Unnamed: 0 int64 0 40.3k | problem stringlengths 10 5.15k | ground_truth stringlengths 1 1.22k | solved_percentage float64 0 100 |
|---|---|---|---|
14,000 | On side \( AB \) of parallelogram \( ABCD \), point \( F \) is selected, and on the extension of side \( BC \) beyond vertex \( B \), point \( H \) is chosen such that \( \frac{AB}{BF} = \frac{BC}{BH} = 5 \). Point \( G \) is selected so that \( BFGH \) is a parallelogram. Line \( GD \) intersects \( AC \) at point \( ... | 40 | 0.78125 |
14,001 | Given points \( A(3,1) \) and \( B\left(\frac{5}{3}, 2\right) \), and the four vertices of quadrilateral \( \square ABCD \) are on the graph of the function \( f(x)=\log _{2} \frac{a x+b}{x-1} \), find the area of \( \square ABCD \). | \frac{26}{3} | 3.90625 |
14,002 | Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \), \( f(1) = 1 \), and \( |f'(x)| \leq 2 \) for all real numbers \( x \). If \( a \) and \( b \) are real numbers such that the set of possible values of \( \int_{0}^{1} f(x) \, dx \) is the open interval \( (a, b) \), de... | 3/4 | 0.78125 |
14,003 | Evaluate the sum of $1001101_2$ and $111000_2$, and then add the decimal equivalent of $1010_2$. Write your final answer in base $10$. | 143 | 64.0625 |
14,004 | How many pairs of positive integer solutions \((x, y)\) satisfy \(\frac{1}{x+1} + \frac{1}{y} + \frac{1}{(x+1) y} = \frac{1}{1991}\)? | 64 | 25.78125 |
14,005 | Let \( a \) be a nonzero real number. In the Cartesian coordinate system \( xOy \), the quadratic curve \( x^2 + ay^2 + a^2 = 0 \) has a focal distance of 4. Determine the value of \( a \). | \frac{1 - \sqrt{17}}{2} | 39.84375 |
14,006 | How many ways can the integers from -7 to 7 be arranged in a sequence such that the absolute value of the numbers in the sequence is nondecreasing? | 128 | 31.25 |
14,007 | Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([4, 6]\). Find \( \underbrace{f(f(\ldots f}_{2017}\left(\frac{9 - \sqrt{19}}{2}\right)) \ldots) \). If necessary, round the answer to two decimal places. | 6.68 | 0 |
14,008 | For which values of \( a \) do the quadratic trinomials \( x^{2} + a x + 1 \) and \( x^{2} + x + a \) have a common root? | -2 | 58.59375 |
14,009 | What is the minimum (identical) number of pencils that need to be placed in each of the 6 boxes so that in any 4 boxes there are pencils of any of the 26 pre-specified colors (assuming there are enough pencils available)? | 13 | 17.1875 |
14,010 | The base of the pyramid is a right triangle with a hypotenuse equal to 6 and an acute angle of $15^{\circ}$. All lateral edges are inclined to the plane of the base at an angle of $45^{\circ}$. Find the volume of the pyramid. | 4.5 | 0.78125 |
14,011 | Find the smallest square in which 5 circles, each with a radius of 1, can be arranged so that no two circles share any interior points. | 2\sqrt{2} + 2 | 2.34375 |
14,012 | Given a defect rate of 3%, products are drawn from the batch without replacement until a non-defective product is found or a maximum of three draws have been made. Let $X$ represent the number of products drawn, and calculate $P(X=3)$. | (0.03)^2 \times 0.97 + (0.03)^3 | 0 |
14,013 | The last 5 digits of $99 \times 10101 \times 111 \times 1001001$ are _____. | 88889 | 79.6875 |
14,014 | During a long voyage of a passenger ship, it was observed that at each dock, a quarter of the passenger composition is renewed, that among the passengers leaving the ship, only one out of ten boarded at the previous dock, and finally, that the ship is always fully loaded.
Determine the proportion of passengers at any ... | 21/40 | 3.125 |
14,015 | For real numbers \( x \), \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). Find the largest positive integer \( n \) such that the following equation holds:
\[
\lfloor \log_2 1 \rfloor + \lfloor \log_2 2 \rfloor + \lfloor \log_2 3 \rfloor + \cdots + \lfloor \log_2 n \rfloor = 1994
\]
... | 312 | 95.3125 |
14,016 | It is known that the number of birch trees in a certain mixed forest plot ranges from $13\%$ to $14\%$ of the total number of trees. Find the minimum possible total number of trees in this plot. | 15 | 64.84375 |
14,017 | In the trapezoid \(ABCD\), the lengths of the bases are \(AD = 24\) and \(BC = 8\), and the lengths of the diagonals are \(AC = 13\) and \(BD = 5\sqrt{17}\). Find the area of the trapezoid. | 80 | 10.9375 |
14,018 | In the rectangular coordinate system $xOy$, the parametric equations of line $l$ are $$\begin{cases} x=2 \sqrt {3}+at \\ y=4+ \sqrt {3}t\end{cases}$$ (where $t$ is the parameter), and in the polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar... | \frac {1}{2} | 7.8125 |
14,019 | Oleg drew an empty 50×50 table and wrote a number above each column and to the left of each row. It turned out that all 100 written numbers are different, 50 of which are rational and the remaining 50 are irrational. Then, in each cell of the table, he recorded the product of the numbers written near its row and its co... | 1250 | 1.5625 |
14,020 | The orthogonal projections of the triangle \(ABC\) onto two mutually perpendicular planes are equilateral triangles with sides of length 1. Find the perimeter of triangle \(ABC\), given that \(AB = \frac{\sqrt{5}}{2}\). | \sqrt{2} + \sqrt{5} | 0 |
14,021 | Let \[\mathbf{N} = \begin{pmatrix} x & y & z \\ y & z & x \\ z & x & y \end{pmatrix}\] be a matrix with real entries such that $\mathbf{N}^3 = \mathbf{I}.$ If $xyz = -1$, find the possible values of $x^3 + y^3 + z^3.$ | -2 | 7.8125 |
14,022 | Given three composite numbers \( A, B, \) and \( C \) which are pairwise coprime, and \( A \times B \times C = 11011 \times 28 \). What is the maximum value of \( A + B + C \)? | 1626 | 25.78125 |
14,023 | Egor wrote a number on the board and encrypted it according to the rules of letter puzzles (different letters correspond to different digits, identical letters - identical digits). The result was the word "GWATEMALA". How many different numbers could Egor have originally written if his number was divisible by 5? | 114240 | 9.375 |
14,024 | A tangent and a secant drawn from the same point to a circle are mutually perpendicular. The length of the tangent is 12, and the internal segment of the secant is 10. Find the radius of the circle. | 13 | 0.78125 |
14,025 | A natural number is written on the board. If its last digit (in the units place) is erased, the remaining non-zero number is divisible by 20. If the first digit is erased, the remaining number is divisible by 21. What is the smallest number that could be on the board if its second digit is not 0? | 1609 | 44.53125 |
14,026 | Given triangle $\triangle ABC$, $A=120^{\circ}$, $D$ is a point on side $BC$, $AD\bot AC$, and $AD=2$. Calculate the possible area of $\triangle ABC$. | \frac{8\sqrt{3}}{3} | 1.5625 |
14,027 | Given two non-zero vectors $\overrightarrow{m}$ and $\overrightarrow{n}$ with an angle of $\frac{\pi}{3}$ between them, and the magnitude of $\overrightarrow{n}$ is a positive scalar multiple of the magnitude of $\overrightarrow{m}$, i.e., $|\overrightarrow{n}| = λ|\overrightarrow{m}| (λ > 0)$. The vector group $\overr... | \frac{8}{3} | 20.3125 |
14,028 | Given a cylinder with height $OO_1 = 12$ and a base radius $r = 5$. There are points $A$ and $B$ on the circumferences of the top and bottom bases respectively, with $AB = 13$. Find the distance between the axis $OO_1$ and line segment $AB$. | \frac{5}{2} \sqrt{3} | 0 |
14,029 | There are a batch of wooden strips with lengths of \(1, 2, 3, 4, 5, 6, 7, 8, 9, 10,\) and 11 centimeters, with an adequate quantity of each length. If you select 3 strips appropriately to form a triangle with the requirement that the base is 11 centimeters long, how many different triangles can be formed? | 36 | 91.40625 |
14,030 | \[\left(\frac{a}{b} + \frac{b}{a} + 2\right) \cdot \left(\frac{a+b}{2a} - \frac{b}{a+b}\right) \div \left(\left(a + 2b + \frac{b^2}{a}\right) \cdot \left(\frac{a}{a+b} + \frac{b}{a-b}\right)\right);\ a = 0.75,\ b = \frac{4}{3}.\] | -\frac{7}{24} | 3.125 |
14,031 | Find the largest real number \(\lambda\) such that for the real-coefficient polynomial \(f(x) = x^3 + ax^2 + bx + c\) with all roots non-negative real numbers, we have \(f(x) \geqslant \lambda(x - a)^3\) for all \(x \geqslant 0\). Also, determine when the equality holds. | -\frac{1}{27} | 4.6875 |
14,032 | If \( p \) and \( q \) are positive integers, \(\max (p, q)\) is the maximum of \( p \) and \( q \) and \(\min (p, q)\) is the minimum of \( p \) and \( q \). For example, \(\max (30,40)=40\) and \(\min (30,40)=30\). Also, \(\max (30,30)=30\) and \(\min (30,30)=30\).
Determine the number of ordered pairs \((x, y)\) th... | 4100 | 43.75 |
14,033 | What is the maximum number of distinct numbers from 1 to 1000 that can be selected so that the difference between any two selected numbers is not equal to 4, 5, or 6? | 400 | 26.5625 |
14,034 | The function \( f(x) \) is defined on the set of real numbers, and satisfies the equations \( f(2+x) = f(2-x) \) and \( f(7+x) = f(7-x) \) for all real numbers \( x \). Let \( x = 0 \) be a root of \( f(x) = 0 \). Denote the number of roots of \( f(x) = 0 \) in the interval \(-1000 \leq x \leq 1000 \) by \( N \). Find ... | 401 | 67.96875 |
14,035 | One material particle entered the opening of a pipe, and after 6.8 minutes, a second particle entered the same opening. Upon entering the pipe, each particle immediately began linear motion along the pipe: the first particle moved uniformly at a speed of 5 meters per minute, while the second particle covered 3 meters i... | 17 | 42.1875 |
14,036 | Given a rectangular box-shaped fort with dimensions 15 feet in length, 12 feet in width, and 6 feet in height, and walls and floor uniformly 1.5 feet thick, calculate the total number of one-foot cubical blocks used to build the fort. | 594 | 29.6875 |
14,037 | A battery of three guns fired a volley, and two shells hit the target. Find the probability that the first gun hit the target, given that the probabilities of hitting the target by the first, second, and third guns are $p_{1}=0,4$, $p_{2}=0,3$, and $p_{3}=0,5$, respectively. | 20/29 | 57.03125 |
14,038 |
It is known that the numbers \( x, y, z \) form an arithmetic progression in the given order with a common difference \( \alpha = \arccos \left(-\frac{3}{7}\right) \), and the numbers \( \frac{1}{\cos x}, \frac{7}{\cos y}, \frac{1}{\cos z} \) also form an arithmetic progression in the given order. Find \( \cos^{2} y \... | \frac{10}{13} | 0 |
14,039 | A regular triangular prism \(A B C A_{1} B_{1} C_{1}\) with base \(A B C\) and lateral edges \(A A_{1}, B B_{1}, C C_{1}\) is inscribed in a sphere. The segment \(C D\) is the diameter of this sphere, and point \(K\) is the midpoint of edge \(A A_{1}\). Find the volume of the prism if \(C K = 2 \sqrt{3}\) and \(D K = 2... | 9\sqrt{2} | 0.78125 |
14,040 | In the diagram, if points $ A$ , $ B$ and $ C$ are points of tangency, then $ x$ equals:
[asy]unitsize(5cm);
defaultpen(linewidth(.8pt)+fontsize(8pt));
dotfactor=3;
pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);
pair O=(0,3/8);
draw((-2/3,9/16)--(2/3,9/16));
draw((-2/3,1/2)--(-sqrt(3)/6,1/2)-... | $\frac{1}{16}$ | 0 |
14,041 | In the cells of an $80 \times 80$ table, pairwise distinct natural numbers are placed. Each number is either prime or the product of two prime numbers (possibly the same). It is known that for any number $a$ in the table, there is a number $b$ in the same row or column such that $a$ and $b$ are not coprime. What is the... | 4266 | 0 |
14,042 | A circle is tangent to two adjacent sides \(AB\) and \(AD\) of square \(ABCD\) and cuts off a segment of length 8 cm from vertices \(B\) and \(D\) at the points of tangency. On the other two sides, the circle cuts off segments of 4 cm and 2 cm respectively from the vertices at the points of intersection. Find the radiu... | 10 | 21.875 |
14,043 | Let \( OP \) be the diameter of the circle \( \Omega \), and let \( \omega \) be a circle with its center at point \( P \) and a radius smaller than that of \( \Omega \). The circles \( \Omega \) and \( \omega \) intersect at points \( C \) and \( D \). The chord \( OB \) of circle \( \Omega \) intersects the second ci... | \sqrt{5} | 71.09375 |
14,044 | In the rectangular coordinate system $xOy$, with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the polar coordinate system is established. The polar coordinate equation of the curve $C$ is $\rho=2\sin\theta+2a\cos\theta$ ($a>0$); the parameter equation of the line $l$ is $$\begin{cases} x=-2+ \fr... | a=2 | 55.46875 |
14,045 | Find the increments of the argument and the function for \( y = 2x^2 + 1 \) when the argument \( x \) changes from 1 to 1.02. | 0.0808 | 4.6875 |
14,046 | If \( a, b, c, d \) are four distinct positive integers such that \( a \times b \times c \times d = 2277 \), what is the maximum value of \( a + b + c + d \)? | 84 | 18.75 |
14,047 | In a convex 13-gon, all diagonals are drawn. They divide it into polygons. Consider the polygon with the largest number of sides among them. What is the greatest number of sides that it can have? | 13 | 10.15625 |
14,048 | Let a three-digit number \( n = \overline{abc} \), where \( a \), \( b \), and \( c \) can form an isosceles (including equilateral) triangle as the lengths of its sides. How many such three-digit numbers \( n \) are there? | 165 | 93.75 |
14,049 | In the diagram, \(C\) lies on \(AE\) and \(AB=BC=CD\). If \(\angle CDE=t^{\circ}, \angle DEC=(2t)^{\circ}\), and \(\angle BCA=\angle BCD=x^{\circ}\), determine the measure of \(\angle ABC\). | 60 | 7.8125 |
14,050 | In a rectangular coordinate system, a circle centered at the point $(1,0)$ with radius $r$ intersects the parabola $y^2 = x$ at four points $A$, $B$, $C$, and $D$. If the intersection point $F$ of diagonals $AC$ and $BD$ is exactly the focus of the parabola, determine $r$. | \frac{\sqrt{15}}{4} | 0 |
14,051 | A necklace consists of 100 beads of red, blue, and green colors. It is known that among any five consecutive beads, there is at least one blue bead, and among any seven consecutive beads, there is at least one red bead. What is the maximum number of green beads that can be in this necklace? (The beads in the necklace a... | 65 | 16.40625 |
14,052 | The radius of the circle inscribed in triangle \(ABC\) is 4, with \(AC = BC\). On the line \(AB\), point \(D\) is chosen such that the distances from \(D\) to the lines \(AC\) and \(BC\) are 11 and 3 respectively. Find the cosine of the angle \(DBC\). | \frac{3}{4} | 0.78125 |
14,053 | Among the numbers $1, 2, 3, \cdots, 50$, if 10 consecutive numbers are selected, what is the probability that exactly 3 of them are prime numbers? | 22/41 | 6.25 |
14,054 | What is the greatest possible value of the expression \(\frac{1}{a+\frac{2010}{b+\frac{1}{c}}}\), where \(a, b, c\) are distinct non-zero digits? | 1/203 | 1.5625 |
14,055 | Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=1$, $|\overrightarrow{b}|= \sqrt {2}$, and $\overrightarrow{a}\perp (\overrightarrow{a}- \overrightarrow{b})$, then the angle between vector $\overrightarrow{a}$ and vector $\overrightarrow{b}$ is ______. | \dfrac {\pi}{4} | 99.21875 |
14,056 | The three different points \(A(x_1, y_1)\), \(B\left(4, \frac{9}{5}\right)\), and \(C(x_2, y_2)\) on the ellipse \(\frac{x^{2}}{25} + \frac{y^{2}}{9} = 1\), along with the focus \(F(4,0)\) have distances that form an arithmetic sequence. If the perpendicular bisector of line segment \(AC\) intersects the x-axis at poin... | 5/4 | 2.34375 |
14,057 | How many triangles exist such that the lengths of the sides are integers not greater than 10? | 125 | 65.625 |
14,058 | Although I am certain that my clock is 5 minutes fast, it is actually 10 minutes slow. On the other hand, my friend's clock is really 5 minutes fast, even though he thinks it is correct. We scheduled a meeting for 10 o'clock and plan to arrive on time. Who will arrive first? After how much time will the other arrive? | 20 | 2.34375 |
14,059 | Let \( S = \{ 1, 2, \cdots, 2005 \} \). Find the smallest number \( n \) such that in any subset of \( n \) pairwise coprime numbers from \( S \), there is at least one prime number. | 16 | 7.8125 |
14,060 | Al and Bert must arrive at a town 22.5 km away. They have one bicycle between them and must arrive at the same time. Bert sets out riding at 8 km/h, leaves the bicycle, and then walks at 5 km/h. Al walks at 4 km/h, reaches the bicycle, and rides at 10 km/h. For how many minutes was the bicycle not in motion? | 75 | 7.8125 |
14,061 | A barcode is composed of alternate strips of black and white, where the leftmost and rightmost strips are always black. Each strip (of either color) has a width of 1 or 2. The total width of the barcode is 12. The barcodes are always read from left to right. How many distinct barcodes are possible? | 116 | 0 |
14,062 | Consider the expanded hexagonal lattice shown below, where each point is one unit from its nearest neighbor. Determine the number of equilateral triangles whose vertices lie on this lattice.
```asy
size(100);
dot(origin);
dot(dir(30) + dir(90));
dot(dir(90));
dot(dir(90) + dir(150));
dot(dir(150));
dot(dir(150) + dir(2... | 28 | 0.78125 |
14,063 | Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases: "My neighbor on the left is a liar" or "My neighbor on the right is a liar." What is the minimum number of knights that can be at the t... | 10 | 13.28125 |
14,064 | Let \( ABCD \) and \( WXYZ \) be two squares that share the same center such that \( WX \parallel AB \) and \( WX < AB \). Lines \( CX \) and \( AB \) intersect at \( P \), and lines \( CZ \) and \( AD \) intersect at \( Q \). If points \( P, W \), and \( Q \) are collinear, compute the ratio \( AB / WX \). | \sqrt{2} + 1 | 0.78125 |
14,065 | Solve the equation \(2 x^{3} + 24 x = 3 - 12 x^{2}\). | \sqrt[3]{\frac{19}{2}} - 2 | 0 |
14,066 | The number of six-digit even numbers formed by 1, 2, 3, 4, 5, 6 without repeating any digit and with neither 1 nor 3 adjacent to 5 can be calculated. | 108 | 60.15625 |
14,067 | The point \( N \) is the center of the face \( ABCD \) of the cube \( ABCDEFGH \). Also, \( M \) is the midpoint of the edge \( AE \). If the area of \(\triangle MNH\) is \( 13 \sqrt{14} \), what is the edge length of the cube? | 2\sqrt{13} | 1.5625 |
14,068 | The sum of the first three terms of an arithmetic progression, as well as the sum of the first six terms, are natural numbers. Additionally, its first term \( d_{1} \) satisfies the inequality \( d_{1} \geqslant \frac{1}{2} \). What is the smallest possible value that \( d_{1} \) can take? | 5/9 | 0.78125 |
14,069 | The density of a body is defined as the ratio of its mass to the volume it occupies. There is a homogeneous cube with a volume of $V=8 \, m^{3}$. As a result of heating, each of its edges increased by 4 mm. By what percentage did the density of this cube change? | 0.6 | 0.78125 |
14,070 | A biased coin lands heads with a probability of $\frac{2}{3}$ and tails with $\frac{1}{3}$. A player can choose between Game C and Game D. In Game C, the player tosses the coin five times and wins if either the first three or the last three outcomes are all the same. In Game D, she tosses the coin five times and wins i... | \frac{29}{81} | 1.5625 |
14,071 | Let a constant $a$ make the equation $\sin x + \sqrt{3}\cos x = a$ have exactly three different solutions $x_{1}$, $x_{2}$, $x_{3}$ in the closed interval $\left[0,2\pi \right]$. The set of real numbers for $a$ is ____. | \{\sqrt{3}\} | 0 |
14,072 | Alex wrote all natural divisors of a natural number \( n \) on the board in ascending order. Dima erased several of the first and several of the last numbers of the resulting sequence so that 151 numbers remained. What is the maximum number of these 151 divisors that could be fifth powers of natural numbers? | 31 | 7.8125 |
14,073 | The sum of sides \( AB \) and \( BC \) of triangle \( ABC \) is 11, angle \( B \) is \( 60^\circ \), and the radius of the inscribed circle is \(\frac{2}{\sqrt{3}}\). It is also known that side \( AB \) is longer than side \( BC \). Find the height of the triangle dropped from vertex \( A \). | 4\sqrt{3} | 6.25 |
14,074 | From $A$ to $B$ it is 999 km. Along the road, there are kilometer markers with distances written to $A$ and to $B$:
$0|999,1|998, \ldots, 999|0$.
How many of these markers have only two different digits? | 40 | 87.5 |
14,075 | Let $k$ be a natural number. For which value of $k$ is $A_k = \frac{19^k + 66^k}{k!}$ maximized? | 65 | 53.125 |
14,076 | Given non-negative real numbers \( a, b, c, x, y, z \) that satisfy \( a + b + c = x + y + z = 1 \), find the minimum value of \( \left(a - x^{2}\right)\left(b - y^{2}\right)\left(c - z^{2}\right) \). | -1/4 | 0 |
14,077 | Find a three-digit number whose square is a six-digit number, such that each subsequent digit from left to right is greater than the previous one. | 367 | 52.34375 |
14,078 | What is the smallest positive integer \(n\) such that \(\frac{n}{n+75}\) is equal to a terminating decimal? | 50 | 2.34375 |
14,079 | Person A starts traveling from point A to point B. Persons B and C start traveling from point B to point A. After person A has traveled 50 kilometers, persons B and C start traveling from point B. Person A and person B meet at point C, and person A and person C meet at point D. It is known that the speed of person A is... | 130 | 6.25 |
14,080 | If three lines from the family of lines given by \( C: x \cos t + (y + 1) \sin t = 2 \) enclose an equilateral triangle \( D \), what is the area of the region \( D \)? | 12\sqrt{3} | 12.5 |
14,081 | Point \(A\) lies on the line \(y = \frac{15}{8}x - 8\), and point \(B\) lies on the parabola \(y = x^2\). What is the minimum length of segment \(AB\)? | 1823/544 | 0 |
14,082 | Mia and Tom jog on a circular track. Mia jogs counterclockwise and completes a lap every 96 seconds, while Tom jogs clockwise and completes a lap every 75 seconds. They both start from the same point at the same time. If a photographer positioned inside the track takes a snapshot sometime between 12 minutes and 13 minu... | \frac{5}{6} | 0 |
14,083 | Given the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$, given that $a^{2} + b^{2} - 3c^{2} = 0$, where $c$ is the semi-latus rectum, find the value of $\frac{a + c}{a - c}$. | 3 + 2\sqrt{2} | 0.78125 |
14,084 | Four cyclists. Four identical circles represent four tracks. The four cyclists start from the center at noon. Each moves along their track at speeds: the first at 6 km/h, the second at 9 km/h, the third at 12 km/h, and the fourth at 15 km/h. They agreed to ride until they all meet again in the center for the fourth tim... | 12:26:40 | 6.25 |
14,085 | Given \( S = [\sqrt{1}] + [\sqrt{2}] + \cdots + [\sqrt{1988}] \), find \( [\sqrt{S}] \). | 241 | 47.65625 |
14,086 | Compute the smallest positive integer $n$ such that $(n + i), (n + i)^3,$ and $(n + i)^4$ are the vertices of a triangle in the complex plane whose area is greater than 3000. | 10 | 5.46875 |
14,087 | Find the largest natural number consisting of distinct digits such that the product of its digits equals 2016. | 876321 | 12.5 |
14,088 | Given a polynomial \( P(x) = a_{2n} x^{2n} + a_{2n-1} x^{2n-1} + \ldots + a_1 x + a_0 \) where each coefficient \( a_i \) belongs to the interval \([100,101]\), what is the smallest natural number \( n \) such that this polynomial can have a real root? | 100 | 2.34375 |
14,089 | In the vertices of a unit square, perpendiculars are erected to its plane. On them, on one side of the plane of the square, points are taken at distances of 3, 4, 6, and 5 from this plane (in order of traversal). Find the volume of the polyhedron whose vertices are the specified points and the vertices of the square. | 4.5 | 13.28125 |
14,090 | Natural numbers \( A \) and \( B \) are divisible by all natural numbers from 1 to 65. What is the smallest natural number that \( A + B \) might not be divisible by? | 67 | 58.59375 |
14,091 | Seven people are seated together around a circular table. Each one will toss a fair coin. If the coin shows a head, then the person will stand. Otherwise, the person will remain seated. The probability that after all of the tosses, no two adjacent people are both standing, can be written in the form \( p / q \), where ... | 81 | 44.53125 |
14,092 | In a certain kingdom, the workforce consists only of a clan of dwarves and a clan of elves. Historically, in this kingdom, dwarves and elves have always worked separately, and no enterprise has ever allowed itself to hire both at the same time. The aggregate labor supply of dwarves is given by the function \( w_{\text{... | 1.25 | 3.90625 |
14,093 | A pyramid \( S A B C D \) has a trapezoid \( A B C D \) as its base, with bases \( B C \) and \( A D \). Points \( P_1, P_2, P_3 \) lie on side \( B C \) such that \( B P_1 < B P_2 < B P_3 < B C \). Points \( Q_1, Q_2, Q_3 \) lie on side \( A D \) such that \( A Q_1 < A Q_2 < A Q_3 < A D \). Let \( R_1, R_2, R_3, \) an... | 2028 | 11.71875 |
14,094 | Line segments \( AB \) and \( CD \) are situated between two parallel planes \( \alpha \) and \( \beta \). \( AC \subset \alpha \) and \( BD \subset \beta \). Given \( AB \perp \alpha \), \( AC = BD = 5 \), \( AB = 12 \), and \( CD = 13 \). Points \( E \) and \( F \) divide \( AB \) and \( CD \) in the ratio \( 1:2 \) ... | \frac{5}{3} \sqrt{7} | 0 |
14,095 | Determine the number of digits in the value of $2^{15} \times 5^{12} - 10^5$. | 12 | 1.5625 |
14,096 | Six IMO competitions are hosted sequentially by two Asian countries, two European countries, and two African countries, where each country hosts once but no continent can host consecutively. How many such arrangements are possible? | 240 | 88.28125 |
14,097 | The sequence \(\left\{a_{n}\right\}\) satisfies \(a_{1} = 1\) and \(a_{n} = \frac{1}{4 a_{n-1}} + \frac{1}{n}\) for \(n \geq 2\). Find the value of \(a_{100}\). | \frac{101}{200} | 0 |
14,098 | Using the trapezoidal rule with an accuracy of 0.01, calculate $\int_{2}^{3} \frac{d x}{x-1}$. | 0.6956 | 0 |
14,099 | How many ordered quadruples \((a, b, c, d)\) of positive odd integers are there that satisfy the equation \(a + b + c + 2d = 15?\) | 34 | 21.875 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.