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stringlengths 33
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|---|---|---|---|
Set $T\subset\{1,2,\dots,n\}^3$ has the property that for any two triplets $(a,b,c)$ and $(x,y,z)$ in $T$ , we have $a<b<c$ , and also, we know that at most one of the equalities $a=x$ , $b=y$ , $c=z$ holds. Maximize $|T|$ .
|
\binom{n}{3}
|
aops_forum
| 0.234375
|
In a certain country, there are exactly 2019 cities and between any two of them, there is exactly one direct flight operated by an airline company, that is, given cities $A$ and $B$, there is either a flight from $A$ to $B$ or a flight from $B$ to $A$. Find the minimum number of airline companies operating in the country, knowing that direct flights between any three distinct cities are operated by different companies.
|
2019
|
olympiads
| 0.09375
|
The expression \( x + \frac{1}{x} \) has a maximum for \( x < 0 \) and a minimum for \( x > 0 \). Find the area of the rectangle whose sides are parallel to the axes and two of whose vertices are the maximum and minimum values of \( x + \frac{1}{x} \).
|
8
|
olympiads
| 0.5
|
Find all polynomials $W$ with real coefficients possessing the following property: if $x+y$ is a rational number, then $W(x)+W(y)$ is rational.
|
W(x) = ax + b where a and b are rational numbers.
|
aops_forum
| 0.0625
|
In the isosceles triangle $ABC$, the angle $A$ at the base is $75^{\circ}$. The angle bisector of angle $A$ intersects the side $BC$ at point $K$. Find the distance from point $K$ to the base $AC$ if $BK = 10$.
|
5
|
olympiads
| 0.125
|
One out of every seven mathematicians is a philosopher, and one out of every nine philosophers is a mathematician. Are there more philosophers or mathematicians?
|
p > m
|
aops_forum
| 0.078125
|
Five consecutive natural numbers were written on the board, and then one number was erased. It turned out that the sum of the remaining four numbers is 2015. Find the smallest of these four numbers.
|
502
|
olympiads
| 0.25
|
Let \( p \) be a prime number such that the next larger number is a perfect square. Find the sum of all such prime numbers.
|
3
|
olympiads
| 0.125
|
A rectangle with dimensions $14 \times 6$ is to be cut into two parts in such a way that it is possible to form a new rectangle with dimensions $21 \times 4$.
|
Solution is Verified
|
olympiads
| 0.0625
|
Let $a$ and $b$ be two real numbers and let $M(a,b)=\max\left \{ 3a^{2}+2b; 3b^{2}+2a\right \}$ . Find the values of $a$ and $b$ for which $M(a,b)$ is minimal.
|
a = -\frac{1}{3}, b = -\frac{1}{3}
|
aops_forum
| 0.15625
|
let $m,n$ be natural number with $m>n$ . find all such pairs of $(m,n) $ such that $gcd(n+1,m+1)=gcd(n+2,m+2) =..........=gcd(m, 2m-n) = 1 $
|
(m, n) = (n+1, n) \text{ for all } n \in \mathbb{N}
|
aops_forum
| 0.078125
|
Sergey Stanislavovich is 36 years, 36 months, 36 weeks, 36 days, and 36 hours old.
How many full years old is Sergey Stanislavovich?
|
39
|
olympiads
| 0.265625
|
Given a natural number $k$ and a real number $a (a>0)$ , find the maximal value of $a^{k_1}+a^{k_2}+\cdots +a^{k_r}$ , where $k_1+k_2+\cdots +k_r=k$ ( $k_i\in \mathbb{N} ,1\le r \le k$ ).
|
a^k
|
aops_forum
| 0.25
|
Simplify the expression:
\[
\frac{\sin 2 \alpha + \cos 2 \alpha - \cos 6 \alpha - \sin 6 \alpha}{\sin 4 \alpha + 2 \sin^2 2 \alpha - 1}
\]
|
2 \sin 2\alpha
|
olympiads
| 0.109375
|
Express \( x \) using the real parameter \( a \) if
$$
\sqrt{\log _{a} (a x) + \log _{x} (a x)} + \sqrt{\log _{a} \left(\frac{x}{a}\right) + \log _{x} \left(\frac{a}{x}\right)} = 2.
$$
|
x = a
|
olympiads
| 0.265625
|
Through how many squares does the diagonal of a 1983 × 999 chessboard pass?
|
2979
|
olympiads
| 0.078125
|
For which values of \(a, b, c,\) and \(d\) does the equality \(5 x^{3}-32 x^{2}+75 x-71 = a(x-2)^{3} + b(x-2)^{2} + c(x-2) + d\) hold as an identity?
|
a = 5, b = -2, c = 7, d = -9
|
olympiads
| 0.28125
|
Find all pairs $(a,b)$ of different positive integers that satisfy the equation $W(a)=W(b)$ , where $W(x)=x^{4}-3x^{3}+5x^{2}-9x$ .
|
(1, 2) and (2, 1)
|
aops_forum
| 0.125
|
Select several numbers from $1, 2, 3, \cdots, 9, 10$ so that every number among $1, 2, 3, \cdots, 19, 20$ can be expressed as either one of the selected numbers or the sum of two selected numbers (which can be the same). What is the minimum number of selections needed?
|
6
|
olympiads
| 0.078125
|
For which $n$ on the plane can you mark $2n$ distinct points such that for any natural $k$ from 1 to $n$, there exists a line containing exactly $k$ marked points?
### Variation
For which $n$ on the plane can you mark $n$ distinct points such that for any natural $k$ from 1 to 17, there exists a line containing exactly $k$ marked points?
|
n=17 \text{ optimal enabling minimum configurations maintaining all necessary lines existence conditions for } k \leq 17
|
olympiads
| 0.265625
|
The points \( M \) and \( N \) are located on one side of a line \( l \). Using a compass and a straightedge, construct a point \( K \) on the line \( l \) such that the sum \( MK + NK \) is minimized.
|
K is the intersection point of l and the line segment MN_1.
|
olympiads
| 0.21875
|
In parallelogram $ABCD$, angle $BAD$ is $60^{\circ}$ and side $AB$ is 3. The angle bisector of angle $A$ intersects side $BC$ at point $E$. Find the area of triangle $ABE$.
|
\frac{9 \sqrt{3}}{4}
|
olympiads
| 0.1875
|
A square area of size $100 \times 100$ is tiled with $1 \times 1$ square tiles in four colors: white, red, black, and gray, such that no two tiles of the same color touch each other (i.e., do not share a side or a vertex). How many red tiles could there be?
|
2500 \text{ tiles}
|
olympiads
| 0.171875
|
The function \( f \) is defined on the set of positive integers and satisfies:
\[ f(x)=\left\{\begin{array}{l}
n-3, \quad n \geqslant 1000 \\
f(f(n+5)), \quad 1 \leqslant n < 1000 .
\end{array}\right. \]
Find \( f(84) \).
|
997
|
olympiads
| 0.15625
|
Let $p=4k+3$ be a prime number. Find the number of different residues mod p of $(x^{2}+y^{2})^{2}$ where $(x,p)=(y,p)=1.$
|
\frac{p-1}{2}
|
aops_forum
| 0.0625
|
A directory has 710 pages. How many digits were needed to number its pages?
|
2022
|
olympiads
| 0.390625
|
Define a new operation: \( A \oplus B = A^2 + B^2 \), and \( A \otimes B \) is the remainder of \( A \) divided by \( B \). Calculate \( (2013 \oplus 2014) \otimes 10 \).
|
5
|
olympiads
| 0.5625
|
Find the greatest integer not exceeding \( 1 + \frac{1}{2^k} + \frac{1}{3^k} + \ldots + \frac{1}{N^k} \), where \( k = \frac{1982}{1983} \) and \( N = 2^{1983} \).
|
1983
|
olympiads
| 0.078125
|
Divide a cube with an edge length of 1 meter into smaller cubes with an edge length of 1 centimeter. Suppose Sun Wukong uses his supernatural power to stack all the small cubes one by one into a rectangular "magic stick" that points up to the sky. Given that the elevation of Mount Everest is 8844 meters, determine how much higher the "magic stick" is compared to Mount Everest.
|
1156
|
olympiads
| 0.125
|
Find the differential \( dy \).
\[ y=\ln \left(\cos ^{2} x+\sqrt{1+\cos ^{4} x}\right) \]
|
dy = -\frac{\sin 2x \cdot dx}{\sqrt{1 + \cos^4 x}}.
|
olympiads
| 0.15625
|
In triangle \(ABC\), the angle bisector \(AD\) divides side \(BC\) in the ratio \(BD : DC = 2 : 1\). In what ratio does the median from vertex \(C\) divide this angle bisector?
|
3:1
|
olympiads
| 0.125
|
Without rearranging the digits in the left side of the equation, insert two plus signs (+) among them to make a correct equation: $8789924=1010$.
|
87+899+24=1010
|
olympiads
| 0.0625
|
A plane-parallel glass plate is normally illuminated by a thin beam of light. Behind the plate, at some distance from it, there is an ideal mirror (with a reflection coefficient of one). The plane of the mirror is parallel to the plate. It is known that the intensity of the beam passing through this system is 256 times less than the intensity of the incident beam. Assume that the reflection coefficient at the glass-air interface is constant regardless of the direction of the beam. Ignore the absorption and scattering of light in the air and glass. Find the reflection coefficient at the glass-air interface under these conditions. (10 points)
|
0.75
|
olympiads
| 0.0625
|
A right circular cone with a base radius \( R \) and height \( H = 3R \sqrt{7} \) is laid sideways on a plane and rolled in such a manner that its apex remains stationary. How many rotations will its base make until the cone returns to its original position?
|
8
|
olympiads
| 0.453125
|
Seven dwarfs sat at a round table, each with a mug containing a total of half a liter of milk. (Some of the mugs might have been empty.) One of the dwarfs stood up and distributed the milk in his mug equally to the others. Then, the dwarfs to his right took turns doing the same. After the seventh dwarf redistributed his milk, each mug had exactly the same amount of milk as before the distributions started. How much milk was originally in each mug?
|
\frac{6}{42}, \,\frac{5}{42}, \,\frac{4}{42}, \,\frac{3}{42}, \,\frac{2}{42}, \,\frac{1}{42},\, 0 \, \text{liters}
|
olympiads
| 0.078125
|
Four people are sitting at four sides of a table, and they are dividing a 32-card Hungarian deck equally among themselves. If one selected player does not receive any aces, what is the probability that the player sitting opposite them also has no aces among their 8 cards?
|
\frac{130}{759}
|
olympiads
| 0.09375
|
The diagonals of a trapezoid are 6 and 8, and the midline is 5. Find the area of the trapezoid.
|
24
|
olympiads
| 0.0625
|
There were cards with numbers from 1 to 20 in a bag. Vlad pulled out 6 cards and asserted that all these cards can be divided into pairs such that the sums of the numbers in each pair were the same. Lena managed to sneak a peek at 5 of Vlad's cards: they had the numbers $2, 4, 9, 17, 19$ written on them. Which number on the sixth card did Lena not manage to see? (It is sufficient to provide one suitable answer.)
|
12
|
olympiads
| 0.078125
|
In the numerical pyramid, place the "+" and "-" signs so that the given equations are satisfied. Between some adjacent digits, no sign can be placed, combining them into a single number.
|
1-2+3+4-5+6 = 7 or 1+2-3-4+5+6 = 7 ; 1+2+3-4+5-6+7 = 8 ; 12-3+4+5+6-7-8=9 or 12+3+4-5-6-7+8=9
|
olympiads
| 0.078125
|
Let the tangent line passing through a point $A$ outside the circle with center $O$ touches the circle at $B$ and $C$ . Let $[BD]$ be the diameter of the circle. Let the lines $CD$ and $AB$ meet at $E$ . If the lines $AD$ and $OE$ meet at $F$ , find $|AF|/|FD|$ .
|
\frac{1}{2}
|
aops_forum
| 0.109375
|
The cards in a stack of $2n$ cards are numbered consecutively from $1$ through $2n$ from top to bottom. The top $n$ cards are removed, kept in order, and form pile $A$ . The remaining cards form pile $B$ . The cards are then restacked by taking cards alternately from the tops of pile $B$ and $A$ , respectively. In this process, card number $(n+1)$ becomes the bottom card of the new stack, card number $1$ is on top of this card, and so on, until piles $A$ and $B$ are exhausted. If, after the restacking process, at least one card from each pile occupies the same position that it occupied in the original stack, the stack is named *magical*. Find the number of cards in the magical stack in which card number $131$ retains its original position.
|
130
|
aops_forum
| 0.078125
|
On the sides $AB$ and $AC$ of triangle $ABC$ with an area of 50, points $M$ and $K$ are taken respectively such that $AM: MB = 1: 5$ and $AK: KC = 3: 2$. Find the area of triangle $AMK$.
|
5
|
olympiads
| 0.125
|
Congruent unit circles intersect in such a way that the center of each circle lies on the circumference of the other. Let $R$ be the region in which two circles overlap. Determine the perimeter of $R.$
|
\frac{2\pi}{3}
|
aops_forum
| 0.109375
|
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \ln \left(\left(e^{x^{2}}-\cos x\right) \cos \left(\frac{1}{x}\right)+\operatorname{tg}\left(x+\frac{\pi}{3}\right)\right)
$$
|
\frac{1}{2} \ln (3)
|
olympiads
| 0.3125
|
In a prison, there are $n$ prisoners. The bored prison guards come up with a game where they place either a red or blue hat on each prisoner's head in the yard such that no one can see the color of their own hat. After the prisoners have had a good look at each other (each prisoner can see all the hats except their own), each of them must write down on a piece of paper the color of the hat on their head. If all of their answers are correct, they will be let out into the yard. What strategy should the prisoners agree upon to maximize their chances of being allowed out?
|
\frac{1}{2}
|
olympiads
| 0.203125
|
A pot weighs $645 \mathrm{~g}$ and another one $237 \mathrm{~g}$. José divides $1 \mathrm{~kg}$ of meat between the two pots so that both, with their contents, have the same weight. How much meat did he put in each pot?
|
296 \text{ g} \; \text{e} \; 704 \text{ g}
|
olympiads
| 0.265625
|
Arrange 8 spheres, each with a radius of 2, in two layers within a cylindrical container such that each sphere is tangent to its four neighboring spheres, as well as to one base and the side surface of the cylinder. Find the height of the cylinder.
|
4 + 2\sqrt[4]{8}
|
olympiads
| 0.125
|
Eleven wise men have their eyes blindfolded and a hat of one of 1000 colors is placed on each of their heads. Afterward, their blindfolds are removed, and each wise man can see all the hats except their own. They must then simultaneously show one of two cards - either white or black - to the others. After this, they must all simultaneously name the color of their own hats. Can they succeed? The wise men can agree on their actions beforehand (before they are blindfolded) and they know the 1000 possible colors of the hats.
|
It will succeed.
|
olympiads
| 0.125
|
Compute the limit of the function:
\[
\lim _{x \rightarrow 0} \frac{2 \sin (\pi(x+1))}{\ln (1+2x)}
\]
|
-\pi
|
olympiads
| 0.296875
|
The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{3}$, and for any $n \in \mathbf{N}^{*}, a_{n+1}=a_{n}^{2}+a_{n}$. Determine the integer part of $\sum_{n=1}^{2016} \frac{1}{a_{n}+1}$.
|
2
|
olympiads
| 0.078125
|
What is the minimum value of the product
\[ \prod_{i=1}^{6} \frac{a_{i} - a_{i+1}}{a_{i+2} - a_{i+3}} \]
given that \((a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6})\) is a permutation of \((1, 2, 3, 4, 5, 6)\)? (Note: \(a_{7} = a_{1}, a_{8} = a_{2}, \ldots\))
|
1
|
olympiads
| 0.09375
|
In an arithmetic sequence \(\{a_{n}\}\) with a non-zero common difference, \(a_{4}=10\), and \(a_{3}\), \(a_{6}\), \(a_{10}\) form a geometric sequence. Find the general term formula for the sequence \(\{a_{n}\}\).
|
n + 6
|
olympiads
| 0.140625
|
For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$ . How many such linear functions $\ell(x)$ can exist?
|
2
|
aops_forum
| 0.484375
|
Two circles have centers that are \( d \) units apart, and each has a diameter \( \sqrt{d} \). For any \( d \), let \( A(d) \) be the area of the smallest circle that contains both of these circles. Find \( \lim _{d \rightarrow \infty} \frac{A(d)}{d^{2}} \).
|
\frac{\pi}{4}
|
olympiads
| 0.53125
|
Given \( a \geqslant 0 \), for any \( m \) and \( x \) (\( 0 \leqslant m \leqslant a, 0 \leqslant x \leqslant \pi \)), it holds that
$$
|\sin x - \sin (x + m)| \leqslant 1.
$$
Find the maximum value of \( a \).
|
\frac{\pi}{3}
|
olympiads
| 0.15625
|
Find the volume of the solid obtained by rotating an equilateral triangle with side length \(a\) around a line parallel to its plane, such that the projection of this line onto the plane of the triangle contains one of the triangle's altitudes.
|
\frac{\pi a^3 \sqrt{3}}{24}
|
olympiads
| 0.078125
|
Express the radius of the inscribed circle in terms of the sides of the triangle if it is right-angled at \( C \).
|
r = \frac{a \cdot b}{a + b + c}
|
olympiads
| 0.296875
|
Determine the maximum value of \(\frac{1+\cos x}{\sin x+\cos x+2}\), where \(x\) ranges over all real numbers.
|
1
|
olympiads
| 0.109375
|
On a circle, there are 2018 points. Each of these points is labeled with an integer. Each integer is greater than the sum of the two integers that immediately precede it in a clockwise direction.
Determine the maximum possible number of positive integers among the 2018 integers.
|
2016
|
olympiads
| 0.109375
|
A group of six people approached the bank of the Nile: three Bedouins, each with his wife. There is a rowing boat at the bank that can only hold two people. A Bedouin cannot allow his wife to be in the company of another man without him. Can the whole group cross to the other side?
|
All can safely cross to the other bank.
|
olympiads
| 0.1875
|
Given the complex numbers \(a, b, c\) that satisfy
\[
\begin{cases}
a^{2}+a b+b^{2}=1, \\
b^{2}+b c+c^{2}=-1, \\
c^{2}+c a+a^{2}=\mathrm{i},
\end{cases}
\]
find the value of \(ab + bc + ca\).
|
\pm \mathrm{i}
|
olympiads
| 0.078125
|
Given a positive integer $d$, define the sequence $\left\{a_{n}\right\}$ :
\[ a_{0} = 1 \]
\[ a_{n+1} = \begin{cases}
\frac{a_{n}}{2}, & \text{if } a_{n} \text{ is even;} \\
a_{n} + d, & \text{if } a_{n} \text{ is odd.}
\end{cases} \]
Find all integers $d$ such that there exists \( n > 0 \) where \( a_{n} = 1 \).
|
All odd integers for d
|
olympiads
| 0.15625
|
Points \( A, B, C, \) and \( D \) are positioned on a line in the given order. It is known that \( BC = 3 \) and \( AB = 2 \cdot CD \). A circle is drawn through points \( A \) and \( C \), and another circle is drawn through points \( B \) and \( D \). Their common chord intersects segment \( BC \) at point \( K \). Find \( BK \).
|
2
|
olympiads
| 0.0625
|
There are 10 different points \( P_{1}(x_{1}, y_{1}), P_{2}(x_{2}, y_{2}), \cdots, P_{10}(x_{10}, y_{10}) \) in the Cartesian plane. If \( x_{i}=x_{j} \) or \( y_{i}=y_{j} \), then \( P_{i} \) and \( P_{j} \) are called a "same-coordinate pair" (the order of \( P_{i} \) and \( P_{j} \) does not matter). It is known that among these 10 different points, the number of "same-coordinate pairs" that each point forms does not exceed \( m \); moreover, it is always possible to partition these points exactly into 5 pairs, where each pair is not a "same-coordinate pair." Find the maximum value of \( m \).
|
4
|
olympiads
| 0.484375
|
Dasha and Tanya live in the same apartment building. Dasha lives on the 6th floor. When leaving Dasha's place, Tanya went up instead of down as she intended. After reaching the top floor, Tanya realized her mistake and went down to her floor. It turned out that Tanya traveled one and a half times more than she would have if she had gone directly down. How many floors are in the building?
|
7
|
olympiads
| 0.109375
|
Two circles are drawn on a plane with centers at $O_{1}$ and $O_{2}$ such that each circle passes through the center of the other. $A$ and $B$ are the two points of their intersection. Find the radius of the circles given that the area of the quadrilateral $O_{1} A O_{2} B$ is $2 \sqrt{3}$.
|
R = 2
|
olympiads
| 0.171875
|
Let \( k, \alpha \) and \( 10k - \alpha \) be positive integers. What is the remainder when the following number is divided by 11?
\[
8^{10k + \alpha} + 6^{10k - \alpha} - 7^{10k - \alpha} - 2^{10k + \alpha}
\]
|
0
|
olympiads
| 0.203125
|
Given the sequence \(\left\{a_{n}\right\}\), which satisfies
\[
a_{1}=0,\left|a_{n+1}\right|=\left|a_{n}-2\right|
\]
Let \(S\) be the sum of the first 2016 terms of the sequence \(\left\{a_{n}\right\}\). Determine the maximum value of \(S\).
|
2016
|
olympiads
| 0.421875
|
For $f(x)=x^4+|x|,$ let $I_1=\int_0^\pi f(\cos x)\ dx,\ I_2=\int_0^\frac{\pi}{2} f(\sin x)\ dx.$
Find the value of $\frac{I_1}{I_2}.$
|
2
|
aops_forum
| 0.0625
|
Let \(a\), \(b\), and \(c\) be the lengths of the three sides of a triangle. Suppose \(a\) and \(b\) are the roots of the equation
\[
x^2 + 4(c + 2) = (c + 4)x,
\]
and the largest angle of the triangle is \(x^\circ\). Find the value of \(x\).
|
90
|
olympiads
| 0.296875
|
How many different draws can be made if different balls are drawn and replaced after each draw without considering the order?
|
\binom{n+k-1}{k}
|
olympiads
| 0.140625
|
On a $4 \times 4$ checkered board, Petya colors several cells. Vasya wins if he is able to cover all these cells with non-overlapping L-shaped tiles, each consisting of three cells, without any part of the tiles extending beyond the border of the square. What is the minimum number of cells Petya needs to color so that Vasya cannot win?
|
16
|
olympiads
| 0.078125
|
Find all prime number $p$ , such that there exist an infinite number of positive integer $n$ satisfying the following condition: $p|n^{ n+1}+(n+1)^n.$ (September 29, 2012, Hohhot)
|
2
|
aops_forum
| 0.46875
|
A bus leaves the station at exactly 7:43 a.m. and arrives at its destination at exactly 8:22 a.m. on the same day. How long, in minutes, was the bus trip?
|
39 \, \text{minutes}
|
olympiads
| 0.46875
|
Find the angle between the planes:
$$
\begin{aligned}
& 3x - y + 2z + 15 = 0 \\
& 5x + 9y - 3z - 1 = 0
\end{aligned}
$$
|
\frac{\pi}{2}
|
olympiads
| 0.59375
|
Two grandmasters take turns placing rooks on a chessboard (one rook per turn) in such a way that no two rooks attack each other. The player who cannot place a rook loses. Who will win with perfect play - the first or the second grandmaster?
|
Second grandmaster wins
|
olympiads
| 0.09375
|
There are 10 digit cards from 0 to 9 on the table. Three people, A, B, and C, each take three cards, and calculate the sum of all possible different three-digit numbers that can be formed with their three cards. The results for A, B, and C are $1554, 1688, 4662$. What is the remaining card on the table? (Note: 6 and 9 cannot be flipped to look like 9 or 6.)
|
9
|
olympiads
| 0.125
|
Calculate the limit of the function:
\[ \lim_{{x \rightarrow \frac{\pi}{2}}} \frac{\ln (\sin x)}{(2x - \pi)^{2}} \]
|
-\frac{1}{8}
|
olympiads
| 0.15625
|
Consider a cube where all edges are colored either red or black in such a way that each face of the cube has at least one black edge. What is the minimum number of black edges?
|
3
|
olympiads
| 0.078125
|
For certain ordered pairs $(a,b)$ of real numbers, the system of equations \begin{eqnarray*} && ax+by =1 &&x^2+y^2=50\end{eqnarray*} has at least one solution, and each solution is an ordered pair $(x,y)$ of integers. How many such ordered pairs $(a,b)$ are there?
|
18
|
aops_forum
| 0.0625
|
The difference of two natural numbers, which are perfect squares, ends with the digit 2. What are the last digits of the minuend and the subtrahend, if the last digit of the minuend is greater than the last digit of the subtrahend?
|
6 \text{ and } 4
|
olympiads
| 0.203125
|
Determine all positive integers \( n \) such that \( n \) equals the square of the sum of the digits of \( n \).
|
1 \text{ and } 81
|
olympiads
| 0.0625
|
A mustache is created by taking the set of points $(x, y)$ in the $xy$ -coordinate plane that satisfy $4 + 4 \cos(\pi x/24) \le y \le 6 + 6\cos(\pi x/24)$ and $-24 \le x \le 24$ . What is the area of the mustache?
|
96
|
aops_forum
| 0.203125
|
An engineer named Paul received for examination an object with a volume of approximately 100 oil samples (a container designed for 100 samples, which was almost full). Each sample is categorized based on sulfur content—either low-sulfur or high-sulfur, and density—either light or heavy. The relative frequency (statistical probability) that a randomly chosen sample is heavy oil is $\frac{1}{7}$. Additionally, the relative frequency that a randomly chosen sample is light low-sulfur oil is $\frac{9}{14}$. How many total samples of high-sulfur oil are there in the object if none of the heavy oil samples are low-sulfur?
|
35
|
olympiads
| 0.09375
|
A capacitor with a capacitance of \(C_{1}=10\) μF is charged to a voltage of \(U_{1}=15\) V. A second capacitor with a capacitance of \(C_{2}=5\) μF is charged to a voltage of \(U_{2}=10\) V. The capacitors are connected with plates of opposite charge. Determine the voltage that will establish itself across the plates.
|
6.67 \text{ V}
|
olympiads
| 0.109375
|
Find the largest positive integer \( k \) such that
$$
1991^{k} \cdot 1990^{1991^{1992}} + 1992^{1991^{1990}}
$$
is divisible by \( k \).
|
1991
|
olympiads
| 0.34375
|
Given two points on a plane and a line parallel to the line segment connecting the two points, along with the angle $\alpha$. Construct the segment on the parallel line that appears at an angle $\alpha$ from both points!
|
CD
|
olympiads
| 0.0625
|
The graph of the quadratic function \( y = ax^2 + c \) intersects the coordinate axes at the vertices of an equilateral triangle. What is the value of \( ac \)?
|
a = -3
|
olympiads
| 0.078125
|
What is the minimum number of tetrahedrons needed to divide a cube?
|
5
|
olympiads
| 0.0625
|
"That" and "this," plus half of "that" and "this" – how much will that be as a percentage of three-quarters of "that" and "this"?
|
200\%
|
olympiads
| 0.5625
|
Three siblings inherited a total of 30000 forints. $B$ received as much less than $A$ as he received more than $C$, and $A$ received as much as $B$ and $C$ together. How many forints did each inherit?
|
A = 15000 \, \text{Ft}, \, B = 10000 \, \text{Ft}, \, C = 5000 \, \text{Ft}
|
olympiads
| 0.328125
|
For every positive integer $n$ , determine the biggest positive integer $k$ so that $2^k |\ 3^n+1$
|
k = 2
|
aops_forum
| 0.171875
|
Ten students (one captain and nine team members) formed a team to participate in a math competition and won first place. The committee decided to award each team member 200 yuan. The captain received 90 yuan more than the average bonus of all ten team members. Determine the amount of bonus the captain received.
|
300
|
olympiads
| 0.578125
|
Let \( f(x) = ax + b \) (where \( a \) and \( b \) are real numbers), \( f_1(x) = f(x) \), \( f_{n+1}(x) = f(f_n(x)) \) for \( n=1, 2, 3, \cdots \). Given that \( 2a + b = -2 \) and \( f_k(x) = -243x + 244 \), find \( k \).
|
5
|
olympiads
| 0.40625
|
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{4 n^{2}-\sqrt[4]{n^{3}}}{\sqrt[3]{n^{6}+n^{3}+1}-5 n}
\]
|
4
|
olympiads
| 0.265625
|
The external angle bisectors of triangle \( \triangle ABC \) determine the triangle \( \triangle A'B'C' \). How should we construct triangle \( \triangle ABC \) if \( \triangle A'B'C' \) is given?
|
\text{This explains how to reconstruct triangle \(ABC\) from given triangle \(A'B'C'\), where the latter's vertices are defined by the external angle bisectors of the former.}
|
olympiads
| 0.09375
|
For which values of the parameter \( a \) is the sum of the squares of the roots of the equation \( x^{2} + 2ax + 2a^{2} + 4a + 3 = 0 \) at its greatest? What is this sum? (The roots are considered with multiplicity.)
|
18
|
olympiads
| 0.125
|
The Haier brothers are on an adventure together. The older brother has 17 more energy bars than the younger brother. After giving 3 energy bars to the younger brother, the older brother still has $\qquad$ more energy bars than the younger brother.
|
11
|
olympiads
| 0.1875
|
If the \( r^{\text{th}} \) day of May in a year is Friday and the \( n^{\text{th}} \) day of May in the same year is Monday, where \( 15 < n < 25 \), find \( n \).
|
20
|
olympiads
| 0.125
|
Bretschneider's theorem (the cosine theorem for a quadrilateral). Let \(a, b, c, d\) be the consecutive sides of a quadrilateral, \(m\) and \(n\) be its diagonals, and \(A\) and \(C\) be two opposite angles. Then the following relation holds:
$$
m^{2} n^{2} = a^{2} c^{2} + b^{2} d^{2} - 2 a b c d \cos (A + C)
$$
|
m^2 n^2 = a^2 c^2 + b^2 d^2 - 2 abcd \cos(A + C)
|
olympiads
| 0.390625
|
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