problem stringlengths 33 2.6k | answer stringlengths 1 359 | source stringclasses 2
values | llama8b_solve_rate float64 0.06 0.59 |
|---|---|---|---|
Given that the maximum element of the set of real numbers $\{1, 2, 3, x\}$ is equal to the sum of all elements in the set, find the value of $x$. | -3 | olympiads | 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 | olympiads | 0.171875 |
Given that \(\frac{x+y}{x-y}+\frac{x-y}{x+y}=3\). Find the value of the expression \(\frac{x^{2}+y^{2}}{x^{2}-y^{2}}+\frac{x^{2}-y^{2}}{x^{2}+y^{2}}\). | \frac{13}{6} | olympiads | 0.40625 |
Solve the following system of equations for \( w \):
\[
\begin{array}{l}
2w + x + y + z = 1 \\
w + 2x + y + z = 2 \\
w + x + 2y + z = 2 \\
w + x + y + 2z = 1
\end{array}
\] | \frac{-1}{5} | olympiads | 0.09375 |
There are 28 identical coins arranged in the shape of a triangle. It is known that the total mass of any trio of coins that touch each other in pairs is 10 g. Find the total mass of all 18 coins on the border of the triangle. | 60 | olympiads | 0.09375 |
Consider a square on the complex plane whose four vertices correspond exactly to the four roots of a monic quartic polynomial with integer coefficients \( x^{4} + p x^{3} + q x^{2} + r x + s = 0 \). Find the minimum possible area of such a square. | 2 | olympiads | 0.0625 |
A number $ x$ is uniformly chosen on the interval $ [0,1]$ , and $ y$ is uniformly randomly chosen on $ [\minus{}1,1]$ . Find the probability that $ x>y$ . | \frac{3}{4} | aops_forum | 0.25 |
Find the largest integer $x$ for which there exists an integer ${y}$ such that the pair $(x, y)$ is a solution to the equation $x^2 - xy - 2y^2 = 9$. | 3 | olympiads | 0.203125 |
Triangle \( G R T \) has \( G R = 5 \), \( R T = 12 \), and \( G T = 13 \). The perpendicular bisector of \( G T \) intersects the extension of \( G R \) at \( O \). Find \( T O \). | \frac{169}{10} | olympiads | 0.140625 |
A geometric sequence has a nonzero first term, distinct terms, and a positive common ratio. If the second, fourth, and fifth terms form an arithmetic sequence, find the common ratio of the geometric sequence. | \frac{1 + \sqrt{5}}{2} | olympiads | 0.09375 |
Alec wishes to construct a string of 6 letters using the letters A, C, G, and N, such that:
- The first three letters are pairwise distinct, and so are the last three letters;
- The first, second, fourth, and fifth letters are pairwise distinct.
In how many ways can he construct the string? | 96 | olympiads | 0.0625 |
There are two types of camels: dromedary camels with one hump on their back and Bactrian camels with two humps. Dromedary camels are taller, with longer limbs, and can walk and run in the desert; Bactrian camels have shorter and thicker limbs, suitable for walking in deserts and snowy areas. In a group of camels that h... | 15 | olympiads | 0.515625 |
At eight o'clock in the evening, two candles of the same height are lit. It is known that one of them burns out in 5 hours, and the other burns out in 4 hours. After some time, the candles were extinguished. The remaining stub of the first candle turned out to be four times longer than that of the second. When were the... | 11:45 \, \text{PM} | olympiads | 0.0625 |
Points \( A \) and \( B \) are located on a straight highway running from west to east. Point \( B \) is 9 km east of \( A \). A car leaves point \( A \) heading east at a speed of 40 km/h. At the same time, a motorcycle leaves point \( B \) in the same direction with a constant acceleration of 32 km/h\(^2\). Determin... | 25 \text{ km} | olympiads | 0.171875 |
Solve the inequality
$$
10^{7x-1} + 6 \cdot 10^{1-7x} - 5 \leq 0
$$ | \left[\frac{1 + \log 2}{7}, \frac{1 + \log 3}{7}\right] | olympiads | 0.0625 |
Find the point of intersection of the line and the plane.
$\frac{x+2}{-1}=\frac{y-1}{1}=\frac{z+4}{-1}$
$2 x - y + 3 z + 23 = 0$ | (-3, 2, -5) | olympiads | 0.3125 |
Solve the equation: \( x^{6} - 19x^{3} = 216 \). | 3 \text{ or } -2 | olympiads | 0.53125 |
Solve the system $\begin{cases} x^3 - y^3 = 26
x^2y - xy^2 = 6
\end{cases}$ <details><summary>other version</summary>solved below
Solve the system $\begin{cases} x^3 - y^3 = 2b
x^2y - xy^2 = b
\end{cases}$</details> | (x, y) = (c, c) | aops_forum | 0.0625 |
The graph of the equation $y = 5x + 24$ intersects the graph of the equation $y = x^2$ at two points. The two points are a distance $\sqrt{N}$ apart. Find $N$ . | 3146 | aops_forum | 0.28125 |
Given a triangle \( ABC \) where \( AB = 15 \text{ cm} \), \( BC = 12 \text{ cm} \), and \( AC = 18 \text{ cm} \). In what ratio does the center of the inscribed circle divide the angle bisector of \( \angle C \)? | 2:1 | olympiads | 0.09375 |
Based on the rule for converting a repeating decimal to a simple fraction, we have:
$$
0.999\ldots = \frac{9}{9} = 1
$$
On the other hand, every decimal fraction with a whole part of zero is less than one. Explain this apparent contradiction. | 0.999\ldots = 1 | olympiads | 0.40625 |
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \),
\[
f\left(x^{714} + y\right) = f\left(x^{2019}\right) + f\left(y^{122}\right)
\] | f(x) = 0 \text{ for all } x \in \mathbb{R} | olympiads | 0.296875 |
On a rectangular piece of paper, a picture in the form of a "cross" is drawn using two rectangles \(ABCD\) and \(EFGH\), the sides of which are parallel to the edges of the sheet. It is known that \(AB=9\), \(BC=5\), \(EF=3\), and \(FG=10\). Find the area of the quadrilateral \(AFCH\). | 52.5 | olympiads | 0.109375 |
The room temperature \( T \) (in degrees Celsius) as a function of time \( t \) (in hours) is given by:
\[ T = a \sin t + b \cos t, \quad t \in (0, +\infty) \]
where \( a \) and \( b \) are positive real numbers. If the maximum temperature difference in the room is 10 degrees Celsius, what is the maximum value of \( a... | 5\sqrt{2} | olympiads | 0.15625 |
Compute the definite integral:
$$
\int_{-3}^{3} x^{2} \cdot \sqrt{9-x^{2}} \, dx
$$ | \frac{81 \pi}{8} | olympiads | 0.15625 |
For an $m \times n$ chessboard, after removing any one small square, it can always be completely covered if and only if $3 \mid mn-1$, $\min(m, n) \neq 1, 2, 5$, or $m=n=2$. | 3 \mid mn-1, \min(m, n) \neq 1, 2, 5 \text{ or } m = n = 2 | olympiads | 0.328125 |
A team of mowers needed to mow two meadows - one twice the size of the other. They spent half a day mowing the larger meadow. After noon, the team split in half: the first half remained to finish mowing the larger meadow by evening, and the second half mowed the smaller meadow, where by the evening a section still rema... | 8 ext{ mowers in the team} | olympiads | 0.09375 |
Nicolas has the same number of tin Indians, Arabs, cowboys, and Eskimos. After a visit from his cousin Sebastian, he was outraged to discover a third of his soldiers were missing. Assume that the number of remaining Eskimos is equal to the number of missing cowboys, and that only two out of every three Indian soldiers ... | 0 | olympiads | 0.0625 |
Find the value of \(2 \times \tan 1^\circ \times \tan 2^\circ \times \tan 3^\circ \times \ldots \times \tan 87^\circ \times \tan 88^\circ \times \tan 89^\circ\). | 2 | olympiads | 0.453125 |
Given 20 identical metal cubes, some (at least 1) are made of aluminum and the rest are made of a heavier aluminum alloy. Using a balance scale without weights, determine the number of alloy cubes within 11 weighings. How should the weighing be conducted? | The number of measurements required is at most 11. | olympiads | 0.34375 |
A baker uses $6\tfrac{2}{3}$ cups of flour when she prepares $\tfrac{5}{3}$ recipes of rolls. She will use $9\tfrac{3}{4}$ cups of flour when she prepares $\tfrac{m}{n}$ recipes of rolls where m and n are relatively prime positive integers. Find $m + n.$ | 55 | aops_forum | 0.453125 |
Motorcyclists Vasya and Petya ride at constant speeds around a circular track 1 km long. Vasya discovered that Petya overtakes him every 2 minutes. Then he doubled his speed and now he himself overtakes Petya every 2 minutes. What were the initial speeds of the motorcyclists? Answer: 1000 and 1500 meters per minute. | 1000 \text{ meters per minute}, 1500 \text{ meters per minute} | olympiads | 0.28125 |
Real numbers \( x \) and \( y \) satisfy the following system of equations:
\[
\begin{cases}
x + \sin y = 2008, \\
x + 2008 \cos y = 2007,
\end{cases}
\]
where \( 0 \leq y \leq \frac{\pi}{2} \). Find \( x + y \). | 2007 + \frac{\pi}{2} | olympiads | 0.21875 |
For which values of the parameter \( p \) does the inequality
$$
\left(1+\frac{1}{\sin x}\right)^{3} \geq \frac{p}{\tan^2 x}
$$
hold for any \( 0 < x < \frac{\pi}{2} \)? | p \leq 8 | olympiads | 0.0625 |
Xiao Ming and Xiao Hua are both in the same class, which has between 20 and 30 students, and all students have different birth dates. Xiao Ming says: "The number of classmates older than me is twice the number of classmates younger than me." Xiao Hua says: "The number of classmates older than me is three times the numb... | 25 | olympiads | 0.125 |
Let \(ABC\) be a triangle with area 1. Let points \(D\) and \(E\) lie on \(AB\) and \(AC\), respectively, such that \(DE\) is parallel to \(BC\) and \(\frac{DE}{BC} = \frac{1}{3}\). If \(F\) is the reflection of \(A\) across \(DE\), find the area of triangle \(FBC\). | \frac{1}{3} | olympiads | 0.09375 |
Given a circle and its center, construct a regular hexagon inscribed in the circle using only a straightedge. The straightedge can be used in the manner customary in Euclidean constructions; the lack of other tools must be compensated by appropriate reasoning, and the correctness of the steps taken must be proven. | A B C D E F | olympiads | 0.09375 |
Find all functions \( f: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R} \) such that
\[ f\left(a^{2}, f(b, c) + 1\right) = a^{2}(bc + 1) \quad \forall a, b, c \in \mathbb{R} \] | f(a, b) = ab | olympiads | 0.078125 |
In the triangular pyramid \( PABC \), the side edge \( PB \) is perpendicular to the base plane \( ABC \), \( PB = 6 \), \( AB = BC = \sqrt{15} \), \( AC = 2\sqrt{3} \). A sphere with center \( O \), which lies on the face \( ABP \), touches the planes of the other faces of the pyramid. Find the distance from the cent... | \frac{24}{6+\sqrt{15}} | olympiads | 0.078125 |
Find the rank of the matrix
$$
A=\left(\begin{array}{rrrrr}
3 & -1 & 1 & 2 & -8 \\
7 & -1 & 2 & 1 & -12 \\
11 & -1 & 3 & 0 & -16 \\
10 & -2 & 3 & 3 & -20
\end{array}\right)
$$ | rank A = 2 | olympiads | 0.078125 |
Let $n> 2$ be a positive integer. Given is a horizontal row of $n$ cells where each cell is painted blue or red. We say that a block is a sequence of consecutive boxes of the same color. Arepito the crab is initially standing at the leftmost cell. On each turn, he counts the number $m$ of cells belonging to the ... | \left\lceil \frac{n+1}{2} \right\rceil | aops_forum | 0.203125 |
Find the maximum possible value of the inradius of a triangle whose vertices lie in the interior, or on the boundary, of a unit square. | \frac{\sqrt{5}-1}{4} | aops_forum | 0.0625 |
Find all positive integers \( n \) such that \( 3^n + 5^n \) is a multiple of \( 3^{n-1} + 5^{n-1} \). | 1 | olympiads | 0.234375 |
The number consists of three digits; the sum of these digits is 11; the units digit is twice the hundreds digit. If you add 297 to the number, you obtain a number with the same digits but in reverse order. What is the number? | 326 | olympiads | 0.296875 |
Using a batch of paper to bind a type of exercise book: If 120 books have been bound, the remaining paper is 40% of the original paper; if 185 books have been bound, 1350 sheets of paper remain. How many sheets of paper are in the batch originally? | 18000 | olympiads | 0.203125 |
Given the formulas provided:
\[
S_{\triangle A O B} = \frac{1}{2} \operatorname{Im}\left(\overline{t_{1}} t_{2}\right) = \frac{1}{4 \mathrm{i}}\left(\overline{t_{1}} t_{2} - t_{1} \overline{t_{2}}\right)
\]
\[
\sin \angle A O B = \frac{\operatorname{Im}\left(\overline{t_{1} t_{2}}\right)}{\left|t_{1} t_{2}\right|} ... | \frac{\overline{t_1} t_2 - t_1 \overline{t_2}}{4i} | olympiads | 0.09375 |
Four friends A, B, C, and D participate in a chess competition where each pair plays one match. The winner gets 3 points, a draw grants each player 1 point, and the loser gets 0 points. D scores 6 points, B scores 4 points, and C scores 2 points. How many points does A have? | 0 | olympiads | 0.09375 |
Given that `a`, `a+1`, and `a+2` are the sides of an obtuse triangle, determine the possible range of values for `a`. | 1 < a < 3 | olympiads | 0.09375 |
Express the complex function \( y = (2x - 5)^{10} \) as a chain of equalities. | y = (2x - 5)^{10} \text{ can be expressed as the chain of equalities: } y = u^{10}, \text{ where } u = 2x - 5 | olympiads | 0.484375 |
Given that \( a \) and \( b \) are real numbers and the sets \( A = \{a, a^{2}, ab\} \) and \( B = \{1, a, b\} \), if \( A = B \), find the value of \( a^{2004} + b^{2004} \). | 1 | olympiads | 0.078125 |
In a $2016 \times 2016$ white grid, some cells are painted black. A natural number $k$ is called lucky if $k \leq 2016$, and in each $k \times k$ square grid within the larger grid, exactly $k$ cells are painted. (For example, if all cells are black, the only lucky number is 1.) What is the maximum number of lucky numb... | 1008 | olympiads | 0.09375 |
King Arthur has two equally wise advisors, Merlin and Percival. Each advisor finds the correct answer to any question with probability \( p \) and the wrong answer with probability \( q = 1 - p \).
If both advisors give the same answer, the king follows their advice. If they give opposite answers, the king makes a dec... | \frac{1}{2} | olympiads | 0.125 |
Let \(\mathcal{A}\) denote the set of all polynomials in three variables \(x, y, z\) with integer coefficients. Let \(\mathcal{B}\) denote the subset of \(\mathcal{A}\) formed by all polynomials which can be expressed as
\[ (x + y + z) P(x, y, z) + (xy + yz + zx) Q(x, y, z) + xyz R(x, y, z) \]
with \(P, Q, R \in \mathc... | 4 | olympiads | 0.328125 |
Paul and Caroline arranged to meet under the Arc de Triomphe between 11 AM and noon. Each of them will arrive at a random time within this interval. According to their agreement, if the other person does not appear within fifteen minutes, the first one to arrive will not wait any longer.
What is the probability that P... | \frac{7}{16} | olympiads | 0.328125 |
At a distance of 6 feet from the shore of a stream stands a tree with a height of 20 feet. The question is: at what height should the tree break so that its top touches the shore? | 9 \frac{1}{10} | olympiads | 0.109375 |
Solve the equation in integers $x^{4} - 2y^{4} - 4z^{4} - 8t^{4} = 0$. | x = y = z = t = 0 | olympiads | 0.15625 |
In rectangle \(ABCD\), \(AB = 10\) cm and \(BC = 20\) cm. Point \(M\) moves from point \(B\) towards point \(A\) along edge \(AB\) at a speed of 1 cm per second, and point \(N\) moves from point \(C\) towards point \(B\) along edge \(BC\) at a speed of 1 cm per second. At the 10th second, the distance traveled by the m... | 5\sqrt{2} \text{ cm} | olympiads | 0.359375 |
Find all integers \( n \) for which \( n^2 + 20n + 11 \) is a perfect square. | 35 | olympiads | 0.0625 |
Let \( p \) be a monic cubic polynomial such that \( p(0)=1 \) and such that all the zeros of \( p^{\prime}(x) \) are also zeros of \( p(x) \). Find \( p \). Note: monic means that the leading coefficient is 1. | (x + 1)^3 | olympiads | 0.125 |
Four resistors, each with the same resistance $R = 5$ MΩ, are connected as shown in the diagram. Determine the overall resistance of the circuit between points $A$ and $B$. The resistance of the connecting wires is negligible. | 5 \text{ M}\Omega | olympiads | 0.21875 |
A wire of length 20 meters is required to fence a flower bed that should have the shape of a circular sector. What radius of the circle should be chosen to maximize the area of the flower bed? | 5 \text{ meters} | olympiads | 0.265625 |
There is a pair of roots for the system
$$
\begin{gathered}
2 x^{2} + 3 x y + y^{2} = 70 \\
6 x^{2} + x y - y^{2} = 50
\end{gathered}
$$
given by \( x_{1} = 3 \) and \( y_{1} = 4 \); find another pair of roots. | (x_2, y_2) = (-3, -4) | olympiads | 0.265625 |
Calculate: $\sum_{k=0}^{2017} \frac{5+\cos \frac{\pi k}{1009}}{26+10 \cos \frac{\pi k}{1009}}$. | \frac{2018 \times 5^{2017}}{5^{2018} - 1} | olympiads | 0.109375 |
On each OMA lottery ticket there is a $9$ -digit number that only uses the digits $1, 2$ and $3$ (not necessarily all three). Each ticket has one of the three colors red, blue or green. It is known that if two banknotes do not match in any of the $9$ figures, then they are of different colors. Bill $122222222$ ... | red | aops_forum | 0.15625 |
Given a square \(ABCD\) on the plane, find the minimum of the ratio \(\frac{OA+OC}{OB+OD}\), where \(O\) is an arbitrary point on the plane. | \frac{1}{\sqrt{2}} | olympiads | 0.0625 |
Given the set \( S = \{1, 2, \cdots, 100\} \), determine the smallest possible value of \( m \) such that in any subset of \( S \) with \( m \) elements, there exists at least one number that is a divisor of the product of the remaining \( m-1 \) numbers. | 26 | olympiads | 0.265625 |
The altitudes of triangle \(ABC\) intersect at point \(O\). It is known that \(OC = AB\). Find the angle at vertex \(C\). | 45^ extcirc | olympiads | 0.109375 |
Find such integer values of \( a \) for which the value of the expression \(\frac{a+9}{a+6}\) is an integer. | -5, -7, -3, -9 | olympiads | 0.453125 |
The areas of the lateral faces are equal to $150, 195, 195$. Find the height of the pyramid. | 6\sqrt{21} | olympiads | 0.0625 |
In triangle \(ABC\), \(BC=4\), \(CA=5\), \(AB=6\), find the value of \(\sin^{6} \frac{A}{2} + \cos^{6} \frac{A}{2}\). | \frac{43}{64} | olympiads | 0.171875 |
Given 8 coins, two of which are counterfeit—one lighter and the other heavier than genuine coins. Using a balance scale without weights, can you determine after 3 weighings whether the combined weight of the two counterfeit coins is less than, greater than, or equal to the combined weight of two genuine coins? | \text{Three weighings suffice to determine the relative weight of counterfeit coins compared to genuine coins.} | olympiads | 0.109375 |
Three congruent circles of radius $2$ are drawn in the plane so that each circle passes through the centers of the other two circles. The region common to all three circles has a boundary consisting of three congruent circular arcs. Let $K$ be the area of the triangle whose vertices are the midpoints of those arcs.... | 300 | aops_forum | 0.0625 |
In the 7a class, $52\%$ of the students are girls. All students in the class can be arranged in a row so that boys and girls alternate. How many students are in the class? | 25 | olympiads | 0.3125 |
What are the values of \( x \) that satisfy the inequality \( \frac{1}{x-2} < 4 \)? | (c) \; x < 2 \; \text{ou} \; x > \frac{9}{4} | olympiads | 0.140625 |
A robot vacuum cleaner called "Orthogonal" decides to clean the coordinate plane. It starts from point $O$, located at the origin of the coordinates, and travels a distance $X_{1}$ along a straight line. At the point where it arrives, the vacuum cleaner turns and describes a circle centered at the origin. After complet... | \pi n \left( d + a^2 \right) | olympiads | 0.171875 |
In a sequence, there are several minus signs. Two players take turns converting one or two adjacent minus signs to plus signs. The player who converts the last minus sign wins. Who will win with optimal play? | First Player | olympiads | 0.09375 |
Let \(\mathbb{Z}_{>0}\) be the set of positive integers. Find all functions \(f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}\) such that
$$
m^{2}+f(n) \mid m f(m)+n
$$
for all positive integers \(m\) and \(n\). | f(n) = n | olympiads | 0.203125 |
Pencils in quantities ranging from 200 to 300 can be packed into red boxes containing 10 each or blue boxes containing 12 each. When packed into red boxes, 7 pencils were left over, and when packed into blue boxes, 3 pencils were short. How many pencils were there to be packed? | a = 237 \text{ or } 297 | olympiads | 0.15625 |
Find all unbounded functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that for all \( x, y \in \mathbb{Z} \),
\[ f(f(x) - y) \mid x - f(y) \] | f(x) = x | olympiads | 0.109375 |
A certain series of books was published at intervals of seven years. When the seventh book was released, the sum of all the years in which the books in this series were published was 13,524. When was the first book of the series published? | 1911 | olympiads | 0.203125 |
Compare the numbers \(\log_{20} 80\) and \(\log_{80} 640\). | \log_{20} 80 < \log_{80} 640 | olympiads | 0.3125 |
$99$ identical balls lie on a table. $50$ balls are made of copper, and $49$ balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to $2$ balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the n... | 98 | aops_forum | 0.09375 |
Given a positive number \( r \), let the set \( T = \left\{(x, y) \mid x, y \in \mathbb{R}, \text{ and } x^{2} + (y-7)^{2} \leq r^{2} \right\} \). This set \( T \) is a subset of the set \( S = \{(x, y) \mid x, y \in \mathbb{R}, \text{ and for any } \theta \in \mathbb{R}, \ \cos 2\theta + x \cos \theta + y \geq 0\} \).... | 4\sqrt{2} | olympiads | 0.078125 |
Compute \( e^{A} \) where \( A \) is defined as
\[
\int_{3 / 4}^{4 / 3} \frac{2 x^{2}+x+1}{x^{3}+x^{2}+x+1} \, dx.
\] | \frac{16}{9} | olympiads | 0.0625 |
Consider a regular polygon with $2^n$ sides, for $n \ge 2$ , inscribed in a circle of radius $1$ . Denote the area of this polygon by $A_n$ . Compute $\prod_{i=2}^{\infty}\frac{A_i}{A_{i+1}}$ | \frac{2}{\pi} | aops_forum | 0.0625 |
Determine whether the system of equations is solvable:
$$
\left\{\begin{aligned}
x_{1}+2 x_{2}-x_{3}+3 x_{4}-x_{5} & =0 \\
2 x_{1}-x_{2}+3 x_{3}+x_{4}-x_{5} & =-1 \\
x_{1}-x_{2}+x_{3}+2 x_{4} & =2 \\
4 x_{1}+3 x_{3}+6 x_{4}-2 x_{5} & =5
\end{aligned}\right.
$$ | The system is inconsistent | olympiads | 0.0625 |
In a class, there are 15 boys and 15 girls. On March 8th, some boys called some girls to congratulate them (no boy called the same girl twice). It turned out that the students can be uniquely paired into 15 pairs such that each pair consists of a boy and a girl whom he called. What is the maximum number of calls that c... | 120 | olympiads | 0.078125 |
Let $a,b,c,d,e,f$ be real numbers such that the polynomial
\[ p(x)=x^8-4x^7+7x^6+ax^5+bx^4+cx^3+dx^2+ex+f \]
factorises into eight linear factors $x-x_i$ , with $x_i>0$ for $i=1,2,\ldots,8$ . Determine all possible values of $f$ . | \frac{1}{256} | aops_forum | 0.109375 |
Given a regular triangular pyramid \( S-ABC \) with a side edge length of \( 6\sqrt{3} \) and a base edge length of 6. Find the surface area of the circumscribed sphere. | \frac{243 \pi}{3} | olympiads | 0.421875 |
What is the smallest area that triangle \( OAB \) can have if its sides \( OA \) and \( OB \) lie on the graph of the function \( y=2|x|-x+1 \), and the line \( AB \) passes through the point \( M(0, 2) \)? | 0.5 | olympiads | 0.171875 |
Given \( m > n \), and the sets \( A = \{1, 2, \cdots, m\} \) and \( B = \{1, 2, \cdots, n\} \), find the number of subsets \( C \subseteq A \) such that \( C \cap B \neq \varnothing \). | 2^m - 2^{m-n} | olympiads | 0.4375 |
Find $$ \inf_{\substack{ n\ge 1 a_1,\ldots ,a_n >0 a_1+\cdots +a_n <\pi }} \left( \sum_{j=1}^n a_j\cos \left( a_1+a_2+\cdots +a_j \right)\right) . $$ | -\pi | aops_forum | 0.0625 |
Feri welded a triangle from thin iron rods of uniform thickness. He wants to determine the centroid, so he places the triangle on a piece of paper and draws the lines connecting the vertices to the midpoints of the opposite sides.
Then Laci arrives and says that this method will not determine the centroid. He suggests... | Laci is correct | olympiads | 0.375 |
What digit can the number \( f(x) = \lfloor x \rfloor + \lfloor 3x \rfloor + \lfloor 6x \rfloor \) end with, where \( x \) is any positive real number? Here, \( \lfloor x \rfloor \) denotes the greatest integer less than or equal to \( x \). | 0, 1, 3, 4, 6, 7 | olympiads | 0.203125 |
In how many ways can a horizontal strip $2 \times n$ be tiled with $1 \times 2$ or $2 \times 1$ bricks? | \mathcal{F}_n | olympiads | 0.28125 |
Two spheres with a radius of 4 are lying on a table, touching each other externally. A cone touches the table with its lateral surface and both spheres externally. The distances from the apex of the cone to the points where the spheres touch the table are 5. Find the vertex angle of the cone. (The vertex angle of the c... | 90^
\circ \text{ or } 2 \operatorname{arcctg} 4 | olympiads | 0.0625 |
There are $15$ people, including Petruk, Gareng, and Bagong, which will be partitioned into $6$ groups, randomly, that consists of $3, 3, 3, 2, 2$ , and $2$ people (orders are ignored). Determine the probability that Petruk, Gareng, and Bagong are in a group. | \frac{3}{455} | aops_forum | 0.0625 |
Given that the positive integers \( a_{1}, a_{2}, a_{3}, a_{4} \) satisfy the condition that in any circular arrangement of these four numbers, there exist two adjacent elements which are not coprime. Find the maximum number of ordered triplets \((i, j, k)\) such that \( i, j, k \in {1, 2, 3, 4} \), \( i \neq j \), \( ... | 16 | olympiads | 0.0625 |
Let $ABC$ be a triangle inscribed in the parabola $y=x^2$ such that the line $AB \parallel$ the axis $Ox$ . Also point $C$ is closer to the axis $Ox$ than the line $AB$ . Given that the length of the segment $AB$ is 1 shorter than the length of the altitude $CH$ (of the triangle $ABC$ ). Determine the ... | \frac{\pi}{4} | aops_forum | 0.078125 |
Given a circle \((x-14)^2 + (y-12)^2 = 36^2\) and a point \(C(4,2)\) inside the circle. Two points \(A\) and \(B\) that lie on the circumference of the circle are such that \(\angle ACB = 90^\circ\). Find the equation of the locus of the midpoint of the hypotenuse \(AB\). | (x-9)^2 + (y-7)^2 = 13 \times 46 | olympiads | 0.078125 |
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