problem stringlengths 33 2.6k | answer stringlengths 1 359 | source stringclasses 2
values | llama8b_solve_rate float64 0.06 0.59 |
|---|---|---|---|
Given the function \( f(x) = \frac{2x^2 + \sqrt{2} \sin \left(x + \frac{\pi}{4}\right)}{2x^2 + \cos x} \), with maximum and minimum values \( a \) and \( b \) respectively, find the value of \( a + b \). | 2 | olympiads | 0.09375 |
\( x, y > 0 \). Let \( S \) denote the smallest of the numbers \( x, \frac{1}{y}, y + \frac{1}{x} \). What is the maximum value that \( S \) can take? | \sqrt{2} | olympiads | 0.109375 |
Place parentheses and operation signs in the sequence 22222 so that the result is 24. | (2+2+2) \times (2+2) | olympiads | 0.25 |
During a break, some students left the lyceum, and some joined it. As a result, the number of students in the lyceum decreased by $10\%$, and the proportion of boys among the students increased from $50\%$ to $55\%$. Did the number of boys increase or decrease? | Уменьшилось | olympiads | 0.09375 |
The numbers $96, 28, 6, 20$ were written on the board. One of them was multiplied, another was divided, another was increased, and another was decreased by the same number. As a result, all the numbers became equal to a single number. What is that number? | 24 | olympiads | 0.0625 |
Given that \( M \) is an arbitrary point on the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0)\), \( F_{1} \) and \( F_{2} \) are the left and right foci of the ellipse, respectively. The lines \( M F_{1} \) and \( M F_{2} \) intersect the ellipse at points \( A \) and \( B \), respectively. Then \(\frac{b... | 4 | olympiads | 0.1875 |
A tourist walked for 3.5 hours, covering exactly 5 km for each 1-hour interval. Does it follow from this that his average speed was 5 km/hour? | It does not follow that the average speed is exactly 5 km/hour. | olympiads | 0.09375 |
In the cube \( A C_{1} \), determine the magnitude of the dihedral angle \( C_{1}-D_{1} B-C \). | 60^{\circ} | olympiads | 0.0625 |
Let \( ABCD \) be a rectangle with area 1, and let \( E \) lie on side \( CD \). What is the area of the triangle formed by the centroids of triangles \( ABE, BCE \), and \( ADE \)? | \frac{1}{9} | olympiads | 0.15625 |
As shown in Figure 1, \( \triangle ABC \) has \( n_1, n_2, n_3 \) points on its sides (excluding vertices \( A, B, C \)). How many distinct triangles can be formed by choosing one point from the points on each of the three sides as the vertices of the triangle? | n_1 n_2 n_3 | olympiads | 0.265625 |
Through the point \( P(1,1) \), draw a line \( l \) such that the midpoint of the chord intercepted by the ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\) on this line is precisely \( P \). Find the equation of the line \( l \). | 4x + 9y = 13 | olympiads | 0.125 |
The maximum value of the function \(y = a \cos x + b\) (where \(a\) and \(b\) are constants) is 1 and the minimum value is -7. Find the maximum value of \(y = 3 + ab \sin x\). | 15 | olympiads | 0.484375 |
Solve the equation \( 2^{x} - 1 = 5^{y} \) in integers. | x=1, y=0 | olympiads | 0.1875 |
On September 1, 2021, Vasya deposited 100,000 rubles in a bank. Each year, the bank accrues 10% annual interest (i.e., increases the amount by 10% of what was currently in the account). For example, on September 2, 2022, Vasya's account will have 110,000 rubles. Find the earliest year in which the amount in Vasya's acc... | 2026 | olympiads | 0.25 |
If $\frac{O}{11}<\frac{7}{\square}<\frac{4}{5}$, then what is the maximum sum of non-zero natural numbers that can be placed in “ $O$ ” and “ $\square$ ”? | 77 | olympiads | 0.0625 |
Three runners move along a circular track at equal constant speeds. When two runners meet, they instantly turn around and start running in the opposite direction.
At a certain moment, the first runner met the second runner. After 20 minutes, the second runner met the third runner for the first time. Another half hour ... | 100 \text{ minutes} | olympiads | 0.09375 |
Suppose $x>1$ is a real number such that $x+\tfrac 1x = \sqrt{22}$ . What is $x^2-\tfrac1{x^2}$ ? | 18 | aops_forum | 0.53125 |
Solve the system of equations:
\[
\begin{aligned}
& 5x + 3y = 65 \\
& 2y - z = 11 \\
& 3x + 4z = 57
\end{aligned}
\] | x = 7, \; y = 10, \; z = 9 | olympiads | 0.359375 |
A Sheik has divided his treasures into nine bags: in the first bag 1 kg, in the second bag 2 kg, in the third bag 3 kg, and so on, with the ninth bag containing 9 kg. A cunning vizier has stolen some treasure from one of the bags. How can the Sheik determine from which bag the treasure was stolen using only two weighin... | Solution using two weighings to identify the tampered bag | olympiads | 0.0625 |
Let \((x_{A}, y_{A}), (x_{B}, y_{B})\) be the coordinates of points \(A\) and \(B\) on a parabola. Then \(y_{A} = x_{A}^2\) and \(y_{B} = x_{B}^2\). Consider the case where \(x_{A} < 0\) and \(x_{B} > 0\). According to the conditions, the lengths of segments \(AC\) and \(CB\) are \(2t\) and \(t\) respectively, so point... | (x_A = -4, x_B = 2), (x_A = 4, x_B = -2) | olympiads | 0.484375 |
Determine the maximum value of the ratio of a three-digit number to the sum of its digits. | 100 | olympiads | 0.078125 |
Given constants $\lambda \neq 0$, $n$, and $m$ as positive integers with $m > n$ and $m \neq 2n$, the function $f(x)$ defined by the equation
$$
f(x+\lambda)+f(x-\lambda)=2 \cos \frac{2n\pi}{m} f(x)
$$
is a periodic function with period $m\lambda$. | m\lambda | olympiads | 0.359375 |
The sum of the first $n$ terms of an arithmetic series is $A$, and the sum of the first $2n$ terms is $B$. Express the sum of the first $3n$ terms using $A$ and $B$. | C = 3(B - A) | olympiads | 0.0625 |
Find all the polynomials \( P \in \mathbb{R}[X] \) such that
$$
\left\{
\begin{array}{l}
P(2) = 2 \\
P\left(X^3\right) = P(X)^3
\end{array}
\right.
$$ | P(X) = X | olympiads | 0.078125 |
A right triangle was cut along a line into two parts and rearranged into a square. What is the length of the shorter leg if the longer leg is 10? | 5 | olympiads | 0.390625 |
A sequence of numbers \( x_1, x_2, x_3, \ldots \) is defined by the rule: \( x_1 = 1 \), \( x_{n+1} = 1 + \frac{x_n^2}{n} \) for \( n = 1, 2, 3, \ldots \). Find \( x_{2019} \). | 2019 | olympiads | 0.421875 |
A polygon $\mathcal{P}$ is drawn on the 2D coordinate plane. Each side of $\mathcal{P}$ is either parallel to the $x$ axis or the $y$ axis (the vertices of $\mathcal{P}$ do not have to be lattice points). Given that the interior of $\mathcal{P}$ includes the interior of the circle \(x^{2}+y^{2}=2022\), find the minimum... | 8 \sqrt{2022} | olympiads | 0.171875 |
Given \( a > 0 \), if the solution set of the inequality \( \sqrt{x+a}+\sqrt{x-a} \leqslant \sqrt{2(x+1)} \) has a length of \(\frac{1}{2}\) on the number line, determine the value of \( a \) that satisfies this condition. | \frac{3}{4} | olympiads | 0.0625 |
Suppose the equation \(| | x - a | - b | = 2008\) has 3 distinct real roots and \( a \neq 0 \). Find the value of \( b \). | 2008 | olympiads | 0.328125 |
Determine the real number $a$ that makes the polynomials $x^{2} + ax + 1$ and $x^{2} + x + a$ have at least one common root. | 1 \text{ or } -2 | olympiads | 0.546875 |
Find the minimum value of the function \( f(x, y) = 6\left(x^{2} + y^{2}\right)(x + y) - 4\left(x^{2} + xy + y^{2}\right) - 3(x + y) + 5 \) in the region \( D = \{(x, y) \mid x > 0, y > 0\} \). | 2 | olympiads | 0.0625 |
Many residents of the city are involved in dancing, many in mathematics, and at least one in both. The number of those who are engaged only in dancing is exactly $p+1$ times more than those who are engaged only in mathematics, where $p$ is some prime number. If you square the number of all mathematicians, you get the n... | 1 | olympiads | 0.1875 |
Find explicit formulas for generating a continuous two-dimensional random variable ( \(X, Y\) ) if the component \(X\) is given by the probability density function \(f_{1}(x)= \frac{x}{2} \) in the interval \((0, 2)\), and the component \(Y\) is uniformly distributed in the interval \(\left(x_{i}, x_{i}+3\right)\) with... | x_i = 2 \sqrt{r}, \quad y = 3r' + 2 \sqrt{r} | olympiads | 0.0625 |
Bretschneider provides an approximation for $\pi$ accurate to the fifth decimal place: $\frac{13}{50} \sqrt{146}$, assuming the radius of the circle is one; construct this approximation. | \pi \approx \frac{13}{50} \sqrt{146} | olympiads | 0.21875 |
What path does the midpoint of a segment between two pedestrians, walking uniformly on straight roads, describe? | A straight line | olympiads | 0.375 |
Let \( S \) be a set of sequences of length 15 formed by using the letters \( a \) and \( b \) such that every pair of sequences in \( S \) differ in at least 3 places. What is the maximum number of sequences in \( S \)? | 2048 | olympiads | 0.0625 |
Calculate
$$
\int \frac{e^{2 z}}{\left(z+\frac{\pi i}{2}\right)^{2}} d z
$$
where \( L \) is the circle \( |z|=1 \). | -4\pi i | olympiads | 0.328125 |
Find the range of the function \( y = |2 \sin x + 3 \cos x + 4| \). | [4 - \sqrt{13}, 4 + \sqrt{13}] | olympiads | 0.453125 |
Vremyankin and Puteykin started simultaneously from Morning Town to Evening Town. The first one walked half the time at a speed of 5 km/h and then at a speed of 4 km/h. The second one walked the first half of the distance at 4 km/h and then at 5 km/h. Who arrived in Evening Town earlier? | Vremyankin \text{ arrives first} | olympiads | 0.234375 |
On graph paper, four grid points are marked to form a 4*4 square. Mark two more grid points and connect them with a closed polyline so that the resulting hexagon (not necessarily convex) has an area of 6 units. | ext{Solution as hexagon of 6 units} | olympiads | 0.0625 |
Given an infinite checkered paper with a cell side of one unit. The distance between two cells is defined as the length of the shortest path of a rook from one cell to another (considering the path from the center of the rook). What is the minimum number of colors needed to color the board (each cell is to be painted w... | 4 \text{ colors} | olympiads | 0.25 |
Jia and Yi are playing a table tennis singles match using a best-of-five format (i.e., the first to win three games wins the championship). For each game, Jia's probability of winning is $\frac{2}{3}$ and Yi's probability of winning is $\frac{1}{3}$. Calculate the probability that Yi wins the championship. | \frac{17}{81} | olympiads | 0.09375 |
Find the number of real values of \( a \) such that for each \( a \), the cubic equation \( x^{3} = ax + a + 1 \) has an even root \( x \) with \( |x| < 1000 \). | 999 | olympiads | 0.125 |
In the figure, $ABCD$ and $DEFG$ are both squares. Given that $CE = 14$ and $AG = 2$, find the sum of the areas of the two squares. | 100 | olympiads | 0.0625 |
Solve the system of inequalities
\[ \left\{\begin{array}{l}
x + 2 < 3 + 2x, \\
4x - 3 < 3x - 1, \\
8 + 5x \geqslant 6x + 7.
\end{array}\right. \] | (-1, 1] | olympiads | 0.078125 |
Find all sequences with distinct natural numbers as members such that \( a_{n} \) is divisible by \( a_{a_{n}} \) for all \( n \). | a_n = n | olympiads | 0.109375 |
Vasya stood at a bus stop for some time. During this time, one bus and two trams passed by. After some time, a Spy came to the same bus stop. While he was there, 10 buses passed by. What is the minimum number of trams that could have passed by during this time? Both buses and trams travel at regular intervals, with bus... | 4 | olympiads | 0.078125 |
The integer sequence \( \left\{a_{n}\right\} \) is defined as follows:
$$
a_{0}=3, a_{1}=4, a_{n+2}=a_{n+1} a_{n}+\left\lceil\sqrt{\left(a_{n+1}^{2}-1\right)\left(a_{n}^{2}-1\right)} \right\rceil \text{ for } n \geq 0.
$$
Find the value of \( \sum_{n=0}^{+\infty}\left(\frac{a_{n+3}}{a_{n+2}}-\frac{a_{n+2}}{a_{n}}+\fra... | 0 | olympiads | 0.09375 |
In triangle \(ABC\), \(AB = 15\), \(BC = 12\), and \(AC = 18\). In what ratio does the center \(O\) of the incircle of the triangle divide the angle bisector \(CM\)? | 2:1 | olympiads | 0.0625 |
Given that \(x\) is a simplest proper fraction, if its numerator is increased by \(a\), it simplifies to \(\frac{1}{3}\); if its denominator is increased by \(a\), it simplifies to \(\frac{1}{4}\). Find \(x\). | \frac{4}{15} | olympiads | 0.46875 |
Let \(ABC\) be a triangle with incenter \(I\) and circumcenter \(O\). Let the circumradius be \(R\). What is the least upper bound of all possible values of \(IO\)? | R | olympiads | 0.34375 |
Let $ P_1$ be a regular $ r$ -gon and $ P_2$ be a regular $ s$ -gon $ (r\geq s\geq 3)$ such that each interior angle of $ P_1$ is $ \frac {59}{58}$ as large as each interior angle of $ P_2$ . What's the largest possible value of $ s$ ? | 117 | aops_forum | 0.1875 |
The height of a parallelogram, drawn from the vertex of the obtuse angle, is $a$ and bisects the opposite side. The acute angle of the parallelogram is $30^{\circ}$.
Find the diagonals of the parallelogram. | 2a \text{ and } 2a \sqrt{3} | olympiads | 0.109375 |
For \( x, y \in (0,1] \), find the maximum value of the expression
\[
A = \frac{\left(x^{2} - y\right) \sqrt{y + x^{3} - x y} + \left(y^{2} - x\right) \sqrt{x + y^{3} - x y} + 1}{(x - y)^{2} + 1}
\] | 1 | olympiads | 0.1875 |
The altitudes of an acute non-isosceles triangle $ABC$ intersect at point $H$. $I$ is the incenter of triangle $ABC$, and $O$ is the circumcenter of triangle $BHC$. It is known that point $I$ lies on the segment $OA$. Find the angle $BAC$. | 60^
| olympiads | 0.0625 |
For which natural values of \( x \) and \( y \) is the equation \( 3x + 7y = 23 \) true? | (x = 3, y = 2) | olympiads | 0.546875 |
The intersection point of two lines \( y_{1} = k_{1} x - 1 \) and \( y_{2} = k_{2} x + 2 \) is on the x-axis. Find the area of the triangle formed by the line \( y = \frac{k_{1}}{\k_{2}} x - \frac{k_{2}}{\k_{1}} \) and the coordinate axes. | 4 | olympiads | 0.296875 |
Eight workers completed $\frac{1}{3}$ of a project in 30 days. Subsequently, 4 more workers were added to complete the rest of the project. How many total days did it take to complete the project? | 70 | olympiads | 0.265625 |
In a certain country, there were two villages located near each other, $A$ and $B$. The inhabitants of village $A$ always told the truth, while the inhabitants of village $B$ always lied. A tourist traveling through this country found himself in one of these villages. To find out in which village he was and from which ... | The minimum number of questions needed is 2. | olympiads | 0.484375 |
What are the prime numbers \( p \) such that \( (p-1)^{p} + 1 \) is a power of \( p \)? | 2 \text{ et } 3 | olympiads | 0.484375 |
The polynomial \( p(x) = x^2 - 3x + 1 \) has zeros \( r \) and \( s \). A quadratic polynomial \( q(x) \) has a leading coefficient of 1 and zeros \( r^3 \) and \( s^3 \). Find \( q(1) \). | -16 | olympiads | 0.46875 |
There are 100 peanuts in total in five bags. The total number of peanuts in the first two bags is 52, in the second and third bags is 43, in the third and fourth bags is 34, and the last two bags together contain 30 peanuts. How many peanuts are in each bag? | 27, 25, 18, 16, 14 | olympiads | 0.21875 |
A $10 \times 10$ square was cut along the grid lines into 17 rectangles, each of which has both sides longer than 1. What is the smallest number of squares that could be among these rectangles? Provide an example of such a cutting. | 1 | olympiads | 0.234375 |
In parallelogram $ABCD$, $EF \parallel AB$ and $HG \parallel AD$. If the area of parallelogram $AHPE$ is 5 square centimeters and the area of parallelogram $PECG$ is 16 square centimeters, what is the area of triangle $PBD$ in square centimeters? | 5.5 \text{ square meters} | olympiads | 0.09375 |
Compute the limit of the function:
\[
\lim _{x \rightarrow 1}(\arcsin x)^{\operatorname{tg} \pi x}
\] | 1 | olympiads | 0.5 |
We have an empty equilateral triangle with length of a side $l$ . We put the triangle, horizontally, over a sphere of radius $r$ . Clearly, if the triangle is small enough, the triangle is held by the sphere. Which is the distance between any vertex of the triangle and the centre of the sphere (as a function of $l$ ... | \sqrt{r^2 + \frac{l^2}{4}} | aops_forum | 0.328125 |
Find all primes $p,q,r$ such that $qr-1$ is divisible by $p$ , $pr-1$ is divisible by $q$ , $pq-1$ is divisible by $r$ . | (2, 3, 5) and its permutations. | aops_forum | 0.109375 |
Triangle \(ABC\) with angle \(\angle ABC = 135^\circ\) is inscribed in a circle \(\omega\). The tangents to \(\omega\) at points \(A\) and \(C\) intersect at point \(D\). Find \(\angle ABD\), given that \(AB\) bisects segment \(CD\). Answer: \(90^\circ\). | 90^
| olympiads | 0.140625 |
Which digit must be substituted instead of the star so that the following large number is divisible by 7?
$$
\underbrace{66 \cdots 66}_{2023\text{ digits}} \times \underbrace{55 \cdots 55}_{2023\text{ digits}}
$$ | 6 | olympiads | 0.09375 |
Find the smallest positive integer \( n (n \geqslant 3) \) such that among any \( n \) points in the plane where no three points are collinear, there are always two points that are vertices of a non-isosceles triangle. | 7 | olympiads | 0.265625 |
The equation
$$
\begin{cases}
x = \frac{1}{2} \left(e^t + e^{-t}\right) \cos \theta, \\
y = \frac{1}{2} \left(e^t - e^{-t}\right) \sin \theta
\end{cases}
$$
where $\theta$ is a constant $\left(\theta \neq \frac{n}{2} \pi, n \in \mathbf{Z}\right)$ and $t$ is a parameter, represents a shape that is $\qquad$. | Hyperbola | olympiads | 0.15625 |
On a \(6 \times 6\) chessboard, we randomly place counters on three different squares. What is the probability that no two counters are in the same row or column? | \frac{40}{119} | olympiads | 0.078125 |
First, let us consider the function \(\varphi(t)=t + \frac{1}{t}\). The function \(\varphi(t)\) is defined for all \(t \neq 0\). We will find the extrema of the function \(\varphi(t)\). To do this, we will find the intervals where the derivative of the function \(\varphi(t)\) is constant: \(\varphi^{\prime}(t)=1-\frac{... | \left[-\frac{8}{65}, 0 \right) \cup \left( 0, \frac{8}{65} \right] | olympiads | 0.15625 |
In the triangle \(ABC\), if the median \(CC_1\) to the side \(AB\) is greater than the median \(AA_1\) to the side \(BC\), then \(\angle CAB\) is less than \(\angle BCA\). Petya, an excellent student, believes this statement is incorrect. Determine who is right. | Petya is correct. | olympiads | 0.34375 |
Find the sum of the even positive divisors of 1000. | 2184 | olympiads | 0.359375 |
Which regular polygons can be obtained (and how) by cutting a cube with a plane? | 3, 4, 6 sided polygons | olympiads | 0.0625 |
Andrey's, Borya's, Vova's, and Gleb's houses are located in some order on a straight street. The distance between Andrey's and Borya's houses, as well as the distance between Vova's and Gleb's houses, is 600 meters. What could be the distance in meters between Andrey's and Gleb's houses if it is known that this distanc... | 900, 1800 | olympiads | 0.34375 |
On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$ . How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible? | 3 | aops_forum | 0.234375 |
A company that produces preparation manuals for exams incurs average costs per manual of $100 + \(\frac{100000}{Q}\), where \(Q\) is the number of manuals produced annually. What annual production volume of the manual corresponds to the break-even point if the planned price of the manual is 300 monetary units? | 500 | olympiads | 0.203125 |
Investigate the power series for the function \( f(x) = a^x \) (where \( a > 0 \) and \( a \neq 1 \)) at the point \( x_0 = 0 \). | a^x = \sum_{n=0}^{\infty} \frac{(\ln a)^n}{n!} x^n | olympiads | 0.40625 |
Solve the equation \(\left(3 \cos 2 x+\frac{9}{4}\right) \cdot \left|1-2 \cos 2 x\right|=\sin x(\sin x-\sin 5 x)\). | x = \pm \frac{\pi}{6} + \frac{k \pi}{2}, \, k \in \mathbb{Z} | olympiads | 0.0625 |
Two parallel lines \( l_{1} \) and \( l_{2} \) have a distance \( d \) between them. They pass through the points \( M(-2, -2) \) and \( N(1, 3) \) respectively, and rotate around \( M \) and \( N \) while remaining parallel. Determine the equations of the lines \( l_{1} \) and \( l_{2} \) when \( d \) reaches its maxi... | l_1: 3x + 5y + 16 = 0, \quad l_2: 3x + 5y - 18 = 0 | olympiads | 0.140625 |
Compute the limit of the function:
$$\lim _{x \rightarrow 2} \frac{\operatorname{tg} x-\operatorname{tg} 2}{\sin (\ln (x-1))}$$ | \frac{1}{\cos^2 (2)} | olympiads | 0.0625 |
Bob writes a random string of 5 letters, where each letter is either \( A \), \( B \), \( C \), or \( D \). The letter in each position is independently chosen, and each of the letters \( A \), \( B \), \( C \), \( D \) is chosen with equal probability. Given that there are at least two \( A \)'s in the string, find th... | \frac{53}{188} | olympiads | 0.09375 |
At certain store, a package of 3 apples and 12 oranges costs 5 dollars, and a package of 20 apples and 5 oranges costs 13 dollars. Given that apples and oranges can only be bought in these two packages, what is the minimum nonzero amount of dollars that must be spent to have an equal number of apples and oranges? | 64 | aops_forum | 0.234375 |
Cinderella was given a bag with a mixture of poppy seeds and millet by her stepmother to sort. When Cinderella was leaving for the ball, she left three bags: one with millet, one with poppy seeds, and one with the unsorted mixture. To avoid confusion, she labeled each bag as "Poppy", "Millet", and "Mixture".
Upon retu... | Cinderella matched the correct contents of each bag using a single seed. | olympiads | 0.140625 |
Find natural numbers $a$ and $b$ such that the number reciprocal to their difference is three times greater than the number reciprocal to their product. | a = 6, \; b = 2 | olympiads | 0.109375 |
In an acute-angled, non-isosceles triangle $ABC$, the altitudes $AA'$ and $BB'$ intersect at point $H$, and the medians of triangle $AHB$ intersect at point $M$. The line $CM$ bisects segment $A'B'$ into two equal parts. Find the angle $C$. | 45^\circ | olympiads | 0.0625 |
Construct 6 equilateral triangles, the first one with a side length of \(1 \text{ cm}\), and the following triangles with sides equal to half the length of the previous triangle, as indicated in the figure. What is the perimeter of this figure? | \frac{127}{32} | olympiads | 0.09375 |
On the Island of Knights and Knaves, knights always tell the truth, and knaves always lie. One day, a traveler questioned seven inhabitants of the island.
- "I am a knight," said the first.
- "Yes, he is a knight," said the second.
- "Among the first two, at least 50% are knaves," said the third.
- "Among the first th... | 5 | olympiads | 0.09375 |
If the minimum degree $\delta(G) \geqslant 3$, then the girth $g(G) < 2 \log |G|$. | g(G) < 2 \log |G| | olympiads | 0.375 |
How to calculate the number of divisors of an integer \( n \)? | \prod_{i=1}^{j} (v_{p_i}(n) + 1) | olympiads | 0.09375 |
Consider a regular cube with side length $2$ . Let $A$ and $B$ be $2$ vertices that are furthest apart. Construct a sequence of points on the surface of the cube $A_1$ , $A_2$ , $\ldots$ , $A_k$ so that $A_1=A$ , $A_k=B$ and for any $i = 1,\ldots, k-1$ , the distance from $A_i$ to $A_{i+1}$ is $3... | 7 | aops_forum | 0.09375 |
Let the strictly increasing sequence $\left\{a_{n}\right\}$ consist of positive integers with $a_{7}=120$ and $a_{n+2}=a_{n}+a_{n+1}$ for $n \in \mathbf{Z}_{+}$. Find $a_{8}=$. | 194 | olympiads | 0.0625 |
In a herd consisting of horses, two-humped camels, and one-humped camels, there are a total of 200 humps. How many animals are in the herd if the number of horses is equal to the number of two-humped camels? | 200 | olympiads | 0.46875 |
In the plane Cartesian coordinate system $xoy$, the function $f(x) = a \sin(ax) + \cos(ax)$ (where $a > 0$) and the function $g(x) = \sqrt{a^2 + 1}$ enclose a region over the interval of the smallest positive period. Find the area of this region. | \frac{2 \pi}{a} \sqrt{a^{2}+1} | olympiads | 0.375 |
$$
\left\{\begin{array} { c }
{ 2 t ^ { 2 } + 3 t - 9 = 0 } \\
{ \sqrt { 2 } \leq t < + \infty }
\end{array} \Leftrightarrow \left\{\begin{array}{c}
(t+3)\left(t-\frac{3}{2}\right)=0 \\
\sqrt{2} \leq t<+\infty
\end{array} \Leftrightarrow t=\frac{3}{2}\right.\right.
$$
Given (*), return to x: $x=\frac{1}{4} ; 1 ; \fr... | \left\{ \frac{1}{4} \right\} \cup \{ 1 \} \cup \left\{ \frac{9}{4} \right\} | olympiads | 0.25 |
In triangle \( ABC \), an incircle is inscribed with center \( I \) and points of tangency \( P, Q, R \) with sides \( BC, CA, AB \) respectively. Using only a ruler, construct the point \( K \) where the circle passing through vertices \( B \) and \( C \) is internally tangent to the incircle. | K | olympiads | 0.5 |
Shelly-Ann normally runs along the Laurel Trail at a constant speed of \( 8 \text{ m/s} \). One day, one-third of the trail is covered in mud, through which Shelly-Ann can only run at one-quarter of her normal speed, and it takes her 12 seconds to run the entire length of the trail. How long is the trail, in meters? | 48 \text{ meters} | olympiads | 0.453125 |
There is infinite sequence of composite numbers $a_1,a_2,...,$ where $a_{n+1}=a_n-p_n+\frac{a_n}{p_n}$ ; $p_n$ is smallest prime divisor of $a_n$ . It is known, that $37|a_n$ for every $n$ .
Find possible values of $a_1$ | 37^2 | aops_forum | 0.109375 |
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