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stringlengths 33
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values | llama8b_solve_rate
float64 0.06
0.59
|
|---|---|---|---|
Find the minimum value of the function \( f(x) = \tan^2 x + 2 \tan x + 6 \cot x + 9 \cot^2 x + 4 \) on the interval \( \left(0, \frac{\pi}{2}\right) \).
|
10 + 4\sqrt{3}
|
olympiads
| 0.078125
|
Suppose \( a_{i}, b_{i} (i=1,2,\ldots,n) \) are rational numbers, such that for any real number \( x \), the following holds:
\[ x^{2}+x+4=\sum_{i=1}^{n}\left(a_{i} x+b_{i}\right)^{2}. \]
Find the minimum possible value of \( n \).
|
5
|
olympiads
| 0.0625
|
Cut the depicted figure (a $9 \times 12$ rectangle with a $1 \times 8$ hole in the center) into two equal parts that can be assembled into a square.
|
Transformation layout: $10 \times 10$ completes
|
olympiads
| 0.0625
|
Let's introduce a coordinate system with the origin at the point where the chute touches the plane, as shown in the figure.
To find the velocity of the sphere's free fall, we use the law of conservation of energy \(V = \sqrt{2 g r \cos \alpha}\). The free fall of the sphere is described by the following equations:
\[
x(t) = R \sin \alpha + V \cos \alpha t, \quad y(t) = R(1 - \cos \alpha) + V \sin \alpha t - \frac{g t^2}{2}
\]
The flight time \(T\) is found from the quadratic equation \(y(T) = 0\):
\[
T = \sqrt{\frac{2 R}{g}} \left[\sin \alpha \sqrt{\cos \alpha} + \sqrt{1 - \cos^3 \alpha}\right]
\]
From this, we find the desired coordinate of the point of impact:
\[
x(T) = R \left[\sin \alpha + \sin 2 \alpha + \sqrt{\cos \alpha \left(1 - \cos^3 \alpha\right)}\right]
\]
|
R \left[ \sin \alpha + \sin 2 \alpha + \sqrt{\cos \alpha (1 - \cos^3 \alpha)} \right]
|
olympiads
| 0.15625
|
A and B are sawing some pieces of wood simultaneously, with each piece having the same length and thickness. A needs to saw each piece into 3 segments, while B needs to saw each piece into 2 segments. After the same period of time, A has 24 segments of wood next to him, and B has 28 segments. Who, $\qquad$ (fill in "A" or "B"), takes less time to saw one piece of wood?
|
A
|
olympiads
| 0.46875
|
Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$ P^n(m)\cdot P^m(n) $$ is a square of an integer for all nonnegative integers $n, m$ .
*Remark:* For a nonnegative integer $k$ and an integer $n$ , $P^k(n)$ is defined as follows: $P^k(n) = n$ if $k = 0$ and $P^k(n)=P(P(^{k-1}(n))$ if $k >0$ .
Proposed by Adrian Beker.
|
P(x) = x + 1
|
aops_forum
| 0.0625
|
Let $p$ a prime number, $p\geq 5$ . Find the number of polynomials of the form
\[ x^p + px^k + p x^l + 1, \quad k > l, \quad k, l \in \left\{1,2,\dots,p-1\right\}, \] which are irreducible in $\mathbb{Z}[X]$ .
|
2 \binom{\frac{p-1}{2}}{2}
|
aops_forum
| 0.125
|
On the hypotenuse \( BC \) of a right triangle \( ABC \), points \( D \) and \( E \) are marked such that \( AD \perp BC \) and \( AD = DE \). On the side \( AC \), a point \( F \) is marked such that \( EF \perp BC \). Find the angle \( ABF \).
|
45^
|
olympiads
| 0.09375
|
Find all the integers \( n > 1 \) with the following property: the numbers \( 1, 2, \ldots, n \) can be arranged in a line so that, of any two adjacent numbers, one is divisible by the other.
|
2, 3, 4, 6
|
olympiads
| 0.125
|
Let \( x \in \mathbf{R} \). Then the maximum value of \( y = \frac{\sin x}{2 - \cos x} \) is
|
\frac{\sqrt{3}}{3}
|
olympiads
| 0.09375
|
Arrange the natural numbers that are divisible by both 5 and 7 in ascending order starting from 105. Take the first 2013 numbers. What is the remainder when the sum of these 2013 numbers is divided by 12?
|
3
|
olympiads
| 0.125
|
Find all a positive integers $a$ and $b$ , such that $$ \frac{a^b+b^a}{a^a-b^b} $$ is an integer
|
(a, b) = (2, 1) } and (a, b) = (1, 2)
|
aops_forum
| 0.09375
|
Given a $2021 \times 2021$ grid, Petya and Vasya play the following game: they take turns placing pieces in empty cells of the grid. The player who makes a move such that every $3 \times 5$ and $5 \times 3$ rectangle contains a piece wins. Petya goes first. Which player can ensure their victory regardless of the opponent's moves?
|
Petya
|
olympiads
| 0.484375
|
After a football match, the coach lined up the team as shown in the figure and commanded: "Run to the locker room if your number is less than any of your neighbors' numbers." After a few players ran off, he repeated his command. The coach continued this until only one player remained. What is Igor's number, given that after he ran off, there were 3 players left in the line? (After each command, one or more players ran off and the line closed up, leaving no empty spaces between the remaining players.)
|
5
|
olympiads
| 0.125
|
Compute
\[
\left\lfloor\frac{2007!+2004!}{2006!+2005!}\right\rfloor .
\]
(Note that \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\).)
|
2006
|
olympiads
| 0.390625
|
Among A, B, C, and D comparing their heights, the sum of the heights of two of them is equal to the sum of the heights of the other two. The average height of A and B is 4 cm more than the average height of A and C. D is 10 cm taller than A. The sum of the heights of B and C is 288 cm. What is the height of A in cm?
|
139
|
olympiads
| 0.25
|
Determine the area enclosed by the parabola $y = x^{2} - 5x + 6$ and the coordinate axes (and adjacent to both axes).
|
4 \frac{2}{3}
|
olympiads
| 0.265625
|
There is a special calculator. When a number is input, the calculator will multiply the number by 2, then reverse the digits of the result. Finally, it will add 2 and display the final result. If you input a two-digit number and the final displayed result is 27, what was the initial input?
|
26
|
olympiads
| 0.234375
|
Two gears are engaged with each other. Gear $A$ has 12 teeth, and gear $B$ has 54 teeth. How many rotations will each gear make before both return to their initial positions?
|
9 \text{ and } 2
|
olympiads
| 0.265625
|
A bank, paying a depositor $p \%$ per annum, in turn, invests the capital at $p_{1} \%$ (of course, $p_{1}>p$). What will be the bank's profit from the entrusted capital \(\boldsymbol{K}\) after \(n\) years, if the interest is compounded annually? Why is the solution: profit \(=K\left(\frac{p_{1}}{100}-\frac{p}{100}\right)^{n}\) incorrect?
|
K \left[ \left(1 + \frac{p_1}{100}\right)^n - \left(1 + \frac{p}{100}\right)^n \right]
|
olympiads
| 0.1875
|
A flock of geese is flying, and a lone goose flies towards them and says, "Hello, a hundred geese!" The leader of the flock responds, "No, we are not a hundred geese! If there were as many of us as there are now, plus the same amount, plus half of that amount, plus a quarter of that amount, plus you, goose, then we would be a hundred geese. But as it is..." How many geese were in the flock?
|
36
|
olympiads
| 0.171875
|
Let \(\alpha, \beta, \gamma\) be real numbers satisfying \(\alpha + \beta + \gamma = 2\) and \(\alpha \beta \gamma = 4\). Let \(v\) be the minimum value of \(|\alpha| + |\beta| + |\gamma|\). Find the value of \(v\).
|
6
|
olympiads
| 0.109375
|
In an abandoned chemistry lab Gerome found a two-pan balance scale and three 1-gram weights, three
5-gram weights, and three 50-gram weights. By placing one pile of chemicals and as many weights as
necessary on the pans of the scale, Gerome can measure out various amounts of the chemicals in the pile.
Find the number of different positive weights of chemicals that Gerome could measure.
|
63
|
aops_forum
| 0.09375
|
Given the following equations:
\[x^{2}-a x+4=0, \quad x^{2}+(a-2) x+4=0, \quad x^{2}+2 a x+a^{2}+1=0\]
find the range of real numbers \(a\) such that at least one of these equations has a real root.
|
a \geq 4 \text{ or } a \leq -2
|
olympiads
| 0.1875
|
A square cake with a perimeter of 100 cm is divided into four equal pieces by two straight cuts, and each piece has a perimeter of 56 cm. The cake, cuts, and equal segments are shown in the picture. What is the length of cut XY?
|
3
|
olympiads
| 0.15625
|
An industrial robot moves from point $A$ to point $B$ following a pre-determined algorithm. The diagram shows a part of its repeating trajectory. Determine how many times faster it would reach point $B$ from point $A$ if it moved in a straight line at three times the speed.
|
9
|
olympiads
| 0.0625
|
Given a regular tetrahedron \( P-ABC \) with edge length 1, point \( D \) is the midpoint of \( PC \). Point \( E \) is a moving point on segment \( AD \). Determine the range of values for the angle formed between line \( BE \) and plane \( ABC \).
|
\left[0, \arctan \left(\frac{\sqrt{14}}{7}\right)\right]
|
olympiads
| 0.0625
|
In parallelogram $ABCD$, diagonal $BD$ is equal to 2, angle $C$ is $45^\circ$, and line $CD$ is tangent to the circumcircle of triangle $ABD$. Find the area of the parallelogram $ABCD$.
|
4
|
olympiads
| 0.25
|
In a committee there are $n$ members. Each pair of members are either friends or enemies. Each committee member has exactly three enemies. It is also known that for each committee member, an enemy of his friend is automatically his own enemy. Find all possible value(s) of $n$
|
n = 4, 6
|
aops_forum
| 0.0625
|
Find the natural number that is equal to the sum of all preceding natural numbers. Does more than one such number exist, or is there only one?
|
3
|
olympiads
| 0.3125
|
Let \( a_0, a_1, \ldots \) be a sequence such that \( a_0 = 3 \), \( a_1 = 2 \), and \( a_{n+2} = a_{n+1} + a_n \) for all \( n \geq 0 \). Find
\[
\sum_{n=0}^{8} \frac{a_n}{a_{n+1} a_{n+2}}.
\]
|
\frac{105}{212}
|
olympiads
| 0.109375
|
The integers \( a \) and \( b \) have the property that the expression
\[ \frac{2n^3 + 3n^2 + an + b}{n^2 + 1} \]
is an integer for every integer \( n \). What is the value of the expression above when \( n = 4 \)?
|
11
|
olympiads
| 0.1875
|
Given vectors \(\vec{a} = \{1,2\}\) and \(\vec{b} = \{-3,2\}\), find the real number \(k\) such that \(k \vec{a} + \vec{b}\) is in the same or opposite direction as \(\vec{a} - 3 \vec{b}\).
|
-\frac{1}{3}
|
olympiads
| 0.125
|
If \(\left(a+\frac{1}{a}\right)^{2}=3\), find \(\left(a+\frac{1}{a}\right)^{3}\) in terms of \(a\).
|
\left(a + \frac{1}{a} \right)^3 = 3 \sqrt{3}
|
olympiads
| 0.140625
|
There are 48 students participating in three sports activities. Each student only participates in one activity, and the number of participants in each activity is different. Each number contains the digit "6". How many students are participating in each of the three sports activities?
|
6, 16, 26
|
olympiads
| 0.1875
|
A turtle and a rabbit are racing on a 1000-meter track. The rabbit's speed is 15 times that of the turtle. However, the rabbit took a break during the race. When the rabbit woke up, the turtle had just reached the finish line, while the rabbit still had 100 meters left to go. How far did the turtle crawl while the rabbit was resting?
|
940 \text{ meters}
|
olympiads
| 0.09375
|
The quadrilateral \(ABCD\) is circumscribed around a circle with a radius of \(1\). Find the greatest possible value of \(\left| \frac{1}{AC^2} + \frac{1}{BD^2} \right|\).
|
\frac{1}{4}
|
olympiads
| 0.0625
|
In 2016, New Year's Day (January 1st) was a Friday. Wang Hua practices calligraphy every Tuesday and Saturday. How many days did Wang Hua practice calligraphy in August 2016?
|
9
|
olympiads
| 0.34375
|
Given a natural number \( x = 9^n - 1 \), where \( n \) is a natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 7. Find \( x \).
|
728
|
olympiads
| 0.296875
|
Three distinct positive integers \( a, b, c \) form a geometric sequence, and their sum is 111. Determine the set \( \{a, b, c\} \).
|
\{1, 10, 100\} \text{ and } \{27, 36, 48\}
|
olympiads
| 0.15625
|
If \( z \) is a complex number with a non-zero real part, find the minimum value of \(\frac{\operatorname{Re}\left(z^{4}\right)}{(\operatorname{Re}(z))^{4}}\).
|
-8
|
olympiads
| 0.1875
|
For n a positive integer, find f(n), the number of pairs of positive integers (a, b) such that \(\frac{ab}{a + b} = n\).
|
The number of pairs (a, b) such that \frac{ab}{a + b} = n \text{ is the number of divisors of } n^2.
|
olympiads
| 0.078125
|
Determine the maximum number of bishops that we can place in a $8 \times 8$ chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop.
Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the $2$ main diagonals and the ones parallel to those ones.
|
16
|
aops_forum
| 0.125
|
It is known that the sequence of numbers \(a_{1}, a_{2}, \ldots\) is an arithmetic progression, and the sequence of products \(a_{1}a_{2}, a_{2}a_{3}, a_{3}a_{4}, \ldots\) is a geometric progression. It is given that \(a_{1} = 1\). Find \(a_{2017}\).
|
1
|
olympiads
| 0.234375
|
Vera, Nina, and Olya were playing with dolls. They dressed their dolls in one item each: either a coat, a jacket, or a dress. When their mother asked what they dressed their dolls in, the girls decided to joke and one of them said: "Vera dressed her doll in a dress, and Olya in a coat." The second girl responded: "Vera dressed her doll in a coat, and Nina in a coat." Then the girls said that in both the first and second responses, one part of the answer is true and the other part is false. Determine whose doll is in a dress and whose doll is in a coat.
|
\text{Vera's doll: Dress, Nina's doll: Coat, Olya's doll: Jacket}
|
olympiads
| 0.109375
|
Determine all real numbers \( x \) for which \( x^{3/5} - 4 = 32 - x^{2/5} \).
|
243
|
olympiads
| 0.21875
|
Formulate the equation of the normal to the given curve at the point with abscissa \( x_0 \).
\[ y = \frac{1 + \sqrt{x}}{1 - \sqrt{x}}, \quad x_{0} = 4 \]
|
y = -2x + 5
|
olympiads
| 0.203125
|
A gardener wants to plant three maple trees, four oak trees, and five birch trees in a row. He randomly determines the arrangement of these trees, and each possible arrangement is equally likely. Let the probability that no two birch trees are adjacent be represented by \(\frac{m}{n}\) in simplest form. Find \(m+n\).
|
106
|
olympiads
| 0.0625
|
Suppose that there exist nonzero complex numbers \(a, b, c,\) and \(d\) such that \(k\) is a root of both the equations \(a x^{3} + b x^{2} + c x + d = 0\) and \(b x^{3} + c x^{2} + d x + a = 0\). Find all possible values of \(k\) (including complex values).
|
1, -1, i, -i
|
olympiads
| 0.28125
|
In a jewelry store display, there are 15 diamonds. Next to them are labels indicating their weights, written as $1, 2, \ldots, 15$ carats. The shopkeeper has a balance scale and four weights with masses $1, 2, 4,$ and $8$ carats. A customer is allowed only one type of weighing: place one of the diamonds on one side of the scale and the weights on the other side to verify that the weight indicated on the corresponding label is correct. However, for each weight taken, the customer must pay the shopkeeper 100 coins. If a weight is removed from the scale and not used in the next weighing, the shopkeeper keeps it. What is the minimum amount that needs to be paid to verify the weights of all the diamonds?
[8 points]
(A. V. Gribalko)
|
800
|
olympiads
| 0.0625
|
Paulinho was studying the Greatest Common Divisor (GCD) at school and decided to practice at home. He called $a, b,$ and $c$ the ages of three people living with him. He then performed operations with their prime factors and obtained the greatest common divisors of the three pairs of numbers. A few days later, he forgot the ages $a, b,$ and $c$, but found the following results noted down:
$$
\begin{aligned}
a \cdot b \cdot c & =2^{4} \cdot 3^{2} \cdot 5^{3} \\
\text{GCD}(a, b) & =15 \\
\text{GCD}(a, c) & =5 \\
\text{GCD}(b, c) & =20
\end{aligned}
$$
Help Paulinho determine the values of $a, b,$ and $c$.
|
(15, 60, 20)
|
olympiads
| 0.09375
|
On a plane, 10 equal line segments are drawn, and all their intersection points are marked. It turns out that each intersection point divides any segment passing through it in the ratio $3: 4$. What is the maximum possible number of marked points?
|
10
|
olympiads
| 0.1875
|
A standard 52-card deck contains cards of 4 suits and 13 numbers, with exactly one card for each pairing of suit and number. If Maya draws two cards with replacement from this deck, what is the probability that the two cards have the same suit or have the same number, but not both?
|
\frac{15}{52}
|
olympiads
| 0.078125
|
Cities \( A \), \( B \), and \( C \) along with the straight roads connecting them form a triangle. It is known that the direct path from \( A \) to \( B \) is 200 km shorter than the detour through \( C \), and the direct path from \( A \) to \( C \) is 300 km shorter than the detour through \( B \). Find the distance between cities \( B \) and \( C \).
|
250 \text{ km}
|
olympiads
| 0.296875
|
Find all functions \( f: \mathbb{Z} \rightarrow \mathbb{Z} \) such that for all \( x, y \) in \( \mathbb{Z} \),
$$
f(x+y)=f(x)+f(y)
$$
|
f(x) = a \cdot x \text{ pour un certain entier } a.
|
olympiads
| 0.5
|
A group of cyclists rode together and decided to share a meal at a tavern. They agreed to split the total bill of four pounds equally among themselves. However, when it was time to pay, two of the cyclists had left without paying. As a result, each of the remaining cyclists had to pay an additional two shillings to cover the total bill. How many cyclists were there originally?
|
10
|
olympiads
| 0.21875
|
For four distinct integers, all pairwise sums and pairwise products were calculated. The obtained sums and products were written on the board. What is the smallest number of distinct numbers that could be on the board?
|
6
|
olympiads
| 0.15625
|
Given that \( 2^{a} \times 3^{b} \times 5^{c} \times 7^{d} = 252000 \), what is the probability that a three-digit number formed by any 3 of the natural numbers \( a, b, c, d \) is divisible by 3 and less than 250?
|
\frac{1}{4}
|
olympiads
| 0.109375
|
The historical Mexican figure Benito Juárez was born in the first half of the 19th century (the 19th century spans from the year 1801 to the year 1900). Knowing that Benito Juárez turned $x$ years old in the year $x^{2}$, what year was he born?
|
1806
|
olympiads
| 0.3125
|
The price of a product was increased by \( p \% \), then during a promotion, it was decreased by \( \frac{p}{2} \% \); thus, the product costs only \( \frac{p}{3} \% \) more than originally. Calculate the value of \( p \).
|
\frac{100}{3} \approx 33.33\%
|
olympiads
| 0.0625
|
A 30% hydrochloric acid solution was mixed with a 10% hydrochloric acid solution to obtain 600 grams of a 15% solution. How many grams of each solution were used?
|
150 \text{ grams of } 30 \% \text{ solution and } 450 \text{ grams of } 10 \% \text{ solution.
|
olympiads
| 0.546875
|
In each square of a $5 \times 5$ board one of the numbers $2, 3, 4$ or $5$ is written so that the the sum of all the numbers in each row, in each column and on each diagonal is always even. How many ways can we fill the board?
Clarification. A $5\times 5$ board has exactly $18$ diagonals of different sizes. In particular, the corners are size $ 1$ diagonals.
|
2^{39}
|
aops_forum
| 0.078125
|
The diagonals of a convex quadrilateral \(ABCD\) intersect at point \(E\). It is known that the area of each of the triangles \(ABE\) and \(DCE\) is equal to 1, and the area of the entire quadrilateral does not exceed 4. Given that \(AD = 3\), find the side \(BC\).
|
3
|
olympiads
| 0.109375
|
Evaluate the given solution and the obtained answer. Identify all errors and shortcomings.
Find all functions \( f(x) \) defined for all \( x \in \mathbb{R} \) such that \( f(x+y) = f(x) + f(y) \) for all \( x, y \in \mathbb{R} \).
|
f(x) = ax, a \in \mathbb{R}
|
olympiads
| 0.203125
|
In the sequence \(\left\{b_{n}\right\}\), if \(b_{n} \cdot b_{n+1} = 3^{n} \left(n \in \mathbf{N}_{+}\right)\), then it can be concluded that ____.
|
b_1, b_3, b_5, \ldots, b_{2n-1}, b_{2n+1}, \ldots \text{ form a geometric sequence with a common ratio of 3.}
|
olympiads
| 0.078125
|
Let $F_n$ be the nth Fibonacci number, defined by $F_1 = F_2 = 1$ and $F_n = F_{n-1} + F_{n-2}$ for $n > 2$ . Let $A_0, A_1, A_2,\cdots$ be a sequence of points on a circle of radius $1$ such that the minor arc from $A_{k-1}$ to $A_k$ runs clockwise and such that
\[\mu(A_{k-1}A_k)=\frac{4F_{2k+1}}{F_{2k+1}^2+1}\]
for $k \geq 1$ , where $\mu(XY )$ denotes the radian measure of the arc $XY$ in the clockwise direction. What is the limit of the radian measure of arc $A_0A_n$ as $n$ approaches infinity?
|
2\pi
|
aops_forum
| 0.0625
|
Each of the thirty sixth graders has one pen, one pencil, and one ruler. After participating in an olympiad, it turned out that 26 students lost their pen, 23 students lost their ruler, and 21 students lost their pencil. Find the smallest possible number of sixth graders who lost all three items.
|
10
|
olympiads
| 0.078125
|
Vasily invented a new operation on the set of positive numbers: \( a \star b = a^{\ln b} \). Find the logarithm of the number \(\frac{(a b) \star(a b)}{(a \star a)(b \star b)}\) to the base \( a \star b \).
|
2
|
olympiads
| 0.09375
|
Find the number of ordered pairs \((x, y)\) of positive integers satisfying \(xy + x = y + 92\).
|
3
|
olympiads
| 0.09375
|
In a circle, chords \( AB \) and \( AC \) are drawn such that \( AB = 2 \), \( AC = 1 \), and \( \angle CAB = 120^\circ \). Find the length of the chord of the circle that bisects the angle \( CAB \).
Answer: 3
|
3
|
olympiads
| 0.25
|
A rhombus $ABCD$ forms the base of a quadrilateral pyramid, where $\angle BAD=60^\circ$. It is known that $SA = SC$, $SD = SB = AB$. Point $E$ is taken on the edge $DC$ such that the area of triangle $BSE$ is the smallest among the areas of all sections of the pyramid containing segment $BS$ and intersecting segment $DC$. Find the ratio $DE:EC$.
|
\frac{2}{3}
|
olympiads
| 0.0625
|
Grisha has 5000 rubles. Chocolate bunnies are sold in a store at a price of 45 rubles each. To carry the bunnies home, Grisha will have to buy several bags at 30 rubles each. One bag can hold no more than 30 chocolate bunnies. Grisha bought the maximum possible number of bunnies and enough bags to carry all the bunnies. How much money does Grisha have left?
|
20 rubles
|
olympiads
| 0.140625
|
The ratio of two exterior angles of a triangle is \(\frac{12}{7}\), and their difference is \(50^\circ\). What are the angles of the triangle?
|
10^
\circ, 60^
\circ, 110^
\circ
|
olympiads
| 0.21875
|
On the side \( BC \) of triangle \( ABC \) a circle is constructed with a radius of 20 cm, using \( BC \) as the diameter. This circle intersects sides \( AB \) and \( AC \) at points \( X \) and \( Y \) respectively. Find \( BX \cdot AB + CY \cdot AC \).
|
1600
|
olympiads
| 0.109375
|
Determine the value of the following expression:
$$
X = \sqrt{x \sqrt{y \sqrt{z \sqrt{x \sqrt{y \sqrt{z \cdots}}}}}}
$$
|
\sqrt[7]{x^4 y^2 z}
|
olympiads
| 0.125
|
The exterior angles of a triangle are proportional to the numbers 5: 7: 8. Find the angle between the altitudes of this triangle drawn from the vertices of its smaller angles.
|
90^\circ
|
olympiads
| 0.0625
|
A fair coin is to be tossed $10$ times. Let $i/j$ , in lowest terms, be the probability that heads never occur on consecutive tosses. Find $i+j$ .
|
73
|
aops_forum
| 0.46875
|
Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2025 \) and \( y = |x - a| + |x - b| + |x - c| \) has exactly one solution. Find the minimum possible value of \( c \).
|
1013
|
olympiads
| 0.078125
|
A mother has two apples and three pears. Each day for five consecutive days, she gives out one fruit. In how many ways can this be done?
|
10
|
olympiads
| 0.34375
|
The route from point A to point B consists only of uphill and downhill sections, with a total distance of 21 kilometers. The uphill speed is 4 km/h, and the downhill speed is 6 km/h. It takes 4.25 hours to travel from point A to point B. How many hours will it take to travel from point B to point A?
|
4.5
|
olympiads
| 0.296875
|
Given that nonzero reals $a,b,c,d$ satisfy $a^b=c^d$ and $\frac{a}{2c}=\frac{b}{d}=2$ , compute $\frac{1}{c}$ .
|
16
|
aops_forum
| 0.21875
|
There is a strip \( 1 \times 99 \) divided into 99 cells of size \( 1 \times 1 \), which are colored alternately in black and white. It is allowed to repaint all the cells of any checkerboard rectangle \( 1 \times k \) at the same time. What is the minimum number of repaintings required to make the entire strip a single color?
|
49
|
olympiads
| 0.140625
|
A sequence \( t_{1}, t_{2}, t_{3}, \ldots \) is defined by
\[ t_{n} = \begin{cases}
\frac{1}{7^{n}} & \text{when } n \text{ is odd} \\
\frac{2}{7^{n}} & \text{when } n \text{ is even}
\end{cases} \]
for each positive integer \( n \). Determine the sum of all of the terms in this sequence, i.e., calculate \( t_{1} + t_{2} + t_{3} + \cdots \).
|
\frac{3}{16}
|
olympiads
| 0.53125
|
The agricultural proverb "Counting in Nines during Winter" refers to dividing the days from the Winter Solstice into segments of nine days each, called One Nine, Two Nine, ..., Nine Nine, with the first day of the Winter Solstice considered as the first day of One Nine. December 21, 2012 is the Winter Solstice. Determine which day of the Nines February 3, 2013 falls on.
|
5, 9
|
olympiads
| 0.0625
|
Two players have a 3 x 3 board. Nine cards, each with a different number, are placed face up in front of the players. Each player in turn takes a card and places it on the board until all the cards have been played. The first player wins if the sum of the numbers in the first and third rows is greater than the sum in the first and third columns, loses if it is less, and draws if the sums are equal. Which player wins and what is the winning strategy?
|
The first player always wins.
|
olympiads
| 0.375
|
A rectangular piece of cardboard was cut along its diagonal. On one of the obtained pieces, two cuts were made parallel to the two shorter sides, at the midpoints of those sides. In the end, a rectangle with a perimeter of $129 \mathrm{~cm}$ remained. The given drawing indicates the sequence of cuts.
What was the perimeter of the original sheet before the cut?
|
258 \text{ cm}
|
olympiads
| 0.1875
|
At the tourist base, the number of two-room cottages is twice the number of one-room cottages. The number of three-room cottages is a multiple of the number of one-room cottages. If the number of three-room cottages is tripled, it will be 25 more than the number of two-room cottages. How many cottages are there in total at the tourist base, given that there are at least 70 cottages?
|
100
|
olympiads
| 0.25
|
Let squares of one kind have a side of \(a\) units, another kind have a side of \(b\) units, and the original square have a side of \(c\) units. Then the area of the original square is given by \(c^{2}=n a^{2}+n b^{2}\).
Numbers satisfying this equation can be obtained by multiplying the equality \(5^{2}=4^{2}+3^{2}\) by \(n=k^{2}\). For \(n=9\), we get \(a=4, b=3, c=15\).
|
15
|
olympiads
| 0.3125
|
The side \( AB \) of triangle \( ABC \) is longer than side \( AC \), and \(\angle A = 40^\circ\). Point \( D \) lies on side \( AB \) such that \( BD = AC \). Points \( M \) and \( N \) are the midpoints of segments \( BC \) and \( AD \) respectively. Find the angle \( \angle BNM \).
|
20^ ext{\circ}
|
olympiads
| 0.078125
|
In $n$ glasses of sufficiently large capacity, an equal amount of water is poured. It is allowed to pour as much water from any glass into any other as the latter contains. For which $n$ is it possible to eventually collect all the water into a single glass in a finite number of steps?
|
For n = 2^k, where k is an integer
|
olympiads
| 0.078125
|
BdMO National 2016 Higher Secondary
<u>**Problem 4:**</u>
Consider the set of integers $ \left \{ 1, 2, ......... , 100 \right \} $ . Let $ \left \{ x_1, x_2, ......... , x_{100} \right \}$ be some arbitrary arrangement of the integers $ \left \{ 1, 2, ......... , 100 \right \}$ , where all of the $x_i$ are different. Find the smallest possible value of the sum, $S = \left | x_2 - x_1 \right | + \left | x_3 - x_2 \right | + ................+ \left |x_{100} - x_{99} \right | + \left |x_1 - x_{100} \right | $ .
|
198
|
aops_forum
| 0.140625
|
Given $\boldsymbol{a} = (\cos \alpha, \sin \alpha)$ and $\boldsymbol{b} = (\cos \beta, \sin \beta)$, the relationship between $\boldsymbol{a}$ and $\boldsymbol{b}$ is given by $|k \boldsymbol{a} + \boldsymbol{b}| - \sqrt{3}|\boldsymbol{a} - k \boldsymbol{b}|$, where $k > 0$. Find the minimum value of $\boldsymbol{a} \cdot \boldsymbol{b}$.
|
\frac{1}{2}
|
olympiads
| 0.078125
|
On every kilometer marker along the highway between the villages of Yolkino and Palkino, there is a post with a sign. On one side of the sign, it states the number of kilometers to Yolkino, and on the other side, the number of kilometers to Palkino. An observant individual noticed that the sum of these two numbers on every post equals 13. What is the distance from Yolkino to Palkino?
|
13 \text{ km}
|
olympiads
| 0.46875
|
Find the number of zeros of the function
$$
F(z)=z^{8}-4z^{5}+z^{2}-1
$$
inside the unit circle \( |z|<1 \).
|
5
|
olympiads
| 0.0625
|
In a movie theater, five friends took seats from 1 to 5 (the leftmost seat is number 1). During the movie, Anya went to get popcorn. When she returned, she found that Varya had moved one seat to the right, Galya had moved two seats to the left, and Diana and Ella had swapped seats, leaving the end seat for Anya. In which seat was Anya sitting before she got up?
|
4
|
olympiads
| 0.171875
|
If you toss a fair coin \( n+1 \) times and I toss it \( n \) times, what is the probability that you get more heads?
|
\frac{1}{2}
|
olympiads
| 0.21875
|
Find the largest five-digit positive integer such that it is not a multiple of 11, and any number obtained by deleting some of its digits is also not divisible by 11.
|
98765
|
olympiads
| 0.109375
|
A rod is broken into three parts; two break points are chosen at random. What is the probability that a triangle can be formed from the three resulting parts?
|
\frac{1}{4}
|
olympiads
| 0.21875
|
Two individuals \( A \) and \( B \) need to travel from point \( M \) to point \( N \), which is 15 km from \( M \). On foot, they can travel at a speed of 6 km/h. Additionally, they have a bicycle that can travel at a speed of 15 km/h. \( A \) starts walking, while \( B \) rides the bicycle until meeting a pedestrian \( C \) walking from \( N \) to \( M \). Then \( B \) continues on foot, and \( C \) rides the bicycle until meeting \( A \), at which point \( C \) hands over the bicycle to \( A \), who then bicycles the rest of the way to \( N \).
When should pedestrian \( C \) leave from \( N \) so that \( A \) and \( B \) arrive at point \( N \) simultaneously, given that \( C \) walks at the same speed as \( A \) and \( B \)?
|
\frac{3}{11} \text{ hours}
|
olympiads
| 0.0625
|
The diagonal of a regular 2006-gon \(P\) is called good if its ends divide the boundary of \(P\) into two parts, each containing an odd number of sides. The sides of \(P\) are also called good. Let \(P\) be divided into triangles by 2003 diagonals, none of which have common points inside \(P\). What is the maximum number of isosceles triangles, each of which has two good sides, that such a division can have?
|
1003
|
olympiads
| 0.1875
|
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