problem
stringlengths 33
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values | llama8b_solve_rate
float64 0.06
0.59
|
|---|---|---|---|
Find the angle of inclination of the $t_{1,2}$ projection axis with a given plane.
|
Inclination Angle
|
olympiads
| 0.0625
|
Given real numbers \(a_{1}, a_{2}, \cdots, a_{18}\) such that:
\[ a_{1} = 0, \left|a_{k+1} - a_{k}\right| = 1 \text{ for } k = 1, 2, \cdots, 17, \]
what is the probability that \(a_{18} = 13\)?
|
\frac{17}{16384}
|
olympiads
| 0.0625
|
On an $8 \times 8$ chessboard, 64 checkers numbered from 1 to 64 are placed. 64 students, one by one, approach the board and flip only those checkers whose numbers are divisible by the ordinal number of the current student. A "king" (or "crowned" checker) is a checker that has been flipped an odd number of times. How many "kings" will there be on the board after the last student steps away?
|
8
|
olympiads
| 0.109375
|
Let $ a,\ b$ be postive real numbers. For a real number $ t$ , denote by $d(t)$ the distance between the origin and the line $ (ae^t)x \plus{} (be^{ \minus{} t})y \equal{} 1$ .
Let $ a,\ b$ vary with $ ab \equal{} 1$ , find the minimum value of $ \int_0^1 \frac {1}{d(t)^2}\ dt$ .
|
e - \frac{1}{e}
|
aops_forum
| 0.078125
|
Determine the interval of convergence for the series
$$
\sum_{n=1}^{\infty} \frac{(3+4 i)^{n}}{(z+2 i)^{n}}+\sum_{n=0}^{\infty}\left(\frac{z+2 i}{6}\right)^{n}
$$
|
5 < |z+2i| < 6
|
olympiads
| 0.3125
|
Find all positive integer solutions to \((n + 1) \cdot m = n! + 1\).
|
(1,1), (2,1), (4,2)
|
olympiads
| 0.078125
|
Do there exist irrational numbers \(a\) and \(b\) such that \(a > 1, b > 1\), and \(\left\lfloor a^m \right\rfloor\) is different from \(\left\lfloor b^n \right\rfloor\) for any natural numbers \(m\) and \(n\)?
|
There exist such irrational numbers
|
olympiads
| 0.09375
|
In one class, there are blondes and brunettes. Blondes lie when answering any question, while brunettes tell the truth. Some children enjoy dyeing their hair and dye it every day to the opposite color! (And others never dye their hair). On Monday, it was Petya's birthday, and all the children were asked: "Were you born this month?" and all 20 students gave an affirmative answer. On Friday, the children were asked again, "Were you born this month?" and only 10 students gave an affirmative answer. When asked again on Monday after the weekend, none of the children gave an affirmative answer. On Friday, it was determined whether there are more blondes or brunettes currently in class. What was the result?
|
There are more brunettes in the class.
|
olympiads
| 0.171875
|
A side of a triangle is equal to \(a\), and the difference between the angles adjacent to this side is \(\frac{\pi}{2}\). Find the angles of the triangle if its area is \(S\).
|
\left( \frac{1}{2} \arctan \frac{4S}{a^2}, \frac{\pi}{2} + \frac{1}{2} \arctan \frac{4S}{a^2}, \frac{\pi}{2} - \arctan \frac{4S}{a^2} \right)
|
olympiads
| 0.0625
|
In a communication system consisting of 2001 subscribers, each subscriber is connected to exactly \( n \) others. Determine all possible values of \( n \).
|
0, 2, 4, 6, 8, \ldots, 2000
|
olympiads
| 0.09375
|
Calculate the area of quadrilateral $S_{MNKP}$ given the area of quadrilateral $S_{ABCD} = \frac{180 + 50\sqrt{3}}{6}$.
|
\frac{90 + 25 \sqrt{3}}{6}
|
olympiads
| 0.390625
|
A group of 5 people is to be selected from 6 men and 4 women. Find \(d\), the number of ways that there are always more men than women.
|
186
|
olympiads
| 0.21875
|
In a ski race, Petya and Vasya started at the same time. Vasya ran the entire race at a constant speed of 12 km/h. Petya ran the first half of the distance at a speed of 9 km/h, falling behind Vasya. What should be Petya's speed on the second half of the distance in order to catch up with Vasya and finish simultaneously with him? Provide the answer in km/h.
|
18 \text{ km/h}
|
olympiads
| 0.484375
|
In an arithmetic sequence \(\{a_{n}\}\), if \(S_{4} \leq 4\) and \(S_{5} \geq 15\), then the minimum value of \(a_{4}\) is
|
7
|
olympiads
| 0.375
|
Let \( f(x) \) be a monotonic function defined on \( (0, +\infty) \), such that for any \( x > 0 \), we have \( f(x) > -\frac{4}{x} \) and \( f\left(f(x) + \frac{4}{x}\right) = 3 \). Find \( f(8) = \qquad \) .
|
\frac{7}{2}
|
olympiads
| 0.09375
|
There are 5 sticks that are 2 cm long, 5 sticks that are 3 cm long, and one stick that is 7 cm long. Is it possible to form a square using all these sticks?
|
It is not possible to form a square with the given sticks.
|
olympiads
| 0.125
|
Compare the integrals \( I_{1}=\int_{0}^{1} x \, dx \) and \( I_{2}=\int_{0}^{1} x^{3} \, dx \) without calculating them.
|
I_1 > I_2
|
olympiads
| 0.09375
|
The production of \( x \) thousand units of products costs \( q = 0.5x^2 - 2x - 10 \) million rubles per year. With a price of \( p \) thousand rubles per unit, the annual profit from selling these products (in million rubles) is \( p x - q \). The factory produces such a quantity of products that the profit is maximized. What is the minimum value of \( p \) for the total profit over three years to be at least 126 million rubles?
|
6
|
olympiads
| 0.109375
|
How many real numbers \( x \) are solutions to the following equation?
\[ |x-1| = |x-2| + |x-3| \]
|
2
|
olympiads
| 0.421875
|
How many real number solutions does the equation \(\frac{1}{3} x^{4}+5|x|=7\) have?
|
2
|
olympiads
| 0.4375
|
Let \( f(x) \) be a real-valued function defined on the set of real numbers, and suppose it satisfies \( 2 f(x-1) - 3 f(1-x) = 5x \). Find \( f(x) \).
|
f(x) = x - 5
|
olympiads
| 0.109375
|
Find all real solutions of the equation: $$ x=\frac{2z^2}{1+z^2} $$ $$ y=\frac{2x^2}{1+x^2} $$ $$ z=\frac{2y^2}{1+y^2} $$
|
(0, 0, 0) or (1, 1, 1)
|
aops_forum
| 0.09375
|
How many people can be in a group where everyone has exactly 3 acquaintances, and two people have a common acquaintance only if they do not know each other?
|
6, 8, \text{or} 10
|
olympiads
| 0.203125
|
How can you cut a 5 × 5 square with straight lines so that the resulting pieces can be assembled into 50 equal squares? It is not allowed to leave unused pieces or to overlap them.
|
50
|
olympiads
| 0.515625
|
Bethany, Chun, Dominic, and Emily go to the movies. They choose a row with four consecutive empty seats. If Dominic and Emily must sit beside each other, in how many different ways can the four friends sit?
|
12
|
olympiads
| 0.421875
|
The distance between every two utility poles along the road is 50 meters. Xiao Wang travels at a constant speed in a car, and sees 41 utility poles in 2 minutes after seeing the first pole. How many meters does the car travel per hour?
|
60000 \text{ meters per hour}
|
olympiads
| 0.59375
|
Three motorcyclists start simultaneously from the same point on a circular highway in the same direction. The first motorcyclist caught up with the second for the first time after completing 4.5 laps from the start, and 30 minutes before that, he caught up with the third motorcyclist for the first time. The second motorcyclist caught up with the third for the first time three hours after the start. How many laps per hour does the first motorcyclist complete?
|
3
|
olympiads
| 0.0625
|
Calculate the limit of the function:
\[
\lim_{x \rightarrow 0} \frac{e^{5x} - e^{3x}}{\sin 2x - \sin x}
\]
|
2
|
olympiads
| 0.203125
|
Given that \(\alpha = \frac{2\pi}{1999}\), find the value of \(\cos \alpha \cos 2 \alpha \cos 3 \alpha \cdots \cos 999 \alpha\).
|
\frac{1}{2^{999}}
|
olympiads
| 0.09375
|
The legs of a right triangle are 36 and 48. Find the distance from the center of the circle inscribed in the triangle to the altitude drawn to the hypotenuse.
|
\frac{12}{5}
|
olympiads
| 0.078125
|
Given the function \( f(x)=\frac{\sin (\pi x)-\cos (\pi x)+2}{\sqrt{x}} \) where \( \frac{1}{4} \leqslant x \leqslant \frac{5}{4} \), determine the minimum value of \( f(x) \).
|
\frac{4 \sqrt{5}}{5}
|
olympiads
| 0.078125
|
Solve problem 5.95 (a) using Menelaus's theorem.
|
\frac{\overline{B A_{1}}}{\overline{C A_{1}}} \cdot \frac{\overline{C B_{1}}}{\overline{A B_{1}}} \cdot \frac{\overline{A C_{1}}}{\overline{B C_{1}}} = 1
|
olympiads
| 0.171875
|
Let $a \geq 0$ be a natural number. Determine all rational $x$ , so that
\[\sqrt{1+(a-1)\sqrt[3]x}=\sqrt{1+(a-1)\sqrt x}\]
All occurring square roots, are not negative.**Note.** It seems the set of natural numbers = $\mathbb N = \{0,1,2,\ldots\}$ in this problem.
|
x = 0, 1
|
aops_forum
| 0.21875
|
A circle contains the points $(0, 11)$ and $(0, -11)$ on its circumference and contains all points $(x, y)$ with $x^2+y^2<1$ in its interior. Compute the largest possible radius of the circle.
|
\sqrt{122}
|
aops_forum
| 0.0625
|
Each of two teams, Team A and Team B, sends 7 players in a predetermined order to participate in a Go contest. The players from both teams compete sequentially starting with Player 1 from each team. The loser of each match is eliminated, and the winner continues to compete with the next player from the opposing team. This process continues until all the players of one team are eliminated, and the other team wins. How many different possible sequences of matches can occur in this contest?
|
3432
|
olympiads
| 0.171875
|
In an isosceles triangle, the sides are 3 and 7. Which side is the base?
|
3
|
olympiads
| 0.46875
|
Let \(D\) be a point on side \(BC\) of \(\triangle ABC\). Points \(E\) and \(F\) are the centroids of \(\triangle ABD\) and \(\triangle ACD\), respectively. The line segment \(EF\) intersects \(AD\) at point \(G\). Find the value of \(\frac{DG}{GA}\).
|
\frac{1}{2}
|
olympiads
| 0.234375
|
A game begins with 7 coins aligned on a table, all showing heads up. To win the game, you need to flip some coins so that in the end, each pair of adjacent coins has different faces up. The rule of the game is: in each move, you must flip two adjacent coins. What is the minimum number of moves required to win the game?
|
4 \text{ jogadas}
|
olympiads
| 0.1875
|
Given two points A and B and a line L in the plane, find the point P on the line for which max(AP, BP) is as short as possible.
|
P
|
olympiads
| 0.140625
|
Let \( z_{1}, z_{2} \in \mathbf{C} \) such that \(\left| z_{1} \right| = \left| z_{1} + z_{2} \right| = 3 \) and \(\left| z_{1} - z_{2} \right| = 3\sqrt{3}\). 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
| 0.140625
|
There are 10 red, 10 yellow, and 10 green balls in a bag, all of the same size. Each red ball is marked with the number "4", each yellow ball with the number "5", and each green ball with the number "6". Xiao Ming draws 8 balls from the bag, and the sum of the numbers on these balls is 39. What is the maximum possible number of red balls among them?
|
4
|
olympiads
| 0.359375
|
There is a plate of fruits. When counted in groups of 3, there are 2 left over; when counted in groups of 4, there are 3 left over; when counted in groups of 5, there are 4 left over; and when counted in groups of 6, there are 5 left over. The minimum number of fruits in this plate is $\qquad$.
|
59
|
olympiads
| 0.1875
|
$$ Problem 4 $$ Straight lines $k,l,m$ intersecting each other in three different points are drawn on a classboard. Bob remembers that in some coordinate system the lines $ k,l,m$ have the equations $y = ax, y = bx$ and $y = c +2\frac{ab}{a+b}x$ (where $ab(a + b)$ is non zero).
Misfortunately, both axes are erased. Also, Bob remembers that there is missing a line $n$ ( $y = -ax + c$ ), but he has forgotten $a,b,c$ . How can he reconstruct the line $n$ ?
|
y = -ax + c
|
aops_forum
| 0.171875
|
What are the integers \(a\) between 1 and 105 such that \(35 \mid a^{3}-1\)?
|
1, 11, 16, 36, 46, 51, 71, 81, 86
|
olympiads
| 0.125
|
Someone took $\frac{1}{13}$ from the treasury. From what was left, another person took $\frac{1}{17}$, leaving 150 in the treasury. We want to find out how much was in the treasury initially.
|
172 \frac{21}{32}
|
olympiads
| 0.5625
|
What is the minimum value of \( t \) such that the inequality \( \sqrt{x y} \leq t(2 x + 3 y) \) holds for all non-negative real numbers \( x \) and \( y \)?
|
\frac{1}{2 \sqrt{6}}
|
olympiads
| 0.0625
|
Define \( a \star = \frac{a-1}{a+1} \) for \( a \neq -1 \). Determine all real values \( N \) for which \( (N \star) \star = \tan 15^\circ \).
|
-2 - \sqrt{3}
|
olympiads
| 0.40625
|
Each of the corner fields of the outer square should be filled with one of the numbers \(2, 4, 6\), and \( 8 \), with different numbers in different fields. In the four fields of the inner square, there should be the products of the numbers from the adjacent fields of the outer square. In the circle, there should be the sum of the numbers from the adjacent fields of the inner square.
Which numbers can be written in the circle? Determine all possibilities.
(M. Dillinger)
Hint: Can different fillings of the corner fields lead to the same sum in the circle?
|
84, 96, 100
|
olympiads
| 0.09375
|
Given triangle \(ABC\) with \(\angle B = 60^\circ\) and \(\angle C = 75^\circ\), an isosceles right triangle \(BDC\) is constructed on side \(BC\) towards the interior of triangle \(ABC\) with \(BC\) as its hypotenuse. What is the measure of \(\angle DAC\)?
|
30^\circ
|
olympiads
| 0.0625
|
Using the method of isolines, construct the integral curves of the equation \(\frac{d y}{d x}=\frac{y-x}{y+x}\).
|
Complete!
|
olympiads
| 0.078125
|
Given \(\sin \left(x+\arccos \frac{4}{5}\right)=\frac{\sqrt{3}}{2}, 0<x<\pi\), find \(\sin x\).
|
\frac{1}{10}(4 \sqrt{3} - 3) \quad \text{or} \quad \frac{1}{10}(4 \sqrt{3} + 3)
|
olympiads
| 0.0625
|
Let \( n > 4 \) be a positive integer. Determine the number of ways to walk from \( (0,0) \) to \( (n, 2) \) using only up and right unit steps such that the path does not meet the lines \( y = x \) or \( y = x - n + 2 \) except at the start and at the end.
|
\frac{1}{2}\left(n^2 - 5n + 2\right)
|
olympiads
| 0.078125
|
The parallelogram \(ABCD\) is folded along the diagonal \(BD\) so that the vertex \(C\) remains in place, while vertex \(A\) moves to position \(A'\). Segments \(BC\) and \(A'D\) intersect at point \(K\), with \(BK:KC = 3:2\). Find the area of triangle \(A'KC\) if the area of parallelogram \(ABCD\) is 27.
|
3.6
|
olympiads
| 0.578125
|
\(\cos \frac{\pi}{11} - \cos \frac{2 \pi}{11} + \cos \frac{3 \pi}{11} - \cos \frac{4 \pi}{11} + \cos \frac{5 \pi}{11} = \) (Answer with a number).
|
\frac{1}{2}
|
olympiads
| 0.078125
|
Determine the positive prime numbers $p$ and $q$ for which the equation $x^{4} + p^{2} x + q = 0$ has a multiple root.
|
(p, q) = (2, 3)
|
olympiads
| 0.0625
|
There are 1000 coins, among which there might be 0, 1, or 2 counterfeit coins. It is known that the counterfeit coins have the same mass, which is different from the mass of the genuine coins. Is it possible, using a balance scale and without weights, to determine in three weighings whether there are counterfeit coins and if they are lighter or heavier than the genuine coins? (The exact number of counterfeit coins does not need to be determined.)
|
Can resolve with three weighings
|
olympiads
| 0.203125
|
Find the sets \( X \) and \( Y \) from the system of equations
\[
\begin{cases}
X \cup Y = A \\
X \cap A = Y
\end{cases}
\]
where \( A \) is a given set.
|
X = Y = A
|
olympiads
| 0.265625
|
Given an integer $n \ge 2$ , solve in real numbers the system of equations \begin{align*}
\max\{1, x_1\} &= x_2
\max\{2, x_2\} &= 2x_3
&\cdots
\max\{n, x_n\} &= nx_1.
\end{align*}
|
(x_1, x_2, \, \ldots, \, x_n) = (1, 1, \, \ldots, \, 1)
|
aops_forum
| 0.203125
|
The non-zero numbers \( a, b, \) and \( c \) are such that the doubled roots of the quadratic polynomial \( x^{2}+a x+b \) are the roots of the polynomial \( x^{2}+b x+c \). What can the ratio \( a / c \) equal?
|
\frac{1}{8}
|
olympiads
| 0.4375
|
We need to find a number which, when multiplied by itself, added to 2, then doubled, added to 3, divided by 5, and finally multiplied by 10, results in 50.
|
3
|
olympiads
| 0.171875
|
Given the sequence \(\{a_{n}\}\) such that
\[
\begin{array}{l}
a_{1}=a_{2}=1, a_{3}=m, \\
a_{n+1}=\frac{k+a_{n} a_{n-1}}{a_{n-2}}(n \geqslant 3),
\end{array}
\]
where \(k, m \in \mathbf{Z}_{+}\), and \((k, m)=1\). What should \(k\) be so that, for any \(n \in \mathbf{Z}_{+}\), \(a_{n}\) is always an integer?
|
k = 1
|
olympiads
| 0.515625
|
There are 7 different positive integers arranged in ascending order to form an arithmetic sequence. It is known that the average of the first three numbers is 20, and the average of the last three numbers is 24. Find the average of the middle three numbers.
|
22
|
olympiads
| 0.40625
|
In an 8a class, 60% of the students on the list are girls. When two boys and one girl were absent due to illness, the percentage of girls present became 62.5%. How many girls and boys are there in the class according to the list?
|
21 \text{ девочка и } 14 \text{ мальчиков}
|
olympiads
| 0.390625
|
In a given circle with a radius of 3, six equal circles are inscribed, each tangent to the given circle and also to two neighboring circles. Find the radii of the inscribed circles.
|
1
|
olympiads
| 0.09375
|
A game begins with seven coins lined up on a table, all showing heads up. To win the game, you need to flip some coins such that, in the end, two adjacent coins always show different faces. The rule of the game is to flip two adjacent coins in each move. What is the minimum number of moves required to win the game?
|
4
|
olympiads
| 0.125
|
Point \( C \) moves along a line segment \( AB \) of length 4. There are two isosceles triangles \( ACD \) and \( BEC \) on the same side of the line passing through \( AB \) such that \( AD = DC \) and \( CE = EB \). Find the minimum length of segment \( DE \).
|
2
|
olympiads
| 0.140625
|
What is the remainder when the sum
$$
1^{5} + 2^{5} + 3^{5} + \cdots + 2007^{5}
$$
is divided by 5?
|
3
|
olympiads
| 0.265625
|
Let ${(a_n)_{n\ge1}} $ be a sequence with ${a_1 = 1} $ and ${a_{n+1} = \lfloor a_n +\sqrt{a_n}+\frac{1}{2}\rfloor }$ for all ${n \ge 1}$ , where ${\lfloor x \rfloor}$ denotes the greatest integer less than or equal to ${x}$ . Find all ${n \le 2013}$ such that ${a_n}$ is a perfect square
|
n = 1
|
aops_forum
| 0.09375
|
Consider the lines with equations \( y = mx \) and \( y = -2x + 28 \) where \( m \) is some real number. The area enclosed by the two lines and the \( x \)-axis in the first quadrant is equal to 98. What is the value of \( m \)?
|
2
|
olympiads
| 0.25
|
Azmi has two fair dice, each with six sides.
The sides of one of the dice are labelled \(1,2,3,4,5,6\).
The sides of the other die are labelled \(t-10, t, t+10, t+20, t+30, t+40\).
When these two dice are rolled, there are 36 different possible values for the sum of the numbers on the top faces. What is the average of these 36 possible sums?
|
30.5
|
olympiads
| 0.0625
|
Let \( A = \{a \mid a = 2^x 3^y 5^z, x, y, z \in \mathbf{N}\} \), and \( B = \{b \mid b \in A, 1 \leq b \leq 10\} \). Find the number of subsets of the set \( B \) that include the elements 1 and 10.
|
128
|
olympiads
| 0.21875
|
Given a fixed point $A$ and a straight line rotating around another fixed point $B$, what will be the geometric locus of the point symmetric to $A$ with respect to the rotating line as the symmetry axis?
|
circle
|
olympiads
| 0.453125
|
12 people sat around a round table; some of them are knights (who always tell the truth), and others are liars (who always lie). Then each of them said, "There is a liar among my neighbors." What is the maximum number of people sitting at the table who can say, "There is a knight among my neighbors?"
|
8
|
olympiads
| 0.0625
|
Calculate the limit of the function:
\[
\lim _{x \rightarrow 0}\left(\frac{\sin 5 x^{2}}{\sin x}\right)^{\frac{1}{x+6}}
\]
|
0
|
olympiads
| 0.078125
|
What can be the product of several distinct prime numbers if it is divisible by each of them minus 1? Find all possible values of this product.
|
6, 42, 1806
|
olympiads
| 0.0625
|
In how many ways can two disjoint subsets be selected from a set with $n$ elements?
|
3^n
|
olympiads
| 0.125
|
Ten children were given 100 pieces of macaroni each on their plates. Some children didn't want to eat and started playing. With one move, one child transfers one piece of macaroni from their plate to each of the other children's plates. What is the minimum number of moves needed such that all the children end up with a different number of pieces of macaroni on their plates?
|
45
|
olympiads
| 0.078125
|
Given \(0 \leq \theta \leq \pi\), find the maximum value of the function \(f(\theta)=\sqrt{1-\cos \theta+\sin \theta}+\sqrt{\cos \theta+2}+\sqrt{3-\sin \theta}\).
|
3\sqrt{2}
|
olympiads
| 0.09375
|
There are four intersecting paths crossing the yard. Plant four apple trees such that there are an equal number of apple trees on both sides of each path.
|
Solution Valid
|
olympiads
| 0.15625
|
Given a right triangular prism \(ABC-A_{1}B_{1}C_{1}\) with the base being a right triangle, \(\angle ACB = 90^{\circ}\), \(AC = 6\), \(BC = CC_{1} = \sqrt{2}\), and \(P\) is a moving point on \(BC_{1}\), find the minimum value of \(CP + PA_{1}\).
|
5\sqrt{2}
|
olympiads
| 0.09375
|
Given a plane intersects all 12 edges of a cube at an angle $\alpha$, find $\sin \alpha$.
|
\frac{\sqrt{3}}{3}
|
olympiads
| 0.140625
|
In parallelogram \(PQRS\), the bisector of the angle at vertex \(P\), which is \(80^{\circ}\), intersects side \(RS\) at point \(L\). Find the radius of the circle that touches segment \(PQ\) and rays \(QR\) and \(PL\), given that \(PQ = 7\).
|
7 \cos 40^\circ \cdot \tan 20^\circ
|
olympiads
| 0.125
|
Find the derivative $y_{x}^{\prime}$.
$$
\left\{\begin{array}{l}
x=\arcsin (\sin t) \\
y=\arccos (\cos t)
\end{array}\right.
$$
|
1
|
olympiads
| 0.328125
|
There are three batches of parts, each containing 30 parts. The number of standard parts in the first, second, and third batches is 20, 15, and 10, respectively. A part is randomly selected from a randomly chosen batch and it turns out to be standard. Then, a second part is randomly selected from the same batch, and it also turns out to be standard. Find the probability that the parts were drawn from the third batch.
|
\frac{9}{68}
|
olympiads
| 0.109375
|
A man married a widow, and each of them had children from their previous marriages. Ten years later, there was a conflict involving all the children, who at that point totaled 12. The mother approached the father exclaiming:
- Hurry! Your children and my children are hitting our children!
Each parent originally had 9 biological children.
How many children were born in those 10 years?
|
6
|
olympiads
| 0.25
|
Given that the real numbers \( a \) and \( b \) satisfy that both quadratic trinomials \( x^2 + ax + b \) and \( x^2 + bx + a \) have two distinct real roots each, and the product of their roots results in exactly three distinct real roots. Find all possible values of the sum of these three distinct roots.
|
0
|
olympiads
| 0.09375
|
Sasha and Vanya were both born on March 19th. Each of them celebrates their birthdays with a cake that has as many candles as their age. The year they met, Sasha had as many candles on his cake as Vanya has today. It is known that the total number of candles on the four cakes (their cakes from the year they met and their cakes today) is 216. How old is Vanya today?
|
54
|
olympiads
| 0.421875
|
For which $\alpha$ does there exist a non-constant function $f: \mathbb{R} \rightarrow \mathbb{R}$ such that
$$
f(\alpha(x+y))=f(x)+f(y)?
$$
|
\alpha = 1
|
olympiads
| 0.375
|
Let $n$ a positive integer. In a $2n\times 2n$ board, $1\times n$ and $n\times 1$ pieces are arranged without overlap.
Call an arrangement **maximal** if it is impossible to put a new piece in the board without overlapping the previous ones.
Find the least $k$ such that there is a **maximal** arrangement that uses $k$ pieces.
|
2n + 1
|
aops_forum
| 0.078125
|
Given that there are \( c \) prime numbers less than 100 such that their unit digits are not square numbers, find the values of \( c \).
|
15
|
olympiads
| 0.125
|
Given that both A and B are either knights or liars, A states: "If B is a knight, then I am a liar". Determine the identities of A and B.
|
A is a knight, and B is a liar
|
olympiads
| 0.15625
|
For which values of \(x\) is the equality \( |x+6| = -(x+6) \) true?
|
x \leq -6
|
olympiads
| 0.078125
|
The sequence $(x_n)_{n\geqslant 0}$ is defined as such: $x_0=1, x_1=2$ and $x_{n+1}=4x_n-x_{n-1}$ , for all $n\geqslant 1$ . Determine all the terms of the sequence which are perfect squares.
|
x_0 = 1
|
aops_forum
| 0.171875
|
The 5-digit number '$XX4XY$' is exactly divisible by 165. What is the value of $X+Y$?
|
14
|
olympiads
| 0.09375
|
Compute the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{\sqrt{1+\operatorname{tg} x}-\sqrt{1+\sin x}}{x^{3}}
$$
|
\frac{1}{4}
|
olympiads
| 0.140625
|
Compute the definite integral:
$$
\int_{0}^{\frac{\pi}{4}} \left( x^{2} + 17.5 \right) \sin 2x \, dx
$$
|
\frac{68 + \pi}{8}
|
olympiads
| 0.265625
|
Let us have $6050$ points in the plane, no three collinear. Find the maximum number $k$ of non-overlapping triangles without common vertices in this plane.
|
2016
|
aops_forum
| 0.09375
|
A wolf noticed a hare 30 meters away and started chasing it when the hare still had 250 meters to reach its place of refuge. Will the wolf catch the hare if the wolf can run 600 meters per minute and the hare can run 550 meters per minute?
|
The wolf does not catch the hare.
|
olympiads
| 0.1875
|
By the first of September, Vlad bought several ballpoint and gel pens. He noticed that if all the purchased pens were gel pens, he would have paid 4 times more than he actually did. Conversely, if all the pens were ballpoint pens, the purchase would have cost him half of the actual amount. How many times more expensive is a gel pen compared to a ballpoint pen?
|
8
|
olympiads
| 0.078125
|
Let \( x \) and \( y \) be real numbers that satisfy the following system of equations:
\[
\begin{cases}
(x-1)^{3} + 2003(x-1) = -1 \\
(y-1)^{3} + 2003(y-1) = 1
\end{cases}
\]
Additionally, it is given that \( x + y = \).
|
2
|
olympiads
| 0.296875
|
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