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|---|---|---|---|
Find the largest constant $k$ such that the inequality $$ a^2+b^2+c^2-ab-bc-ca \ge k \left|\frac{a^3-b^3}{a+b}+\frac{b^3-c^3}{b+c}+\frac{c^3-a^3}{c+a}\right| $$ holds for any for non negative real numbers $a,b,c$ with $(a+b)(b+c)(c+a)>0$ .
|
k = \frac{1}{2}
|
aops_forum
| 0.0625
|
We consider a chessboard of size $8 \times 8$ where the squares are alternately colored white and black. An infernal rook is a piece that can attack the squares of its color located on its row, as well as the squares of the other color located in its column. What is the maximum number of infernal rooks that can be placed on the chessboard such that no two infernal rooks can ever attack each other?
|
16
|
olympiads
| 0.203125
|
Given a triangle \( \triangle ABC \) with circumradius \( R \), if the expression \( \frac{a \cos A + b \cos B + c \cos C}{a \sin B + b \sin C + c \sin A} = \frac{a + b + c}{9R} \), where \( a, b, c \) are the lengths of the sides opposite to angles \( \angle A, \angle B, \angle C \) respectively, determine the measures of the three internal angles of \( \triangle ABC \).
|
A = B = C = 60^
|
olympiads
| 0.078125
|
For which $n$ can an $n \times n$ grid be divided into one $2 \times 2$ square and some number of strips consisting of five cells, in such a way that the square touches the side of the board?
|
n \equiv 2 \pmod{5}
|
olympiads
| 0.171875
|
Three cans of juice fill $\frac{2}{3}$ of a one-litre jug. How many cans of juice are needed to completely fill 8 one-litre jugs?
|
36
|
olympiads
| 0.484375
|
The equation $\left\{\begin{array}{l}x=\frac{1}{2}\left(e^{t}+e^{-t}\right) \cos \theta, \\ y=\frac{1}{2}\left(e^{t}-e^{-t}\right) \sin \theta\end{array}\right.$ where $\theta$ is a constant $\left(\theta \neq \frac{n}{2} \pi, n \in \mathbf{Z}\right)$ and $t$ is a parameter, represents what shape?
|
Hyperbola
|
olympiads
| 0.21875
|
A full can of milk weighs 34 kg, and a half-filled can weighs 17.5 kg. How much does the empty can weigh?
|
1 \, \text{kg}
|
olympiads
| 0.078125
|
Let \(a\) and \(b\) be real numbers such that the function \(f(x) = ax + b\) satisfies \(|f(x)| \leq 1\) for any \(x \in [0,1]\). Determine the maximum value of \(ab\).
|
\frac{1}{4}
|
olympiads
| 0.1875
|
Given the even numbers arranged in the pattern shown below, determine in which column the number 2008 will appear.
```
2 4 6 8
16 14 12 10
18 20 22 24
32 30 28 26
...
```
|
4
|
olympiads
| 0.0625
|
In a regular quadrilateral pyramid, the dihedral angle at the lateral edge is $120^{\circ}$. Find the lateral surface area of the pyramid, if the area of its diagonal section is $S$.
|
4S
|
olympiads
| 0.0625
|
Let $P(n) = (n + 1)(n + 3)(n + 5)(n + 7)(n + 9)$ . What is the largest integer that is a divisor of $P(n)$ for all positive even integers $n$ ?
|
15
|
aops_forum
| 0.28125
|
Given an arithmetic sequence \(a_1, a_2, \cdots, a_{1000}\), the sum of the first 100 terms is 100, and the sum of the last 100 terms is 1000. Find \(a_1\).
|
\frac{101}{200}
|
olympiads
| 0.203125
|
Find all natural numbers \( n \) such that every natural number formed by one digit 7 and \( n-1 \) digits 1 in its decimal representation is a prime number.
|
1, 2
|
olympiads
| 0.109375
|
There is an urn containing 9 slips of paper numbered from 1 to 9. We draw one slip, record the number, and put it back in the urn. This process is repeated 8 more times, writing the drawn single-digit numbers side by side from left to right. What is the probability that in the resulting 9-digit number, at least one digit will appear in its own position? (Own position; counting from left to right, the $n$-th digit being in the $n$-th position).
|
0.653
|
olympiads
| 0.0625
|
A circle with radius $r$ is inscribed in a triangle with perimeter $p$ and area $S$. How are these three quantities related?
|
S = \frac{1}{2} r p
|
olympiads
| 0.296875
|
Given a set of points in space, a *jump* consists of taking two points, $P$ and $Q,$ and replacing $P$ with the reflection of $P$ over $Q$ . Find the smallest number $n$ such that for any set of $n$ lattice points in $10$ -dimensional-space, it is possible to perform a finite number of jumps so that some two points coincide.
|
1025
|
aops_forum
| 0.0625
|
Let $x$ and $y$ be real numbers satisfying the equation $x^2-4x+y^2+3=0$ . If the maximum and minimum values of $x^2+y^2$ are $M$ and $m$ respectively, compute the numerical value of $M-m$ .
|
8
|
aops_forum
| 0.21875
|
Let $A,B,C,...,Z$ be $26$ nonzero real numbers. Suppose that $T=TNYWR$ . Compute the smallest possible value of \[
\left\lceil A^2+B^2+\cdots+Z^2\right\rceil .
\] (The notation $\left\lceil x\right\rceil$ denotes the least integer $n$ such that $n\geq x$ .)
|
26
|
aops_forum
| 0.140625
|
There is an urn with 10 balls, among which there are 2 red, 5 blue, and 3 white balls. Find the probability that a randomly drawn ball will be a colored one (event $A$).
|
0.7
|
olympiads
| 0.140625
|
A parallelepiped is inscribed in a sphere of radius $\sqrt{3}$, and its volume is 8. Find the total surface area of the parallelepiped.
|
24
|
olympiads
| 0.078125
|
Find the number of pairs $(n, q)$ , where $n$ is a positive integer and $q$ a non-integer rational number with $0 < q < 2000$ , that satisfy $\{q^2\}=\left\{\frac{n!}{2000}\right\}$
|
1
|
aops_forum
| 0.125
|
Alina and Masha wanted to create an interesting version of the school tour of the Olympiad. Masha proposed several problems and rejected every second problem of Alina (exactly half). Alina also proposed several problems and only accepted every third problem of Masha (exactly one-third). In the end, there were 10 problems, and initially, 27 problems were proposed. How many more problems did Masha propose compared to Alina?
|
15
|
olympiads
| 0.25
|
Find the number of ordered pairs of integers $(x, y)$ such that $$ \frac{x^2}{y}- \frac{y^2}{x}= 3 \left( 2+ \frac{1}{xy}\right) $$
|
0
|
aops_forum
| 0.09375
|
On an island, there are only knights, who always tell the truth, and liars, who always lie. Once, 99 inhabitants of this island stood in a circle, and each of them said: "All ten people following me in a clockwise direction are liars." How many knights could there be among those who stood in the circle?
|
9 knights
|
olympiads
| 0.09375
|
Nine integers from 1 to 5 are written on a board. It is known that seven of them are at least 2, six are greater than 2, three are at least 4, and one is at least 5. Find the sum of all the numbers.
|
26
|
olympiads
| 0.0625
|
Angry reviews about the operation of an online store are left by $80\%$ of dissatisfied customers (those who were poorly served in the store). Among satisfied customers, only $15\%$ leave a positive review. A certain online store received 60 angry and 20 positive reviews. Using this statistics, estimate the probability that the next customer will be satisfied with the service in this online store.
|
0.64
|
olympiads
| 0.078125
|
On an island, there live knights, liars, and yes-men; each knows who is who among them. In a row, 2018 island inhabitants were asked the question: "Are there more knights than liars on the island?" Each inhabitant answered either "Yes" or "No" in turn such that everyone else could hear their response. Knights always tell the truth, liars always lie. Each yes-man answered according to the majority of responses given before their turn, or gave either response if there was a tie. It turned out that exactly 1009 answered "Yes." What is the maximum number of yes-men that could be among the island inhabitants?
|
1009 sycophants
|
olympiads
| 0.15625
|
Calculate: $2468 \times 629 \div (1234 \times 37)=$
|
34
|
olympiads
| 0.21875
|
Find the smallest positive integer \( n \) such that for any 5 vertices colored red in an \( n \)-sided polygon \( S \), there is always an axis of symmetry \( l \) of \( S \) for which the symmetric point of each red vertex with respect to \( l \) is not red.
|
14
|
olympiads
| 0.265625
|
From an $m \times n$ chessboard, remove an $L$ shape consisting of three squares (as shown in the right figure). How many different ways can this be done?
|
4(m-1)(n-1)
|
olympiads
| 0.109375
|
Two friends agree to meet at a specific place between 12:00 PM and 12:30 PM. The first one to arrive waits for the other for 20 minutes before leaving. Find the probability that the friends will meet, assuming each chooses their arrival time randomly (between 12:00 PM and 12:30 PM) and independently.
|
\frac{8}{9}
|
olympiads
| 0.09375
|
An isosceles triangle $ ABC$ with $ AB \equal{} AC$ is given on the plane. A variable circle $ (O)$ with center $ O$ on the line $ BC$ passes through $ A$ and does not touch either of the lines $ AB$ and $ AC$ . Let $ M$ and $ N$ be the second points of intersection of $ (O)$ with lines $ AB$ and $ AC$ , respectively. Find the locus of the orthocenter of triangle $ AMN$ .
|
BC
|
aops_forum
| 0.21875
|
Calculate the sum of the following series:
$$
\frac{1}{1}+\frac{2}{2}+\frac{3}{4}+\frac{4}{8}+\frac{5}{16}+\ldots+\frac{n}{2^{n-1}}
$$
What does the sum approach as the number of terms increases without bound?
|
4
|
olympiads
| 0.515625
|
For any positive integer \( k \), let \( f_1(k) \) be defined as the square of the sum of the digits of \( k \). For \( n \geq 2 \), let \( f_n(k) = f_1(f_{n-1}(k)) \). Find \( f_{1988}(11) \).
|
169
|
olympiads
| 0.171875
|
Find the minimum value of the function
$$
f(x) = 7 \sin^2(x) + 5 \cos^2(x) + 2 \sin(x)
$$
|
4.5
|
olympiads
| 0.5
|
In the 6 plots of land shown in Figure 3-11, plant either type A or type B vegetables (you can plant only one type or both). The requirement is that two adjacent plots should not both have type A vegetables. Determine the total number of possible planting schemes.
|
21
|
olympiads
| 0.09375
|
Positive real numbers $a$ and $b$ verify $a^5+b^5=a^3+b^3$ . Find the greatest possible value of the expression $E=a^2-ab+b^2$ .
|
1
|
aops_forum
| 0.3125
|
Mrs. Lígia has a square-shaped plot. She decides to divide it into five regions: four rectangles and one square, as illustrated in the figure below:
In the figure above:
- The central square has an area of \(64 \, \text{m}^2\);
- The longer sides of the four rectangles have the same length;
- The five regions have the same perimeter.
Determine the area of Mrs. Lígia's plot.
|
256 \text{ m}^2
|
olympiads
| 0.109375
|
Water makes up 80 per cent of fresh mushrooms. However, water makes up only 20 per cent of dried mushrooms. By what percentage does the mass of a fresh mushroom decrease during drying?
|
75\%
|
olympiads
| 0.1875
|
In a unit cube \(ABCDA_1B_1C_1D_1\), eight planes \(AB_1C, BC_1D, CD_1A, DA_1B, A_1BC_1, B_1CD_1, C_1DA_1,\) and \(D_1AB_1\) intersect the cube. What is the volume of the part that contains the center of the cube?
|
\frac{1}{6}
|
olympiads
| 0.125
|
The angle $A$ in triangle $ABC$ is $\alpha$. A circle passing through $A$ and $B$ and tangent to $BC$ intersects the median to side $BC$ (or its extension) at point $M$, distinct from $A$. Find $\angle BMC$.
|
180^
\circ - \alpha
|
olympiads
| 0.0625
|
The lines \( y = x \) and \( y = mx - 4 \) intersect at the point \( P \). What is the sum of the positive integer values of \( m \) for which the coordinates of \( P \) are also positive integers?
|
10
|
olympiads
| 0.140625
|
Transform the expression
\[ x^{2}+2xy+5y^{2}-6xz-22yz+16z^{2} \]
into the algebraic sum of the complete squares of a trinomial, a binomial, and a monomial.
|
\left[ x + (y - 3z) \right]^2 + (2y - 4z)^2 - (3z)^2
|
olympiads
| 0.125
|
Determine the area inside the circle with center \(\left(1, -\frac{\sqrt{3}}{2}\right)\) and radius 1 that lies inside the first quadrant.
|
\frac{1}{6}\pi - \frac{\sqrt{3}}{4}
|
olympiads
| 0.0625
|
$\textbf{Problem C.1}$ There are two piles of coins, each containing $2010$ pieces. Two players $A$ and $B$ play a game taking turns ( $A$ plays first). At each turn, the player on play has to take one or more coins from one pile or exactly one coin from each pile. Whoever takes the last coin is the winner. Which player will win if they both play in the best possible way?
|
B
|
aops_forum
| 0.3125
|
Misha wrote on the board 2004 pluses and 2005 minuses in some order. From time to time, Yura comes to the board, erases any two signs, and writes one in their place. If he erases two identical signs, he writes a plus; if the signs are different, he writes a minus. After several such actions, only one sign remains on the board. What is the final sign?
|
-
|
olympiads
| 0.390625
|
Using eight $2 \times 1$ small rectangles as shown on the right, you can form a $4 \times 4$ square. If one formed square is considered the same as another formed square after rotation, then the two formed squares are considered the same. In all possible formed square patterns, how many of these patterns are vertically symmetrical and have two adjacent blank small squares in the first row?
|
5
|
olympiads
| 0.109375
|
In $\triangle \mathrm{ABC}$, $\mathrm{M}$ is the midpoint of side $\mathrm{AB}$, and $\mathrm{N}$ is the trisection point of side $\mathrm{AC}$. $\mathrm{CM}$ and $\mathrm{BN}$ intersect at point K. If the area of $\triangle \mathrm{BCK}$ is equal to 1, then what is the area of $\triangle \mathrm{ABC}$?
|
4
|
olympiads
| 0.0625
|
For how many ordered pairs of positive integers \((a, b)\) is \(1 < a + b < 22\)?
|
210
|
olympiads
| 0.21875
|
Calculate the sum
$$
S=\frac{2013}{2 \cdot 6}+\frac{2013}{6 \cdot 10}+\frac{2013}{10 \cdot 14}+\ldots+\frac{2013}{2010 \cdot 2014}
$$
Provide the remainder when the even number nearest to the value of $S$ is divided by 5.
|
2
|
olympiads
| 0.140625
|
A construction rope is tied to two trees. It is straight and taut. It is then vibrated at a constant velocity $v_1$ . The tension in the rope is then halved. Again, the rope is vibrated at a constant velocity $v_2$ . The tension in the rope is then halved again. And, for the third time, the rope is vibrated at a constant velocity, this time $v_3$ . The value of $\frac{v_1}{v_3}+\frac{v_3}{v_1}$ can be expressed as a positive number $\frac{m\sqrt{r}}{n}$ , where $m$ and $n$ are relatively prime, and $r$ is not divisible by the square of any prime. Find $m+n+r$ . If the number is rational, let $r=1$ .
|
8
|
aops_forum
| 0.328125
|
With how many dice is the probability the greatest that exactly one six will be rolled when the dice are thrown simultaneously?
|
n=5 \text{ or } n=6
|
olympiads
| 0.5
|
Let $[x]$ denote the greatest integer less than or equal to the real number $x$. Then the solutions to the equation $[\tan x] = 2 \cos^{2} x$ are $\qquad$
|
\frac{\pi}{4}
|
olympiads
| 0.0625
|
Find the sum of the squares of the distances from a point \( M \), taken on the diameter of some circle, to the ends of any chord parallel to this diameter, if the radius of the circle is \( R \) and the distance from point \( M \) to the center of the circle is \( a \).
|
2(a^2 + R^2)
|
olympiads
| 0.203125
|
## Problem
A triangle $AMB$ is circumscribed by a circle whose center is 10 units away from side $AM$. The extension of side $AM$ beyond vertex $M$ intersects the tangent to the circle through vertex $B$, creating a segment $CB$ with a length of 29. Find the area of triangle $CMB$, given that angle $ACB$ is $\operatorname{arctg} \frac{20}{21}$.
|
105
|
olympiads
| 0.0625
|
In the pyramid \( A B C D \), the angle \( A B C \) is \( \alpha \). The orthogonal projection of point \( D \) onto the plane \( A B C \) is point \( B \). Find the angle between the planes \( A B D \) and \( C B D \).
|
\alpha
|
olympiads
| 0.109375
|
A and B are jogging on a 300-meter track. They start at the same time and in the same direction, with A starting behind B. A overtakes B for the first time after 12 minutes, and for the second time after 32 minutes. Assuming both maintain constant speeds, how many meters behind B did A start?
|
180 \text{ meters}
|
olympiads
| 0.0625
|
What is the smallest positive integer $n$ such that $2^n - 1$ is a multiple of $2015$ ?
|
60
|
aops_forum
| 0.34375
|
Two unit squares are placed in the plane such that the center of one square is at the vertex of the other square. What is the area of the intersection of the two squares?
|
\frac{1}{4}
|
olympiads
| 0.53125
|
A bag contains nine blue marbles, ten ugly marbles, and one special marble. Ryan picks marbles randomly from this bag with replacement until he draws the special marble. He notices that none of the marbles he drew were ugly. Given this information, what is the expected value of the number of total marbles he drew?
|
\frac{20}{11}
|
olympiads
| 0.09375
|
If set \( A = \{1, 2, \cdots, n\} \) is partitioned arbitrarily into 63 pairwise disjoint subsets (they are non-empty, and their union is \( A \)), \( A_1, A_2, \cdots, A_{63} \), then there always exist two positive integers \( x \) and \( y \) belonging to the same subset \( A_i \) (\(1 \leqslant i \leqslant 63\)) such that \( x > y \) and \( 31x \leqslant 32y \). Find the smallest positive integer \( n \) that satisfies the condition.
|
2016
|
olympiads
| 0.078125
|
One of the angles of a parallelogram is $50^{\circ}$ less than the other. Find the angles of the parallelogram.
|
65^\circ, 115^\circ, 65^\circ, 115^\circ
|
olympiads
| 0.40625
|
Find the coordinates of point \( A \), which is equidistant from points \( B \) and \( C \).
\[ A(x; 0; 0) \]
\[ B(4; 6; 8) \]
\[ C(2; 4; 6) \]
|
(15, 0, 0)
|
olympiads
| 0.3125
|
Calculate the areas of the figures bounded by the lines given in polar coordinates.
$$
r=\cos \phi, \quad r=\sin \phi, \quad \left(0 \leq \phi \leq \frac{\pi}{2}\right)
$$
|
\frac{1}{2}
|
olympiads
| 0.09375
|
The natural number $n>1$ is called “heavy”, if it is coprime with the sum of its divisors. What’s the maximal number of consecutive “heavy” numbers?
|
4
|
aops_forum
| 0.109375
|
The coefficients \( p \) and \( q \) of the quadratic equation \( x^{2}+p x+q=0 \) are chosen at random in the interval \( (0, 2) \). What is the probability that the roots of this equation will be real numbers?
|
\frac{1}{6}
|
olympiads
| 0.484375
|
Find all solutions to the equation \(x^{2} - 8 \lfloor x \rfloor + 7 = 0\), where \(\lfloor x \rfloor\) denotes the greatest integer less than or equal to \(x\).
|
1, \sqrt{33}, \sqrt{41}, 7
|
olympiads
| 0.0625
|
Given four points on a plane such that the pairwise distances between them take exactly two different values, $a$ and $b$. Determine the possible values of $a / b$, given that $a > b$.
|
\sqrt{2}, \frac{1 + \sqrt{5}}{2}, \sqrt{3}, \sqrt{2 + \sqrt{3}}
|
olympiads
| 0.234375
|
Let \( p(z) \equiv z^6 + 6z + 10 \). How many roots lie in each quadrant of the complex plane?
|
1 \text{ root in each of the first and fourth quadrants, and 2 in each of the second and third quadrants.}
|
olympiads
| 0.1875
|
After a ballroom dancing competition involving 7 boys and 8 girls, each participant named the number of their partners: $3,3,3,3,3,5,6,6,6,6,6,6,6,6,6$. Did anyone make a mistake?
|
Someone made a mistake.
|
olympiads
| 0.078125
|
H is the orthocenter of the acute-angled triangle ABC. It is known that HC = AB. Find the angle ACB.
|
heta ACB = 45^\circ
|
olympiads
| 0.28125
|
All natural numbers from 999 to 1 were written consecutively without spaces in descending order: 999998 ...321. Which digit is in the 2710th position in the resulting long number?
|
9
|
olympiads
| 0.25
|
Two players play on a $19 \times 94$ grid. Players take turns marking a square along the grid lines (of any possible size) and shading it. The one who shades the last cell wins. Cells cannot be shaded more than once. Who will win with optimal play and how should they play?
|
The first player wins.
|
olympiads
| 0.078125
|
The room temperature $T$ (in degrees Celsius) as a function of time $t$ (in hours) is given by: $T = a \sin t + b \cos t, \ t \in (0, +\infty)$, where $a$ and $b$ are positive real numbers. If the maximum temperature difference in the room is 10 degrees Celsius, what is the maximum value of $a + b$?
|
5\sqrt{2}
|
olympiads
| 0.203125
|
Compute the limit of the numerical sequence:
$$
\lim _{n \rightarrow \infty}\left(\frac{n^{3}+n+1}{n^{3}+2}\right)^{2 n^{2}}
$$
|
e^2
|
olympiads
| 0.28125
|
The Group of Twenty (G20) is an international economic cooperation forum with 20 member countries. These members come from Asia, Europe, Africa, Oceania, and America. The number of members from Asia is the highest, and the numbers from Africa and Oceania are equal and the least. The number of members from America, Europe, and Asia are consecutive natural numbers. How many members of the G20 are from Asia?
|
7
|
olympiads
| 0.34375
|
For any positive integer \( n \), the sequence \( \{a_n\} \) satisfies \(\sum_{i=1}^{n} a_i = n^3\). Find the value of \(\sum_{i=2}^{2009} \frac{1}{a_i - 1}\).
|
\frac{2008}{6027}
|
olympiads
| 0.203125
|
The number 6 has 4 divisors: 1, 2, 3, and 6. Determine the sum of all numbers between 1 and 1000 that have exactly 7 divisors.
|
793
|
olympiads
| 0.09375
|
In a certain kingdom, there were 32 knights. Some of them were vassals to others (a vassal can only have one liege, and the liege is always wealthier than the vassal). A knight with at least four vassals held the title of baron. What is the maximum number of barons that could exist under these conditions?
(In the kingdom, the law stated: "the vassal of my vassal is not my vassal.")
|
7
|
olympiads
| 0.0625
|
In Cologne, there were three brothers who had 9 containers of wine. The first container had a capacity of 1 quart, the second contained 2 quarts, with each subsequent container holding one more quart than the previous one, so the last container held 9 quarts. The task is to divide the wine equally among the three brothers without transferring wine between the containers.
|
15 \text{ quarts per brother}
|
olympiads
| 0.484375
|
- On the table are a banana, an orange, a watermelon, a kiwi, and an apple.
- The watermelon weighs more than the remaining four items.
- The orange and the kiwi together weigh the same as the banana and the apple together.
- The orange weighs more than the banana but less than the kiwi.
Match the items to their weights in grams: 210, 180, 200, 170, 1400.
Items: banana, orange, watermelon, kiwi, and apple.
|
Apple: 210g, Orange: 180g, Watermelon: 1400g, Kiwi: 200g, Banana: 170g
|
olympiads
| 0.09375
|
After completing a test, students Volodya, Sasha, and Petya reported their grades at home:
Volodya: "I got a 5."
Sasha: "I got a 3."
Petya: "I did not get a 5."
After verification, it was found that one of the boys received a grade of "4", another got "3", and the third got "5". What grade did each boy receive, given that two of them correctly reported their grade and one was mistaken?
|
\text{Volodya} = 5, \\ \text{Sasha} = 4, \\ \text{Petya} = 3
|
olympiads
| 0.25
|
Given that \(\sum_{y=1}^{n} \frac{1}{y}=\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}\), find \(\sum_{y=3}^{10} \frac{1}{y-2}-\sum_{y=3}^{10} \frac{1}{y-1}\). (Express your answer in fraction.)
|
\frac{8}{9}
|
olympiads
| 0.359375
|
Simplify the expression $\log_{a+b} m + \log_{a-b} m - 2 \log_{a+b} m \cdot \log_{a-b} m$, given that $m^2 = a^2 - b^2$.
|
0
|
olympiads
| 0.15625
|
Find all natural numbers \(a\) for which the number \(a^2 - 10a + 21\) is prime.
|
2, 8
|
olympiads
| 0.546875
|
Represent the number 1944 as the sum of the cubes of four integers.
|
1944=325^3+323^3+(-324)^3+(-324)^3
|
olympiads
| 0.15625
|
A sleeping passenger. When a passenger had traveled half of the entire journey, he fell asleep and slept until half of the distance he traveled while sleeping remained. What fraction of the journey did he travel while sleeping?
|
\frac{1}{3}
|
olympiads
| 0.140625
|
At nine o'clock in the morning, Paul set off on his bicycle from point $A$ to point $B$ at a speed of 15 km/h. At a quarter to ten, Peter set off from $B$ to $A$ at a speed of 20 km/h. They agreed to have a picnic halfway and fulfilled this condition exactly. At what time did they meet?
|
12:00 \text{ PM}
|
olympiads
| 0.109375
|
Given the quadratic polynomial \( f(x) = ax^2 - ax + 1 \), it is known that \( |f(x)| \leq 1 \) for all \( x \in [0, 1] \). What is the maximum value that \( a \) can take?
|
8
|
olympiads
| 0.265625
|
Let the real numbers \(a_{1}, a_{2}, \cdots, a_{2016}\) satisfy
$$
\begin{array}{l}
9 a_{i}>11 a_{i+1}^{2} \text{ for } i=1,2, \cdots, 2015.
\end{array}
$$
Find the maximum value of
$$
\left(a_{1}-a_{2}^{2}\right)\left(a_{2}-a_{3}^{2}\right) \cdots\left(a_{2015}-a_{2016}^{2}\right)\left(a_{2016}-a_{1}^{2}\right).
$$
|
\frac{1}{4^{2016}}
|
olympiads
| 0.0625
|
Howard chooses \( n \) different numbers from the list \{2, 3, 4, 5, 6, 7, 8, 9, 10, 11\}, so that no two of his choices add up to a square. What is the largest possible value of \( n \)?
|
7
|
olympiads
| 0.109375
|
What is the smallest natural number \( k \) for which the expression \( 2019 \cdot 2020 \cdot 2021 \cdot 2022 + k \) is a square of a natural number?
|
1
|
olympiads
| 0.09375
|
A cross-section of a river is a trapezoid with bases 10 and 16 and slanted sides of length 5. At this section, the water is flowing at \(\pi \mathrm{mph}\). A little ways downstream is a dam where the water flows through 4 identical circular holes at \(16 \mathrm{mph}\). What is the radius of the holes?
|
\frac{\sqrt{13}}{4}
|
olympiads
| 0.21875
|
An integer has two prime divisors. It has 6 divisors in total and the sum of its divisors is 28. What is this number?
|
12
|
olympiads
| 0.171875
|
The base of a triangular pyramid is a right triangle with legs \(a\) and \(b\). The lateral edges are equal to \(l\). Find the height of the pyramid.
|
\frac{1}{2} \sqrt{4l^2 - a^2 - b^2}
|
olympiads
| 0.3125
|
Given the sequence $\left\{a_{n}\right\}$:
$$
a_{0}=1, \quad a_{n}=2 \sum_{i=0}^{n-1} a_{i} \quad (n \geqslant 1).
$$
Find the largest positive integer $n$ such that $a_{n} \leqslant 2018$.
|
7
|
olympiads
| 0.125
|
Find all numbers $x \in \mathbb Z$ for which the number
\[x^4 + x^3 + x^2 + x + 1\]
is a perfect square.
|
-1, 0, 3
|
aops_forum
| 0.0625
|
A sequence \( a_{1}, a_{2}, a_{3}, \ldots \) of positive reals satisfies \( a_{n+1} = \sqrt{\frac{1 + a_{n}}{2}} \). Determine all \( a_{1} \) such that \( a_{i} = \frac{\sqrt{6} + \sqrt{2}}{4} \) for some positive integer \( i \).
|
\frac{\sqrt{2} + \sqrt{6}}{2}, \frac{\sqrt{3}}{2}, \frac{1}{2}
|
olympiads
| 0.078125
|
Determine all pairs \((a, b)\) of real numbers such that \(10, a, b, ab\) is an arithmetic progression.
|
(4, -2), (2.5, -5)
|
olympiads
| 0.3125
|
Six cottages are situated along a 27 km road that surrounds an abandoned and deserted area. These cottages are positioned such that the distances between them range from 1 km up to 26 km inclusively. For example, Brown could be 1 km away from Stiggins, Jones could be 2 km away from Rogers, Wilson could be 3 km away from Jones, and so forth. The inhabitants can travel to each other either clockwise or counterclockwise. Can you determine the positions of the cottages so that the conditions of the problem are satisfied? The drawing is intentionally not to be used as a hint.
|
1,1,4,4,3,14
|
olympiads
| 0.078125
|
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