Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
15,600 | Point \(A\) lies on the line \(y=\frac{12}{5} x-9\), and point \(B\) lies on the parabola \(y=x^{2}\). What is the minimum length of segment \(AB\)? | 189/65 | 0 |
15,601 |
The quadratic equations \(x^{2} + px + q\) and \(x^{2} + ax + b\) each have one root. Among the numbers \(p, q, a, b\) there are 16, 64, and 1024. What can the fourth number be?
If there are multiple possible answers, input the larger one into the system, and specify all of them in the written solution. | 262144 | 32.8125 |
15,602 | The height of a rhombus, drawn from the vertex of its obtuse angle, divides the side of the rhombus in the ratio $1:3$, measured from the vertex of its acute angle. What fraction of the area of the rhombus is occupied by the area of a circle inscribed in it? | \frac{\pi \sqrt{15}}{16} | 16.40625 |
15,603 | Private Petrov took a bucket of unpeeled potatoes and peeled them in 1 hour. During this process, 25% of the potatoes went to peels. How long did it take him to collect half a bucket of peeled potatoes? | 40 | 43.75 |
15,604 |
Compute the definite integral:
$$
\int_{-\pi}^{0} 2^{8} \sin ^{6} x \cos ^{2} x \, dx
$$ | 10\pi | 60.15625 |
15,605 | The number \( N = 3^{16} - 1 \) has a divisor of 193. It also has some divisors between 75 and 85 inclusive. What is the sum of these divisors? | 247 | 30.46875 |
15,606 | The numbers \(a, b, c, d\) belong to the interval \([-5.5, 5.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 132 | 3.125 |
15,607 | How many distinct right triangles exist with one leg equal to \( \sqrt{2016} \), and the other leg and hypotenuse expressed as natural numbers? | 12 | 69.53125 |
15,608 | Find a number such that when it is multiplied by its reverse, the product is 78445. | 145 | 32.8125 |
15,609 | Distribute 61 books to a class of students. If at least one person must receive at least 3 books, what is the maximum number of students in the class? | 30 | 42.1875 |
15,610 | Nathaniel and Obediah play a game in which they take turns rolling a fair six-sided die and keep a running tally of the sum of the results of all rolls made. A player wins if, after he rolls, the number on the running tally is a multiple of 7. Play continues until either player wins, or else indefinitely. If Nathaniel ... | 5/11 | 2.34375 |
15,611 | By how much did the dollar exchange rate change over the course of 2014 (from January 1, 2014, to December 31, 2014)? Provide the answer in rubles, rounded to the nearest whole number (answer - whole number). | 24 | 80.46875 |
15,612 | At each vertex of a cube with an edge length of 1, there is the center of a sphere. All the spheres are identical, and each touches three neighboring spheres. Find the length of the part of the space diagonal of the cube that lies outside the spheres. | \sqrt{3} - 1 | 47.65625 |
15,613 | For which values of \(a\) does the equation \(|x-3| = a x - 1\) have two solutions? Enter the midpoint of the interval of parameter \(a\) in the provided field. Round the answer to three significant digits according to rounding rules and enter it in the provided field. | 0.667 | 28.125 |
15,614 | The diagonals of a trapezoid are mutually perpendicular, and one of them is equal to 17. Find the area of the trapezoid if its height is 15. | 4335/16 | 1.5625 |
15,615 | A triangle is divided into 1000 smaller triangles. What is the minimum number of distinct points that can be the vertices of these triangles? | 503 | 0.78125 |
15,616 | Five integers are written on a board. By summing them in pairs, the following set of 10 numbers was obtained: $-1, 5, 8, 9, 11, 12, 14, 18, 20, 24$. Determine which numbers are written on the board. Provide their product as the answer. | -2002 | 20.3125 |
15,617 | Let the set \( S \) contain 2012 elements, where the ratio of any two elements is not an integer. An element \( x \) in \( S \) is called a "good element" if there exist distinct elements \( y \) and \( z \) in \( S \) such that \( x^2 \) divides \( y \cdot z \). Find the maximum possible number of good elements in \( ... | 2010 | 17.96875 |
15,618 | The curve given by the equation \( y = 2^p x^2 + 5px - 2^{p^2} \) intersects the \( Ox \) axis at points \( A \) and \( B \), and the \( Oy \) axis at point \( C \). Find the sum of all values of the parameter \( p \) for which the center of the circle circumscribed around triangle \( ABC \) lies on the \( Ox \) axis. | -1 | 20.3125 |
15,619 | On the board, two sums are written:
\[1+22+333+4444+55555+666666+7777777+88888888+999999999\]
\[9+98+987+9876+98765+987654+9876543+98765432+987654321\]
Determine which one is greater (or if they are equal). | 1097393685 | 0 |
15,620 | Let \( a \) and \( b \) be the roots of \( x^2 + 2000x + 1 = 0 \) and let \( c \) and \( d \) be the roots of \( x^2 - 2008x + 1 = 0 \). Find the value of \( (a+c)(b+c)(a-d)(b-d) \). | 32064 | 28.125 |
15,621 | Given a linear function \( f(x) \). It is known that the distance between the points of intersection of the graphs \( y = x^2 - 1 \) and \( y = f(x) + 1 \) is \( 3\sqrt{10} \), and the distance between the points of intersection of the graphs \( y = x^2 \) and \( y = f(x) + 3 \) is \( 3\sqrt{14} \). Find the distance b... | 3\sqrt{2} | 21.875 |
15,622 | A team of fishermen planned to catch 1800 centners of fish within a certain timeframe. During one-third of this period, there was a storm, causing them to fall short of their daily target by 20 centners each day. However, on the remaining days, the team managed to catch 20 centners more than the daily norm and complete... | 100 | 20.3125 |
15,623 | In how many ways can two rooks be arranged on a chessboard such that one cannot capture the other? (A rook can capture another if it is on the same row or column of the chessboard). | 3136 | 13.28125 |
15,624 | In triangle ABC, angle CAB is 30 degrees, and angle ABC is 80 degrees. The point M lies inside the triangle such that angle MAC is 10 degrees and angle MCA is 30 degrees. Find angle BMC in degrees. | 110 | 6.25 |
15,625 | Given that the side lengths of triangle \( \triangle ABC \) are 6, \( x \), and \( 2x \), find the maximum value of its area \( S \). | 12 | 51.5625 |
15,626 | Take a clay sphere of radius 13, and drill a circular hole of radius 5 through its center. Take the remaining "bead" and mold it into a new sphere. What is this sphere's radius? | 12 | 19.53125 |
15,627 | Teams A, B, and C need to complete two projects, $A$ and $B$. The workload of project $B$ is $\frac{1}{4}$ more than the workload of project $A$. If teams A, B, and C work alone, they can finish project $A$ in 20 days, 24 days, and 30 days respectively. To complete these two projects simultaneously, team A is assigned ... | 15 | 2.34375 |
15,628 | Given that the positive real numbers \( u, v, \) and \( w \) are all not equal to 1, if \(\log _{u} (v w)+\log _{v} w=5\) and \(\log _{v} u+\log _{w} v=3\), then find the value of \(\log _{w} u\). | 4/5 | 20.3125 |
15,629 | A triangular pyramid \( S-ABC \) has a base that is an equilateral triangle with a side length of 4. It is given that \( AS = BS = \sqrt{19} \) and \( CS = 3 \). Find the surface area of the circumscribed sphere of the triangular pyramid \( S-ABC \). | \frac{268\pi}{11} | 0 |
15,630 | Three shepherds met on a large road, each driving their respective herds. Jack says to Jim:
- If I give you 6 pigs for one horse, your herd will have twice as many heads as mine.
And Dan remarks to Jack:
- If I give you 14 sheep for one horse, your herd will have three times as many heads as mine.
Jim, in turn, say... | 39 | 15.625 |
15,631 | Edward stopped to rest at a place 1,875 feet from the prison and was spotted by a guard with a crossbow. The guard fired an arrow with an initial velocity of \( 100 \, \mathrm{ft/s} \). At the same time, Edward started running away with an acceleration of \( 1 \, \mathrm{ft/s^2} \). Assuming that air resistance causes ... | 75 | 72.65625 |
15,632 | Let \( a, b, c, d, e \) be positive integers whose sum is 2018. Let \( M = \max (a+b, b+c, c+d, d+e) \). Find the smallest possible value of \( M \). | 673 | 26.5625 |
15,633 | A stalker throws a small nut from the Earth's surface at an angle of $\alpha=30^{\circ}$ to the horizontal with an initial speed $v_{0}=10 \, \mathrm{m}/\mathrm{s}$. The normal acceleration due to gravity is $g=10 \, \mathrm{m}/\mathrm{s}^{2}$. At the highest point of its trajectory, the nut enters a zone of gravitatio... | 250 | 10.15625 |
15,634 | Positive real numbers \( x, y, z \) satisfy
\[
\left\{
\begin{array}{l}
\frac{2}{5} \leqslant z \leqslant \min \{x, y\}, \\
xz \geqslant \frac{4}{15}, \\
yz \geqslant \frac{1}{5}.
\end{array}
\right.
\]
Find the maximum value of \( \frac{1}{x} + \frac{2}{y} + \frac{3}{z} \). | 13 | 80.46875 |
15,635 |
We traveled by train from Anglchester to Klinkerton. But an hour after the train started, a locomotive malfunction was discovered. We had to continue the journey at a speed that was $\frac{3}{5}$ of the original speed. As a result, we arrived in Klinkerton with a delay of 2 hours, and the driver said that if the break... | 200 | 14.0625 |
15,636 | In the final round of a giraffe beauty contest, two giraffes named Tall and Spotted have made it to this stage. There are 105 voters divided into 5 districts, each district divided into 7 sections, with each section having 3 voters. Voters select the winner in their section by majority vote; in a district, the giraffe ... | 24 | 57.03125 |
15,637 | In a certain kingdom, the king has decided to build 25 new towns on 13 uninhabited islands so that on each island there will be at least one town. Direct ferry connections will be established between any pair of new towns which are on different islands. Determine the least possible number of these connections. | 222 | 1.5625 |
15,638 | On side \(BC\) and on the extension of side \(AB\) through vertex \(B\) of triangle \(ABC\), points \(M\) and \(K\) are located, respectively, such that \(BM: MC = 4: 5\) and \(BK: AB = 1: 5\). Line \(KM\) intersects side \(AC\) at point \(N\). Find the ratio \(CN: AN\). | 5/24 | 2.34375 |
15,639 | The sum
$$
\frac{1}{1 \times 2 \times 3}+\frac{1}{2 \times 3 \times 4}+\frac{1}{3 \times 4 \times 5}+\cdots+\frac{1}{100 \times 101 \times 102}
$$
can be expressed as $\frac{a}{b}$, a fraction in its simplest form. Find $a+b$. | 12877 | 27.34375 |
15,640 | Refer to the diagram, $P$ is any point inside the square $O A B C$ and $b$ is the minimum value of $P O + P A + P B + P C$. Find $b$. | 2\sqrt{2} | 23.4375 |
15,641 | The price of an item is an integer number of yuan. With 100 yuan, you can buy up to 3 items. Person A and Person B each have a certain number of 100-yuan bills. The amount of money Person A has can buy at most 7 items, and the amount of money Person B has can buy at most 14 items. Together, they can buy 1 more item tha... | 27 | 0 |
15,642 | Entrepreneurs Vasiliy Petrovich and Petr Gennadievich opened a clothing factory "ViP." Vasiliy Petrovich invested 200 thousand rubles, while Petr Gennadievich invested 350 thousand rubles. The factory was successful, and after a year, Anastasia Alekseevna approached them with an offer to buy part of the shares. They ag... | 1000000 | 7.8125 |
15,643 | The lengths of the diagonals of a rhombus and the length of its side form a geometric progression. Find the sine of the angle between the side of the rhombus and its longer diagonal, given that it is greater than \( \frac{1}{2} \). | \sqrt{\frac{\sqrt{17}-1}{8}} | 0 |
15,644 | It is known that when 2008 is divided by certain natural numbers, the remainder is always 10. How many such natural numbers are there? | 11 | 69.53125 |
15,645 | How many parallelograms with sides 1 and 2, and angles \(60^{\circ}\) and \(120^{\circ}\), can be placed inside a regular hexagon with side length 3? | 12 | 25.78125 |
15,646 | Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(2, 3, 1)$
$M_{2}(4, 1, -2)$
$M_{3}(6, 3, 7)$
$M_{0}(-5, -4, 8)$ | 11 | 88.28125 |
15,647 | A bot is held in orbit by the gravitational force of attraction to the planet, which creates a centripetal acceleration equal to $4 \pi^{2}(2 R) / T^{2}$, where $R$ is the radius of the planet and $T$ is the period of revolution of the bot. From Newton's second law, we have:
$$
m \frac{4 \pi^{2}(2 R)}{T^{2}}=G \frac{m... | 6000 | 9.375 |
15,648 | Given that \( x + y + z = xy + yz + zx \), find the minimum value of \( \frac{x}{x^2 + 1} + \frac{y}{y^2 + 1} + \frac{z}{z^2 + 1} \). | -1/2 | 0 |
15,649 | A smaller square was cut out from a larger square in such a way that one side of the smaller square lies on a side of the original square. The perimeter of the resulting octagon is $40\%$ greater than the perimeter of the original square. By what percentage is the area of the octagon less than the area of the original ... | 64 | 24.21875 |
15,650 | Inside an angle of $60^{\circ}$, there is a point located at distances $\sqrt{7}$ and $2 \sqrt{7}$ from the sides of the angle. Find the distance of this point from the vertex of the angle. | \frac{14 \sqrt{3}}{3} | 27.34375 |
15,651 | In the quadrilateral \(ABCD\), it is known that \(\angle BAC = \angle CAD = 60^\circ\), and \(AB + AD = AC\). Additionally, it is known that \(\angle ACD = 23^\circ\). What is the measure of angle \(ABC\) in degrees? | 83 | 71.875 |
15,652 | In a $4 \times 4$ grid, place candies according to the following requirements: (1) Each cell must contain candies; (2) In adjacent cells, the left cell has 1 fewer candy than the right cell and the upper cell has 2 fewer candies than the lower cell; (3) The bottom-right cell contains 20 candies. How many candies are th... | 248 | 2.34375 |
15,653 | The numbers \(a, b, c, d\) belong to the interval \([-7.5, 7.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 240 | 4.6875 |
15,654 | A line passing through the intersection point of the medians of triangle \(ABC\) intersects the sides \(BA\) and \(BC\) at points \(A^{\prime}\) and \(C_1\) respectively. Given that:
\(BA^{\prime} < BA = 3\), \(BC = 2\), and \(BA^{\prime} \cdot BC_1 = 3\). Find \(BA^{\prime}\). | \frac{3}{2} | 24.21875 |
15,655 | On a clock, there are two hands: the hour hand and the minute hand. At a random moment in time, the clock stops. Find the probability that the angle between the hands on the stopped clock is acute. | 1/2 | 8.59375 |
15,656 | A natural number, when raised to the sixth power, has digits which, when arranged in ascending order, are:
$$
0,2,3,4,4,7,8,8,9
$$
What is this number? | 27 | 89.0625 |
15,657 | How many ordered pairs \((b, g)\) of positive integers with \(4 \leq b \leq g \leq 2007\) are there such that when \(b\) black balls and \(g\) gold balls are randomly arranged in a row, the probability that the balls on each end have the same colour is \(\frac{1}{2}\)? | 59 | 0.78125 |
15,658 | What is the repeating sequence? Determine what is the repeating sequence in the decimal expansion of the fraction \(\frac{1}{49}\). | 020408163265306122448979591836734693877551 | 80.46875 |
15,659 | Given that point \( P \) lies on the hyperbola \(\frac{x^{2}}{16} - \frac{y^{2}}{9} = 1\), and the distance from \( P \) to the right directrix of the hyperbola is the arithmetic mean of the distances from \( P \) to the two foci of the hyperbola, find the x-coordinate of point \( P \). | -\frac{64}{5} | 0.78125 |
15,660 | Each edge of a regular tetrahedron is divided into three equal parts. Through each point of division, two planes are drawn, each parallel to one of the two faces of the tetrahedron that do not pass through this point. Into how many parts do the constructed planes divide the tetrahedron? | 27 | 69.53125 |
15,661 | On graph paper, two right triangles are drawn. Find the sum of the angles BCA and \(\mathrm{B}_{1} \mathrm{C}_{1} \mathrm{~A}_{1}\). | 90 | 36.71875 |
15,662 | The number of solutions to the equation $\sin |x| = |\cos x|$ in the closed interval $[-10\pi, 10\pi]$ is __. | 20 | 0 |
15,663 | At a tribal council meeting, 60 people spoke in turn. Each of them said only one phrase. The first three speakers all said the same thing: "I always tell the truth!" The next 57 speakers also said the same phrase: "Among the previous three speakers, exactly two of them told the truth." What is the maximum number of spe... | 45 | 43.75 |
15,664 | A number is the product of four prime numbers. What is this number if the sum of the squares of the four prime numbers is 476? | 1989 | 50.78125 |
15,665 | Petrov writes down odd numbers: \(1, 3, 5, \ldots, 2013\), and Vasechkin writes down even numbers: \(2, 4, \ldots, 2012\). Each of them calculates the sum of all the digits of all their numbers and tells it to the star student Masha. Masha subtracts Vasechkin's result from Petrov's result. What is the outcome? | 1007 | 94.53125 |
15,666 | If the centroid of the inscribed triangle \( ABC \) of the curve \( y^{2}=4 \sqrt{2} x \) is its focus \( F \), then
\[ |FA|^{2} + |FB|^{2} + |FC|^{2} = \] | 27 | 3.90625 |
15,667 | A positive integer is said to be bi-digital if it uses two different digits, with each digit used exactly twice. For example, 1331 is bi-digital, whereas 1113, 1111, 1333, and 303 are not. Determine the exact value of the integer \( b \), the number of bi-digital positive integers. | 243 | 12.5 |
15,668 | A heavy concrete platform anchored to the seabed in the North Sea supported an oil rig that stood 40 m above the calm water surface. During a severe storm, the rig toppled over. The catastrophe was captured from a nearby platform, and it was observed that the top of the rig disappeared into the depths 84 m from the poi... | 68.2 | 4.6875 |
15,669 | Compute the sum of all possible distinct values of \( m+n \) if \( m \) and \( n \) are positive integers such that
$$
\operatorname{lcm}(m, n) + \operatorname{gcd}(m, n) = 2(m+n) + 11
$$ | 32 | 16.40625 |
15,670 | Find the smallest multiple of 9 that does not contain any odd digits. | 288 | 90.625 |
15,671 | A square is constructed on one side of a regular octagon, outward. In the octagon, two diagonals intersect at point \( B \) (see the drawing). Find the measure of angle \( ABC \).
(A polygon is called regular if all its sides are equal and all its angles are equal.) | 22.5 | 22.65625 |
15,672 | Let \( A, B, C \) be points on the same plane with \( \angle ACB = 120^\circ \). There is a sequence of circles \( \omega_0, \omega_1, \omega_2, \ldots \) on the same plane (with corresponding radii \( r_0, r_1, r_2, \ldots \) where \( r_0 > r_1 > r_2 > \cdots \)) such that each circle is tangent to both segments \( CA... | \frac{3}{2} + \sqrt{3} | 0 |
15,673 | There are two coal mines, Mine A and Mine B. Each gram of coal from Mine A releases 4 calories of heat when burned, and each gram of coal from Mine B releases 6 calories of heat when burned. The price per ton of coal at the production site is 20 yuan for Mine A and 24 yuan for Mine B. It is known that the transportatio... | 18 | 10.9375 |
15,674 | In triangle \(ABC\), side \(AC = 42\). The angle bisector \(CL\) is divided by the point of intersection of the angle bisectors of the triangle in the ratio \(2:1\) from the vertex. Find the length of side \(AB\) if the radius of the circle inscribed in triangle \(ABC\) is 14. | 56 | 17.1875 |
15,675 | Let \( x_{1}, x_{2}, \ldots, x_{100} \) be natural numbers greater than 1 (not necessarily distinct). In a \( 100 \times 100 \) table, numbers are placed as follows: at the intersection of the \( i \)-th row and the \( k \)-th column, the number \( \log _{x_{k}} \frac{x_{i}}{4} \) is written. Find the smallest possible... | -10000 | 88.28125 |
15,676 | The numbers \(a, b, c, d\) belong to the interval \([-8.5, 8.5]\). Find the maximum value of the expression \(a + 2b + c + 2d - ab - bc - cd - da\). | 306 | 1.5625 |
15,677 | In $\triangle ABC$, $\angle ABC = \angle ACB = 40^\circ$, and $P$ is a point inside the triangle such that $\angle PAC = 20^\circ$ and $\angle PCB = 30^\circ$. Find the measure of $\angle PBC$. | 20 | 7.03125 |
15,678 | What is the greatest common divisor of all the numbers $7^{n+2} + 8^{2n+1}$ for $n \in \mathbb{N}$? | 57 | 42.1875 |
15,679 | A car license plate contains three letters and three digits, for example, A123BE. The allowed letters are А, В, Е, К, М, Н, О, Р, С, Т, У, Х (a total of 12 letters), and all digits except for the combination 000. Tanya considers a license plate happy if the first letter is a consonant, the second letter is also a conso... | 384000 | 14.84375 |
15,680 | Given a cube with its eight vertices labeled with the numbers $1, 2, 3, \cdots, 8$ in any order, define the number on each edge as $|i-j|$, where $i$ and $j$ are the labels of the edge’s endpoints. Let $S$ be the sum of the numbers on all the edges. Find the minimum value of $S$. | 28 | 5.46875 |
15,681 | Given \( n \) numbers \( a_{1}, a_{2}, \cdots, a_{n} \), their root mean square is defined as \(\left(\frac{a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}}{n}\right)^{\frac{1}{2}} \). Let \( M \) be the set of values of \( n \) (for \( n > 1 \)) such that the root mean square of the first \( n \) positive integers is an integer.... | 337 | 100 |
15,682 | Given 1 coin of 0.1 yuan, 1 coin of 0.2 yuan, 1 coin of 0.5 yuan, 4 coins of 1 yuan, and 2 coins of 5 yuan, how many different amounts of money can be paid using any combination of these coins? | 120 | 0 |
15,683 | Two distinct natural numbers end with 8 zeros and have exactly 90 divisors. Find their sum. | 700000000 | 4.6875 |
15,684 | Starting with 1, alternately add 4 and 3 to obtain the sequence $1, 5, 8, 12, 15, 19, 22, \ldots \ldots$. In this sequence, the number closest to 2013 is $\qquad$ . | 2014 | 62.5 |
15,685 | Someone says that 7 times their birth year divided by 13 gives a remainder of 11, and 13 times their birth year divided by 11 gives a remainder of 7. How old will this person be in the year 1954? | 86 | 3.90625 |
15,686 | Given that 5 students each specialize in one subject (Chinese, Mathematics, Physics, Chemistry, History) and there are 5 test papers (one for each subject: Chinese, Mathematics, Physics, Chemistry, History), a teacher randomly distributes one test paper to each student. Calculate the probability that at least 4 student... | 89/120 | 6.25 |
15,687 | Determine the maximum possible value of the expression
$$
27abc + a\sqrt{a^2 + 2bc} + b\sqrt{b^2 + 2ca} + c\sqrt{c^2 + 2ab}
$$
where \(a, b, c\) are positive real numbers such that \(a + b + c = \frac{1}{\sqrt{3}}\). | \frac{2}{3 \sqrt{3}} | 0 |
15,688 | Find the maximum value of the expression \((\sqrt{36-4 \sqrt{5}} \sin x-\sqrt{2(1+\cos 2 x)}-2) \cdot (3+2 \sqrt{10-\sqrt{5}} \cos y-\cos 2 y)\). If the answer is not an integer, round it to the nearest whole number. | 27 | 1.5625 |
15,689 | There are 36 students in a club. If any 33 of them attend a session, girls will always be in the majority. However, if 31 students attend, it might happen that boys are in the majority. How many girls are in the club? | 20 | 25.78125 |
15,690 | On 40 squares of an $8 \times 8$ chessboard, a stone was placed on each square. The product of the number of stones on white squares and the number of stones on black squares was calculated. Find the minimum possible value of this product. | 256 | 0 |
15,691 | Add 3 digits after 325 to make a six-digit number such that it is divisible by 3, 4, and 5, and make this number as small as possible. What is the new six-digit number? | 325020 | 64.84375 |
15,692 | Given a sequence where each term is either 1 or 2, begins with the term 1, and between the $k$-th term 1 and the $(k+1)$-th term 1 there are $2^{k-1}$ terms of 2 (i.e., $1,2,1,2,2,1,2,2,2,2,1,2,2,2,2,2,2,2,2,1, \cdots$), what is the sum of the first 1998 terms in this sequence? | 3985 | 68.75 |
15,693 |
Tetrahedron \(ABCD\) has base \( \triangle ABC \). Point \( E \) is the midpoint of \( AB \). Point \( F \) is on \( AD \) so that \( FD = 2AF \), point \( G \) is on \( BD \) so that \( GD = 2BG \), and point \( H \) is on \( CD \) so that \( HD = 2CH \). Point \( M \) is the midpoint of \( FG \) and point \( P \) is... | 1/10 | 10.15625 |
15,694 | Vasya remembers that his friend Petya lives on Kurchatovskaya street in building number 8, but he forgot the apartment number. When asked for clarification, Petya replied: "The number of my apartment is a three-digit number. If you rearrange its digits, you get five other three-digit numbers. The sum of these five numb... | 425 | 8.59375 |
15,695 | Vasya thought of a four-digit number and wrote down the product of each pair of its adjacent digits on the board. After that, he erased one product, and the numbers 20 and 21 remained on the board. What is the smallest number Vasya could have in mind? | 3745 | 25.78125 |
15,696 | Calculate the definite integral
$$
\int_{0}^{\pi / 2} \frac{\sin x}{2+\sin x} \, dx
$$ | \frac{\pi}{2} - \frac{2 \pi}{3 \sqrt{3}} | 5.46875 |
15,697 | In the parliament of the island nation Promenade-and-Tornado, only native inhabitants, who are divided into knights and liars, can be elected. Knights always tell the truth, and liars always lie. In the latest term, 2020 deputies were elected. At the first plenary session of the parliament, 1011 deputies declared: "If ... | 1010 | 75 |
15,698 | Two groups have an equal number of students. Each student studies at least one language: English or French. It is known that 5 people in the first group and 5 in the second group study both languages. The number of students studying French in the first group is three times less than in the second group. The number of s... | 28 | 21.09375 |
15,699 | There is a target on the wall consisting of five zones: a central circle (bullseye) and four colored rings. The width of each ring is equal to the radius of the bullseye. It is known that the number of points for hitting each zone is inversely proportional to the probability of hitting that zone and that hitting the bu... | 45 | 31.25 |
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