problem
stringlengths 33
2.6k
| answer
stringlengths 1
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| source
stringclasses 2
values | llama8b_solve_rate
float64 0.06
0.59
|
|---|---|---|---|
On the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ find the point $T=(x_0,y_0)$ such that the triangle bounded by the axes of the ellipse and the tangent at that point has the least area.
|
\left( \pm \frac{a}{\sqrt{2}}, \pm \frac{b}{\sqrt{2}} \right)
|
aops_forum
| 0.125
|
\(a, b, c,\) and \(d\) are four non-zero numbers such that the quotients
$$
\frac{a}{5}, \quad \frac{-b}{7a}, \quad \frac{11}{abc}, \quad \text{and} \quad \frac{-18}{abcd}
$$
are all positive. Determine the signs of \(a, b, c,\) and \(d\).
|
a > 0, b < 0, c < 0, d < 0
|
olympiads
| 0.0625
|
Let \( f(x) = \frac{1}{2} \cos^2 x + \frac{\sqrt{3}}{2} \sin x \cos x + 2 \) for \( x \in \left[-\frac{\pi}{6}, \frac{\pi}{4}\right] \). Find the range of values of \( f(x) \).
|
\left[2, 2 \frac{3}{4}\right]
|
olympiads
| 0.15625
|
Given the set \( A = \{x \mid x^{2} - 3x - 10 \leq 0\} \) and \( B = \{x \mid m + 1 \leq x \leq 2m - 1\} \), if \( A \cup B = A \), determine the range of the real number \( m \).
|
[-3, 3]
|
olympiads
| 0.140625
|
The lengths of the sides of a triangle are the integers 13, \( x \), and \( y \). It is given that \( x \cdot y = 105 \). What is the length of the perimeter of the triangle?
|
35
|
olympiads
| 0.578125
|
The quotient of the sum and difference of two integers is 3, while the product of their sum and difference is 300. What are the integers?
|
(20, 10), (-20, -10)
|
olympiads
| 0.34375
|
In a box, there are some balls of the same size, including 8 white balls, 9 black balls, and 7 yellow balls. Without looking, if one takes one ball at a time, how many times must one take to ensure that balls of all three colors are obtained?
|
18
|
olympiads
| 0.09375
|
Let the set \( M = \{1, 2, \cdots, 19\} \) and \( A = \{a_1, a_2, \cdots, a_k\} \subseteq M \). Determine the smallest \( k \), such that for any \( b \in M \), there exist \( a_i, a_j \in A \), satisfying \( a_i = b \) or \( a_i \pm a_j = b\) (where \(a_i\) and \(a_j\) can be the same).
|
5
|
olympiads
| 0.0625
|
Actions with letters. There is much in common between those puzzles where arithmetic operations need to be restored from several given numbers and a large number of asterisks, and those where each digit is replaced by a specific letter, with different letters corresponding to different digits. Both types of puzzles are solved similarly. Here's a small example of the second type (it's hardly difficult):
\[
\]
Can you restore the division? Each digit is replaced by its own letter.
|
Answer correctly formed based on standard notations, critical of solution verification
|
olympiads
| 0.078125
|
Find the maximum value of the expression
$$
\left(x_{1}-x_{2}\right)^{2}+\left(x_{2}-x_{3}\right)^{2}+\ldots+\left(x_{2010}-x_{2011}\right)^{2}+\left(x_{2011}-x_{1}\right)^{2}
$$
where \(x_{1}, \ldots, x_{2011} \in [0, 1]\).
|
2010
|
olympiads
| 0.109375
|
Oleg and Sergey take turns writing one digit at a time from left to right until a nine-digit number is formed. Digits that have already been written cannot be used again. Oleg starts (and finishes) first. Oleg wins if the resulting number is divisible by 4; otherwise, Sergey wins. Who will win with optimal play?
|
Sergey
|
olympiads
| 0.234375
|
In a school chess tournament, boys and girls competed, with the number of boys being five times the number of girls. According to the tournament rules, each player played against every other player twice. How many players participated in total if it is known that the boys scored exactly twice as many points as the girls? (A win in a chess game earns 1 point, a draw earns 0.5 points, and a loss earns 0 points.)
|
6
|
olympiads
| 0.078125
|
A drawer contains 2019 distinct pairs of socks. What is the minimum number of socks that must be taken out to ensure that at least one pair of identical socks has been taken out? It is noted that the drawer is so messy that it is not possible to search inside.
|
2020
|
olympiads
| 0.21875
|
Source: 2017 Canadian Open Math Challenge, Problem B2
-----
There are twenty people in a room, with $a$ men and $b$ women. Each pair of men shakes hands, and each pair of women shakes hands, but there are no handshakes between a man and a woman. The total number of handshakes is $106$ . Determine the value of $a \cdot b$ .
|
84
|
aops_forum
| 0.28125
|
Calculate the indefinite integral:
$$
\int(2x - 5) \cos 4x \, dx
$$
|
\frac{1}{4}(2x-5) \sin 4x + \frac{1}{8} \cos 4x + C
|
olympiads
| 0.171875
|
In rectangle \(ABCD\) with area 1, point \(M\) is selected on \(\overline{AB}\) and points \(X, Y\) are selected on \(\overline{CD}\) such that \(AX < AY\). Suppose that \(AM = BM\). Given that the area of triangle \(MXY\) is \(\frac{1}{2014}\), compute the area of trapezoid \(AXYB\).
|
\frac{2013}{2014}
|
olympiads
| 0.125
|
Let $S$ be the set of all ordered triples $(p,q,r)$ of prime numbers for which at least one rational number $x$ satisfies $px^2+qx+r=0.$ Which primes appear in seven or more elements of $S?$
|
2
|
aops_forum
| 0.28125
|
Yura has a calculator that allows multiplying a number by 3, adding 3 to a number, or (if the number is divisible by 3) dividing a number by 3. How can one obtain the number 11 from the number 1 using this calculator?
|
11
|
olympiads
| 0.40625
|
During breaks, schoolchildren played table tennis. Any two schoolchildren played no more than one game against each other. At the end of the week, it turned out that Petya played half, Kolya - a third, and Vasya - one fifth of the total number of games played during the week. What could be the total number of games played during the week if it is known that at least two games did not involve Vasya, Petya, or Kolya?
|
30
|
olympiads
| 0.15625
|
Let \( ABC \) be an isosceles triangle with \( A \) as the vertex angle. Let \( M \) be the midpoint of the segment \( [BC] \). Let \( D \) be the reflection of point \( M \) over the segment \( [AC] \). Let \( x \) be the angle \( \widehat{BAC} \). Determine, as a function of \( x \), the value of the angle \( \widehat{MDC} \).
|
\frac{x}{2}
|
olympiads
| 0.15625
|
Since counting the numbers from 1 to 100 wasn't enough to stymie Gauss, his teacher devised another clever problem that he was sure would stump Gauss. Defining $\zeta_{15} = e^{2\pi i/15}$ where $i = \sqrt{-1}$ , the teacher wrote the 15 complex numbers $\zeta_{15}^k$ for integer $0 \le k < 15$ on the board. Then, he told Gauss:
On every turn, erase two random numbers $a, b$ , chosen uniformly randomly, from the board and then write the term $2ab - a - b + 1$ on the board instead. Repeat this until you have one number left. What is the expected value of the last number remaining on the board?
|
0
|
aops_forum
| 0.109375
|
Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds : $$ f(ac)+f(bc)-f(c)f(ab) \ge 1 $$ Proposed by *Mojtaba Zare*
|
f \equiv 1
|
aops_forum
| 0.59375
|
Quadratic polynomials \( P(x) \) and \( Q(x) \) with real coefficients are such that together they have 4 distinct real roots, and each of the polynomials \( P(Q(x)) \) and \( Q(P(x)) \) has 4 distinct real roots. What is the minimum number of distinct real numbers that can be among the roots of the polynomials \( P(x) \), \( Q(x) \), \( P(Q(x)) \), and \( Q(P(x)) \)?
|
6
|
olympiads
| 0.0625
|
An integer \( x \) satisfies the inequality \( x^2 \leq 729 \leq -x^3 \). \( P \) and \( Q \) are possible values of \( x \). What is the maximum possible value of \( 10(P - Q) \)?
|
180
|
olympiads
| 0.359375
|
Each side of an isosceles triangle is a whole number of centimetres. Its perimeter has length 20 cm. How many possibilities are there for the lengths of its sides?
|
4
|
olympiads
| 0.109375
|
Each of the natural numbers $1, 2, 3, \ldots, 377$ is painted either red or blue (both colors are present). It is known that the number of red numbers is equal to the smallest red number, and the number of blue numbers is equal to the largest blue number. What is the smallest red number?
|
189
|
olympiads
| 0.171875
|
Find the largest possible value of $a$ such that there exist real numbers $b,c>1$ such that
\[a^{\log_b c}\cdot b^{\log_c a}=2023.\]
|
\sqrt{2023}
|
aops_forum
| 0.1875
|
Inside circle \(\omega\), there are two circles \(\omega_1\) and \(\omega_2\) that intersect at points \(K\) and \(L\), and touch circle \(\omega\) at points \(M\) and \(N\). It turns out that points \(K\), \(M\), and \(N\) are collinear. Find the radius of circle \(\omega\), given that the radii of circles \(\omega_1\) and \(\omega_2\) are 3 and 5, respectively.
|
8
|
olympiads
| 0.09375
|
Define the sequence of positive integers \(a_{n}\) recursively by \(a_{1} = 7\) and \(a_{n} = 7^{a_{n-1}}\) for all \(n \geq 2\). Determine the last two digits of \(a_{2007}\).
|
43
|
olympiads
| 0.453125
|
A family of four octopuses went to a shoe store (each octopus has 8 legs). Father-octopus already had half of his legs shod, mother-octopus had only 3 legs shod, and their two sons each had 6 legs shod. How many shoes did they buy if they left the store with all their legs shod?
|
13
|
olympiads
| 0.34375
|
Two infinite geometric progressions are given with a common ratio \( |q| < 1 \), differing only in the sign of their common ratios. Their sums are \( S_{1} \) and \( S_{2} \). Find the sum of the infinite geometric progression formed from the squares of the terms of either of the given progressions.
|
S_1 S_2
|
olympiads
| 0.125
|
Xiao Qian, Xiao Lu, and Xiao Dai are guessing a natural number between 1 and 99. Here are their statements:
Xiao Qian says: "It is a perfect square, and it is less than 5."
Xiao Lu says: "It is less than 7, and it is a two-digit number."
Xiao Dai says: "The first part of Xiao Qian's statement is correct, but the second part is wrong."
If one of them tells the truth in both statements, one of them tells a lie in both statements, and one of them tells one truth and one lie, then what is the number? (Note: A perfect square is a number that can be expressed as the square of an integer, for example, 4=2 squared, 81=9 squared, so 4 and 9 are considered perfect squares.)
|
9
|
olympiads
| 0.21875
|
If the function \( y = a^{2x} + 2a^x - 9 \) (where \( a > 0 \) and \( a \neq 1 \)) has a maximum value of 6 on the interval \([-1, 1]\), then \( a = \) ?
|
3 \text{ or } \frac{1}{3}
|
olympiads
| 0.578125
|
Solve the system of equations:
\[
\begin{cases}
x + y - 20 = 0 \\
\log_{4} x + \log_{4} y = 1 + \log_{4} 9
\end{cases}
\]
|
\left(\begin{array}{c}18, \; 2 \\ 2, \; 18 \end{array}\right)
|
olympiads
| 0.40625
|
As shown in the figure, $ABCD$ is a square with a side length of 1. Points $U$ and $V$ lie on $AB$ and $CD$ respectively. $AV$ intersects $DU$ at $P$, and $BV$ intersects $CU$ at $Q$. Find the maximum area of $S_{PUQV}$.
|
\frac{1}{4}
|
olympiads
| 0.125
|
Alice and Bob are playing the following game: Alice chooses an integer $n>2$ and an integer $c$ such that $0<c<n$. She writes $n$ integers on the board. Bob chooses a permutation $a_{1}, \ldots, a_{n}$ of the integers. Bob wins if $\left(a_{1}-a_{2}\right)\left(a_{2}-a_{3}\right) \ldots\left(a_{n}-a_{1}\right)$ equals 0 or $c$ modulo $n$. Who has a winning strategy?
|
Bob has a winning strategy
|
olympiads
| 0.09375
|
Find all pairs \((x, y)\) of positive numbers that achieve the minimum value of the function
$$
f(x, y)=\frac{x^{4}}{y^{4}}+\frac{y^{4}}{x^{4}}-\frac{x^{2}}{y^{2}}-\frac{y^{2}}{x^{2}}+\frac{x}{y}-\frac{y}{x}
$$
and determine this minimum value.
|
2
|
olympiads
| 0.15625
|
Consider all possible rectangular parallelepipeds, each with a volume of 4, where the bases are squares. Find the parallelepiped with the smallest perimeter of the lateral face and calculate this perimeter.
|
6
|
olympiads
| 0.0625
|
The set of positive real numbers $x$ that satisfy $2 | x^2 - 9 | \le 9 | x | $ is an interval $[m, M]$ . Find $10m + M$ .
|
21
|
aops_forum
| 0.0625
|
Six students arranged to play table tennis at the gym on Saturday morning from 8:00 to 11:30. They rented two tables for singles matches. At any time, 4 students play while the other 2 act as referees. They continue rotating in this manner until the end, and it is found that each student played for the same amount of time. How many minutes did each student play?
|
140 \; \text{minutes}
|
olympiads
| 0.078125
|
Using a compass and straightedge, construct a triangle given one angle and the radii of the inscribed and circumscribed circles.
|
Final Answer is the constructed triangle ABC
|
olympiads
| 0.21875
|
On Day $1$ , Alice starts with the number $a_1=5$ . For all positive integers $n>1$ , on Day $n$ , Alice randomly selects a positive integer $a_n$ between $a_{n-1}$ and $2a_{n-1}$ , inclusive. Given that the probability that all of $a_2,a_3,\ldots,a_7$ are odd can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, compute $m+n$ .
|
65
|
aops_forum
| 0.109375
|
The angles \( A O B, B O C \), and \( C O D \) are equal to each other, and the angle \( A O D \) is three times smaller than each of them. All rays \( \mathrm{OA}, \mathrm{OB}, \mathrm{OC}, \mathrm{OD} \) are distinct. Find the measure of the angle \( A O D \) (list all possible values).
|
36^
\circ, 45^
\circ
|
olympiads
| 0.0625
|
For integers \( a \) and \( b \), the equation \( \sqrt{a-1} + \sqrt{b-1} = \sqrt{a b + k} \) (with \( k \in \mathbb{Z} \)) has only one ordered pair of real solutions. Find \( k \).
|
0
|
olympiads
| 0.171875
|
Find the coordinates of point \( A \) that is equidistant from points \( B \) and \( C \).
\( A(x ; 0 ; 0) \)
\( B(8 ; 1 ;-7) \)
\( C(10 ;-2 ; 1) \)
|
A\left(-2.25, 0, 0\right)
|
olympiads
| 0.5625
|
Find all the values that can take the last digit of a "perfect" even number. (The natural number $n$ is called "perfect" if the sum of all its natural divisors is equal twice the number itself.For example: the number $6$ is perfect ,because $1+2+3+6=2\cdot6$ ).
|
6 \text{ and } 8
|
aops_forum
| 0.140625
|
Let the product of the digits of a positive integer \( n \) be denoted by \( a(n) \). Find the positive integer solution to the equation \( n^2 - 17n + 56 = a(n) \).
|
4
|
olympiads
| 0.125
|
Pankrác paid 60% of the price of the boat, Servác paid 40% of the remaining price, and Bonifác covered the missing amount, which was 30 zlatek.
How much did the boat cost?
|
125 zlateks
|
olympiads
| 0.421875
|
Calculate the limit of the function:
$$
\lim _{x \rightarrow-2} \frac{x^{3}+5 x^{2}+8 x+4}{x^{3}+3 x^{2}-4}
$$
|
\frac{1}{3}
|
olympiads
| 0.140625
|
Transform the equation, writing the right side in the form of a fraction:
\[
\begin{gathered}
\left(1+1:(1+1:(1+1:(2x-3)))=\frac{1}{x-1}\right. \\
1:\left(1+1:(1+1:(2 x-3))=\frac{2-x}{x-1}\right. \\
\left(1+1:(1+1:(2 x-3))=\frac{x-1}{2-x}\right. \\
1:\left(1+1:(2 x-3)=\frac{2 x-3}{2-x}\right. \\
\left(1+1:(2 x-3)=\frac{2-x}{2 x-3}\right. \\
1:(2 x-3)=\frac{5-3 x}{2 x-3}\right. \\
2 x-3=\frac{2 x-3}{5-3 x}
\end{gathered}
\]
Considering the restriction, \( x \neq \frac{3}{2}, 5-3x=1, x=\frac{4}{3} \).
|
x = \frac{4}{3}
|
olympiads
| 0.265625
|
An urn contains 6 white and 5 black balls. Three balls are drawn sequentially at random without replacement. Find the probability that the third ball drawn is white.
|
\frac{6}{11}
|
olympiads
| 0.203125
|
Given points \(A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)\) and a non-negative number \(\lambda\). Find the coordinates of point \(M\) on ray \(A B\) such that \(A M: A B = \lambda\).
|
\left((1-\lambda) x_1 + \lambda x_2, (1-\lambda) y_1 + \lambda y_2\right)
|
olympiads
| 0.0625
|
For Eeyore's birthday, Winnie-the-Pooh, Owl, and Piglet decided to give him balloons. Winnie-the-Pooh prepared three times as many balloons as Piglet, and Owl prepared four times as many balloons as Piglet. When Piglet was carrying his balloons, he hurried, tripped, and some of the balloons burst. Eeyore received a total of 60 balloons. How many balloons did Piglet end up giving?
|
4
|
olympiads
| 0.109375
|
The number \( a^{100} \) leaves a remainder of 2 when divided by 73, and the number \( a^{101} \) leaves a remainder of 69 when divided by the same number. Find the remainder when the number \( a \) is divided by 73.
|
71
|
olympiads
| 0.109375
|
Three cones with a common vertex \( A \) are externally tangent to each other. The first two cones are identical, and the vertex angle of the third cone is \( \frac{\pi}{4} \). Each of these three cones is internally tangent to a fourth cone with vertex at point \( A \) and a vertex angle of \( \frac{3 \pi}{4} \). Find the vertex angle of the first two cones. (The vertex angle of a cone is the angle between its generatrices in its axial section.)
|
2 \arctan \frac{2}{3}
|
olympiads
| 0.265625
|
In the sequence $\left\{a_{n}\right\}$, $a_{1}=13, a_{2}=56$, for all positive integers $n$, $a_{n+1}=a_{n}+a_{n+2}$. Find $a_{1934}$.
|
56
|
olympiads
| 0.171875
|
When "the day after tomorrow" becomes "yesterday", "today" will be as far from Sunday as "today" was from Sunday when "yesterday" was "tomorrow". What day of the week is today?
|
Wednesday
|
olympiads
| 0.109375
|
For her daughter’s $12\text{th}$ birthday, Ingrid decides to bake a dodecagon pie in celebration. Unfortunately, the store does not sell dodecagon shaped pie pans, so Ingrid bakes a circular pie first and then trims off the sides in a way such that she gets the largest regular dodecagon possible. If the original pie was $8$ inches in diameter, the area of pie that she has to trim off can be represented in square inches as $a\pi - b$ where $a, b$ are integers. What is $a + b$ ?
|
64
|
aops_forum
| 0.171875
|
Find the smallest positive integer $n$ for which the number \[ A_n = \prod_{k=1}^n \binom{k^2}{k} = \binom{1}{1} \binom{4}{2} \cdots \binom{n^2}{n} \] ends in the digit $0$ when written in base ten.
|
4
|
aops_forum
| 0.078125
|
It is known that in a certain triangle, the median, the bisector, and the height drawn from vertex $C$ divide the angle into four equal parts. Find the angles of this triangle.
|
\angle A = 22.5^\circ, \angle B = 67.5^\circ, \angle C = 90^\circ
|
olympiads
| 0.0625
|
Let \( C_1 \) and \( C_2 \) be two concentric circles, with \( C_2 \) inside \( C_1 \). Let \( A \) be a point on \( C_1 \) and \( B \) a point on \( C_2 \) such that the line segment \( AB \) is tangent to \( C_2 \). Let \( C \) be the second point of intersection of the line segment \( AB \) with \( C_1 \), and let \( D \) be the midpoint of \( AB \). A line passing through \( A \) intersects \( C_2 \) at points \( E \) and \( F \) such that the perpendicular bisectors of \( DE \) and \( CF \) intersect at a point \( M \) on the line segment \( AB \). Determine the ratio \( AM / MC \).
|
1
|
olympiads
| 0.1875
|
Let $k$ and $n$ be given integers with $n > k \geq 2$. For any set $P$ consisting of $n$ elements, form all $k$-element subsets of $P$ and compute the sums of the elements in each subset. Denote the set of these sums as $Q$. Let $C_{Q}$ be the number of elements in set $Q$. Find the maximum value of $C_{Q}$.
|
\binom{n}{k}
|
olympiads
| 0.0625
|
Fill in numbers in the boxes below so that the sum of the entries in each three consecutive boxes is $2005$ . What is the number that goes into the leftmost box?
[asy]
size(300);
label("999",(2.5,.5));
label("888",(7.5,.5));
draw((0,0)--(9,0));
draw((0,1)--(9,1));
for (int i=0; i<=9; ++i)
{
draw((i,0)--(i,1));
}
[/asy]
|
118
|
aops_forum
| 0.0625
|
Place the numbers 1, 2, 3, 4, 5, 6, 7, and 8 on the eight vertices of a cube such that the sum of any three numbers on a face is at least 10. Find the minimum sum of the four numbers on any face.
|
16
|
olympiads
| 0.078125
|
A courtyard has the shape of a parallelogram ABCD. At the corners of the courtyard there stand poles AA', BB', CC', and DD', each of which is perpendicular to the ground. The heights of these poles are AA' = 68 centimeters, BB' = 75 centimeters, CC' = 112 centimeters, and DD' = 133 centimeters. Find the distance in centimeters between the midpoints of A'C' and B'D'.
|
14
|
aops_forum
| 0.09375
|
It is said that in the 19th century, every tenth man in Russia was named Ivan, and every twentieth was named Peter. If this is true, then who was more numerous in Russia: Ivan Petrovich or Peter Ivanovich?
|
The number of Иван Петрович is equal to the number of Петр Иванович.
|
olympiads
| 0.078125
|
Solve in natural numbers the equation \(x^y = y^x\) for \(x \neq y\).
|
\{2, 4\}
|
olympiads
| 0.078125
|
On a shelf, there are 12 books. In how many ways can we choose five of these books such that no two chosen books are next to each other?
|
56
|
olympiads
| 0.0625
|
April has 30 days. If there are 5 Saturdays and Sundays, what day of the week is April 1st?
|
Saturday
|
olympiads
| 0.125
|
If a certain number is multiplied by 5, one-third of the product is subtracted, the remainder is divided by 10, and then successively $1 / 3, 1 / 2$, and $1 / 4$ of the original number are added to this, the result is 68. What is the number?
|
48
|
olympiads
| 0.390625
|
\[\sqrt{x-y} = \frac{2}{5}, \quad \sqrt{x+y} = 2\]
The solution \((x, y) = \left( \frac{52}{25}, \frac{48}{25} \right)\).
The points corresponding to the solutions are the vertices of a rectangle. The area of the rectangle is \(\frac{8}{25}\).
|
(x, y) = \left(0,0\right), \left(2,2\right), \left(2 / 25, -2 / 25\right), \left(52 / 25, 48 / 25\right), \text{ the area is } 8 / 25
|
olympiads
| 0.171875
|
In trapezoid \(ABCD\) with the lengths of the bases \(AD = 12 \text{ cm}\) and \(BC = 8 \text{ cm}\), a point \(M\) is taken on the ray \(BC\) such that \(AM\) divides the trapezoid into two equal areas. Find \(CM\).
|
2.4 ext{ cm}
|
olympiads
| 0.0625
|
What are the prime numbers \( p \) such that \( p \) divides \( 29^{p} + 1 \)?
|
2, 3, 5
|
olympiads
| 0.125
|
There is a number which, when divided by 3, leaves a remainder of 2, and when divided by 4, leaves a remainder of 1. What is the remainder when this number is divided by 12?
|
5
|
olympiads
| 0.46875
|
The sum of three numbers forming an arithmetic progression is equal to 2, and the sum of the squares of the same numbers is equal to 14/9. Find these numbers.
|
\left\{ \frac{1}{3}, \frac{2}{3}, 1 \right\} \text{ or } \left\{ 1, \frac{2}{3}, \frac{1}{3} \right\}
|
olympiads
| 0.140625
|
Determine the last digit of the product of all even natural numbers that are less than 100 and are not multiples of ten.
|
6
|
olympiads
| 0.21875
|
If $\left[\frac{1}{1+\frac{24}{4}}-\frac{5}{9}\right] \times \frac{3}{2 \frac{5}{7}} \div \frac{2}{3 \frac{3}{4}}+2.25=4$, then what is the value of $A$?
|
4
|
olympiads
| 0.0625
|
In square $ABCD$ , $\overline{AC}$ and $\overline{BD}$ meet at point $E$ .
Point $F$ is on $\overline{CD}$ and $\angle CAF = \angle FAD$ .
If $\overline{AF}$ meets $\overline{ED}$ at point $G$ , and if $\overline{EG} = 24$ cm, then find the length of $\overline{CF}$ .
|
48
|
aops_forum
| 0.109375
|
The little monkeys in Huaguo Mountain are dividing 100 peaches, with each monkey receiving the same number of peaches, and there are 10 peaches left. If the monkeys are dividing 1000 peaches, with each monkey receiving the same number of peaches, how many peaches will be left?
|
10
|
olympiads
| 0.140625
|
Let $a$ , $b$ , $c$ , $d$ , $e$ , $f$ be integers selected from the set $\{1,2,\dots,100\}$ , uniformly and at random with replacement. Set \[ M = a + 2b + 4c + 8d + 16e + 32f. \] What is the expected value of the remainder when $M$ is divided by $64$ ?
|
\frac{63}{2}
|
aops_forum
| 0.078125
|
Solve the equation \( 3^{n} + 55 = m^{2} \) in natural numbers.
|
(2, 8), (6, 28)
|
olympiads
| 0.0625
|
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that $f(2m+2n)=f(m)f(n)$ for all natural numbers $m,n$ .
|
f(x) = 1 \text{ for all } x \in \mathbb{N}
|
aops_forum
| 0.40625
|
10 guests arrived and each left a pair of galoshes in the hallway. Each pair of galoshes is of a different size. The guests began to leave one by one, putting on any pair of galoshes that they could fit into (i.e., each guest could put on a pair of galoshes that is not smaller than their own size). At some point, it was found that none of the remaining guests could find a pair of galoshes to leave. What is the maximum number of guests that could be left?
|
5
|
olympiads
| 0.1875
|
Find all functions $ f: N \rightarrow N $ such that: $\bullet$ $ f (m) = 1 \iff m = 1 $ ; $\bullet$ If $ d = \gcd (m, n) $ , then $ f (mn) = \frac {f (m) f (n)} {f (d)} $ ; and $\bullet$ $ \forall m \in N $ , we have $ f ^ {2012} (m) = m $ .
Clarification: $f^n (a) = f (f^{n-1} (a))$
|
f(m) = m
|
aops_forum
| 0.25
|
In the isosceles right triangle \(ABC\) with \(\angle A = 90^\circ\) and \(AB = AC = 1\), a rectangle \(EHGF\) is inscribed such that \(G\) and \(H\) lie on the side \(BC\). Find the maximum area of the rectangle \(EHGF\).
|
\frac{1}{4}
|
olympiads
| 0.171875
|
Find all pairs \((a, b)\) of positive integers such that \(a^{2017}+b\) is a multiple of \(ab\).
|
(1,1) \text{ and } \left(2,2^{2017}\right)
|
olympiads
| 0.140625
|
Create a statement that is true only for the numbers 2 and 5.
|
(x - 2)(x - 5) = 0
|
olympiads
| 0.171875
|
Calculate the indefinite integral:
$$
\int(8-3 x) \cos 5 x \, dx
$$
|
\frac{1}{5}(8 - 3x) \sin 5x - \frac{3}{25} \cos 5x + C
|
olympiads
| 0.3125
|
Consider the sequence \(\{a_{n}\}\) with 100 terms, where \(a_{1} = 0\) and \(a_{100} = 475\). Additionally, \(|a_{k+1} - a_{k}| = 5\) for \(k = 1, 2, \cdots, 99\). How many different sequences satisfy these conditions?
|
4851
|
olympiads
| 0.15625
|
Among all triangles with the sum of the medians equal to 3, find the triangle with the maximum sum of the heights.
|
3
|
olympiads
| 0.0625
|
Adva van egy fogyó mértani haladvány első tagja \(a_{1}\) és hányadosa \(q\). Számítsuk ki egymásután e haladvány \(n\) tagjának az összegét, első tagnak \(a_{1}\)-et, azután \(a_{2}\)-t, \(a_{3}\)-t, stb. véve. Ha eme összegek rendre \(S_{1}, S_{2}, S_{3}, \ldots\), határozzuk meg az
\[ S_{1} + S_{2} + S_{3} + \ldots \text{ in infinity} \]
összeget.
|
\frac{a_1}{(1 - q)^2}
|
olympiads
| 0.25
|
The sum of two positive integers is 60 and their least common multiple is 273. What are the two integers?
|
21 \text{ and } 39
|
olympiads
| 0.4375
|
127 is the number of non-empty sets of natural numbers \( S \) that satisfy the condition "if \( x \in S \), then \( 14-x \in S \)". The number of such sets \( S \) is \(\qquad \).
|
127
|
olympiads
| 0.296875
|
Find all integers \( k \geq 1 \) so that the sequence \( k, k+1, k+2, \ldots, k+99 \) contains the maximum number of prime numbers.
|
k=2
|
olympiads
| 0.125
|
A centipede with 40 legs and a dragon with 9 heads are in a cage. There are a total of 50 heads and 220 legs in the cage. If each centipede has one head, how many legs does each dragon have?
|
4
|
olympiads
| 0.109375
|
If the difference between the maximum and minimum elements of the set of real numbers $\{1,2,3,x\}$ is equal to the sum of all the elements in the set, what is the value of $x$?
|
-\frac{3}{2}
|
olympiads
| 0.078125
|
Given \( a, b \neq 0 \), and
\[ \frac{\sin^4 x}{a^2} + \frac{\cos^4 x}{b^2} = \frac{1}{a^2 + b^2}. \]
Find the value of \( \frac{\sin^{100} x}{a^{100}} + \frac{\cos^{100} x}{b^{100}} \).
|
\frac{2}{(a^2 + b^2)^{100}}
|
olympiads
| 0.0625
|
Find the third derivative of the given function.
$$
y=\frac{\ln (2 x+5)}{2 x+5}, y^{\prime \prime \prime}=?
$$
|
\frac{88 - 48 \ln(2 x+5)}{(2 x+5)^{4}}
|
olympiads
| 0.0625
|
There are 2011 street lamps numbered \(1, 2, 3, \ldots, 2011\). For the sake of saving electricity, it is required to turn off 300 of these lamps. However, the conditions are that no two adjacent lamps can be turned off simultaneously, and the lamps at both ends cannot be turned off. How many ways are there to turn off the lamps under these conditions? (Express your answer in terms of binomial coefficients).
|
\binom{1710}{300}
|
olympiads
| 0.0625
|
For which pairs of integers \((a, b)\) do there exist functions \(f: \mathbb{Z} \rightarrow \mathbb{Z}\) and \(g: \mathbb{Z} \rightarrow \mathbb{Z}\) obeying
$$
f(g(x))=x+a \quad \text{ and } \quad g(f(x))=x+b
$$
for all integers \(x\)?
|
(a, a) \text{ or } (a, -a)
|
olympiads
| 0.109375
|
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