points \( A \) and \( B \) on side \( XY \), \( C \) and \( D \) on side \( YZ \), and \( E \) and \( F \) on side \( ZX \). Suppose \( AB = CD = EF \). Find \( \angle XPY \) in degrees. | 120 | 65.625 |
11,785 | In how many different ways can four couples sit around a circular table such that no couple sits next to each other? | 1488 | 76.5625 |
11,786 | A \(4 \times 4\) Sudoku grid is filled with digits so that each column, each row, and each of the four \(2 \times 2\) sub-grids that compose the grid contains all of the digits from 1 to 4.
Find the total number of possible \(4 \times 4\) Sudoku grids. | 288 | 63.28125 |
11,787 | Suppose the product $\dfrac{4}{3}\cdot \dfrac{5}{4}\cdot \dfrac{6}{5}\cdot \ldots\cdot \dfrac{c}{d} = 16$, find the sum of $c$ and $d$. | 95 | 91.40625 |
11,788 | Given a positive real number \(\alpha\), determine the greatest real number \(C\) such that the inequality
$$
\left(1+\frac{\alpha}{x^{2}}\right)\left(1+\frac{\alpha}{y^{2}}\right)\left(1+\frac{\alpha}{z^{2}}\right) \geq C \cdot\left(\frac{x}{z}+\frac{z}{x}+2\right)
$$
holds for all positive real numbers \(x, y\), an... | 16 | 95.3125 |
11,789 | At a physical education lesson, 29 seventh graders attended, some of whom brought one ball each. During the lesson, sometimes one seventh grader would give their ball to another seventh grader who did not have a ball.
At the end of the lesson, $N$ seventh graders said, "I received balls less often than I gave them awa... | 14 | 51.5625 |
11,790 | What is the lowest prime number that is thirteen more than a cube? | 229 | 21.875 |
11,791 | Let $M$ be the midpoint of side $AC$ of the triangle $ABC$ . Let $P$ be a point on the side $BC$ . If $O$ is the point of intersection of $AP$ and $BM$ and $BO = BP$ , determine the ratio $\frac{OM}{PC}$ . | 1/2 | 75.78125 |
11,792 | How many students are there in our city? The number expressing the quantity of students is the largest of all numbers where any two adjacent digits form a number that is divisible by 23. | 46923 | 0 |
11,793 | The digits $1,2,3,4,5,$ and $6$ can be arranged to form many different $6$-digit positive integers with six distinct digits. In how many such integers is the digit $1$ to the left of both the digits $2$ and $3$? | 240 | 25 |
11,794 | Given the regression equation $y = 0.849x - 85.712$, where $x$ represents the height in cm and $y$ represents the weight in kg, determine the predicted weight of a female student who is 172 cm tall. | 60.316 | 0 |
11,795 | Given the random variables \( X \sim N(1,2) \) and \( Y \sim N(3,4) \), if \( P(X < 0) = P(Y > a) \), find the value of \( a \). | 3 + \sqrt{2} | 67.1875 |
11,796 |
The numerical sequence \(a_{0}, a_{1}, a_{2}, \ldots \) is such that for all non-negative \(m\) and \(n\) (where \(m \geq n\)), the following relation holds:
\[a_{m+n} + a_{m-n} = \frac{1}{2} (a_{2m} + a_{2n})\]
Find \(a_{1995}\) if \(a_{1} = 1\). | 1995^2 | 0 |
11,797 | If \( a + b + c = 1 \), what is the maximum value of \( \sqrt{3a+1} + \sqrt{3b+1} + \sqrt{3c+1} \)? | 3\sqrt{2} | 97.65625 |
11,798 | In the production of a certain item, its weight \( X \) is subject to random fluctuations. The standard weight of the item is 30 g, its standard deviation is 0.7, and the random variable \( X \) follows a normal distribution. Find the probability that the weight of a randomly selected item is within the range from 28 t... | 0.9215 | 2.34375 |
11,799 | Let $S-ABC$ be a triangular prism with circumscribed sphere centered at $O$. The midpoints of $SB$ and $AC$ are $N$ and $M$, respectively. The midpoint of line segment $MN$ is $P$, and it is given that $SA^{2} + SB^{2} + SC^{2} = AB^{2} + BC^{2} + AC^{2}$. If $SP = 3\sqrt{7}$ and $OP = \sqrt{21}$, find the radius of sp... | 2 \sqrt{21} | 25 |
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