problem stringlengths 33 2.6k | answer stringlengths 1 359 | source stringclasses 2
values | llama8b_solve_rate float64 0.06 0.59 |
|---|---|---|---|
Six consecutive numbers were written on a board. When one of them was crossed out and the remaining were summed, the result was 10085. What number could have been crossed out? Specify all possible options. | 2020 | olympiads | 0.0625 |
As shown in the figure, given that \( BE = 3EC \) and \( ED = 3AD \), and the area of triangle \( BDE \) is 18 square centimeters, find the area of triangle \( ABC \) in square centimeters. | 32 | olympiads | 0.0625 |
Let complex numbers \( z_1 \) and \( z_2 \) satisfy \( \left|z_1\right| = \left|z_1 + z_2\right| = 3, \left|z_1 - z_2\right| = 3 \sqrt{3} \). Find the value of \( \log_3 \left| \left( z_1 \overline{z_2} \right)^{2000} + \left( \overline{z_1} z_2 \right)^{2000} \right|. \) | 4000 | olympiads | 0.234375 |
A mother rabbit bought seven drums of different sizes and seven pairs of drumsticks of different lengths for her seven bunnies. If a bunny sees that both its drum is bigger and its drumsticks are longer compared to those of any of its siblings, it starts drumming loudly. What is the maximum number of bunnies that can s... | 6 | olympiads | 0.1875 |
Find all triples of natural numbers \((A, B, C)\) such that
\[ A^2 + B - C = 100 \]
\[ A + B^2 - C = 124 \] | (12, 13, 57) | olympiads | 0.078125 |
Each face of a hexahedron and each face of a regular octahedron are squares with side length $a$. The ratio of the radii of the inscribed spheres of these two polyhedra is an irreducible fraction $\frac{m}{n}$. What is the product $m \cdot n$? | 6 | olympiads | 0.21875 |
The king decided to test his one hundred wise men by conducting the following experiment: he will line them up with their eyes blindfolded, then place a black or white cap on each of their heads. After the blindfolds are removed, each wise man, starting from the last in the line, must state the color of his cap. If he ... | 99 | olympiads | 0.15625 |
Given \(\frac{\sin ^{4} \alpha}{\sin ^{2} \beta}+\frac{\cos ^{4} \alpha}{\cos ^{2} \beta}=1\), \(\sin \alpha \cos \alpha \sin \beta \cos \beta \neq 0\), find the value of \(\frac{\sin ^{4} \beta}{\sin ^{2} \alpha}+\frac{\cos ^{4} \beta}{\cos ^{2} \alpha}\). | 1 | olympiads | 0.4375 |
Two material particles, initially located 295 meters apart, start moving towards each other simultaneously. The first particle moves uniformly with a speed of \(15 \mathrm{~m}/\mathrm{s}\), while the second particle covers 1 meter in the first second and increases its distance covered by 3 meters more each subsequent s... | 60^
\circ | olympiads | 0.140625 |
On a line, one hundred points were placed at equal intervals, covering a segment of length \( a \). Then ten thousand points were placed at the same intervals on the line, covering a segment of length \( b \). By what factor is \( b \) greater than \( a \)? | 101 | olympiads | 0.40625 |
Find all functions \( f: \mathbb{Q} \rightarrow \mathbb{R} \) such that \( f(x + y) = f(x) + f(y) + 2xy \) for all \( x, y \) in \( \mathbb{Q} \) (the rationals). | f(x) = x^2 + kx | olympiads | 0.09375 |
A bagel is a loop of \( 2a + 2b + 4 \) unit squares which can be obtained by cutting a concentric \( a \times b \) hole out of an \( (a+2) \times (b+2) \) rectangle, for some positive integers \( a \) and \( b \). (The side of length \( a \) of the hole is parallel to the side of length \( a+2 \) of the rectangle.)
Co... | \alpha = \frac{3}{2} | olympiads | 0.078125 |
Given that \( F \) is the right focus of the hyperbola \( x^{2} - y^{2} = 1 \), \( l \) is the right directrix of the hyperbola, and \( A \) and \( B \) are two moving points on the right branch of the hyperbola such that \( A F \perp B F \). The projection of the midpoint \( M \) of line segment \( AB \) onto \( l \) ... | \frac{1}{2} | olympiads | 0.25 |
We wrote distinct real numbers at the vertices of an $n$-sided polygon such that any number is equal to the product of the numbers written at the two neighboring vertices.
Determine the value of $n$. | 6 | olympiads | 0.09375 |
In parallelogram \(ABCD\), the angle \( \angle ACD \) is \(30^{\circ}\). It is known that the centers of the circumcircles of triangles \(ABD\) and \(BCD\) are located on the diagonal \(AC\). Find the angle \( \angle ABD \). | 60^\circ \text{ or } 30^\circ | olympiads | 0.359375 |
The numbers \( x \), \( y \), and \( z \) satisfy the equations \( 9x + 3y - 5z = -4 \) and \( 5x + 2y - 2z = 13 \). What is the mean of \( x \), \( y \), and \( z \)? | 10 | olympiads | 0.125 |
We want to make an open-top rectangular prism-shaped box with a square base from a square sheet of paper with a side length of $20 \mathrm{~cm}$. To do this, we cut congruent squares from each of the four corners of the paper and fold up the resulting "flaps." What is the maximum possible volume of the box? | \frac{16000}{27} \, \text{cm}^3 | olympiads | 0.25 |
The Cucumber River, flowing through Flower Town, has several islands with a total perimeter of 8 meters in the dock area. Znayka claims that it is possible to depart from the dock in a boat and cross to the other side of the river while traveling less than 3 meters. The riverbanks near the dock are parallel, and the wi... | Znayka is correct. | olympiads | 0.09375 |
A $23 \times 23$ square is tiled with $1 \times 1, 2 \times 2$ and $3 \times 3$ squares. What is the smallest possible number of $1 \times 1$ squares? | 1 | aops_forum | 0.09375 |
List all the combinations that can be obtained by rearranging the letters in the word MAMA. | \{ \text{MAMA, MAM, MAAM, AMAM, AAMM, AMMA} \} | olympiads | 0.15625 |
In the equation on the right, the letters \( a, b, c, d \) and " \( \square \) " each represent one of the ten digits from 0 to 9. The four letters \( a, b, c, d \) represent different digits. Find the sum of the digits represented by \( a, b, c, d \). | 10, 18, 19 | olympiads | 0.15625 |
Given the sets
$$
\begin{array}{l}
A=\{(x, y) \mid |x| + |y| = a, a > 0\}, \\
B=\{(x, y) \mid |xy| + 1 = |x| + |y|\}
\end{array}
$$
If $A \cap B$ forms the vertices of a regular octagon in the plane, find the value of $a$. | \sqrt{2} | olympiads | 0.109375 |
In the parallelepiped $ABCD A_1B_1C_1D_1$, the face $ABCD$ is a square with side length 5, the edge $AA_1$ is also equal to 5, and this edge forms angles of $60^\circ$ with the edges $AB$ and $AD$. Find the length of the diagonal $BD_1$. | 5\sqrt{3} | olympiads | 0.09375 |
In an isosceles trapezoid, the angle at the base is $50^{\circ}$, and the angle between the diagonals, adjacent to the lateral side, is $40^{\circ}$. Where is the center of the circumscribed circle located, inside or outside the trapezoid? | Outside | olympiads | 0.328125 |
Suppose someone fills out all possible lottery tickets (13 matches, each filled with 1, 2, or $x$) in a week and then groups their tickets by the number of correct predictions after the matches. In which group (based on the number of correct predictions) will most of the tickets fall? (It is not necessary to calculate ... | 4 | olympiads | 0.15625 |
Given a frustum-shaped glass full of water, with the lower and upper inner diameters being 4 and 14 units respectively, and an inner height of 12 units. First, a solid sphere A is placed at the bottom such that it touches the cup walls (i.e., it is tangent). Then, a second solid sphere B is placed such that it touches ... | 61 \pi | olympiads | 0.125 |
Determine the largest possible value of the expression $ab+bc+ 2ac$ for non-negative real numbers $a, b, c$ whose sum is $1$ . | \frac{1}{2} | aops_forum | 0.171875 |
The number of common terms (terms with the same value) in the arithmetic sequences $2, 5, 8, \cdots, 2015$ and $4, 9, 14, \cdots, 2014$ is $\qquad$ . | 134 | olympiads | 0.0625 |
Which functions \( f \) satisfy the inequality
\[ |f(x)-f(y)| \leq (x-y)^{2} \]
for all pairs of numbers \( x \) and \( y \)? | f \text{ is constant} | olympiads | 0.21875 |
A motorcyclist traveled the first half of the distance at a speed of 30 km/h and the second half at a speed of 60 km/h. Find the motorcyclist's average speed. | 40 km/h | olympiads | 0.15625 |
In the three words "尽心尽力", "力可拔山", and "山穷水尽", each Chinese character represents a digit from 1 to 8. Identical characters represent the same digit, and different characters represent different digits. If the sum of the digits represented by the characters in each word is 19, and "尽" > "山" > "力", what is the maximum po... | 7 | olympiads | 0.125 |
Circle \(\omega_{1}\) with center \(O\) intersects with circle \(\omega_{2}\) at points \(K\) and \(L\). Circle \(\omega_{2}\) passes through point \(O\). A line through point \(O\) intersects circle \(\omega_{2}\) again at point \(A\). Segment \(OA\) intersects circle \(\omega_{1}\) at point \(B\). Find the ratio of t... | 1:1 | olympiads | 0.203125 |
Find all natural numbers $n$ for which the number $n^2 + 3n$ is a perfect square. | n = 1 | olympiads | 0.125 |
The function \( f \) maps the set of positive integers onto itself and satisfies the equation
\[
f(f(n)) + f(n) = 2n + 6
\]
What could this function be? | f(n) = n + 2 | olympiads | 0.1875 |
Define the determinant $D_1$ = $|1|$ , the determinant $D_2$ = $|1 1|$ $|1 3|$
, and the determinant $D_3=$
|1 1 1|
|1 3 3|
|1 3 5|
.
In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its first row and first
colu... | 12 | aops_forum | 0.09375 |
Solve the following equation over the set of real numbers:
$$
\sqrt{\lg ^{2} x+\lg x^{2}+1}+\lg x+1=0
$$ | 0 < x \leq \frac{1}{10} | olympiads | 0.09375 |
A figure in the plane has exactly two axes of symmetry. Find the angle between these axes. | 90^
| olympiads | 0.0625 |
Let $n$ be a positive integer. Each number $1, 2, ..., 1000$ has been colored with one of $n$ colours. Each two numbers , such that one is a divisor of second of them, are colored with different colours. Determine minimal number $n$ for which it is possible. | n = 10 | aops_forum | 0.078125 |
There is a pocket containing 5 coins with a face value of 1 yuan and 6 coins with a face value of 0.5 yuan. Xiao Ming randomly takes 6 coins from the pocket. How many possible sums can the face values of these 6 coins have? | 7 | olympiads | 0.5 |
Given the real numbers \(a, b, c, x, y, z\):
$$
\begin{aligned}
& a^{2}+b^{2}+c^{2}=25 \\
& x^{2}+y^{2}+z^{2}=36 \\
& a x+b y+c z=30
\end{aligned}
$$
What is the value of the quotient \(\frac{a+b+c}{x+y+z}\)? | \frac{5}{6} | olympiads | 0.578125 |
We form a decimal code of $21$ digits. the code may start with $0$ . Determine the probability that the fragment $0123456789$ appears in the code. | \frac{12 \cdot 10^{11} - 30}{10^{21}} | aops_forum | 0.125 |
Given are the lines $l_1,l_2,\ldots ,l_k$ in the plane, no two of which are parallel and no three of which are concurrent. For which $k$ can one label the intersection points of these lines by $1, 2,\ldots , k-1$ so that in each of the given lines all the labels appear exactly once? | k such that 2 \mid k | aops_forum | 0.09375 |
Let the set \( A = \{ 1, 2, \cdots, n \} \). Let \( S_n \) denote the sum of all elements in the non-empty proper subsets of \( A \), and \( B_n \) denote the number of subsets of \( A \). Find the limit \(\lim_{{n \to \infty}} \frac{S_n}{n^2 B_n}\). | \frac{1}{4} | olympiads | 0.0625 |
Fill in the blanks with “+” or “-” signs in the expression $1 \square 2 \square 3 \square 6 \square 12$ to obtain different natural number results. How many different natural number results can be obtained? | 9 | olympiads | 0.09375 |
25 oranges cost as many rubles as can be bought for 1 ruble. How many oranges can be bought for 3 rubles? | 15 | olympiads | 0.125 |
Given real numbers \(x_{1}, \ldots, x_{n}\), find the maximum value of the expression:
$$
A=\left(\sin x_{1}+\ldots+\sin x_{n}\right) \cdot\left(\cos x_{1}+\ldots+\cos x_{n}\right)
$$ | \frac{n^2}{2} | olympiads | 0.171875 |
At the parade, drummers stand in a perfect square formation of 50 rows with 50 drummers each. The drummers are dressed either in blue or red uniforms. What is the maximum number of drummers that can be dressed in blue uniforms so that each drummer in blue sees only red drummers? Consider the drummers as looking in all ... | 625 | olympiads | 0.109375 |
Compute the positive difference between the two real solutions to the equation $$ (x-1)(x-4)(x-2)(x-8)(x-5)(x-7)+48\sqrt 3 = 0. $$ | \sqrt{25 + 8\sqrt{3}} | aops_forum | 0.078125 |
Calculate: $1+3 \frac{1}{6}+5 \frac{1}{12}+7 \frac{1}{20}+9 \frac{1}{30}+11 \frac{1}{42}+13 \frac{1}{56}+15 \frac{1}{72}+17 \frac{1}{90}$ | 81 \frac{2}{5} | olympiads | 0.296875 |
The sum of the first two digits of a four-digit number is equal to the sum of the last two digits. The sum of the first and the last digits equals the third digit. The sum of the second and fourth digits is twice the sum of the first and third digits. What is this four-digit number? | 1854 | olympiads | 0.078125 |
A merchant accidentally mixed first-grade candies (priced at 3 rubles per pound) with second-grade candies (priced at 2 rubles per pound). At what price should this mixture be sold to yield the same total amount, given that the initial total cost of all first-grade candies was equal to the total cost of all second-grad... | 2 \text{ rubles } 40 \text{ kopeks} | olympiads | 0.40625 |
Given that \( a, b, c \) are the lengths of the sides of a right triangle, and for any natural number \( n > 2 \), the equation \(\left(a^{n} + b^{n} + c^{n}\right)^{2} = 2\left(a^{2n} + b^{2n} + c^{2n}\right)\) holds, find \( n \). | n = 4 | olympiads | 0.0625 |
Two delegations are scheduled to meet on the top floor of a tower that has several elevators, each with a capacity of nine people. The first delegation used a certain number of elevators filled to capacity, plus one last elevator with five vacant spots. Then, the second delegation did the same, with the last elevator h... | 3 | olympiads | 0.15625 |
How can you express with a single equality sign that at least one of the three numbers \( a, b, c \) is equal to zero? | abc = 0 | olympiads | 0.09375 |
How many $k$-configurations that have $m$ elements are there of a set that has $n$ elements? | \binom{n}{k} | olympiads | 0.21875 |
Suppose \(a, b\), and \(c\) are integers such that the greatest common divisor of \(x^{2} + a x + b\) and \(x^{2} + b x + c\) is \(x + 1\) (in the ring of polynomials in \(x\) with integer coefficients), and the least common multiple of \(x^{2} + a x + b\) and \(x^{2} + b x + c\) is \(x^{3} - 4 x^{2} + x + 6\). Find \(... | -6 | olympiads | 0.109375 |
It is known that a sphere can be inscribed in a certain prism. Find the lateral surface area of the prism if the area of its base is \( S \). | 4S | olympiads | 0.109375 |
Two circles with radii \(R\) and \(r\) are tangent to the sides of a given angle and to each other. Find the radius of a third circle that is tangent to the sides of the same angle and whose center is at the point of tangency of the two circles. | \frac{2 r R}{R + r} | olympiads | 0.0625 |
Let $S=\left(x_{1}, x_{2}, \cdots, x_{n}\right)$ be a permutation of the first $n$ natural numbers $1,2, \cdots, n$ in any order. Define $f(S)$ to be the minimum of the absolute values of the differences between every two adjacent elements in $S$. Find the maximum value of $f(S)$. | \left\lfloor \frac{n}{2} \right\rfloor | olympiads | 0.09375 |
Find $\lim _{x \rightarrow 0} \frac{x \sin 6 x}{(\operatorname{arctg} 2 x)^{2}}$. | \frac{3}{2} | olympiads | 0.109375 |
On a sheet of paper, several non-zero numbers are written, each of which is equal to the half-sum of the others. How many numbers are written? | 3 | olympiads | 0.234375 |
Fishermen caught several crucian carps and pikes. Each fisherman caught as many crucian carps as the pikes caught by all the others. How many fishermen were there in total, if the number of crucian carps caught was 10 times the number of pikes caught? | 11 | olympiads | 0.3125 |
How many alphabetic sequences (that is, sequences containing only letters from $a\cdots z$ ) of length $2013$ have letters in alphabetic order? | \binom{2038}{25} | aops_forum | 0.140625 |
Ivan wanted to buy nails. In one store, where 100 grams of nails cost 180 rubles, he couldn't buy the required amount because he was short 1430 rubles. Then he went to another store where 100 grams cost 120 rubles. He bought the required amount and received 490 rubles in change. How many kilograms of nails did Ivan buy... | 3.2 \, \text{kg} | olympiads | 0.078125 |
Define \( f_1(x) = \sqrt{x^2 + 48} \) and \( f_n(x) = \sqrt{x^2 + 6f_{n-1}(x)} \). Find all real solutions to \( f_n(x) = 2x \). | 4 | olympiads | 0.125 |
Place a cube inside a sphere with radius \( R \) and find its volume. | \frac{8R^3}{3\sqrt{3}} | olympiads | 0.140625 |
The equation \( x^2 + ax + 6 = 0 \) has two distinct roots \( x_1 \) and \( x_2 \). It is given that
\[ x_1 - \frac{72}{25 x_2^3} = x_2 - \frac{72}{25 x_1^3} \]
Find all possible values of \( a \). | a = \pm 9 | olympiads | 0.0625 |
Find the volume of a cube if the distance from its diagonal to the non-intersecting edge is equal to \(d\). | 2 d^3 \sqrt{2} | olympiads | 0.234375 |
(My problem. :D)
Call the number of times that the digits of a number change from increasing to decreasing, or vice versa, from the left to right while ignoring consecutive digits that are equal the *flux* of the number. For example, the flux of 123 is 0 (since the digits are always increasing from left to right) and t... | \frac{175}{333} | aops_forum | 0.09375 |
Find the smallest number \( a \) such that a square of side \( a \) can contain five disks of radius 1 so that no two of the disks have a common interior point. | 2 + 2\sqrt{2} | olympiads | 0.09375 |
A coin is tossed 10 times. Find the probability that no two heads appear consecutively. | \frac{9}{64} | olympiads | 0.46875 |
Straight-A student Polycarp and failing student Kolka were composing the largest five-digit number that consists of distinct odd digits. Polycarp composed his number correctly, but Kolka made a mistake—he did not notice the word "distinct" in the condition and was very happy that his number turned out to be larger than... | 97531 \text{ and } 99999 | olympiads | 0.171875 |
In a drawer, there are 5 distinct pairs of socks. Four socks are drawn at random. The probability of drawing two pairs is one in $n$. Determine the value of $n$. | 21 | olympiads | 0.296875 |
The popular variable-speed bicycle installs gears with different numbers of teeth on the driving and rear axles. By connecting different gear combinations with a chain, several different speeds can be achieved through different transmission ratios.
The "Hope" variable-speed bicycle has three gears on the driving axle ... | 8 | olympiads | 0.125 |
Find the volume of an oblique triangular prism, the base of which is an equilateral triangle with side length \(a\), if the lateral edge of the prism equals the side of the base and is inclined to the base plane at an angle of \(60^{\circ}\). | 0.375 a^3 | olympiads | 0.453125 |
Find the value of $$ \sum_{1\le a<b<c} \frac{1}{2^a3^b5^c} $$ (i.e. the sum of $\frac{1}{2^a3^b5^c}$ over all triples of positive integers $(a, b, c)$ satisfying $a<b<c$ ) | \frac{1}{406} | aops_forum | 0.0625 |
A strip is divided into 30 cells in a single row. Each end cell has one token. Two players take turns to move their tokens one or two cells in any direction. You cannot move your token past your opponent's token. The player who cannot make a move loses. How should the starting player play to win? | 2 | olympiads | 0.078125 |
Using 3 red beads and 2 blue beads to inlay on a circular ring, how many different arrangements are possible? | 2 | olympiads | 0.203125 |
In the figure, triangle \(ABC\) has \(\angle ABC = 2\beta^\circ\), \(AB = AD\), and \(CB = CE\). If \(\gamma^\circ = \angle DBE\), determine the value of \(\gamma\). |
\gamma = 45^
| olympiads | 0.09375 |
Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$ . | (a_1, b_1) = (1, 2), (1, 3), (2, 1), (3, 1) \text{ and the explicit formula is } 5a_n - b_n = a_{n+1}, b_n = a_{n-1} | aops_forum | 0.0625 |
In what ratio does the line $TH$ divide the side $BC$? | 1:1 | olympiads | 0.203125 |
What is the maximum possible value for the sum of the squares of the roots of $x^4+ax^3+bx^2+cx+d$ where $a$ , $b$ , $c$ , and $d$ are $2$ , $0$ , $1$ , and $7$ in some order? | 49 | aops_forum | 0.25 |
The diagonals of quadrilateral \( ABCD \) intersect at point \( O \). Points \( M \) and \( N \) are the midpoints of sides \( BC \) and \( AD \) respectively. The segment \( MN \) divides the area of the quadrilateral into two equal parts. Find the ratio \( OM: ON \) given that \( AD = 2BC \). | 1 : 2 | olympiads | 0.203125 |
Let \( f(m, n) = 3m + n + (m + n)^2 \). Find \( \sum_{m=0}^{\infty} \sum_{n=0}^{\infty} 2^{-f(m, n)} \). | \frac{4}{3} | olympiads | 0.078125 |
How many ways are there to shuffle a deck of 32 cards? | 32! | olympiads | 0.28125 |
Find the initial moments of the random variable \( X \) with the probability density function
\[
p(x) =
\begin{cases}
0 & \text{if } x \leq 1 \\
\frac{5}{x^{6}} & \text{if } x > 1
\end{cases}
\] |
v_k = \frac{5}{5 - k} \text{ for } k < 5
| olympiads | 0.078125 |
Given a plane \( P \) and two points \( A \) and \( B \) on opposite sides of it. Construct a sphere that passes through these points and intersects \( P \), forming the smallest possible circle. | \sqrt{MA \cdot MB} | olympiads | 0.09375 |
Cut an isosceles right triangle into several similar triangles such that any two of them differ in size. | The decomposition is correct as per the steps mentioned above. | olympiads | 0.109375 |
Explain the exact meaning of the following statement encountered in the study of a quadratic equation: "If in the equation \( a x^{2} + b x + c = 0 \) the coefficient \( a = 0 \), then one of the roots is \( \infty \), the other is \( -\frac{c}{b} \)". | -\frac{c}{b} | olympiads | 0.296875 |
In triangle $ABC$ , points $M$ and $N$ are on segments $AB$ and $AC$ respectively such that $AM = MC$ and $AN = NB$ . Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $ABC$ . Given that the perimeters of $PMN$ and $BCNM$ are $21$ and $29$ respectively, and that ... | \frac{200}{21} | aops_forum | 0.078125 |
Let the function \( f(x) = \frac{1}{2} + \log_2 \frac{x}{1-x} \) and \( S_n = \sum_{i=1}^{n-1} f\left(\frac{i}{n}\right) \), where \( n \in \mathbf{N}^{*} \) and \( n \geq 2 \). Find \( S_n \). | \frac{n-1}{2} | olympiads | 0.3125 |
Angles \(\alpha\) and \(\beta\) are such that \(\operatorname{tg} \alpha+\operatorname{tg} \beta=2\) and \(\operatorname{ctg} \alpha+\operatorname{ctg} \beta=5\). Find the value of \(\operatorname{tg}(\alpha+\beta)\). | \frac{10}{3} | olympiads | 0.546875 |
What is the maximum value of the expression
$$
a e k - a f h + b f g - b d k + c d h - c e g
$$
if each of the numbers \( a, \ldots, k \) is either 1 or -1? | 4 | olympiads | 0.25 |
Suppose $ A_1,\dots, A_6$ are six sets each with four elements and $ B_1,\dots,B_n$ are $ n$ sets each with two elements, Let $ S \equal{} A_1 \cup A_2 \cup \cdots \cup A_6 \equal{} B_1 \cup \cdots \cup B_n$ . Given that each elements of $ S$ belogs to exactly four of the $ A$ 's and to exactly three of the ... | 9 | aops_forum | 0.125 |
Compute the sum of all positive integers \( n \) for which the expression
$$
\frac{n+7}{\sqrt{n-1}}
$$
is an integer. | 89 | olympiads | 0.171875 |
Find the ratio of the volume of a cube to the volume of a regular tetrahedron, whose edge is equal to the diagonal of a face of the cube. | 3 | olympiads | 0.234375 |
Find all prime numbers $p$ such that there exists a unique $a \in \mathbb{Z}_p$ for which $a^3 - 3a + 1 = 0.$ | p = 3 | aops_forum | 0.125 |
Write the canonical equations of the line.
\(6x - 7y - z - 2 = 0\)
\(x + 7y - 4z - 5 = 0\) | \frac{x - 1}{35} = \frac{y - \frac{4}{7}}{23} = \frac{z}{49} | olympiads | 0.0625 |
Twelve chairs are arranged in a row. Occasionally, a person sits on one of the free chairs. At that moment, exactly one of their neighbors (if they exist) stands up and leaves. What is the maximum number of people that can be seated simultaneously? | 11 | olympiads | 0.140625 |
Find the sum $\sin x+\sin y+\sin z$, given that $\sin x=\operatorname{tg} y$, $\sin y=\operatorname{tg} z$, and $\sin z=\operatorname{tg} x$. | 0 | olympiads | 0.21875 |
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