Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
15,300 | In a tetrahedron \(A B C D\), a plane is drawn through the midpoint \(M\) of edge \(A D\), vertex \(C\), and point \(N\) on edge \(B D\) such that \(B N: N D = 2: 1\). In what ratio does this plane divide the segment \(K P\), where \(K\) and \(P\) are the midpoints of edges \(A B\) and \(C D\) respectively? | 1:3 | 6.25 |
15,301 | The arithmetic mean of several consecutive natural numbers is 5 times greater than the smallest of them. How many times is the arithmetic mean smaller than the largest of these numbers? | 1.8 | 0 |
15,302 | Some unit squares in an infinite sheet of squared paper are colored red so that every 2 x 3 and 3 x 2 rectangle contains exactly two red squares. How many red squares are there in a 9 x 11 rectangle? | 33 | 20.3125 |
15,303 | The sum of three positive numbers is 1, and none of the numbers is greater than twice any other number. What is the minimum product of the three numbers? | 1/32 | 12.5 |
15,304 | The number 2015 is split into 12 terms, and then all the numbers that can be obtained by adding some of these terms (from one to nine) are listed. What is the minimum number of numbers that could have been listed? | 10 | 3.90625 |
15,305 | In triangle \( ABC \), the angle bisector \( BL \) is drawn. Find the area of the triangle, given that \( AL=2 \), \( BL=3\sqrt{10} \), and \( CL=3 \). | \frac{15\sqrt{15}}{4} | 28.90625 |
15,306 | In the center of a circular field, there is a geologist's cabin. From it extend 6 straight roads, dividing the field into 6 equal sectors. Two geologists start a journey from their cabin at a speed of 4 km/h each on a randomly chosen road. Determine the probability that the distance between them will be at least 6 km a... | 0.5 | 32.8125 |
15,307 | Let's call two positive integers almost neighbors if each of them is divisible (without remainder) by their difference. In a math lesson, Vova was asked to write down in his notebook all the numbers that are almost neighbors with \(2^{10}\). How many numbers will he have to write down? | 21 | 15.625 |
15,308 | In the right triangle \(ABC\), the leg \(AB = 3\), and the leg \(AC = 6\). The centers of circles with radii 1, 2, and 3 are located at points \(A\), \(B\), and \(C\) respectively. Find the radius of the circle that is externally tangent to each of these three given circles. | \frac{8 \sqrt{11} - 19}{7} | 0 |
15,309 | Find the minimum value of the expression \(\frac{5 x^{2}-8 x y+5 y^{2}-10 x+14 y+55}{\left(9-25 x^{2}+10 x y-y^{2}\right)^{5 / 2}}\). If necessary, round the answer to hundredths. | 0.19 | 6.25 |
15,310 | All dwarves are either liars or knights. Liars always lie, while knights always tell the truth. Each cell of a $4 \times 4$ board contains one dwarf. It is known that among them there are both liars and knights. Each dwarf stated: "Among my neighbors (by edge), there are an equal number of liars and knights." How many ... | 12 | 0.78125 |
15,311 | Point \(D\) is the midpoint of the hypotenuse \(AB\) in the right triangle \(ABC\) with legs measuring 3 and 4.
Find the distance between the centers of the inscribed circles of triangles \(ACD\) and \(BCD\). | \frac{5 \sqrt{13}}{12} | 3.125 |
15,312 | The numbers \( a, b, c, d \) belong to the interval \([-7, 7]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 210 | 97.65625 |
15,313 | Remove all perfect squares and perfect cubes from the set
$$
A=\left\{n \mid n \leqslant 10000, n \in \mathbf{Z}_{+}\right\}
$$
and arrange the remaining elements in ascending order. What is the 2014th element of this sequence? | 2068 | 85.15625 |
15,314 | Box $A$ contains 1 red ball and 5 white balls, and box $B$ contains 3 white balls. Three balls are randomly taken from box $A$ and placed into box $B$. After mixing thoroughly, three balls are then randomly taken from box $B$ and placed back into box $A$. What is the probability that the red ball moves from box $A$ to ... | 1/4 | 31.25 |
15,315 | Find the number of positive integers \( x \), where \( x \neq 9 \), such that
\[
\log _{\frac{x}{9}}\left(\frac{x^{2}}{3}\right)<6+\log _{3}\left(\frac{9}{x}\right) .
\] | 223 | 9.375 |
15,316 | Given \(x\) and \(y\) are real numbers, and:
\[
z_1 = x + (y + 2)i,
\]
\[
z_2 = (x - 2) + yi,
\]
with the condition \(\left|z_1\right| + \left|z_2\right| = 4\),
Find the maximum value of \(|x + y|\). | 2\sqrt{2} | 11.71875 |
15,317 | Determine the largest prime factor of the sum \(\sum_{k=1}^{11} k^{5}\). | 263 | 3.125 |
15,318 | The sum of two sides of a rectangle is 11, and the sum of three sides is 19.5. Find the product of all possible distinct values of the perimeter of such a rectangle. | 15400 | 2.34375 |
15,319 | Given 8 planes in space, for each pair of planes, the line of their intersection is noted. For each pair of these noted lines, the point of their intersection is noted (if the lines intersect). What is the maximum number of noted points that could be obtained? | 56 | 0 |
15,320 | In a right square pyramid $O-ABCD$, $\angle AOB=30^{\circ}$, the dihedral angle between plane $OAB$ and plane $OBC$ is $\theta$, and $\cos \theta = a \sqrt{b} - c$, where $a, b, c \in \mathbf{N}$, and $b$ is not divisible by the square of any prime number. Find $a+b+c=$ _______. | 14 | 6.25 |
15,321 | The numbers \( a, b, c, d \) belong to the interval \([-13.5, 13.5]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | 756 | 4.6875 |
15,322 | In triangle \( ABC \), the sides \( AC = 14 \) and \( AB = 6 \) are known. A circle with center \( O \), constructed on side \( AC \) as its diameter, intersects side \( BC \) at point \( K \). It is given that \(\angle BAK = \angle ACB\). Find the area of triangle \( BOC \). | 21 | 27.34375 |
15,323 | Given the ellipse \(\frac{x^2}{2} + y^2 = 1\) with two foci \(F_1\) and \(F_2\), a chord \(AB\) passing through the right focus \(F_2\) has an inclination of \(45^\circ\). Determine the area of the triangle \(\triangle ABF_1\). | \frac{4}{3} | 96.875 |
15,324 | What is the maximum number of angles less than $150^{\circ}$ that a non-self-intersecting 2017-sided polygon can have if all its angles are strictly less than $180^{\circ}$? | 12 | 15.625 |
15,325 | Out of three hundred eleventh-grade students, 77% received excellent and good grades on the first exam, 71% on the second exam, and 61% on the third exam. What is the minimum number of participants who received excellent and good grades on all three exams? | 27 | 40.625 |
15,326 | A function \( f(x) \) defined on the interval \([1,2017]\) satisfies \( f(1)=f(2017) \), and for any \( x, y \in [1,2017] \), \( |f(x) - f(y)| \leqslant 2|x - y| \). If the real number \( m \) satisfies \( |f(x) - f(y)| \leqslant m \) for any \( x, y \in [1,2017] \), find the minimum value of \( m \). | 2016 | 35.9375 |
15,327 | If the complex number \( z \) satisfies \( |z| = 2 \), then the maximum value of \( \frac{\left|z^{2}-z+1\right|}{|2z-1-\sqrt{3}i|} \) is _______. | \frac{3}{2} | 3.90625 |
15,328 |
Compute the definite integral:
$$
\int_{0}^{\frac{\pi}{2}} \frac{\sin x \, dx}{(1+\cos x+\sin x)^{2}}
$$ | \ln 2 - \frac{1}{2} | 16.40625 |
15,329 | Let \( z \) be a complex number. If \( \frac{z-2}{z-\mathrm{i}} \) (where \( \mathrm{i} \) is the imaginary unit) is a real number, then the minimum value of \( |z+3| \) is \(\quad\). | \sqrt{5} | 64.84375 |
15,330 | If the intended number is multiplied by 6, then 382 is added to the product, the result is the largest three-digit number written with two identical even digits and one odd digit. Find the intended number. | 101 | 0.78125 |
15,331 | Let \( M \) be the centroid of \( \triangle ABC \), and \( AM = 3 \), \( BM = 4 \), \( CM = 5 \). Find the area of \( \triangle ABC \). | 18 | 59.375 |
15,332 | The radius of the circumcircle of an acute triangle \( ABC \) is 1. It is known that the center of the circle passing through the vertices \( A \), \( C \), and the orthocenter of triangle \( ABC \) lies on this circumcircle. Find the length of side \( AC \). | \sqrt{3} | 27.34375 |
15,333 | Given that $\sin ^{10} x+\cos ^{10} x=\frac{11}{36}$, find the value of $\sin ^{14} x+\cos ^{14} x$. | \frac{41}{216} | 13.28125 |
15,334 | The distance between Luga and Volkhov is 194 km, between Volkhov and Lodeynoe Pole is 116 km, between Lodeynoe Pole and Pskov is 451 km, and between Pskov and Luga is 141 km. What is the distance between Pskov and Volkhov? | 335 | 13.28125 |
15,335 | Given that $\tan (3 \alpha-2 \beta)=\frac{1}{2}$ and $\tan (5 \alpha-4 \beta)=\frac{1}{4}$, find $\tan \alpha$. | \frac{13}{16} | 7.03125 |
15,336 | Let \( a < b < c < d < e \) be real numbers. We calculate all the possible sums of two distinct numbers among these five numbers. The three smallest sums are 32, 36, and 37, and the two largest sums are 48 and 51. Find all possible values of \( e \). | \frac{55}{2} | 10.15625 |
15,337 | The length of the curve given by the parametric equations \(\left\{\begin{array}{l}x=2 \cos ^{2} \theta \\ y=3 \sin ^{2} \theta\end{array}\right.\) (where \(\theta\) is the parameter) is? | \sqrt{13} | 20.3125 |
15,338 | In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively, and
$$
a=5, \quad b=4, \quad \cos(A-B)=\frac{31}{32}.
$$
Find the area of $\triangle ABC$. | \frac{15 \sqrt{7}}{4} | 22.65625 |
15,339 | From Moscow to city \( N \), a passenger can travel by train, taking 20 hours. If the passenger waits for a flight (waiting will take more than 5 hours after the train departs), they will reach city \( N \) in 10 hours, including the waiting time. By how many times is the plane’s speed greater than the train’s speed, g... | 10 | 2.34375 |
15,340 | Vasya has a stick that is 22 cm long. He wants to break it into three parts with integer lengths and form a triangle from the resulting pieces. How many ways can he do this? (Methods that result in identical triangles are considered the same). | 10 | 87.5 |
15,341 | The decreasing sequence $a, b, c$ is a geometric progression, and the sequence $19a, \frac{124b}{13}, \frac{c}{13}$ is an arithmetic progression. Find the common ratio of the geometric progression. | 247 | 0.78125 |
15,342 | What two digits need to be added to the right of the number 2013 to make the resulting six-digit number divisible by 101? Find all possible answers. | 94 | 54.6875 |
15,343 | Given point \( A(4,0) \) and \( B(2,2) \), while \( M \) is a moving point on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), the maximum value of \( |MA| + |MB| \) is ______. | 10 + 2\sqrt{10} | 30.46875 |
15,344 | Given two arbitrary positive integers \( n \) and \( k \), let \( f(n, k) \) denote the number of unit squares that one of the diagonals of an \( n \times k \) grid rectangle passes through. How many such pairs \( (n, k) \) are there where \( n \geq k \) and \( f(n, k) = 2018 \)? | 874 | 5.46875 |
15,345 | Let \( n \) be a natural number. Define \( 1 = d_{1} < d_{2} < d_{3} < \cdots < d_{k} = n \) as its divisors. It is noted that \( n = d_{2}^{2} + d_{3}^{3} \). Determine all possible values of \( n \). | 68 | 98.4375 |
15,346 | Along the southern shore of a boundless sea stretches an archipelago of an infinite number of islands. The islands are connected by an endless chain of bridges, and each island is connected by a bridge to the shore. In the event of a strong earthquake, each bridge independently has a probability $p=0.5$ of being destro... | 2/3 | 0 |
15,347 | From the integers 1 to 2020, there are a total of 1616 integers that are not multiples of 5. These 1616 numbers need to be divided into groups (each group may have a different number of elements), such that the difference (larger number minus smaller number) between any two numbers in the same group is a prime number. ... | 404 | 0.78125 |
15,348 | Inside the tetrahedron \( ABCD \), points \( X \) and \( Y \) are given. The distances from point \( X \) to the faces \( ABC, ABD, ACD, BCD \) are \( 14, 11, 29, 8 \) respectively. The distances from point \( Y \) to the faces \( ABC, ABD, ACD, BCD \) are \( 15, 13, 25, 11 \) respectively. Find the radius of the inscr... | 17 | 1.5625 |
15,349 | At night, there was a heavy snowfall. In the morning, Xiao Long and his father measured the length of a circular path in the garden by walking. They started from the same point and walked in the same direction. Xiao Long's step length is 54 cm, and his father's step length is 72 cm. Each of them walked one complete lap... | 21.6 | 4.6875 |
15,350 | If a four-digit number $\overline{a b c d}$ meets the condition $a + b = c + d$, it is called a "good number." For instance, 2011 is a "good number." How many "good numbers" are there? | 615 | 100 |
15,351 | Transport Teams A and B need to deliver a batch of relief supplies to an earthquake-stricken area. Team A can transport 64.4 tons per day, which is 75% more than Team B can transport per day. If both teams transport the supplies simultaneously, when Team A has transported half of the total supplies, it has transported ... | 644 | 28.125 |
15,352 | In the Cartesian coordinate plane $xOy$, the equation of the ellipse $C$ is $\frac{x^{2}}{9}+\frac{y^{2}}{10}=1$. Let $F$ be the upper focus of $C$, $A$ be the right vertex of $C$, and $P$ be a moving point on $C$ located in the first quadrant. Find the maximum value of the area of the quadrilateral $OAPF$. | \frac{3 \sqrt{11}}{2} | 56.25 |
15,353 | In what ratio does the point \( P \) divide the perpendicular segment dropped from vertex \( A \) of a regular tetrahedron \( ABCD \) to the face \( BCD \), given that the lines \( PB \), \( PC \), and \( PD \) are mutually perpendicular to each other? | 1:1 | 17.96875 |
15,354 | The legs of a right triangle are 3 and 4. Find the area of the triangle with vertices at the points of tangency of the incircle with the sides of the triangle. | 6/5 | 3.125 |
15,355 | On New Year's Eve, Santa Claus gave the children the following task: by using all nine digits from 1 to 9 exactly once, insert either "+" or "-" between each pair of adjacent digits so that the result yields all possible two-digit prime numbers. How many such numbers can be obtained? | 10 | 0 |
15,356 | Let \( m = \min \left\{ x + 2y + 3z \mid x^{3} y^{2} z = 1 \right\} \). What is the value of \( m^{3} \)? | 72 | 2.34375 |
15,357 | Eight consecutive three-digit positive integers have the following property: each of them is divisible by its last digit. What is the sum of the digits of the smallest of these eight integers? | 13 | 41.40625 |
15,358 | Find the distance between the midpoints of the non-parallel sides of different bases of a regular triangular prism, each of whose edges is 2. | \sqrt{5} | 9.375 |
15,359 | Let $\left\{a_{n}\right\}$ be the number of subsets of the set $\{1,2, \ldots, n\}$ with the following properties:
- Each subset contains at least two elements.
- The absolute value of the difference between any two elements in the subset is greater than 1.
Find $\boldsymbol{a}_{10}$. | 133 | 36.71875 |
15,360 | A regular 2017-gon \( A_1 A_2 \cdots A_{2017} \) is inscribed in a unit circle \( O \). If two different vertices \( A_i \) and \( A_j \) are chosen randomly, what is the probability that \( \overrightarrow{O A_i} \cdot \overrightarrow{O A_j} > \frac{1}{2} \)? | 1/3 | 23.4375 |
15,361 | Point \( M \) lies on the side of a regular hexagon with side length 12. Find the sum of the distances from point \( M \) to the lines containing the remaining sides of the hexagon. | 36\sqrt{3} | 30.46875 |
15,362 | If the 3-digit decimal number \( n = \overline{abc} \) satisfies that \( a \), \( b \), and \( c \) form an arithmetic sequence, then what is the maximum possible value of a prime factor of \( n \)? | 317 | 17.96875 |
15,363 | Person A and person B start simultaneously from points A and B, respectively, and move towards each other. When person A reaches the midpoint C of A and B, person B is still 240 meters away from point C. When person B reaches point C, person A has already moved 360 meters past point C. What is the distance between poin... | 144 | 18.75 |
15,364 | A train has 18 identical cars. In some of the cars, half of the seats are free, in others, one third of the seats are free, and in the remaining cars, all the seats are occupied. In the entire train, exactly one ninth of all seats are free. How many cars have all seats occupied? | 13 | 3.90625 |
15,365 | Find the measure of the angle
$$
\delta=\arccos \left(\left(\sin 2903^{\circ}+\sin 2904^{\circ}+\cdots+\sin 6503^{\circ}\right)^{\cos 2880^{\circ}+\cos 2881^{\circ}+\cdots+\cos 6480^{\circ}}\right)
$$ | 67 | 2.34375 |
15,366 | In rectangle \( ABCD \), a circle \(\omega\) is constructed using side \( AB \) as its diameter. Let \( P \) be the second intersection point of segment \( AC \) with circle \(\omega\). The tangent to \(\omega\) at point \( P \) intersects segment \( BC \) at point \( K \) and passes through point \( D \). Find \( AD \... | 24 | 0.78125 |
15,367 | Let \( m \in \mathbf{N}^{*} \), and let \( F(m) \) represent the integer part of \( \log_{2} m \). Determine the value of \( F(1) + F(2) + \cdots + F(1024) \). | 8204 | 89.84375 |
15,368 | Let the set \( T \) consist of integers between 1 and \( 2^{30} \) whose binary representations contain exactly two 1s. If one number is randomly selected from the set \( T \), what is the probability that it is divisible by 9? | 5/29 | 12.5 |
15,369 | Find the number of pairs of integers \((x ; y)\) that satisfy the equation \(y^{2} - xy = 700000000\). | 324 | 13.28125 |
15,370 | A rectangle of size $1000 \times 1979$ is divided into cells. Into how many parts will it be divided if one diagonal is drawn in it? | 2978 | 39.0625 |
15,371 | In the vertices of a convex 2020-gon, numbers are placed such that among any three consecutive vertices, there is both a vertex with the number 7 and a vertex with the number 6. On each segment connecting two vertices, the product of the numbers at these two vertices is written. Andrey calculated the sum of the numbers... | 1010 | 50.78125 |
15,372 | The area of a triangle is $4 \sqrt{21}$, its perimeter is 24, and the segment of the angle bisector from one of the vertices to the center of the inscribed circle is $\frac{\sqrt{30}}{3}$. Find the longest side of the triangle. | 11 | 14.84375 |
15,373 | In the country of Draconia, there are red, green, and blue dragons. Each dragon has three heads; each head always tells the truth or always lies. Each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each dragon's heads made the following statements:
- 1st head: "To... | 176 | 65.625 |
15,374 | A circle inscribed in an isosceles trapezoid divides its lateral side into segments equal to 4 and 9. Find the area of the trapezoid. | 156 | 8.59375 |
15,375 | Find all three-digit numbers $\overline{МГУ}$, comprised of different digits $M$, $\Gamma$, and $Y$, for which the equality $\overline{\text{МГУ}} = (M + \Gamma + Y) \times (M + \Gamma + Y - 2)$ holds. | 195 | 87.5 |
15,376 | A circle, whose center lies on the line \( y = b \), intersects the parabola \( y = \frac{12}{5} x^2 \) at least at three points; one of these points is the origin, and two of the remaining points lie on the line \( y = \frac{12}{5} x + b \). Find all values of \( b \) for which this configuration is possible. | 169/60 | 0 |
15,377 | In the tetrahedron \( OABC \), \(\angle AOB = 45^\circ\), \(\angle AOC = \angle BOC = 30^\circ\). Find the cosine value of the dihedral angle \(\alpha\) between the planes \( AOC \) and \( BOC \). | 2\sqrt{2} - 3 | 4.6875 |
15,378 | A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people we... | 10 | 46.09375 |
15,379 | Let \( m \) and \( n \) be positive integers satisfying
\[ m n^{2} + 876 = 4 m n + 217 n. \]
Find the sum of all possible values of \( m \). | 93 | 11.71875 |
15,380 | An urn contains $k$ balls labeled with $k$, for all $k = 1, 2, \ldots, 2016$. What is the minimum number of balls we must draw, without replacement and without looking at the balls, to ensure that we have 12 balls with the same number? | 22122 | 0 |
15,381 | A four-digit number $\overline{a b c d}$ (where digits can repeat and are non-zero) is called a "good number" if it satisfies the conditions $\overline{a b}<20$, $\overline{b c}<21$, and $\overline{c d}<22$. How many such "good numbers" are there? | 10 | 57.03125 |
15,382 | In a triangle with sides \(AB = 4\), \(BC = 2\), and \(AC = 3\), an incircle is inscribed. Find the area of triangle \(AMN\), where \(M\) and \(N\) are the points of tangency of this incircle with sides \(AB\) and \(AC\) respectively. | \frac{25 \sqrt{15}}{64} | 1.5625 |
15,383 | In a kindergarten class, there are two (small) Christmas trees and five children. The teachers want to divide the children into two groups to form a ring around each tree, with at least one child in each group. The teachers distinguish the children but do not distinguish the trees: two configurations are considered ide... | 50 | 6.25 |
15,384 | In the plane Cartesian coordinate system \( xOy \), the circle \( \Omega \) intersects the parabola \( \Gamma: y^{2} = 4x \) at exactly one point, and the circle \( \Omega \) is tangent to the x-axis at the focus \( F \) of \( \Gamma \). Find the radius of the circle \( \Omega \). | \frac{4 \sqrt{3}}{9} | 0.78125 |
15,385 | Suppose the edge length of a regular tetrahedron $ABC D$ is 1 meter. A bug starts at point $A$ and moves according to the following rule: at each vertex, it chooses one of the three edges connected to this vertex with equal probability and crawls along this edge to the next vertex. What is the probability that the bug ... | 7/27 | 85.9375 |
15,386 | Two ferries travel between two opposite banks of a river at constant speeds. Upon reaching a bank, each immediately starts moving back in the opposite direction. The ferries departed from opposite banks simultaneously, met for the first time 700 meters from one of the banks, continued onward to their respective banks, ... | 1700 | 63.28125 |
15,387 | Let \( S_{n} = 1 + \frac{1}{2} + \cdots + \frac{1}{n} \) for \( n = 1, 2, \cdots \). Find the smallest positive integer \( n \) such that \( S_{n} > 10 \). | 12367 | 100 |
15,388 | Given that $\left\{a_{n}\right\}$ is an arithmetic sequence with a nonzero common difference and $\left\{b_{n}\right\}$ is a geometric sequence where $a_{1}=3, b_{1}=1, a_{2}=b_{2}, 3a_{5}=b_{3}$, and there exist constants $\alpha$ and $\beta$ such that for every positive integer $n$, $a_{n} = \log_{\alpha} b_{n} + \be... | \sqrt[3]{3} + 3 | 3.125 |
15,389 | Petrov booked an apartment in a newly-built house which has five identical entrances. Initially, the entrances were numbered from left to right, and Petrov's apartment had the number 636. Then the builder changed the numbering to the opposite direction (from right to left). After that, Petrov's apartment received the n... | 985 | 8.59375 |
15,390 | Let \( LOVER \) be a convex pentagon such that \( LOVE \) is a rectangle. Given that \( OV = 20 \) and \( LO = VE = RE = RL = 23 \), compute the radius of the circle passing through \( R, O \), and \( V \). | 23 | 8.59375 |
15,391 | From the 20 numbers 11, 12, 13, 14, ... 30, how many numbers at a minimum must be taken to ensure that among the selected numbers, there will definitely be a pair whose sum is a multiple of ten? | 11 | 42.96875 |
15,392 | Santa Claus has 36 identical gifts distributed among 8 bags. Each bag contains at least 1 gift, and the number of gifts in each of the 8 bags is unique. From these bags, select some bags such that the total number of gifts in the selected bags can be evenly divided among 8 children, with each child receiving at least o... | 31 | 23.4375 |
15,393 | The product $8 \cdot 78$ and the increasing sequence of integers ${3, 15, 24, 48, \cdots}$, which consists of numbers that are divisible by 3 and are one less than a perfect square, are given. What is the remainder when the 1994th term in this sequence is divided by 1000? | 63 | 9.375 |
15,394 | Given $x, y \in \mathbb{N}$, find the maximum value of $y$ such that there exists a unique value of $x$ satisfying the following inequality:
$$
\frac{9}{17}<\frac{x}{x+y}<\frac{8}{15}.
$$ | 112 | 25.78125 |
15,395 | Find the numerical value of the monomial \(0.007 a^{7} b^{9}\) if \(a = -5\) and \(b = 2\). | -280000 | 7.8125 |
15,396 | Find the area enclosed by the graph \( x^{2}+y^{2}=|x|+|y| \) on the \( xy \)-plane. | \pi + 2 | 0 |
15,397 | A natural number $N$ greater than 20 is a palindromic number in both base 14 and base 20 (A palindromic number is a number that reads the same forwards and backwards, such as 12321 and 3443, but not 12331). What is the smallest value of $N$ (expressed in decimal)? | 105 | 67.96875 |
15,398 | There is a magical tree with 63 fruits. On the first day, 1 fruit will fall from the tree. Starting from the second day, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number that should fall on that day, then the sequence... | 15 | 93.75 |
15,399 | From the 16 vertices of a $3 \times 3$ grid comprised of 9 smaller unit squares, what is the probability that any three chosen vertices form a right triangle? | 9/35 | 1.5625 |
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