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stringlengths 33
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values | llama8b_solve_rate
float64 0.06
0.59
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|---|---|---|---|
\(ABCD\) is a square with side length 9. Let \(P\) be a point on \(AB\) such that \(AP: PB = 7:2\). Using \(C\) as the center and \(CB\) as the radius, a quarter circle is drawn inside the square. The tangent from \(P\) meets the circle at \(E\) and \(AD\) at \(Q\). The segments \(CE\) and \(DB\) meet at \(K\), while \(AK\) and \(PQ\) meet at \(M\). Find the length of \(AM\).
|
\frac{85}{22}
|
olympiads
| 0.109375
|
Let $n$ be a positive integer. A frog starts on the number line at $0$ . Suppose it makes a finite sequence of hops, subject to two conditions:
- The frog visits only points in $\{1, 2, \dots, 2^n-1\}$ , each at most once.
- The length of each hop is in $\{2^0, 2^1, 2^2, \dots\}$ . (The hops may be either direction, left or right.)
Let $S$ be the sum of the (positive) lengths of all hops in the sequence. What is the maximum possible value of $S$ ?
|
\frac{4^n - 1}{3}
|
aops_forum
| 0.265625
|
What is the smallest natural number by which 720 must be multiplied to obtain the cube of a natural number?
|
300
|
olympiads
| 0.109375
|
The arithmetic mean of several consecutive natural numbers is 5 times greater than the smallest of them. How many times is the arithmetic mean smaller than the largest of these numbers?
|
1.8
|
olympiads
| 0.296875
|
Simplify the expression \(\left(\frac{2-n}{n-1}+4 \cdot \frac{m-1}{m-2}\right):\left(n^{2} \cdot \frac{m-1}{n-1}+m^{2} \cdot \frac{2-n}{m-2}\right)\) given that \(m=\sqrt[4]{400}\) and \(n=\sqrt{5}\).
|
\frac{\sqrt{5}}{5}
|
olympiads
| 0.078125
|
A bullet with a mass of \( m = 10 \) g, flying horizontally with a speed of \( v_{1} = 400 \) m/s, passes through a massive board and emerges from it with a speed of \( v_{2} = 100 \) m/s. Find the amount of work done on the bullet by the resistive force of the board.
|
750 \, \text{J}
|
olympiads
| 0.40625
|
\[
\begin{cases}
\lg (x+y) - \lg 5 = \lg x + \lg y - \lg 6 \\
\frac{\lg x}{\lg (y+6) - (\lg y + \lg 6)} = -1
\end{cases}
\]
|
(2, 3)
|
olympiads
| 0.140625
|
Let $[x]$ denote the greatest integer less than or equal to the real number $x$. The sequence $\left\{a_{n}\right\}$ satisfies:
$$
x_{1}=1, x_{n+1}=4 x_{n}+\left[\sqrt{11} x_{n}\right] \text {. }
$$
Find the units digit of $x_{2021}$.
|
9
|
olympiads
| 0.078125
|
Alexey plans to buy one of two car brands: "A" costing 900,000 rubles or "B" costing 600,000 rubles. On average, Alexey drives 15,000 km per year. The cost of gasoline is 40 rubles per liter. The cars use the same type of gasoline. The car is planned to be used for 5 years, after which the car of brand "A" can be sold for 500,000 rubles, and the car of brand "B" for 350,000 rubles.
| Car Brand | Fuel Consumption (L/100km) | Annual Insurance Cost (rubles) | Average Annual Maintenance Cost (rubles) |
| :--- | :--- | :--- | :--- |
| "A" | 9 | 35,000 | 25,000 |
| "B" | 10 | 32,000 | 20,000 |
Using the data from the table, answer the question: How much more expensive will it be for Alexey to buy and own the more expensive car?
|
160000
|
olympiads
| 0.296875
|
Several circles are inscribed in an angle, with their radii increasing. Each subsequent circle touches the previous circle. Find the sum of the circumferences of the second and third circles if the radius of the first circle is 1, and the area of the circle bounded by the fourth circle is $64\pi$.
|
12\pi
|
olympiads
| 0.25
|
Let $p$ be a prime number such that $\frac{28^p-1}{2p^2+2p+1}$ is an integer. Find all possible values of number of divisors of $2p^2+2p+1$ .
|
2
|
aops_forum
| 0.203125
|
In square \(ABCD\) with a side length of 10, points \(P\) and \(Q\) lie on the segment joining the midpoints of sides \(AD\) and \(BC\). Connecting \(PA\), \(PC\), \(QA\), and \(QC\) divides the square into three regions of equal area. Find the length of segment \(PQ\).
|
\frac{20}{3}
|
olympiads
| 0.0625
|
Given a fixed triangle \( \triangle ABC \) and a point \( P \), find the maximum value of
\[
\frac{AB^{2} + BC^{2} + CA^{2}}{PA^{2} + PB^{2} + PC^{2}}
\]
|
3
|
olympiads
| 0.140625
|
Given the circle \( C: (x-1)^2 + (y-2)^2 = 25 \) and the line \( l: (2m+1)x + (m+1)y - 7m - 4 = 0 \) intersect, find the equation of line \( l \) when the chord length intercepted by the circle \( C \) is minimized.
|
2x - y - 5 = 0
|
olympiads
| 0.09375
|
Tanya's clock is 5 minutes slow for every hour. The guests will arrive at noon. It is now 6 AM. What time should she set the clock to display the correct time at noon?
|
6:30 AM
|
olympiads
| 0.3125
|
In how many ways can four black balls, four white balls, and four blue balls be distributed into six different boxes?
|
\left( \binom{9}{4} \right)^3
|
olympiads
| 0.296875
|
Determine the numeral system in which the following multiplication is performed: \(352 \cdot 31 = 20152\).
|
6
|
olympiads
| 0.171875
|
The numbers \(a\), \(b\), \(c\) are such that \(a > b\) and \((a-b)(b-c)(c-a) > 0\). Which is greater: \(a\) or \(c\)?
|
a > c
|
olympiads
| 0.203125
|
Victor was driving to the airport in a neighboring city. Half an hour into the drive at a speed of 60 km/h, he realized that if he did not change his speed, he would be 15 minutes late. So he increased his speed, covering the remaining distance at an average speed of 80 km/h, and arrived at the airport 15 minutes earlier than planned initially. What is the distance from Victor's home to the airport?
|
s = 150 \text{ km}
|
olympiads
| 0.0625
|
By writing successive natural numbers, we get the sequence
$$
12345678910111213141516171819202122 \ldots
$$
What is the digit that is in the $2009^{th}$ position of this sequence?
|
0
|
olympiads
| 0.25
|
Find all integer triplets $(x, y, z)$ such that $x^{2}+y^{2}+z^{2}-2xyz=0$.
|
(0, 0, 0)
|
olympiads
| 0.25
|
Katya placed a square with a perimeter of 40 cm next to a square with a perimeter of 100 cm as shown in the picture. What is the perimeter of the resulting figure in centimeters?
|
120 \, \text{cm}
|
olympiads
| 0.125
|
Four vertices of a square are marked. Mark four more points so that each perpendicular bisector of the segments with endpoints at the marked points has exactly two marked points on it.
|
The configurations presented are accurate and serve the problem constraints.
|
olympiads
| 0.296875
|
A team of several workers can complete a task in 7 full days, the same task that this same team minus two workers can complete in a certain number of full days, and the same task that this team without six workers can complete in another certain number of full days. How many workers are in the team? (Assuming the productivity of the workers is the same.)
|
9
|
olympiads
| 0.0625
|
Teacher Li and three students (Xiao Ma, Xiao Lu, and Xiao Zhou) depart from school one after another and walk the same route to a cinema. The walking speed of the three students is equal, and Teacher Li's walking speed is 1.5 times that of the students. Currently, Teacher Li is 235 meters from the school, Xiao Ma is 87 meters from the school, Xiao Lu is 59 meters from the school, and Xiao Zhou is 26 meters from the school. After they walk a certain number of meters, the distance of Teacher Li from the school will be equal to the sum of the distances of the three students from the school. Find the number of meters they need to walk for this condition to be true.
|
x = 42
|
olympiads
| 0.078125
|
In how many ways can we place two rooks of different colors on a chessboard such that they do not attack each other?
|
3136
|
olympiads
| 0.15625
|
The triangle \(PQR\) is isosceles with \(PR = QR\). Angle \(PRQ = 90^\circ\) and length \(PQ = 2 \text{ cm}\). Two arcs of radius \(1 \text{ cm}\) are drawn inside triangle \(PQR\). One arc has its center at \(P\) and intersects \(PR\) and \(PQ\). The other arc has its center at \(Q\) and intersects \(QR\) and \(PQ\). What is the area of the shaded region, in \(\text{cm}^2\)?
|
1 - \frac{\pi}{4}
|
olympiads
| 0.078125
|
Given that \(x^{2} + a x + b\) is a common factor of \(2x^{3} + 5x^{2} + 24x + 11\) and \(x^{3} + Px - 22\), and if \(Q = a + b\), find the value of \(Q\).
|
13
|
olympiads
| 0.0625
|
Compute the limit of the function:
$$\lim _{x \rightarrow \pi} \frac{\cos 5 x-\cos 3 x}{\sin ^{2} x}$$
|
8
|
olympiads
| 0.1875
|
Perpendiculars \( AP \) and \( AK \) are dropped from vertex \( A \) of triangle \( ABC \) onto the angle bisectors of the external angles at \( B \) and \( C \) respectively. Find the length of segment \( PK \), given that the perimeter of triangle \( ABC \) is \( P \).
|
\frac{P}{2}
|
olympiads
| 0.09375
|
Given the sequence \(\left\{a_{n}\right\}\) where \(a_{1} = 1\) and \(a_{n+1} = \frac{\sqrt{3} a_{n} + 1}{\sqrt{3} - a_{n}}\), find the value of \(\sum_{n=1}^{2022} a_{n}\).
|
0
|
olympiads
| 0.0625
|
In a regular tetrahedron \( S-ABC \) with side length \( a \), \( E \) and \( F \) are the midpoints of \( SA \) and \( BC \) respectively. Find the angle between the skew lines \( BE \) and \( SF \).
|
\arccos \left( -\frac{2}{3} \right)
|
olympiads
| 0.0625
|
Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any real numbers \( x \) and \( y \), the following holds:
$$
f\left(x^{2}\right) f\left(y^{2}\right) + |x| f\left(-x y^{2}\right) = 3|y| f\left(x^{2} y\right).
$$
|
f(x) = 0 \text{ and } f(x) = 2|x|
|
olympiads
| 0.203125
|
The length of the bus route is 16 km. During peak hours, the bus switches to express mode, significantly reducing the number of stops. As a result, the travel time from the beginning to the end of the route is shortened by 4 minutes, and the average speed of the bus increases by 8 km/h. What is the speed of the bus in express mode?
|
48 \text{ km/h}
|
olympiads
| 0.140625
|
Solve the system of equations:
$$
\begin{aligned}
& xy = 1 \\
& yz = 2 \\
& zx = 8
\end{aligned}
$$
|
(2, \frac{1}{2}, 4), (-2, -\frac{1}{2}, -4)
|
olympiads
| 0.296875
|
This year is 2015, and Xiaoming says: "My current age is exactly equal to the sum of the four digits of the year I was born." How old is Xiaoming now?
|
22 \text{ or } 4
|
olympiads
| 0.109375
|
Find the number of rectangles with sides parallel to the axes whose vertices are all of the form \((a, b)\) with \(a\) and \(b\) being integers such that \(0 \le a, b \le n\).
|
\frac{(n+1)^2 \times n^2}{4}
|
olympiads
| 0.578125
|
The equation \( x^{3} - 9x^{2} + 8x + 2 = 0 \) has three real roots \( p, q, r \). Find \( \frac{1}{p^{2}} + \frac{1}{q^{2}} + \frac{1}{r^{2}} \).
|
25
|
olympiads
| 0.46875
|
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{n^{2}-\sqrt{3 n^{5}-7}}{\left(n^{2}-n \cos n+1\right) \sqrt{n}}
\]
|
-
\sqrt{3}
|
olympiads
| 0.109375
|
We shuffle a deck of 52 French playing cards, and then draw one by one from the pile until we find an ace of black color. In what position is it most likely for the first black ace to appear?
|
1
|
olympiads
| 0.109375
|
A $7 \times 7$ table is filled with zeroes. In one operation, the minimum number in the table is found (if there are several such numbers, any one of them is chosen), and one is added to it as well as to all the numbers in the cells adjacent to it by side or corner. What is the largest number that can appear in one of the cells of the table after 90 operations?
Answer: 40.
|
40
|
olympiads
| 0.484375
|
For which value of $\lambda$ is there a pair of values $x$ and $y$ that satisfies the following system of equations:
$$
x^{2}+y^{2}=8 x+6 y ; \quad 9 x^{2}+y^{2}=6 y ; \quad y^{2}+9=\lambda x+6 y+\lambda
$$
|
9
|
olympiads
| 0.1875
|
For which values of \( x \) can the numerator and the denominator of the fraction \(\frac{a}{b}\) be multiplied by the expression \( x - m \) without changing the value of the fraction?
|
x \neq m
|
olympiads
| 0.484375
|
Suppose that $ABCD$ is a rectangle with sides of length $12$ and $18$ . Let $S$ be the region of points contained in $ABCD$ which are closer to the center of the rectangle than to any of its vertices. Find the area of $S$ .
|
54
|
aops_forum
| 0.09375
|
Given \( x \) and \( y \) are the values to be determined, solve for \( x \) and \( y \) given the equation \( x + y = xy \). This can be rewritten as \( (x-1)(y-1) = 1 \).
|
(2,2), (0,0)
|
olympiads
| 0.171875
|
The lateral edges of a triangular pyramid are pairwise perpendicular and have lengths $a, b,$ and $c$. Find the volume of the pyramid.
|
\frac{a b c}{6}
|
olympiads
| 0.0625
|
Determine the sign of the number $\log_{1.7}\left(\frac{1}{2}\left(1-\log_{7} 3\right)\right)$.
|
negative
|
olympiads
| 0.59375
|
Two cars started at the same time from two different locations that are 60 km apart and traveled in the same direction. The first car, moving at a speed of 90 km/h, caught up with the second car after three hours. What is the speed of the second car?
|
70 \, \text{km/h}
|
olympiads
| 0.59375
|
In a rectangular parallelepiped $\mathrm{ABCDA}_{1} \mathrm{~B}_{1} \mathrm{C}_{1} \mathrm{D}_{1}$, with $\mathrm{AB}=1$ cm, $\mathrm{AD}=2$, and $\mathrm{AA}_{1}=1$, find the minimum area of triangle $\mathrm{PA}_{1} \mathrm{C}$, where vertex $\mathrm{P}$ lies on the line $\mathrm{AB}_{1}$.
|
\frac{\sqrt{2}}{2}
|
olympiads
| 0.078125
|
Transform the expression:
$$
\frac{x^{3}+4 x^{2}-x-4}{(x+2)(x+4)-3 x-12}=\frac{x^{2} (x+4)-(x+4)}{(x+2)(x+4)-3(x+4)}=\frac{(x+4)(x-1)(x+1)}{(x+4)(x-1)}=x+1
$$
Given \( f(x)=\left|\frac{x^{3}+4 x^{2}-x-4}{(x+2)(x+4)-3 x-12}\right|=|x+1| \), with the domain \( D(f): x \neq-4, x \neq 1 \),
and
\[ p(x)=\sqrt{x^{2}-8 x+16}+a=|x-4|+a \],
we obtain the equation: \( |x+1| = |x-4| + a \).
Solve it graphically:
For \( x < -1 \):
\[
-x-1 = -x+4+a \Rightarrow a = -5
\]
For \( -1 \leq x < 4 \):
\[
x+1 = -x+4+a \Rightarrow a = 2x - 3
\]
For \( x \geq 4 \):
\[
x+1 = x+4+a \Rightarrow a = 5
\]
The equation has one solution when \( a \in (-5, -1) \cup (-1, 5) \).
|
a \in (-5, -1) \cup (-1, 5)
|
olympiads
| 0.546875
|
Calculate the limit of the function:
$\lim _{x \rightarrow \frac{\pi}{2}} \frac{1-\sin ^{3} x}{\cos ^{2} x}$
|
\frac{3}{2}
|
olympiads
| 0.09375
|
Determine all pairs \((a, n)\) of positive integers such that
$$
3^n = a^2 - 16
$$
|
(5, 2)
|
olympiads
| 0.25
|
Write the decomposition of vector $x$ in terms of vectors $p, q, r$:
$x = \{-15 ; 5 ; 6\}$
$p = \{0 ; 5 ; 1\}$
$q = \{3 ; 2 ; -1\}$
$r = \{-1 ; 1 ; 0\}$
|
x = 2p - 4q + 3r
|
olympiads
| 0.0625
|
A domino has a left end and a right end, each of a certain color. Alice has four dominos, colored red-red, red-blue, blue-red, and blue-blue. Find the number of ways to arrange the dominos in a row end-to-end such that adjacent ends have the same color. The dominos cannot be rotated.
|
4
|
olympiads
| 0.171875
|
Determine all integers \( x \) and \( y \) such that:
\[ \frac{1}{x} + \frac{1}{y} = \frac{1}{19} \]
|
(20, 380), (380, 20), (18, -342), (-342, 18), (38, 38)
|
olympiads
| 0.09375
|
Determine the integers $n\geqslant 2$ for which the equation $x^2-\hat{3}\cdot x+\hat{5}=\hat{0}$ has a unique solution in $(\mathbb{Z}_n,+,\cdot).$
|
n = 11
|
aops_forum
| 0.140625
|
Find all the values of the parameter \( a \) for which the system
\[
\begin{cases}
y = |x - \sqrt{a}| + \sqrt{a} - 2 \\
(|x| - 4)^{2} + (|y| - 3)^{2} = 25
\end{cases}
\]
has exactly three solutions.
|
a \in \{1, 16, \left(\frac{5\sqrt{2} + 1}{2}\right)^{2}\}
|
olympiads
| 0.0625
|
Let $f$ be a function such that $f(0)=1$ , $f^\prime (0)=2$ , and \[f^{\prime\prime}(t)=4f^\prime(t)-3f(t)+1\] for all $t$ . Compute the $4$ th derivative of $f$ , evaluated at $0$ .
|
54
|
aops_forum
| 0.265625
|
In an equilateral triangle \(ABC\), a point \(F\) lies on the side \(BC\). The area of triangle \(ABF\) is three times the area of triangle \(ACF\), and the difference in their perimeters is \(5 \, \text{cm}\).
Determine the length of a side of triangle \(ABC\).
|
10 \, \text{cm}
|
olympiads
| 0.0625
|
Find the number such that when $1108, 1453, 1844,$ and $2281$ are divided by it, they give the same remainder.
|
23
|
olympiads
| 0.140625
|
Carlson was given a bag of candies: chocolate and caramel. In the first 10 minutes, Carlson ate 20\% of all the candies, of which 25\% were caramels. After that, Carlson ate three more chocolate candies, and the proportion of caramels among the candies Carlson had eaten decreased to 20\%. How many candies were in the bag given to Carlson?
|
60
|
olympiads
| 0.3125
|
Sandra has 5 pairs of shoes in a drawer, each pair a different color. Every day for 5 days, Sandra takes two shoes out and throws them out the window. If they are the same color, she treats herself to a practice problem from a past HMMT. What is the expected value (average number) of practice problems she gets to do?
|
\frac{5}{9}
|
olympiads
| 0.1875
|
Given \( A = \{|a+1|, 3, 5\} \) and \( B = \{2a + 1, a^{2a + 2}, a^2 + 2a - 1\} \), if \( A \cap B = \{2, 3\} \), find \( A \cup B \).
|
A \cup B = \{1, 2, 3, 5\}
|
olympiads
| 0.390625
|
Let \( f: \mathbf{N}^{*} \rightarrow \mathbf{N}^{\top} \), and for all \( m, n \in \mathbf{N}^{\top} \), \( f(f(m) + f(n)) = m + n \). Find \( f(2005) \).
|
2005
|
olympiads
| 0.4375
|
Let $\mathcal M$ be the set of $n\times n$ matrices with integer entries. Find all $A\in\mathcal M$ such that $\det(A+B)+\det(B)$ is even for all $B\in\mathcal M$ .
|
A \equiv 0 \pmod{2}
|
aops_forum
| 0.140625
|
For various nonzero real numbers \(a, b, c\), the following equalities hold:
\[ a + \frac{2}{b} = b + \frac{2}{c} = c + \frac{2}{a} = d \]
Find all possible values of the number \(d\).
|
\pm \sqrt{2}
|
olympiads
| 0.0625
|
For which values of \( k \) does the polynomial \( X^{2017} - X^{2016} + X^{2} + kX + 1 \) have a rational root?
|
k = -2 \text{ and } k = 0
|
olympiads
| 0.40625
|
A prism $A B C A_1 B_1 C_1$ is given. Construct the image of the point $M$ where the planes $A_1 B C$, $A B_1 C$, and $A B C_1$ intersect. Let the height of the prism be $h$. Find the distance from point $M$ to the bases of the prism.
|
\frac{1}{3}h \text{ and } \frac{2}{3}h
|
olympiads
| 0.078125
|
Calculate the limit of the function:
$$\lim _{x \rightarrow 1} \frac{1 - x^{2}}{\sin (\pi x)}$$
|
\frac{2}{\pi}
|
olympiads
| 0.515625
|
Find the particular solution of the differential equation
$$
y^{\prime \prime}+4 y^{\prime}+5 y=8 \cos x
$$
that is bounded as $x \rightarrow -\infty$.
|
y = 2 (\cos x + \sin x)
|
olympiads
| 0.15625
|
Investigate the extremum of the function \( z = x^3 + y^3 - 3xy \).
|
-1
|
olympiads
| 0.078125
|
The numbers \( p \) and \( q \) are such that the parabolas \( y=-2x^{2} \) and \( y=x^{2}+px+q \) intersect at two points, enclosing a certain region.
Find the equation of the vertical line that divides the area of this region in half.
|
x = -\frac{p}{6}
|
olympiads
| 0.328125
|
Simplify the expression \(\frac{\log _{a} \sqrt{a^{2}-1} \cdot \log _{1 / a}^{2} \sqrt{a^{2}-1}}{\log _{a^{2}}\left(a^{2}-1\right) \cdot \log _{\sqrt[3]{a}} \sqrt[6]{a^{2}-1}}\).
|
\log _{a} \sqrt{a^{2}-1}
|
olympiads
| 0.125
|
On an island, there are magical sheep. There are 22 blue, 18 red, and 15 green sheep. When two sheep of different colors meet, they both change to the third color. Is it possible that after a certain finite number of meetings, all the sheep will be the same color? If yes, what can this color be?
|
The only color that all sheep can turn into is blue
|
olympiads
| 0.125
|
In $ xy$ plane, find the minimum volume of the solid by rotating the region boubded by the parabola $ y \equal{} x^2 \plus{} ax \plus{} b$ passing through the point $ (1,\ \minus{} 1)$ and the $ x$ axis about the $ x$ axis
|
\frac{16\pi}{15}
|
aops_forum
| 0.125
|
Find the least positive integer $k$ such that for any $a \in [0, 1]$ and any positive integer $n,$ \[a^k(1 - a)^n < \frac{1}{(n+1)^3}.\]
|
k = 4
|
aops_forum
| 0.140625
|
The lines \( y = kx + b \), \( y = 2kx + 2b \), and \( y = bx + k \) are distinct and intersect at one point. What could be the coordinates of this point?
|
(1, 0)
|
olympiads
| 0.234375
|
Which two consecutive odd numbers in the number series have a sum of squares in the form $\frac{n(n+1)}{2}$, where $n$ is a natural number?
|
1 \text{ and } 3
|
olympiads
| 0.125
|
Let \(\mathcal{P}\) be a regular 17-gon. We draw in the \(\binom{17}{2}\) diagonals and sides of \(\mathcal{P}\) and paint each side or diagonal one of eight different colors. Suppose that there is no triangle (with vertices among vertices of \(\mathcal{P}\)) whose three edges all have the same color. What is the maximum possible number of triangles, all of whose edges have different colors?
|
544
|
olympiads
| 0.0625
|
Let \(\alpha, \beta\) be a pair of conjugate complex numbers. If \(|\alpha - \beta| = 2 \sqrt{3}\) and \(\frac{\alpha}{\beta^{2}}\) is a real number, what is \(|\alpha|\)?
|
2
|
olympiads
| 0.59375
|
An unfair coin comes up heads with probability \(\frac{4}{7}\) and tails with probability \(\frac{3}{7}\). Aerith and Bob take turns flipping the coin until one of them flips tails, with Aerith going first. What is the probability that Aerith wins the game?
|
\frac{7}{11}
|
olympiads
| 0.15625
|
The chord $[CD]$ is parallel to the diameter $[AB]$ of a circle with center $O$ . The tangent line at $A$ meet $BC$ and $BD$ at $E$ and $F$ . If $|AB|=10$ , calculate $|AE|\cdot |AF|$ .
|
100
|
aops_forum
| 0.125
|
Calculate the value of \(\sin \left(-\frac{5 \pi}{3}\right) + \cos \left(-\frac{5 \pi}{4}\right) + \tan \left(-\frac{11 \pi}{6}\right) + \cot \left(-\frac{4 \pi}{3}\right)\).
|
\frac{\sqrt{3} - \sqrt{2}}{2}
|
olympiads
| 0.125
|
Each of the lateral edges of a pyramid is $269 / 32$ cm. The base of the pyramid is a triangle with sides of 13 cm, 14 cm, and 15 cm. Find the volume of the pyramid.
|
60.375 \, \text{cm}^3
|
olympiads
| 0.234375
|
Find the third term of an infinite geometric series with a common ratio \( |q| < 1 \), whose sum is \( \frac{8}{5} \) and whose second term is \( -\frac{1}{2} \).
|
\frac{1}{8}
|
olympiads
| 0.390625
|
Using a compass and a ruler, construct a square based on four points that lie on its four sides.
|
Square Constructed
|
olympiads
| 0.28125
|
The beaver is chess piece that move to $2$ cells by horizontal or vertical. Every cell of $100 \times 100$ chessboard colored in some color,such that we can not get from one cell to another with same color with one move of beaver or knight. What minimal color do we need?
|
4
|
aops_forum
| 0.375
|
Which is greater: \(\frac{10\ldots 01}{10\ldots 01}\) (with 1984 zeros in the numerator and 1985 in the denominator) or \(\frac{10\ldots 01}{10\ldots .01}\) (with 1985 zeros in the numerator and 1986 in the denominator)?
|
The first fraction
|
olympiads
| 0.265625
|
Find the minimum value of the function
$$
f(x)=4^{x}+4^{-x}-2^{x+1}-2^{1-x}+5
$$
|
3
|
olympiads
| 0.09375
|
A team 50 meters long is marching from east to west at a speed of 3 meters per second. Another team 60 meters long is marching from west to east at a speed of 2 meters per second. How many seconds will it take from the time the first members meet until the last members pass each other?
|
22
|
olympiads
| 0.234375
|
The exradii of a certain triangle are 2, 3, and 6 cm. Find the radius of the inscribed circle of this triangle.
|
1
|
olympiads
| 0.1875
|
Four points are marked on a straight line and one point is marked outside the line. There are six triangles that can be formed with vertices at these points.
What is the maximum number of these triangles that can be isosceles?
|
6
|
olympiads
| 0.109375
|
Given a ruler with parallel edges and without markings. Construct the center of a circle, a segment of which is given in the diagram.
|
Center of the circle
|
olympiads
| 0.109375
|
I live on the ground floor of a ten-story building. Each friend of mine lives on a different floor. One day, I put the numbers $1, 2, \ldots, 9$ into a hat and drew them randomly, one by one. I visited my friends in the order in which I drew their floor numbers. On average, how many meters did I travel by elevator, if the distance between each floor is 4 meters, and I took the elevator from each floor to the next one drawn?
|
\frac{440}{3} \text{ meters}
|
olympiads
| 0.078125
|
A two-digit number is given in which the first digit is greater than the second. Its digits were rearranged to obtain a new two-digit number. The difference between the first and the second number turned out to be a perfect square. How many possible two-digit numbers satisfy this condition?
|
13
|
olympiads
| 0.0625
|
Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations:
$x^3-4x^2-16x+60=y$ ;
$y^3-4y^2-16y+60=z$ ;
$z^3-4z^2-16z+60=x$ .
|
(3, 3, 3), (-4, -4, -4), (5, 5, 5)
|
aops_forum
| 0.15625
|
Find all positive integers \( k \) such that the equation
\[
x^2 + y^2 + z^2 = kxyz
\]
has positive integer solutions \((x, y, z)\).
|
1, 3
|
olympiads
| 0.140625
|
132. A is a consequence of B. Write this as a conditional proposition.
|
If B, then A.
|
olympiads
| 0.5625
|
Find the smallest value of the parameter \( a \) for which the system of equations
\[
\left\{
\begin{array}{c}
\sqrt{(x-6)^{2}+(y-13)^{2}}+\sqrt{(x-18)^{2}+(y-4)^{2}}=15 \\
(x-2a)^{2}+(y-4a)^{2}=\frac{1}{4}
\end{array}
\right.
\]
has a unique solution.
|
\frac{135}{44}
|
olympiads
| 0.125
|
A ladder sliding along a wall has its top end's slide distance from the starting point denoted as $A T$, and the bottom end's slide distance from the wall corner denoted as $B O$. During the motion, which one of the two quantities, the slide distance (at the top) or the slip distance (at the bottom), is greater?
|
O B > T A
|
olympiads
| 0.15625
|
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