problem
stringlengths 18
4.46k
| answer
stringlengths 1
942
| pass_at_n
float64 0.08
0.92
|
|---|---|---|
A number, when added to 3, is a multiple of 7, and when subtracted by 5, is a multiple of 8. What is the smallest such number? | 53 | 0.875 |
Find the maximum value of $S$ such that any finite number of small squares with a total area of $S$ can be placed inside a unit square $T$ with side length 1, in such a way that no two squares overlap. | \frac{1}{2} | 0.375 |
127 is the number of non-empty sets of natural numbers \( S \) that satisfy the condition "if \( x \in S \), then \( 14-x \in S \)". The number of such sets \( S \) is \(\qquad \). | 127 | 0.875 |
Three positive integers differ from each other by at most 6. The product of these three integers is 2808. What is the smallest integer among them? | 12 | 0.625 |
An irrigation canal has the shape of an isosceles trapezoid, where the legs are equal to the smaller base. At what angle of inclination of the legs is the cross-sectional area of the canal the greatest? | 60^\circ | 0.875 |
In triangle $ABC$, a median $BM$ is drawn. It is given that $\angle ABM = 40^\circ$, and $\angle MBC = 70^\circ$. Find the ratio $AB:BM$. Justify your answer. | 2 | 0.875 |
Investigate the convergence of the functional series \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+3} \cdot\left(\frac{x+1}{x-1}\right)^{n}\). | (-\infty, 0] | 0.5 |
Given a square \(ABCD\) with point \(P\) inside such that \(PA = 1\), \(PB = 2\), and \(PC = 3\), calculate the angle \(\widehat{APB}\). | 135^\circ | 0.75 |
Let \( a < b < c < d < e \) be real numbers. Among the 10 sums of the pairs of these numbers, the least three are 32, 36, and 37, while the largest two are 48 and 51. Find all possible values of \( e \). | 27.5 | 0.75 |
Compute the limit of the function:
$$
\lim _{x \rightarrow \pi} \frac{\ln (2+\cos x)}{\left(3^{\sin x}-1\right)^{2}}
$$ | \frac{1}{2 \ln^2 3} | 0.25 |
Given that \( P_{0}(1, f(1)) \) is a point on the curve \( C: f(x) = \mathrm{e}^{x} \), the tangent line \( l_{1} \) to \( C \) at \( P_{0} \) intersects the \( x \)-axis at point \( Q_{1}(x_{1}, 0) \). A vertical line through \( Q_{1} \) intersects \( C \) at \( P_{1} \), and the tangent line \( l_{2} \) to \( C \) at \( P_{1} \) intersects the \( x \)-axis at \( Q_{2}(x_{2}, 0) \). A vertical line through \( Q_{2} \) intersects \( C \) at \( P_{2} \), and the tangent line \( l_{3} \) to \( C \) at \( P_{2} \) intersects the \( x \)-axis at \( Q_{3}(x_{3}, 0) \). By repeating this operation, points \( Q_{4}(x_{4}, 0), Q_{5}(x_{5}, 0), \cdots \) are obtained in succession. Find \( x_{2023} \). | -2022 | 0.75 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow \frac{\pi}{2}} \frac{e^{\operatorname{tg} 2 x} - e^{-\sin 2 x}}{\sin x - 1}
\] | 0 | 0.625 |
In the United States, dates are written as: month number, day number, and year. In Europe, however, the format is day number, month number, and year. How many days in a year are there where the date cannot be read unambiguously without knowing which format is being used? | 132 | 0.25 |
If the integers \( a, b, \) and \( c \) satisfy:
\[
a + b + c = 3, \quad a^3 + b^3 + c^3 = 3,
\]
then what is the maximum value of \( a^2 + b^2 + c^2 \)? | 57 | 0.125 |
On the diagonals $AC$ and $CE$ of a regular hexagon $ABCDEF$, points $M$ and $N$ are taken respectively, such that $\frac{AM}{AC} = \frac{CN}{CE} = \lambda$. It is known that points $B, M$, and $N$ lie on one line. Find $\lambda$. | \frac{\sqrt{3}}{3} | 0.5 |
Given the system of equations for positive numbers \( x, y, z \):
\[
\left\{
\begin{array}{l}
x^{2}+x y+y^{2}=48 \\
y^{2}+y z+z^{2}=9 \\
z^{2}+x z+x^{2}=57
\end{array}
\right.
\]
Find the value of the expression \( x y + y z + x z \). | 24 | 0.375 |
Given the equation of circle $C_{0}$ as $x^{2}+y^{2}=r^{2}$, find the equation of the tangent line passing through a point $M\left(x_{0}, y_{0}\right)$ on circle $C_{0}$. | x_0 x + y_0 y = r^2 | 0.875 |
Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1. | 11111111100000000 | 0.5 |
A homeowner currently pays a 10% income tax on rental income. By what percentage should the homeowner increase the rent to maintain the same income if the income tax is raised to 20%? | 12.5\% | 0.75 |
On a \(3 \times 3\) grid of 9 squares, each square is to be painted with either Red or Blue. If \(\alpha\) is the total number of possible colorings in which no \(2 \times 2\) grid consists of only Red squares, determine the value of \(\alpha\). | 417 | 0.25 |
Given the sets
$$
\begin{array}{l}
A=\left\{(x, y) \mid x=m, y=-3m+2, m \in \mathbf{Z}_{+}\right\}, \\
B=\left\{(x, y) \mid x=n, y=a\left(a^{2}-n+1\right), n \in \mathbf{Z}_{+}\right\},
\end{array}
$$
find the total number of integers $a$ such that $A \cap B \neq \varnothing$. | 10 | 0.625 |
Calculate the area of the figure bounded by the graphs of the following functions:
$$
y=\frac{e^{1 / x}}{x^{2}}, y=0, x=2, x=1
$$ | e - \sqrt{e} | 0.875 |
How many roots does the equation \(\sqrt{14-x^{2}}(\sin x-\cos 2x)=0\) have? | 6 | 0.5 |
There is a reservoir A and a town B connected by a river. When the reservoir does not release water, the water in the river is stationary; when the reservoir releases water, the water in the river flows at a constant speed. When the reservoir was not releasing water, speedboat M traveled for 50 minutes from A towards B and covered $\frac{1}{3}$ of the river's length. At this moment, the reservoir started releasing water, and the speedboat took only 20 minutes to travel another $\frac{1}{3}$ of the river's length. The driver then turned off the speedboat's engine and allowed it to drift with the current, taking $\quad$ minutes for the speedboat to reach B. | \frac{100}{3} | 0.75 |
How many integer solutions \((x, y)\) does the inequality \(|x| + |y| < n\) have? | 2n^2 - 2n + 1 | 0.375 |
Compute the sum of all positive integers \( a \leq 26 \) for which there exist integers \( b \) and \( c \) such that \( a + 23b + 15c - 2 \) and \( 2a + 5b + 14c - 8 \) are both multiples of 26. | 31 | 0.625 |
An easel in a corner hosts three $30 \text{ cm} \times 40 \text{ cm}$ shelves, with equal distances between neighboring shelves. Three spiders resided where the two walls and the middle shelf meet. One spider climbed diagonally up to the corner of the top shelf on one wall, another climbed diagonally down to the corner of the lower shelf on the other wall. The third spider stayed in place and observed that from its position, the other two spiders appeared at an angle of $120^\circ$. What is the distance between the shelves? (The distance between neighboring shelves is the same.) | 35 \text{ cm} | 0.625 |
Given the function \( f(x) = -x^2 + x + m + 2 \), if the inequality \( f(x) \geq |x| \) has exactly one integer in its solution set, determine the range of the real number \( m \). | [-2, -1) | 0.875 |
Petya chooses non-negative numbers \(x_{1}, x_{2}, \ldots, x_{11}\) such that their sum is 1. Vasya arranges them in a row at his discretion, calculates the products of adjacent numbers, and writes on the board the largest of the resulting ten products. Petya wants the number on the board to be as large as possible, while Vasya wants it to be as small as possible. What number will be on the board with the best play from both Petya and Vasya? | \frac{1}{40} | 0.125 |
Let $P$ be a moving point on the parabola $y^2 = 2x$. A tangent line is drawn at $P$ to the parabola, which intersects the circle $x^2 + y^2 = 1$ at points $M$ and $N$. The tangents to the circle at points $M$ and $N$ intersect at point $Q$. Find the equation of the locus of point $Q$. | y^2 = -2 x | 0.625 |
In $\triangle ABC$, \(BC = 5\), \(AC = 12\), \(AB = 13\). Points \(D\) and \(E\) are located on sides \(AB\) and \(AC\) respectively, such that line segment \(DE\) divides $\triangle ABC$ into two regions of equal area. Determine the minimum length of such a line segment \(DE\). | 2\sqrt{3} | 0.5 |
Two-headed and seven-headed dragons came to a meeting. At the very beginning of the meeting, one of the heads of one of the seven-headed dragons counted all the other heads. There were 25 of them. How many dragons in total came to the meeting? | 8 | 0.875 |
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 resets and starts falling 1 fruit again each day, following the original pattern. On which day will all the fruits be gone from the tree? | 15 | 0.25 |
Among the 200 natural numbers from 1 to 200, list the numbers that are neither multiples of 3 nor multiples of 5 in ascending order. What is the 100th number in this list? | 187 | 0.375 |
Construct a point \( O \) in the scalene triangle \( ABC \) such that the angles \( AOB \), \( BOC \), and \( COA \) are equal.
| O | 0.75 |
Calculate the areas of the figures bounded by the lines given in polar coordinates.
$$
r=\cos \phi, \quad r=2 \cos \phi
$$ | \frac{3\pi}{4} | 0.75 |
Let $\mathcal{P}$ and $\mathcal{P}^{\prime}$ be two convex quadrilaterals with $\mathcal{P}^{\prime}$ inside $\mathcal{P}$, and let $d$ (resp. $d^{\prime}$) be the sum of the lengths of the diagonals of $\mathcal{P}$ (resp. $\mathcal{P}^{\prime}$). Show that:
$$
d^{\prime} < 2d
$$ | d' < 2d | 0.875 |
Which is greater, \(\log_{2008}(2009)\) or \(\log_{2009}(2010)\)? | \log_{2008}(2009) | 0.625 |
Let \( AB \) be a focal chord of the parabola \( y^2 = 2px \) (where \( p > 0 \)), and let \( O \) be the origin. Then:
1. The minimum area of triangle \( OAB \) is \( \frac{p^2}{2} \).
2. The tangents at points \( A \) and \( B \) intersect at point \( M \), and the minimum area of triangle \( MAB \) is \( p^2 \). | p^2 | 0.125 |
There are $n$ balls that look identical, among which one ball is lighter than the others (all other balls have equal weight). If using an unweighted balance scale as a tool, it takes at least 5 weighings to find the lighter ball, then the maximum value of $n$ is ___. | 243 | 0.875 |
Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), find the maximum value of \(A = \frac{\left(1 - \sqrt{\tan \frac{\alpha}{2} \tan \frac{\beta}{2}}\right)^2}{\cot \alpha + \cot \beta}\). | 3 - 2\sqrt{2} | 0.875 |
Three cones are standing on their bases on a table, touching each other. The radii of their bases are 23, 46, and 69. A truncated cone is placed on the table with its smaller base down, sharing a common slant height with each of the other cones. Find the radius of the smaller base of the truncated cone. | 6 | 0.125 |
Find a nine-digit number in which all the digits are different and do not include zero, and the square root of this number has the form $\overline{a b a b c}$, where $\overline{a b} = c^3$. | 743816529 | 0.375 |
Given a convex quadrilateral with area \( S \). A point inside it is chosen and reflected symmetrically with respect to the midpoints of its sides. This results in four vertices of a new quadrilateral. Find the area of the new quadrilateral. | 2S | 0.875 |
A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. This four-digit number has a prime factor such that the prime factor minus 5 times another prime factor equals twice the third prime factor. What is this number? | 1221 | 0.75 |
The plane figure $W$ consists of all points whose coordinates $(x, y)$ satisfy the inequality $|2 - |x| - ||y| - 2|| \leqslant 1$. Draw the figure $W$ and find its area. | 30 | 0.125 |
The plane vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ satisfy $|\boldsymbol{a}|=1, \boldsymbol{a} \cdot \boldsymbol{b}=\boldsymbol{b} \cdot \boldsymbol{c}=1,$ and $|\boldsymbol{a}-\boldsymbol{b}+\boldsymbol{c}| \leqslant 2 \sqrt{2}$. Find the maximum value of $\boldsymbol{a} \cdot \boldsymbol{c}$. | 2 | 0.375 |
On the hypotenuse \( A B \) of an isosceles right triangle \( A B C \), points \( K \) and \( L \) are marked such that \( A K : K L : L B = 1 : 2 : \sqrt{3} \). Determine \( \angle K C L \). | 45^{\circ} | 0.625 |
There is an oak tree growing by the road from Yolkino to Palkino, and it is twice as close to Yolkino as it is to Palkino. Fedya, traveling at a constant (and positive) speed from Yolkino to Palkino, was twice as close to the oak tree as he was to Yolkino at 12:00. At 12:40, it turned out again that Fedya was twice as close to the oak tree as he was to Yolkino. When will Fedya arrive in Palkino? | 13:10 | 0.75 |
Find all positive integers \( n \) such that \( 3^n + 5^n \) is a multiple of \( 3^{n-1} + 5^{n-1} \). | 1 | 0.875 |
Find the infinite sum of \(\frac{1^{3}}{3^{1}}+\frac{2^{3}}{3^{2}}+\frac{3^{3}}{3^{3}}+\frac{4^{3}}{3^{4}}+\cdots\).
求 \(\frac{1^{3}}{3^{1}}+\frac{2^{3}}{3^{2}}+\frac{3^{3}}{3^{3}}+\frac{4^{3}}{3^{4}}+\cdots\) 無限項之和。 | \frac{33}{8} | 0.125 |
Assume there is a bag with red, yellow, blue, and green balls. Each time, one ball is taken from the bag, the color is confirmed, and then the ball is put back into the bag. This process continues until two consecutive red balls are drawn. Let $\zeta$ be the total number of draws. If each ball is equally likely to be drawn, what is the expected value of $\zeta$? | 20 | 0.625 |
ABC is a triangle with M the midpoint of BC. The segment XY with X on AB and Y on AC has its midpoint on AM. Show that it is parallel to BC. | XY \parallel BC | 0.875 |
How many Pythagorean triangles are there in which one of the legs is equal to 2013? (A Pythagorean triangle is a right triangle with integer sides. Identical triangles count as one.). | 13 | 0.25 |
Can we find a line normal to the curves \( y = \cosh x \) and \( y = \sinh x \)? | \text{No} | 0.75 |
Let the parabola \( C: y^2 = 2x \) have its directrix intersect the \( x \)-axis at point \( A \). A line \( l \) through point \( B(-1, 0) \) is tangent to the parabola \( C \) at point \( K \). Draw a line through point \( A \) parallel to \( l \), which intersects the parabola \( C \) at points \( M \) and \( N \). Find the area of triangle \( \triangle KMN \). | \frac{1}{2} | 0.875 |
Connecting point \( M \) with the vertices of triangle \( ABC \). Show that the resultant of the forces \( MA, MB \), and \( MC \) passes through the centroid \( G \) of the triangle and is equal to \( 3MG \). | MH = 3 MG | 0.875 |
The UEFA Champions League playoffs is a 16-team soccer tournament in which Spanish teams always win against non-Spanish teams. In each of 4 rounds, each remaining team is randomly paired against one other team; the winner advances to the next round, and the loser is permanently knocked out of the tournament. If 3 of the 16 teams are Spanish, what is the probability that there are 2 Spanish teams in the final round? | \frac{4}{5} | 0.125 |
Factory A and Factory B both produce the same type of clothing. Factory A produces 2700 sets of clothing per month, with the time ratio of producing tops to pants being 2:1. Factory B produces 3600 sets of clothing per month, with the time ratio of producing tops to pants being 3:2. If the two factories work together for one month, what is the maximum number of sets of clothing they can produce? | 6700 | 0.25 |
Let point \( O \) be a point inside triangle \( ABC \) that satisfies the equation
\[
\overrightarrow{OA} + 2 \overrightarrow{OB} + 3 \overrightarrow{OC} = 3 \overrightarrow{AB} + 2 \overrightarrow{BC} + \overrightarrow{CA}.
\]
Then, find the value of \(\frac{S_{\triangle AOB} + 2 S_{\triangle BOC} + 3 S_{\triangle COA}}{S_{\triangle ABC}}\). | \frac{11}{6} | 0.875 |
Among all fractions of the form \(\frac{m}{n}\), where \(m\) and \(n\) are four-digit numbers with the same sum of digits, the largest one was chosen. Find it and write your answer as an improper fraction (do not round or simplify it!). | \frac{9900}{1089} | 0.125 |
Let \(AB\) be the diameter of a circle, and \(CD\) a chord that is not perpendicular to it. If perpendiculars \(AE\) and \(BF\) are dropped from the ends of the diameter onto the chord, then the segments \(CF\) and \(DE\) are equal. | CF = DE | 0.875 |
How many integers (1) have 5 decimal digits, (2) have the last digit 6, and (3) are divisible by 3? | 3000 | 0.5 |
Let {a_n} be a sequence of positive numbers with the sum of its first n terms being b_n. Let {b_n} be such that the product of its first n terms is c_n, and b_n + c_n = 1. The number in the sequence {1/a_n} that is closest to 2002 is ___ . | 1980 | 0.625 |
After the teacher Maria Ivanovna moved Vovochka from the first row to the second, Vanechka from the second row to the third, and Masha from the third row to the first, the average age of students in the first row increased by one week, in the second row increased by two weeks, and in the third row decreased by four weeks. It is known that there are 12 students each in the first and second rows. How many students are sitting in the third row? | 9 | 0.875 |
Given that \(a, b, c\) are positive integers such that the roots of the three quadratic equations
\[
x^2 - 2ax + b = 0, \quad x^2 - 2bx + c = 0, \quad x^2 - 2cx + a = 0
\]
are all positive integers. Determine the maximum value of the product \(abc\). | 1 | 0.75 |
Complex numbers \(a, b, c\) form an equilateral triangle with side length 18 in the complex plane. If \(|a+b+c|=36\), find \(|bc + ca + ab|\). | 432 | 0.625 |
A cyclist traveled from point A to point B, stayed there for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and met him again 2 hours later on his way back. The pedestrian arrived at point B at the same time the cyclist returned to point A. How much time did it take the pedestrian to travel from A to B if his speed is four times less than the speed of the cyclist? | 10 \text{ hours} | 0.125 |
A hollow glass sphere with uniform wall thickness and an outer diameter of $16 \mathrm{~cm}$ floats in water in such a way that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness, given that the specific gravity of the glass is $s = 2.523$? | 0.8 \, \text{cm} | 0.125 |
Find a six-digit number where the first digit is 6 times less than the sum of all the digits to its right, and the second digit is 6 times less than the sum of all the digits to its right. | 769999 | 0.5 |
In space, there are \( n \) (\( n \geqslant 3 \)) planes, where any three planes do not share a common perpendicular plane. There are the following four assertions:
(1) No two planes are parallel to each other;
(2) No three planes intersect in a single line;
(3) Any two intersection lines between the planes are not parallel;
(4) Each intersection line between the planes intersects with \( n-2 \) other planes.
Determine how many of these assertions are correct. | 4 | 0.375 |
Let \( m=30030=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \) and let \( M \) be the set of its positive divisors which have exactly two prime factors. Determine the minimal integer \( n \) with the following property: for any choice of \( n \) numbers from \( M \), there exist three numbers \( a, b, c \) among them satisfying \( a \cdot b \cdot c = m \). | 11 | 0.25 |
Six members of a study group decide on a real number \( b \), which their incoming seventh member needs to guess. Upon entering, the following statements are provided sequentially:
I. \( b \) is an even number.
II. Adding the cube of \( b \) to \( b \) results in a positive number smaller than 8000.
III. \( b \) is a real number whose square is 13.
IV. \( b \) is an integer divisible by 7.
V. \( b \) is a rational number.
VI. \( b \) is an integer divisible by 14.
Additionally, it is stated that out of I and II, III and IV, V and VI, one statement is always true and the other is always false. - How did the seventh group member figure out the number? | 7 | 0.875 |
A gardener wants to plant 3 maple trees, 4 oak trees, and 5 birch trees in a row. He will randomly determine the order of these trees. What is the probability that no two birch trees are adjacent? | \frac{7}{99} | 0.875 |
Find the angle $B$ of triangle $ABC$ if the length of the altitude $CH$ is half the length of side $AB$, and $\angle BAC = 75^\circ$. | 30^\circ | 0.75 |
Given that the positive real number \(a\) satisfies \(a^{a}=(9 a)^{8 a}\), find the value of \(\log _{a}(3 a)\). | \frac{9}{16} | 0.875 |
Let's call a natural number "remarkable" if it is the smallest among all natural numbers with the same sum of digits as it.
How many three-digit remarkable numbers exist?
| 9 | 0.125 |
Two scientific production enterprises supply substrates for growing orchids to the market. In the "Orchid-1" substrate, there is three times more pine bark than sand, and twice as much peat as sand. In the "Orchid-2" substrate, there is half as much bark as there is peat, and one and a half times more sand than peat. In what ratio should the substrates be mixed so that the new mixed composition has equal parts of bark, peat, and sand? | 1:1 | 0.5 |
Let \( T = \left\{9^{k} \mid k \right. \) be integers, \(\left.0 \leqslant k \leqslant 4000\right\} \). Given that \(9^{4000}\) has 3817 digits and its most significant digit is 9, how many elements in \(T\) have 9 as their most significant digit?
(The 8th American Mathematics Invitational, 1990) | 184 | 0.5 |
In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $\cos A=\frac{1}{3}$.
1. Find the value of $\sin^{2} \frac{B+C}{2}+\cos 2A$.
2. If $a=\sqrt{3}$, find the maximum value of $b \cdot c$. | \frac{9}{4} | 0.75 |
The sides of triangle $ABC$ are extended as shown in Figure 5, with $AA' = 3AB$, $BB' = 5BC$, and $CC' = 8CA$. By what factor is the area of triangle $ABC$ smaller than the area of triangle $A'B'C'$? | 64 | 0.75 |
Given a cyclic quadrilateral \(A B C D\), the rays \(A B\) and \(D C\) intersect at point \(K\). It is found that the points \(B\), \(D\), and the midpoints of segments \(A C\) and \(K C\) lie on the same circle. What values can the angle \(A D C\) take? | 90^\circ | 0.625 |
In a $10 \times 10$ grid square with the centers of all unit squares marked (a total of 100 points), what is the minimum number of lines, not parallel to the sides of the square, that need to be drawn to cross out all the marked points? | 18 | 0.375 |
Initially, the number 1 and two positive numbers \( x \) and \( y \) are written on a blackboard. In each move, a player can choose any two numbers on the board, not necessarily distinct, and write their sum or their difference on the board. Additionally, they can choose any non-zero number on the board and write its inverse. After a finite number of moves, describe how we can obtain the following numbers:
a) \( x^{2} \).
b) \( x y \). | xy | 0.125 |
Find the smallest positive integer \( n \) such that there exists a sequence of \( n+1 \) terms \( a_0, a_1, \ldots, a_n \) with the properties \( a_0 = 0 \), \( a_n = 2008 \), and \( |a_i - a_{i-1}| = i^2 \) for \( i = 1, 2, \ldots, n \). | 19 | 0.625 |
Find the number of solutions in natural numbers for the equation \((x-4)^{2}-35=(y-3)^{2}\). | 3 | 0.625 |
Find the sum of the first 10 elements that appear in both the arithmetic progression $\{4,7,10,13, \ldots\}$ and the geometric progression $\{20,40,80,160, \ldots\}$. | 13981000 | 0.125 |
In a castle, there are 16 identical square rooms arranged in a $4 \times 4$ grid. Each of these rooms is occupied by one person, making a total of 16 people, who are either liars or knights (liars always lie, knights always tell the truth). Each of these 16 people said: "There is at least one liar in one of the rooms adjacent to mine." Two rooms are considered adjacent if they share a wall. What is the maximum number of liars that could be among these 16 people? | 8 | 0.625 |
On a road, there are three locations $A$, $O$, and $B$. $O$ is between $A$ and $B$, and $A$ is 1360 meters away from $O$. Two individuals, Jia and Yi, start simultaneously from points $A$ and $O$ towards point $B$. At the 10th minute after departure, both Jia and Yi are equidistant from point $O$. In the 40th minute, Jia and Yi meet at point $B$. What is the distance between points $O$ and $B$ in meters? | 2040 \text{ meters} | 0.875 |
A tetrahedron \(ABCD\) has edge lengths 7, 13, 18, 27, 36, 41, with \(AB = 41\). Determine the length of \(CD\). | 13 | 0.875 |
The notebook lists all the irreducible fractions with numerator 15 that are greater than $\frac{1}{16}$ and less than $\frac{1}{15}$. How many such fractions are listed in the notebook? | 8 | 0.75 |
Find the last non-zero digit of \(50! = 1 \times 2 \times \cdots \times 50\). | 2 | 0.625 |
In the sequence \(\left\{a_{n}\right\}\), for \(1 \leqslant n \leqslant 5\), we have \(a_{n}=n^{2}\). Additionally, for all positive integers \(n\), the following holds:
\[ a_{n+5} + a_{n+1} = a_{n+4} + a_{n}. \]
Determine the value of \(a_{2023}\). | 17 | 0.5 |
In an isosceles trapezoid \(ABCD\), the side \(AB\) and the shorter base \(BC\) are both equal to 2, and \(BD\) is perpendicular to \(AB\). Find the area of this trapezoid. | 3\sqrt{3} | 0.5 |
The distance between the midpoints of mutually perpendicular chords \(AC\) and \(BC\) of a circle is 10. Find the diameter of the circle. | 20 | 0.625 |
Solve in integers: \(6x^{2} + 5xy + y^{2} = 6x + 2y + 7\)
Indicate the answer for which the value \(|x| + |y|\) is the largest. Write the answer in the form \((x; y)\). | (-8; 25) | 0.5 |
In a six-digit number, the first digit, which is 2, was moved to the last position while leaving the other digits in the same order. The resulting number turned out to be three times the original number. Find the original number. | 285714 | 0.875 |
Given the sequence \(\left\{a_{n}\right\}\) where \(a_{1} = 1\) and \(a_{n+1} = \frac{\sqrt{3} a_{n} + 1}{\sqrt{3} - a_{n}}\), find the value of \(\sum_{n=1}^{2022} a_{n}\). | 0 | 0.75 |
In the middle of a square lake with a side length of 10 feet, there is a reed extending 1 foot above the water. If the reed is bent, its tip reaches the shore. How deep is the lake? | 12 | 0.875 |
Two people agree to meet between 7:00 AM and 8:00 AM, and the first person to arrive will wait for the other person for 20 minutes. The times at which they arrive at the designated meeting place between 7:00 AM and 8:00 AM are random and independent. What is the probability that they will meet? | \frac{5}{9} | 0.875 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.