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|---|---|---|---|
There are three simplest proper fractions with their numerators in the ratio $3: 2: 4$ and their denominators in the ratio $5: 9: 15$. When these three fractions are added together and then reduced to simplest form, the result is $\frac{28}{45}$. What is the sum of the denominators of these three fractions?
|
203
|
olympiads
| 0.125
|
The diagonals of an isosceles trapezoid are mutually perpendicular, and its area is $a^2$. Determine the height of the trapezoid.
|
a
|
olympiads
| 0.09375
|
For which values of \( n \) is the expression \( 2^{n} + 1 \) a nontrivial power of a natural number?
|
3
|
olympiads
| 0.265625
|
A square steel plate has had a rectangular piece of width 3 cm and another rectangular piece of width 5 cm cut off (as shown in the figure). The total area removed is 81 square centimeters. What was the area of the original square? $\qquad$ square centimeters.
|
144
|
olympiads
| 0.0625
|
The Fibonacci sequence is defined recursively as follows: \(a_{1} = a_{2} = 1\), and \(a_{n+2} = a_{n+1} + a_{n}\) for all natural numbers \(n\). What is the last digit of \(a_{2020}\)?
|
5
|
olympiads
| 0.140625
|
Winnie-the-Pooh and Tigger are climbing two identical fir trees. Winnie-the-Pooh climbs up twice as slowly as Tigger, and he descends three times faster than Tigger. They started and finished at the same time. By how many times is Tigger faster when climbing up compared to climbing down?
|
1.5
|
olympiads
| 0.15625
|
Let \( a \) be a strictly positive integer. Suppose that \( 4(a^n + 1) \) is the cube of an integer for every positive integer \( n \). Find \( a \).
|
1
|
olympiads
| 0.34375
|
Which theorems are expressed by the equalities:
$$
\sin (-x)=-\sin x \quad \text{and} \quad \cos (-x)=\cos x ?
$$
|
122. y = \sin(x) \text{ — odd function}; y = \cos(x) \text{ — even function.
|
olympiads
| 0.125
|
Given the sequence $\left\{a_{n}\right\}$ which satisfies
$$
a_{1}=\frac{1}{2}, a_{1}+a_{2}+\cdots+a_{n}=n^{2} a_{n}, n \geqslant 1,
$$
find the general term formula for $a_{n}$.
|
a_{n} = rac{1}{n(n+1)}, ext{ } n=1, 2, 3, ext{ } ext{...}
|
olympiads
| 0.125
|
In triangle \( \triangle ABC \), \( AB = 8 \), \( BC = 11 \), and \( AC = 6 \). The points \( P \) and \( Q \) are on \( BC \) such that \( \triangle PBA \) and \( \triangle QAC \) are each similar to \( \triangle ABC \). What is the length of \( PQ \)?
|
\frac{21}{11}
|
olympiads
| 0.09375
|
According to the proverb "Counting Nines in Winter," starting from the Winter Solstice, every nine days are grouped together and referred to as "First Nine," "Second Nine," and so on, up to "Ninth Nine." The Winter Solstice is the first day of the First Nine. December 21, 2012, was the Winter Solstice. Which "Nine" and which day within that "Nine" is New Year's Day 2013?
|
3rd day of the second nine-day period
|
olympiads
| 0.140625
|
Calculate the limit of the function:
$$\lim _{x \rightarrow 1} \frac{\sqrt{x^{2}-x+1}-1}{\operatorname{tg} \pi x}$$
|
\frac{1}{2\pi}
|
olympiads
| 0.109375
|
In a modified episode of Deal or No Deal, there are sixteen cases numbered 1 through 16 with dollar amounts ranging from $2^{1}=2$ to $2^{16}=65536$ in random order. The game has eight turns in which you claim one case per turn without opening it, after which a random remaining case is opened and removed from the game. At the end, the total amount of money in your claimed cases is revealed and you win this amount. Given that you can see the amount of money in each case, what is the expected amount of money you will win if you play optimally?
|
\frac{7 \cdot 2^{18} + 4}{15} \text{ or } \frac{1835012}{15}
|
olympiads
| 0.140625
|
If $\frac{6}{b} < x < \frac{10}{b}$, determine the value of $c = \sqrt{x^{2} - 2x + 1} + \sqrt{x^{2} - 6x + 9}$.
|
2
|
olympiads
| 0.46875
|
On the given segment \( AB \), construct an equilateral triangle using a compass with a fixed opening (not equal to \( AB \)).
|
\triangle ABE \text{ is an equilateral triangle}
|
olympiads
| 0.09375
|
Solve the equation \( x^y + 1 = z \) in prime numbers \( x, y, \) and \( z \).
|
(2, 2, 5)
|
olympiads
| 0.390625
|
If $ a\equal{}2b\plus{}c$ , $ b\equal{}2c\plus{}d$ , $ 2c\equal{}d\plus{}a\minus{}1$ , $ d\equal{}a\minus{}c$ , what is $ b$ ?
|
b = \frac{2}{9}
|
aops_forum
| 0.3125
|
A peasant, while buying goods, paid the first vendor half of his money plus 1 ruble; then he paid the second vendor half of the remaining money plus 2 rubles; and lastly, he paid the third vendor half of the remaining money plus 1 ruble. After these transactions, the peasant had no money left. How many rubles did he initially have?
|
18 \text{ rubles}
|
olympiads
| 0.25
|
In summer, Ponchik eats honey cakes four times a day: instead of morning exercise, instead of a daytime walk, instead of an evening run, and instead of a nighttime swim. The quantities of cakes eaten instead of exercise and instead of a walk are in the ratio $3:2$; instead of a walk and instead of a run - in the ratio $5:3$; and instead of a run and instead of a swim - in the ratio $6:5$. How many more or fewer cakes did Ponchik eat instead of exercise compared to the swim on the day when a total of 216 cakes were eaten?
|
60
|
olympiads
| 0.0625
|
Calculate the area of the parallelogram formed by the vectors $a$ and $b$.
$a = 6p - q$
$b = p + 2q$
$|p| = 8$
$|q| = \frac{1}{2}$
$\angle(p, q) = \frac{\pi}{3}$
|
26 \sqrt{3}
|
olympiads
| 0.078125
|
On a certain day, there were some nuts in a bag. On the next day, the same number of nuts was added to the bag, but eight nuts were taken out. On the third day, the same thing happened: the same number of nuts as were already in the bag was added, but eight were taken out. The same process occurred on the fourth day, and after this, there were no nuts left in the bag. How many nuts were in the bag at the very beginning?
|
7
|
olympiads
| 0.078125
|
The grasshopper can only jump exactly 50 cm. It wants to visit 8 points marked on a grid (the side of each cell is 10 cm). What is the minimum number of jumps it has to make? (It is allowed to visit other points on the plane, including non-grid points. The starting and ending points can be any of these points.)
|
8 \text{ jumps}
|
olympiads
| 0.140625
|
If the ten-digit number \(2016 \mathrm{ab} 2017\) is a multiple of 99, what is \(a+b\)?
|
8
|
olympiads
| 0.140625
|
A roulette can land on any number from 0 to 2007 with equal probability. The roulette is spun repeatedly. Let $P_{k}$ be the probability that at some point the sum of the numbers that have appeared in all spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008}$?
|
P_{2007}
|
olympiads
| 0.15625
|
Given a positive integer \( n \), find the number of ordered quadruples of integers \( (a, b, c, d) \) such that \( 0 \leq a \leq b \leq c \leq d \leq n \).
|
\binom{n+4}{4}
|
olympiads
| 0.40625
|
Let \( A \) be an angle such that \( \tan 8A = \frac{\cos A - \sin A}{\cos A + \sin A} \). Suppose \( A = x^\circ \) for some positive real number \( x \). Find the smallest possible value of \( x \).
|
5
|
olympiads
| 0.171875
|
It is known that the expressions \( 4k+5 \) and \( 9k+4 \) for some natural values of \( k \) are both perfect squares. What values can the expression \( 7k+4 \) take for the same values of \( k \)?
|
39
|
olympiads
| 0.234375
|
An angle \(ACB\) is constructed on a grid of equilateral triangles (see the figure). Find its measure.
|
90^
|
olympiads
| 0.0625
|
Let \( S = \left\{ p_1 p_2 \cdots p_n \mid p_1, p_2, \ldots, p_n \text{ are distinct primes and } p_1, \ldots, p_n < 30 \right\} \). Assume 1 is in \( S \). Let \( a_1 \) be an element of \( S \). We define, for all positive integers \( n \):
\[ a_{n+1} = \frac{a_n}{n+1} \quad \text{if } a_n \text{ is divisible by } n+1; \]
\[ a_{n+1} = (n+2) a_n \quad \text{if } a_n \text{ is not divisible by } n+1. \]
How many distinct possible values of \( a_1 \) are there such that \( a_j = a_1 \) for infinitely many \( j \)'s?
|
512
|
olympiads
| 0.0625
|
Three runners started from the same point on a circular track and ran in the same direction. The first runner overtook the second one in 6 minutes and the third one in 10 minutes. In how many minutes did the third runner overtake the second? (Assume that the runners' speeds are uniform and ignore the width of the track.)
|
15
|
olympiads
| 0.109375
|
The equation with integer coefficients \( x^{4} + a x^{3} + b x^{2} + c x + d = 0 \) has four positive roots considering their multiplicities.
Find the smallest possible value of the coefficient \( b \) given these conditions.
|
b = 6
|
olympiads
| 0.4375
|
Tom Sawyer needs to paint a very long fence such that any two boards spaced exactly two, three, or five boards apart from one another must be painted different colors. What is the minimum number of colors Tom requires for this task?
|
3
|
olympiads
| 0.21875
|
Find all real numbers \( x, y, z \) such that \( x^{2} + y^{2} + z^{2} = x - z = 2 \).
|
(x, y, z) = (1, 0, -1)
|
olympiads
| 0.296875
|
A convex hexagon has pairs of opposite sides that are parallel. The lengths of the consecutive sides are $5, 6, 7, 5+x$, $6-x$, and $7+x$ units, where $x$ satisfies the inequality $-5 < x < 6$. Given a particular $x$, what is the side length of the largest regular hexagon that can be cut out from the original hexagon in one piece?
|
\frac{11}{2}
|
olympiads
| 0.09375
|
The pattern in the fi gure below continues inward in finitely. The base of the biggest triangle is 1. All triangles are equilateral. Find the shaded area.
[asy]
defaultpen(linewidth(0.8));
pen blu = rgb(0,112,191);
real r=sqrt(3);
fill((8,0)--(0,8r)--(-8,0)--cycle, blu);
fill(origin--(4,4r)--(-4,4r)--cycle, white);
fill((2,2r)--(0,4r)--(-2,2r)--cycle, blu);
fill((0,2r)--(1,3r)--(-1,3r)--cycle, white);[/asy]
|
\frac{\sqrt{3}}{4}
|
aops_forum
| 0.0625
|
Given a point $P$ on the inscribed circle of a square $ABCD$, considering the angles $\angle APC = \alpha$ and $\angle BPD = \beta$, find the value of $\tan^2 \alpha + $\tan^2 \beta$.
|
8
|
olympiads
| 0.078125
|
Find all real $x$ in the interval $[0, 2\pi)$ such that
\[27 \cdot 3^{3 \sin x} = 9^{\cos^2 x}.\]
|
x \in \left\{ \frac{7\pi}{6}, \frac{3\pi}{2}, \frac{11\pi}{6} \right\}
|
aops_forum
| 0.1875
|
Arrange the positive integers in the following number matrix:
\begin{tabular}{lllll}
1 & 2 & 5 & 10 & $\ldots$ \\
4 & 3 & 6 & 11 & $\ldots$ \\
9 & 8 & 7 & 12 & $\ldots$ \\
16 & 15 & 14 & 13 & $\ldots$ \\
$\ldots$ & $\ldots$ & $\ldots$ & $\ldots$ & $\ldots$
\end{tabular}
What is the number in the 21st row and 21st column?
|
421
|
olympiads
| 0.078125
|
Find all natural integers \( m, n \) such that \( m^{2} - 8 = 3^{n} \).
|
(m, n) = (3, 0)
|
olympiads
| 0.171875
|
In a racing tournament with 12 stages and \( n \) participants, after each stage, all participants receive points \( a_{k} \) depending on their place \( k \) (the numbers \( a_{k} \) are natural, and \( a_{1}>a_{2}>\ldots>a_{n} \)). For what smallest \( n \) can the tournament organizer choose the numbers \( a_{1}, \ldots, a_{n} \) so that after the penultimate stage, regardless of the distribution of places, at least two participants have a chance to take the first place?
|
n = 13
|
olympiads
| 0.296875
|
Let $ABCD$ be a tetrahedron and $O$ its incenter, and let the line $OD$ be perpendicular to $AD$ . Find the angle between the planes $DOB$ and $DOC.$
|
90^
|
aops_forum
| 0.078125
|
The kindergarten received flashcards for reading: some say "MA", and the others say "NYA". Each child took three cards to form words. It turned out that 20 children can form the word "MAMA" from their cards, 30 children can form the word "NYANYA", and 40 children can form the word "MANYA". How many children have all three cards the same?
|
10
|
olympiads
| 0.15625
|
Let \( D \) be the set of divisors of 100. Let \( Z \) be the set of integers between 1 and 100, inclusive. Mark chooses an element \( d \) of \( D \) and an element \( z \) of \( Z \) uniformly at random. What is the probability that \( d \) divides \( z \)?
|
\frac{217}{900}
|
olympiads
| 0.40625
|
On the diagonals \(AC\) and \(BD\) of the trapezoid \(ABCD\), points \(M\) and \(N\) are taken respectively such that \(AM : MC = DN : NB = 1 : 4\). Find \(MN\), if the bases \(AD = a\) and \(BC = b\) with \(a > b\).
|
\frac{1}{5}(4a - b)
|
olympiads
| 0.078125
|
The volume of a parallelepiped is \( V \). Find the volume of the polyhedron whose vertices are the centers of the faces of this parallelepiped.
|
\frac{1}{6} V
|
olympiads
| 0.34375
|
Find the volume of a right square pyramid, if the side of its base is \( a \), and the dihedral angle at the base is \( \alpha \).
|
\frac{a^3}{6} \tan(\alpha)
|
olympiads
| 0.0625
|
Who participated in the robbery? It is known that out of the six gangsters, exactly two participated in the robbery. When asked who participated in the robbery, they gave the following answers:
- Harry: Charlie and George.
- James: Donald and Tom.
- Donald: Tom and Charlie.
- George: Harry and Charlie.
- Charlie: Donald and James.
Tom could not be caught. Who participated in the robbery if it is known that four of the gangsters correctly named one participant in the robbery, while one named both incorrectly?
|
Charlie and James
|
olympiads
| 0.078125
|
Each of the 25 students received a grade of either "3", "4", or "5" for their test. How many more 5s were there than 3s if the total sum of all grades is 106?
|
6
|
olympiads
| 0.421875
|
What is the last digit of the difference
$$
1 \cdot 2 \cdot 3 \cdot 4 \ldots 13 - 1 \cdot 3 \cdot 5 \cdot 7 \ldots 13 ?
$$
|
5
|
olympiads
| 0.3125
|
Using a 34-meter long rope to form a rectangle where the side lengths are integers, there are ___ different ways to form such a rectangle (rectangles with the same side lengths are considered the same way).
|
8
|
olympiads
| 0.4375
|
If the complex number \( z \) satisfies \( z^2 - 2|z| + 3 = 0 \), then \( |z| = \) .
|
1
|
olympiads
| 0.25
|
Factor the following polynomial into a product of two cubic polynomials with integer coefficients:
$$
x^{6}-x^{5}+x^{4}-x^{3}-x^{2}-x+1
$$
|
\left(x^3 - x^2 + 2x - 1\right)\left(x^3 - x - 1\right)
|
olympiads
| 0.078125
|
Yukihira is counting the minimum number of lines $m$ , that can be drawn on the plane so that they intersect in exactly $200$ distinct points.What is $m$ ?
|
21
|
aops_forum
| 0.390625
|
There are 20 cards, each with a number from 1 to 20. These cards are placed in a box, and 4 people each draw one card without replacement. The two people who draw the smaller numbers form one group, and the two people who draw the larger numbers form another group. If two people draw the numbers 5 and 14, what is the probability that these two people are in the same group? Answer in the simplest fraction form.
|
\frac{7}{51}
|
olympiads
| 0.09375
|
The number of solutions of the trigonometric equation \(\cos 7x = \cos 5x\) in the interval \([0, \pi]\) is \(\qquad\).
|
7
|
olympiads
| 0.25
|
Let $ABCDEF$ be a regular hexagon with side length $2$ . Calculate the area of $ABDE$ .
|
4\sqrt{3}
|
aops_forum
| 0.140625
|
There are nine cards, each with the numbers $2, 3, 4, 5, 6, 7, 8, 9, 10$. Four people, A, B, C, and D, each draw two of these cards.
Person A says: "The two numbers I drew are relatively prime because they are consecutive."
Person B says: "The two numbers I drew are not relatively prime and are not multiples of each other."
Person C says: "The two numbers I drew are both composite numbers and they are relatively prime."
Person D says: "The two numbers I drew are in a multiple relationship and they are not relatively prime."
Assuming all four people are telling the truth, what is the number on the remaining card? $\quad \quad$
|
7
|
olympiads
| 0.28125
|
What relationship between the properties of a parallelogram is established by the theorem: "In a rectangle, the diagonals are equal"?
|
"In a rectangle, the diagonals are equal."
|
olympiads
| 0.125
|
The difference between the minimum value of the sum of the squares of ten different odd numbers and the remainder of this minimum value divided by 4 is $\qquad$. (Note: The product of a natural number with itself is called the square of the number, such as $1 \times 1 = 1^2$, $2 \times 2 = 2^2$, $3 \times 3 = 3^3$, and so on).
|
1328
|
olympiads
| 0.171875
|
From a sphere of what is the smallest radius can a regular quadrilateral pyramid with a base edge of 14 and an apothem of 12 be cut?
|
7\sqrt{2}
|
olympiads
| 0.0625
|
Which triangle has a larger inscribed circle: the triangle with sides 17, 25, and 26, or the triangle with sides 17, 25, and 28?
|
6
|
olympiads
| 0.109375
|
In the regular tetrahedron \(ABCD\), the midpoint \(M\) is taken on the altitude \(AH\), connecting \(BM\) and \(CM\), then \(\angle BMC =\)
|
90^{\circ}
|
olympiads
| 0.0625
|
Six people, A, B, C, D, E, and F are each assigned a unique number.
A says: "The six of our numbers exactly form an arithmetic sequence."
B says: "The smallest number in this arithmetic sequence is 2."
C says: "The sum of our six numbers is 42."
D says: "The sum of A, C, and E's numbers is twice the sum of B, D, and F's numbers."
E says: "The sum of B and F's numbers is twice A's number."
What is D's number? ( )
|
D = 2
|
olympiads
| 0.0625
|
Jarris the triangle is playing in the \((x, y)\) plane. Let his maximum \(y\) coordinate be \(k\). Given that he has side lengths 6, 8, and 10 and that no part of him is below the \(x\)-axis, find the minimum possible value of \(k\).
|
\frac{24}{5}
|
olympiads
| 0.171875
|
Given \( S = \frac{2}{1 \times 3} + \frac{2^2}{3 \times 5} + \frac{2^3}{5 \times 7} + \cdots + \frac{2^{49}}{97 \times 99} \) and \( T = \frac{1}{3} + \frac{2}{5} + \frac{2^2}{7} + \cdots + \frac{2^{48}}{99} \), find \( S - T \).
|
1 - \frac{2^{49}}{99}
|
olympiads
| 0.109375
|
The natural numbers \(1, 2, 3, 4, 5, 6, 7, 8, 9\) are repeatedly written in sequence to form a 2012-digit integer. What is the remainder when this number is divided by 9?
|
6
|
olympiads
| 0.15625
|
The median $AA_{0}$ of triangle $ABC$ is extended from point $A_{0}$ perpendicularly to side $BC$ outside of the triangle. Let the other end of the constructed segment be denoted as $A_{1}$. Similarly, points $B_{1}$ and $C_{1}$ are constructed. Find the angles of triangle $A_{1}B_{1}C_{1}$, given that the angles of triangle $ABC$ are $30^{\circ}, 30^{\circ}$, and $120^{\circ}$.
|
60^
ightarrow
|
olympiads
| 0.0625
|
In a class, there are $a_{1}$ students who received at least one failing grade during the year, $a_{2}$ students who received at least two failing grades, ..., $a_{k}$ students who received at least $k$ failing grades. How many failing grades are there in total in this class?
(It is assumed that no student received more than $k$ failing grades.)
|
a_1 + a_2 + \ldots + a_k
|
olympiads
| 0.140625
|
Engineers always tell the truth, and merchants always lie. \( F \) and \( G \) are engineers. \( A \) announces that \( B \) claims that \( C \) assures that \( D \) says that \( E \) insists that \( F \) denies that \( G \) is an engineer. \( C \) also announces that \( D \) is a merchant. If \( A \) is a merchant, how many merchants are there in this company?
|
Total businessmen in the company is 3
|
olympiads
| 0.375
|
For which values of $x$ does the following inequality hold?
$$
x^{2}-2 \sqrt{x^{2}+1}-2 \geq 0 .
$$
|
x \geq 2\sqrt{2} \text{ or } x \leq -2\sqrt{2}
|
olympiads
| 0.078125
|
The side length of the base of a regular quadrilateral pyramid is \(a\). The lateral face forms an angle of \(60^\circ\) with the plane of the base. Find the radius of the inscribed sphere.
|
\frac{a \sqrt{3}}{6}
|
olympiads
| 0.09375
|
Find all positive integers \( a, b, c \) such that \( 2^a 3^b + 9 = c^2 \).
|
(4,0,5), (4,5,51), (3,3,15), (4,3,21), (3,2,9)
|
olympiads
| 0.09375
|
Vasya, whom you are familiar with from the first round, came up with $n$ consecutive natural numbers. He wrote down the sum of the digits for each number, and as a result, he also got $n$ consecutive numbers (possibly not in order). What is the maximum possible value of $n$ for which this is possible?
|
18
|
olympiads
| 0.0625
|
The number $201212200619$ has a factor $m$ such that $6 \cdot 10^9 <m <6.5 \cdot 10^9$ . Find $m$ .
|
6490716149
|
aops_forum
| 0.0625
|
Five distinct digits from 1 to 9 are given. Arnaldo forms the largest possible number using three of these 5 digits. Then, Bernaldo writes the smallest possible number using three of these 5 digits. What is the units digit of the difference between Arnaldo's number and Bernaldo's number?
|
0
|
olympiads
| 0.109375
|
Given the set \( T = \{ 9^k \mid k \text{ is an integer}, 0 \leq k \leq 4000 \} \), and knowing that \( 9^{4000} \) has 3817 digits with its leftmost digit being 9, how many elements in \( T \) have 9 as their leftmost digit?
|
184
|
olympiads
| 0.328125
|
Given $\alpha \in\left(0, \frac{\pi}{2}\right)$, find the minimum value of $\frac{\sin ^{3} \alpha}{\cos \alpha} + \frac{\cos ^{3} \alpha}{\sin \alpha}$.
|
1
|
olympiads
| 0.0625
|
Express the number $\frac{201920192019}{191719171917}$ as an irreducible fraction. In the answer, write down the denominator of the resulting fraction.
|
639
|
olympiads
| 0.140625
|
Let's call a number greater than 25 semi-prime if it is the sum of two distinct prime numbers. What is the maximum number of consecutive natural numbers that can be semi-prime?
|
5
|
olympiads
| 0.0625
|
How can two trains pass each other using the switch shown in the image and continue moving forward with locomotives in front? A small side track is sufficient only to accept either a locomotive or one wagon at a time. No tricks with ropes or flying are allowed. Each change in direction made by one locomotive is considered one move. What is the minimum number of moves?
|
14
|
olympiads
| 0.0625
|
Given a grid of squares on a plane ("graph paper"), is it possible to construct an equilateral triangle with vertices at the intersections of the grid lines?
|
\text{No, it is not possible to construct an equilateral triangle with its vertices at the intersection points of the grid.}
|
olympiads
| 0.40625
|
On the sides \( BC \) and \( AD \) of the convex quadrilateral \( ABCD \), the midpoints \( M \) and \( N \) are marked, respectively. The segments \( MN \) and \( AC \) intersect at point \( O \), such that \( MO = ON \). It is known that the area of triangle \( ABC \) is 2019. Find the area of the quadrilateral \( ABCD \).
|
4038
|
olympiads
| 0.296875
|
A water tank with a capacity of $120 \mathrm{hl}$ has three pipes: the first two fill the tank, and the third one drains it. The first pipe fills the tank one hour faster than the second. When the first and third pipes are open, the tank fills in $7 \frac{1}{2}$ hours. When the second and third pipes are open, the tank fills in 20 hours. How many hours does it take to fill the tank if all three pipes are open simultaneously?
|
\frac{60}{23} \, \text{hours}
|
olympiads
| 0.078125
|
Find the least positive integer $N$ which is both a multiple of 19 and whose digits add to 23.
|
779
|
aops_forum
| 0.21875
|
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty} \sqrt{n+2}(\sqrt{n+3}-\sqrt{n-4})$$
|
\frac{7}{2}
|
olympiads
| 0.390625
|
If the ellipse $x^{2} + 4(y-a)^{2} = 4$ and the parabola $x^{2} = 2y$ have common points, what is the range of the real number $a$?
|
-1 \leq a \leq \frac{17}{8}
|
olympiads
| 0.125
|
In the cells of an $8 \times 8$ chessboard, there are 8 white and 8 black pieces arranged such that no two pieces are in the same cell. Additionally, no pieces of the same color are in the same row or column. For each white piece, the distance to the black piece in the same column is calculated. What is the maximum value that the sum of these distances can take? The distance between the pieces is considered to be the distance between the centers of the cells they occupy.
|
32
|
olympiads
| 0.0625
|
Lucía multiplies some positive one-digit numbers (not necessarily distinct) and obtains a number $n$ greater than 10. Then, she multiplies all the digits of $n$ and obtains an odd number. Find all possible values of the units digit of $n$ . $\textit{Proposed by Pablo Serrano, Ecuador}$
|
5
|
aops_forum
| 0.0625
|
Calculate the limit of the function:
$$\lim _{x \rightarrow \frac{\pi}{2}}(\sin x)^{\frac{18 \sin x}{\operatorname{ctg} x}}$$
|
1
|
olympiads
| 0.53125
|
The class studies 12 subjects. In how many ways can 5 different subjects be scheduled on Monday?
|
95040
|
olympiads
| 0.125
|
Given an integer \( n \geqslant 2 \). Let \( a_{1}, a_{2}, \cdots, a_{n} \) and \( b_{1}, b_{2}, \cdots, b_{n} \) be positive numbers that satisfy
\[ a_{1} + a_{2} + \cdots + a_{n} = b_{1} + b_{2} + \cdots + b_{n}, \]
and for any \( i, j \) ( \( 1 \leqslant i < j \leqslant n \)), it holds that \( a_{i}a_{j} \geqslant b_{i} + b_{j} \). Find the minimum value of \( a_{1} + a_{2} + \cdots + a_{n} \).
|
2n
|
olympiads
| 0.125
|
In an equilateral triangle, 3 points are marked on each side so that they divide the side into 4 equal segments. These points and the vertices of the triangle are colored. How many isosceles triangles with vertices at the colored points exist?
|
18
|
olympiads
| 0.0625
|
The plane vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ satisfy $|\boldsymbol{a}|=1, \boldsymbol{a} \cdot \boldsymbol{b}=\boldsymbol{b} \cdot \boldsymbol{c}=1,$ and $|\boldsymbol{a}-\boldsymbol{b}+\boldsymbol{c}| \leqslant 2 \sqrt{2}$. Find the maximum value of $\boldsymbol{a} \cdot \boldsymbol{c}$.
|
2
|
olympiads
| 0.078125
|
The binomial expansion of $\left(\sqrt{x}+\frac{1}{2 \sqrt[4]{x}}\right)^{n}$ is arranged in descending powers of $x$. If the coefficients of the first three terms form an arithmetic sequence, then there are ___ terms in the expansion in which the exponent of $x$ is an integer.
|
3
|
olympiads
| 0.1875
|
The radius of the circumscribed circle around a right triangle is \( 15 \) cm, and the radius of the inscribed circle inside it is \( 6 \) cm. Find the sides of the triangle.
|
18, 24, 30 \, \text{cm}
|
olympiads
| 0.140625
|
The minimum value of the maximum value $M(a)$ of the function $f(x)=\left|x^{2}-a\right|$ on the interval $-1 \leq x \leq 1$ is
|
\frac{1}{2}
|
olympiads
| 0.171875
|
Kolya, Lenya, and Misha pooled their money and bought a soccer ball. The amount of money each of them contributed does not exceed half of the total amount contributed by the other two. How much money did Misha contribute if the ball cost 6 rubles?
|
2
|
olympiads
| 0.46875
|
The equation \(x^{2} + px > 4x + p - 3\) holds for all \(0 \leq p \leq 4\). Find the range of the real number \(x\).
|
x < 1 \text{ or } x > 3
|
olympiads
| 0.078125
|
In a triangle, at which interior point is the product of the distances to the sides maximized?
|
Centroid
|
olympiads
| 0.09375
|
Two parabolas with distinct vertices are the graphs of quadratic polynomials with leading coefficients $p$ and $q$. It is known that the vertex of each parabola lies on the other parabola. What could be the value of $p+q$?
|
0
|
olympiads
| 0.34375
|
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