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stringlengths 33
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|---|---|---|---|
Solve the equation with one unknown and verify the solution by substitution. By doing this, are you only ensuring that no gross errors were made during the calculations, or could there be a more significant goal?
|
x = 2 \text{ or } x = 3
|
olympiads
| 0.296875
|
Calculate the area of the parallelogram formed by vectors \( a \) and \( b \).
Given:
\[ a = 3p + q \]
\[ b = p - 2q \]
\[ |p| = 4 \]
\[ |q| = 1 \]
\[ \widehat{(\widehat{p}, q}) = \frac{\pi}{4} \]
|
14\sqrt{2}
|
olympiads
| 0.1875
|
Can you place the numbers from 1 to 8 inside the circles, without repeating them, so that the calculations horizontally and vertically are correct?
Hint: What are the possibilities for multiplication? What are the possible places for the number 1?
|
Option\: 1\ OR \ Option\: 2
|
olympiads
| 0.0625
|
In $\triangle ABC$, find the value of $a^{3} \sin (B-C) + b^{3} \sin (C-A) + c^{3} \sin (A-B)$.
|
0
|
olympiads
| 0.34375
|
Let \(\lambda=\{x \mid x=a+b \sqrt{3}, a, b \in \mathbf{Z}\}\). If \(x=7+a \sqrt{3} \in \lambda\), and \(\frac{1}{x} \in \lambda\), find the value of \(a\).
|
\pm 4
|
olympiads
| 0.09375
|
In the plane, three lines are given. How many circles exist that are tangent to all three lines? Explore all possible cases.
|
4
|
olympiads
| 0.0625
|
For which \( n \) is the polynomial \( 1 + x^2 + x^4 + \ldots + x^{2n-2} \) divisible by \( 1 + x + x^2 + \ldots + x^{n-1} \)?
|
For odd n
|
olympiads
| 0.078125
|
There are 39 non-zero numbers written in a row. The sum of each pair of neighboring numbers is positive, and the sum of all the numbers is negative.
What is the sign of the product of all the numbers?
|
The product of all 39 numbers is positive.
|
olympiads
| 0.1875
|
Diana wrote a two-digit number and then wrote another two-digit number which is a permutation of the digits of the first number. It turned out that the difference between the first and the second number is equal to the sum of the digits of the first number. What is the four-digit number written?
|
5445
|
olympiads
| 0.09375
|
A rectangular prism $ABCD-A^{\prime}B^{\prime}C^{\prime}D^{\prime}$ has dimensions $a$, $b$, $c$ (length, width, height). The intersection of two tetrahedra $AB^{\prime}CD^{\prime}$ and $A^{\prime}DC^{\prime}B$ is a polyhedron $V$. What is the volume of $V$?
|
\frac{1}{6}abc
|
olympiads
| 0.234375
|
A two-digit number, when multiplied by 109, yields a four-digit number. It is divisible by 23, and the quotient is a one-digit number. The maximum value of this two-digit number is $\qquad$.
|
69
|
olympiads
| 0.125
|
$\overline{1 \mathrm{abc}}$ is a four-digit number, and this four-digit number can be divided by $2, 3,$ and $5$ without leaving a remainder. What is the minimum value of $\overline{1 \mathrm{abc}}$?
|
1020
|
olympiads
| 0.265625
|
In triangle \(ABC\), \(\angle A = 60^\circ\). Points \(M\) and \(N\) are on sides \(AB\) and \(AC\) respectively, such that the circumcenter of triangle \(ABC\) bisects segment \(MN\). Find the ratio \(AN:MB\).
|
2:1
|
olympiads
| 0.140625
|
Find all the extrema of the function \( y = \frac{2}{3} \cos \left(3x - \frac{\pi}{6}\right) \) on the interval \( (0, \frac{\pi}{2}) \).
|
x_{\text{max}} = \frac{\pi}{18}, \quad y_{\text{max}} = \frac{2}{3}, \quad x_{\text{min}} = \frac{7\pi}{18}, \quad y_{\text{min}} = -\frac{2}{3}
|
olympiads
| 0.0625
|
Find the largest constant $K>0$ such that for any $0\le k\le K$ and non-negative reals $a,b,c$ satisfying $a^2+b^2+c^2+kabc=k+3$ we have $a+b+c\le 3$ .
|
K = 1
|
aops_forum
| 0.125
|
Find all real numbers $x, y, z$ that satisfy the following system $$ \sqrt{x^3 - y} = z - 1 $$ $$ \sqrt{y^3 - z} = x - 1 $$ $$ \sqrt{z^3 - x} = y - 1 $$
|
(1, 1, 1)
|
aops_forum
| 0.140625
|
In a rectangular frame made of metal wires with dimensions \(3 \times 4 \times 7\), what is the maximum radius of a sphere placed inside?
|
\frac{5}{2}
|
olympiads
| 0.078125
|
The takeoff run time of an airplane from the start to the moment of lift-off is 15 seconds. Find the length of the takeoff run if the lift-off speed for this airplane model is 100 km/h. Assume the airplane's motion during takeoff is uniformly accelerated. Provide the answer in meters, rounding to the nearest whole number if necessary.
|
208
|
olympiads
| 0.40625
|
Given that the equation \( x^3 + a x^2 + b x + c = 0 \) has three non-zero real roots that form a geometric progression, find the value of \( a^3 c - b^3 \).
|
0
|
olympiads
| 0.1875
|
In a regular quadrangular pyramid, the side of the base is 6 dm, and the height is 4 dm. Find the lateral surface area of the truncated pyramid that is cut off from the original by a plane parallel to its base and 1 dm away from it.
|
26.25 \, \text{dm}^2
|
olympiads
| 0.109375
|
Our Slovak grandmother shopped at a store where they had only apples, bananas, and pears. Apples were 50 cents each, pears were 60 cents each, and bananas were cheaper than pears. Grandmother bought five pieces of fruit, with exactly one banana among them, and paid 2 euros and 75 cents.
How many cents could one banana cost? Determine all possibilities.
|
35, 45, 55
|
olympiads
| 0.234375
|
Given real numbers \( a_{1}, a_{2}, \cdots, a_{2016} \) satisfying \( 9 a_{i} > 11 a_{i+1}^{2} \) for \( i = 1, 2, \cdots, 2015 \). 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
| 0.125
|
Find all positive integers $n$ for which there exist non-negative integers $a_1, a_2, \ldots, a_n$ such that
\[
\frac{1}{2^{a_1}} + \frac{1}{2^{a_2}} + \cdots + \frac{1}{2^{a_n}} =
\frac{1}{3^{a_1}} + \frac{2}{3^{a_2}} + \cdots + \frac{n}{3^{a_n}} = 1.
\]
|
n \equiv 1, 2 \pmod{4}
|
aops_forum
| 0.078125
|
Let $f(x)$ be a generalized convex function on the interval $I \subseteq \mathbb{R}^{+}$. Then for any $x_{i} \in I$ and $p_{i}>0$ $(i=1,2, \cdots, n)$, $m \in \mathbb{R}, r \in \mathbb{R}$ with $m \geqslant r$, we have $M_{n}^{m}\left[f\left(x_{i}\right), p_{i}\right] \geqslant f\left[M_{n}^{r}\left(x_{i}, p_{i}\right)\right]$.
If $f(x)$ is generalized strictly convex on $I$, the equality in equation $(7-15)$ holds if and only if $x_{1}=x_{2}=\cdots=x_{n}$.
If $f(x)$ is generalized (strictly) concave on $I$, the inequality in equation $(7-15)$ reverses.
|
M_{n}^{m}\left[ f(x_{i}), p_{i} \right] \geqslant f \left[ M_{n}^{r}\left( x_{i}, p_{i} \right) \right]
|
olympiads
| 0.421875
|
Points \( A_1, B_1 \), and \( C_1 \) are taken on the sides \( BC, CA, \) and \( AB \) of triangle \( ABC \) respectively, such that the segments \( AA_1, BB_1, \) and \( CC_1 \) intersect at a single point \( M \). For which position of point \( M \) is the value of \( \frac{MA_1}{AA_1} \cdot \frac{MB_1}{BB_1} \cdot \frac{MC_1}{CC_1} \) maximized?
|
M is the centroid of triangle ABC.
|
olympiads
| 0.109375
|
Rong Rong transferred from Class One to Class Two, and Lei Lei transferred from Class Two to Class One. As a result, the average height of students in Class One increased by 2 centimeters, and the average height of students in Class Two decreased by 3 centimeters. If Lei Lei's height is 158 centimeters and Rong Rong's height is 140 centimeters, determine the total number of students in the two classes.
|
15
|
olympiads
| 0.078125
|
On Valentine's Day, each boy in the school gave a valentine to each girl in the school. It was found that the number of valentines was 40 more than the total number of students. How many valentines were given?
|
84
|
olympiads
| 0.203125
|
In a room, there is a group of 11 people with an average age of exactly 25 years. A second group of 7 people arrives, and the overall average age of the room then becomes exactly 32 years. What is the average age of the second group?
|
43
|
olympiads
| 0.46875
|
In how many ways can a horizontal strip of size $2 \times n$ be covered with bricks of size $1 \times 2$ or $2 \times 1$?
|
\frac{1}{\sqrt{5}} \left( \varphi^{n+1} - \bar{\varphi}^{n+1} \right)
|
olympiads
| 0.171875
|
Given that \( a \), \( b \), and \( c \) are the sides of triangle \( \triangle ABC \), and they satisfy the inequality
\[
\sum \frac{1}{a} \sqrt{\frac{1}{b}+\frac{1}{c}} \geqslant \frac{3}{2} \sqrt{\Pi\left(\frac{1}{a}+\frac{1}{b}\right)},
\]
where \( \sum \) and \( \Pi \) represent the cyclic symmetric sum and cyclic symmetric product, respectively. Determine the type of triangle \( \triangle ABC \).
|
Equilateral Triangle
|
olympiads
| 0.203125
|
In triangle \(ABC\), the angle at vertex \(B\) is \(120^{\circ}\). The angle bisector of this angle intersects side \(AC\) at point \(P\). The external angle bisector from vertex \(C\) intersects the extension of side \(AB\) at point \(Q\), and the segment \(PQ\) intersects side \(BC\) at point \(R\). What is the measure of angle \(PRA\)?
|
30^ ext{circ}
|
olympiads
| 0.140625
|
Simplify the expression \(\frac{\sin 4\alpha + \sin 5\alpha + \sin 6\alpha}{\cos 4\alpha + \cos 5\alpha + \cos 6\alpha}\).
|
\tan 5\alpha
|
olympiads
| 0.0625
|
Archimedes has a two-pan balance with arms of different lengths. To weigh two kilograms of sugar, he proceeded as follows: he placed a one-kilogram weight on the left pan and sugar on the other side until the balance was even. Then, he emptied both pans, placed the one-kilogram weight on the right pan, and sugar on the left pan until the balance was even again. Does the total amount of sugar from these two weighings add up to less than two kilograms, more than two kilograms, or exactly two kilograms? Note: For the balance to be even, the weights on the pans must be inversely proportional to the corresponding arm lengths.
|
mais de dois quilos
|
olympiads
| 0.078125
|
We have three containers of equal volume and a pipette. The first container is half-filled with a water-soluble liquid, while the second and third containers are empty. From the first container, we transfer a full pipette of liquid into the second container. Then, we add water to the second container to make it half-full. After mixing, we transfer a full pipette of this solution to the third container, fill it halfway with water, and mix it. We then use this new mixture to fill the first container halfway, and the remaining mixture is poured into the second container. Now, the first and second containers contain mixtures with equal dilution. What fraction of the containers' volume is the volume of the pipette?
|
\frac{-1 + \sqrt{17}}{4}
|
olympiads
| 0.0625
|
In an equilateral triangle $(t)$, we draw the medial triangle $\left(t_{1}\right)$. In the medial triangle $\left(t_{1}\right)$, we again draw the medial triangle $\left(t_{2}\right)$; the question is, if this process is continued indefinitely, what will be the value of the sum
$$
S=t+t-1+t-2+\ldots+\operatorname{in} \inf
$$
|
\frac{4t}{3}
|
olympiads
| 0.109375
|
Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}^+$ such that if $a,b,c$ are the length sides of a triangle, and $r$ is the radius of its incircle, then $f(a),f(b),f(c)$ also form a triangle where its radius of the incircle is $f(r)$ .
|
f(x) = kx
|
aops_forum
| 0.203125
|
Meredith drives 5 miles to the northeast, then 15 miles to the southeast, then 25 miles to the southwest, then 35 miles to the northwest, and finally 20 miles to the northeast. How many miles is Meredith from where she started?
|
20
|
aops_forum
| 0.09375
|
For every positive integeer $n>1$ , let $k(n)$ the largest positive integer $k$ such that there exists a positive integer $m$ such that $n = m^k$ .
Find $$ lim_{n \rightarrow \infty} \frac{\sum_{j=2}^{j=n+1}{k(j)}}{n} $$
|
1
|
aops_forum
| 0.1875
|
Given the sets \(A=\left\{1, \frac{x+y}{2}-1\right\}\) and \(B=\{-\ln(xy), x\}\), if \(A = B\) and \(0 < y < 2\), find the value of
\[
\left(x^{2} - \frac{1}{y^{2}}\right) + \left(x^{4} - \frac{1}{y^{4}}\right) + \cdots + \left(x^{2022} - \frac{1}{y^{2022}}\right).
\]
|
0
|
olympiads
| 0.140625
|
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all real numbers \( x \) and \( y \), the following equation holds:
\[ f\left(f(y)+x^{2}+1\right)+2x = y + f^{2}(x+1). \]
|
f(x)=x
|
olympiads
| 0.0625
|
The sequence $\left\{x_{n}\right\}$ is defined as follows: $x_{1}=\frac{1}{2}, x_{n+1}=x_{n}^{2}+x_{n}$. Find the integer part of the following sum: $\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{200}}$.
|
1
|
olympiads
| 0.171875
|
Find the area of an isosceles trapezoid if its diagonal is $l$ and the angle between this diagonal and the larger base is $\alpha$.
|
\frac{1}{2} l^2 \sin 2 \alpha
|
olympiads
| 0.1875
|
The sequence \(a_{n}\) is defined as follows:
\(a_{1}=2, a_{n+1}=a_{n}+\frac{2 a_{n}}{n}\), for \(n \geq 1\). Find \(a_{100}\).
|
10100
|
olympiads
| 0.09375
|
Solve the system
$$
\left\{\begin{array}{l}
4 \log _{2}^{2} x+1=2 \log _{2} y \\
\log _{2} x^{2} \geqslant \log _{2} y
\end{array}\right.
$$
|
(\sqrt{2}, 2)
|
olympiads
| 0.078125
|
Triangle $ABC$ is isosceles with $AB=AC$ . The bisectors of angles $ABC$ and $ACB$ meet at $I$ . If the measure of angle $CIA$ is $130^\circ$ , compute the measure of angle $CAB$ .
|
80^ extcirc
|
aops_forum
| 0.21875
|
The toy factory produces wireframe cubes, with small multicolored balls placed at the vertices. According to the standard, each cube must use balls of all eight colors (white and the seven colors of the rainbow). How many different models of cubes can the factory produce?
|
1680
|
olympiads
| 0.25
|
Given \( x, y, z \geqslant 3 \), find the minimum value of the expression
$$
A=\frac{\left(x^{3}-24\right) \sqrt[3]{x+24}+\left(y^{3}-24\right) \sqrt[3]{y+24}+\left(z^{3}-24\right) \sqrt[3]{z+24}}{x y+y z+z x}
$$
|
1
|
olympiads
| 0.0625
|
One day, a truck driver drove through a tunnel and measured the time taken between the moment at which the truck started entering the tunnel and the moment at which the truck left the tunnel completely. The next day, a container was added and the length of the truck was increased from $6 \mathrm{~m}$ to $12 \mathrm{~m}$. The driver reduced the speed by $20 \%$ and measured the time again. He found that the time taken was increased by half. Find the length of the tunnel (in metres).
|
24 ext{ meters}
|
olympiads
| 0.171875
|
Find all triples \((a, b, p)\) of natural numbers such that \( p \) is a prime number and the equation
\[ (a+b)^p = p^a + p^b \]
is fulfilled.
|
(1, 1, 2)
|
olympiads
| 0.1875
|
A tile, in the shape of a regular polygon, was removed from its place in a panel. It was observed that if this tile were rotated by \( 40^\circ \) or \( 60^\circ \) around its center, it could fit perfectly into the spot that was left vacant on the panel. What is the smallest number of sides that this polygon can have?
|
18
|
olympiads
| 0.09375
|
Are the vectors \( c_1 \) and \( c_2 \), constructed from the vectors \( a \) and \( b \), collinear?
Given:
\[ a = \{-1, 4, 2\} \]
\[ b = \{3, -2, 6\} \]
\[ c_1 = 2a - b \]
\[ c_2 = 3b - 6a \]
|
The vectors \ c_{1} \ text{ and } \ c_{2} \ text{ are collinear.
|
olympiads
| 0.203125
|
For each positive integer $k$ , let $d(k)$ be the number of positive divisors of $k$ and $\sigma(k)$ be the sum of positive divisors of $k$ . Let $\mathbb N$ be the set of all positive integers. Find all functions $f: \mathbb{N} \to \mathbb N$ such that \begin{align*}
f(d(n+1)) &= d(f(n)+1)\quad \text{and}
f(\sigma(n+1)) &= \sigma(f(n)+1)
\end{align*}
for all positive integers $n$ .
|
f(n) = n
|
aops_forum
| 0.46875
|
Let $\alpha ,\ \beta$ be the distinct positive roots of the equation of $2x=\tan x$ .
Evaluate the following definite integral.
\[\int_{0}^{1}\sin \alpha x\sin \beta x\ dx \]
|
0
|
aops_forum
| 0.140625
|
There are 5 girls sitting in a row on five chairs, and opposite them, on five chairs, there are 5 boys sitting. It was decided that the boys would switch places with the girls. In how many ways can this be done?
|
14400
|
olympiads
| 0.296875
|
In the Cartesian plane, given three points \( M(0,0) \), \( P(0,5) \), \( Q\left(\frac{12}{5}, \frac{9}{5}\right) \), let \( A \) be the set of circles passing through point \( M \), \( B \) be the set of circles passing through point \( P \), and \( C \) be the set of circles passing through point \( Q \). Determine the equation of the curve represented by the point set \( A \cap B \cap C \).
|
x^2 + \left(y - \frac{5}{2}\right)^2 = \frac{25}{4}
|
olympiads
| 0.140625
|
A $9 \times 9$ table is divided into nine $3 \times 3$ squares. Petya and Vasya take turns writing numbers from 1 to 9 in the cells of the table according to the rules of sudoku, meaning that no row, column, or any of the nine $3 \times 3$ squares may contain duplicate numbers. Petya starts the game; the player who cannot make a move loses. Which player can guarantee a win, no matter how the opponent plays?
|
\text{Petya}
|
olympiads
| 0.140625
|
Two places A and B are 66 kilometers apart. Person A and Person C start walking from A to B, and Person B starts walking from B to A. Person A walks 12 km per hour, Person B walks 10 km per hour, and Person C walks 8 km per hour. If all three start at the same time, after how many hours will Person B be exactly at the midpoint of the distance between Person A and Person C?
|
3.3 \text{ hours}
|
olympiads
| 0.40625
|
Write the smallest four-digit number in which all digits are different.
|
1023
|
olympiads
| 0.109375
|
It is known that when 2008 is divided by certain natural numbers, the remainder is always 10. How many such natural numbers are there?
|
11
|
olympiads
| 0.09375
|
In an acute triangle $ABC$ , the segment $CD$ is an altitude and $H$ is the orthocentre. Given that the circumcentre of the triangle lies on the line containing the bisector of the angle $DHB$ , determine all possible values of $\angle CAB$ .
|
60^{\circ}
|
aops_forum
| 0.078125
|
Ada and Luisa train every day, each always running at the same speed, for the big race that will happen at the end of the year at school. The training starts at point $A$ and ends at point $B$, 3000 meters apart. They start at the same time, but when Luisa finishes the race, Ada still has 120 meters left to reach point $B$. Yesterday, Luisa gave Ada a chance: "We start at the same time, but I start some meters before point A so that we arrive together." How many meters before point $A$ should Luisa start to finish together with Ada?
|
125 \, \text{m}
|
olympiads
| 0.0625
|
As shown in the figure, \(ABCD\) is a rectangle and \(AEFG\) is a square. If \(AB = 6\), \(AD = 4\), and the area of \(\triangle ADE\) is 2, find the area of \(\triangle ABG\).
|
3
|
olympiads
| 0.25
|
Let $p=2^{16}+1$ be a prime. A sequence of $2^{16}$ positive integers $\{a_n\}$ is *monotonically bounded* if $1\leq a_i\leq i$ for all $1\leq i\leq 2^{16}$ . We say that a term $a_k$ in the sequence with $2\leq k\leq 2^{16}-1$ is a *mountain* if $a_k$ is greater than both $a_{k-1}$ and $a_{k+1}$ . Evan writes out all possible monotonically bounded sequences. Let $N$ be the total number of mountain terms over all such sequences he writes. Find the remainder when $N$ is divided by $p$ .
|
49153
|
aops_forum
| 0.078125
|
Galya thought of a number, multiplied it by \( \mathrm{N} \), then added \( \mathrm{N} \) to the result, divided the resulting number by \( \mathrm{N} \), and subtracted \( \mathrm{N} \). As a result, she obtained a number that is 7729 less than the original thought number. What is \( \mathrm{N} \)?
|
7730
|
olympiads
| 0.515625
|
Teacher Li brought a stack of craft paper, just enough to evenly distribute among 24 students. Later, 8 more students arrived, and each student received 2 fewer sheets than originally planned. How many sheets of craft paper did Teacher Li bring in total?
|
192
|
olympiads
| 0.5625
|
Given $m$ and $n$ are non-negative integers, the sets $A=\{1, n\}$ and $B=\{2, 4, m\}$, and the set $C=\{c \mid c=xy, x \in A, y \in B\}$. If $|C|=6$ and the sum of all elements in $C$ is 42, find the value of $m+n$.
|
6
|
olympiads
| 0.140625
|
In how many ways can the natural numbers from 1 to 14 (each used once) be placed in a $2 \times 7$ table so that the sum of the numbers in each of the seven columns is odd?
|
2^7 \cdot (7!)^2
|
olympiads
| 0.34375
|
Divide a regular pentagon into several triangles so that each triangle borders exactly three others. To border means to have a common boundary segment.
|
Correct
|
olympiads
| 0.265625
|
Solve the equation \(\sin 3x + 3\cos x = 2\sin 2x(\sin x + \cos x)\).
|
x = \frac{\pi}{4} + k\pi, \quad k \in \mathbb{Z}
|
olympiads
| 0.0625
|
The sequence \(\left\{x_{n}\right\}\) is defined by \(x_{1}=\frac{1}{2}\) and \(x_{k+1}=x_{k}^{2}+x_{k}\). Find the integer part of \(\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{2009}+1}\).
|
1
|
olympiads
| 0.125
|
A deck of cards contains 52 cards (excluding two jokers). The cards are dealt to 4 people, each person receiving 13 cards. In how many ways can one person receive exactly 13 cards with all four suits represented? (Express the answer using combinations.)
|
\binom{52}{13} - 4 \binom{39}{13} + 6 \binom{26}{13} - 4 \binom{13}{13}
|
olympiads
| 0.109375
|
In a regular triangular pyramid (not a regular tetrahedron), the area of the base is twice the area of a lateral face. The height of the pyramid is 50 cm.
Construct the following (infinite) sequence of spheres. Let $S_{1}$ be the sphere inscribed in this pyramid. Then $S_{2}$ is the sphere that touches the lateral faces of the pyramid and the sphere $S_{1}$; $S_{3}$ is the sphere that touches the lateral faces of the pyramid and the sphere $S_{2}$, different from $S_{1}$, and so on; $S_{n+1}$ is the sphere that touches the lateral faces of the pyramid and the sphere $S_{n}$, different from $S_{n-1}$.
Find the total volume of all these spheres.
|
\frac{1000}{93} \pi \, \text{dm}^3
|
olympiads
| 0.0625
|
Calculate the definite integral:
$$
\int_{0}^{1} x^{2} \cdot e^{3 x} \, dx
$$
|
\frac{5e^3 - 2}{27}
|
olympiads
| 0.3125
|
Write the number 1 and the first 10 prime numbers into the small circles in the illustration so that the sum of any three numbers connected by a single line equals the 15th prime number; and the sum of any three numbers connected by a double line equals the product of the 2nd and 5th prime numbers. How many different ways can this be done, considering that arrangements that are mirror images across the vertical axis are not different?
|
1
|
olympiads
| 0.09375
|
For \(\theta < \pi\), find the largest real number \(\theta\) such that
$$
\prod_{k=0}^{10} \cos 2^{k} \theta \neq 0, \prod_{k=0}^{10}\left(1+\frac{1}{\cos 2^{k} \theta}\right)=1.
$$
|
\frac{2046 \pi}{2047}
|
olympiads
| 0.140625
|
It is known that for three consecutive natural values of the argument, the quadratic function \( f(x) \) takes the values 6, 5, and 5, respectively. Find the minimum possible value of \( f(x) \).
|
5
|
olympiads
| 0.171875
|
Calculate the limit of the function:
$$\lim_{x \rightarrow 0} \frac{7^{3x} - 3^{2x}}{\tan{x} + x^3}$$
|
\ln \frac{343}{9}
|
olympiads
| 0.59375
|
In triangle \( ABC \), it is known that \( AB = 3 \), the height \( CD = \sqrt{3} \). The base \( D \) of the height \( CD \) lies on the side \( AB \) and \( AD = BC \). Find \( AC \).
|
\sqrt{7}
|
olympiads
| 0.125
|
Upon seeing a fox several meters away, a dog started chasing it down a straight countryside road. The dog's jump is $23\%$ longer than the fox's jump. There is a time interval during which both the fox and the dog make an integer number of jumps, and during this interval, the dog consistently manages to make $t\%$ fewer jumps than the fox, where $t$ is an integer. Assuming that all jumps are the same for both the dog and the fox, find the minimum value of $t$ at which the fox can escape from the dog.
|
19
|
olympiads
| 0.09375
|
The regular hexagon \(ABCDEF\) has diagonals \(AC\) and \(CE\). The internal points \(M\) and \(N\) divide these diagonals such that \(AM: AC = CN: CE = r\). Determine \(r\) if it is known that points \(B\), \(M\), and \(N\) are collinear.
|
\frac{1}{\sqrt{3}}
|
olympiads
| 0.0625
|
Determine all triples (x, y, z) of non-negative real numbers that satisfy the following system of equations $\begin{cases} x^2 - y = (z - 1)^2
y^2 - z = (x - 1)^2
z^2 - x = (y -1)^2 \end{cases}$ .
|
(x, y, z) = (1, 1, 1)
|
aops_forum
| 0.1875
|
Guilherme wrote 0 or 1 in each cell of a \(4 \times 4\) board. He placed the numbers such that the sum of the numbers in the neighboring cells of each cell of the board is equal to 1.
For example, in the figure, considering the cell marked with \(\bullet\), the sum of the numbers in the shaded cells is equal to 1.
Determine the sum of all 16 numbers on the board.
|
8
|
olympiads
| 0.296875
|
Calculate the definite integral:
$$
\int_{0}^{2 \pi}\left(2 x^{2}-15\right) \cos 3 x \, dx
$$
|
\frac{8\pi}{9}
|
olympiads
| 0.171875
|
Find the minimum value of \((a+b)\left(\frac{1}{a}+\frac{4}{b}\right)\), where \(a\) and \(b\) range over all positive real numbers.
|
9
|
olympiads
| 0.140625
|
Professor Rackbrain recently asked his young friends to find all five-digit perfect squares for which the sum of the numbers formed by the first two digits and the last two digits equals a perfect cube. For example, if we take the square of the number 141, which is equal to 19881, and add 81 to 19, we get 100 – a number that unfortunately is not a perfect cube.
How many such solutions exist?
|
3
|
olympiads
| 0.09375
|
In the plane Cartesian coordinate system $xOy$, a parabola $C: x^{2} = y$ is given. Suppose there exists a line $l$ passing through the point $P(0, t)$ that intersects the parabola $C$ at points $A$ and $B$. Additionally, it is given that there always exists a point $M$ on the parabola $C$ such that the line $l$ intersects the circumcircle of $\triangle MOP$ at another point $Q$, satisfying $\angle PMQ = \angle AMB = 90^{\circ}$. Determine the range of positive values for $t$.
|
t \in \left( \frac{3 - \sqrt{5}}{2}, +\infty \right)
|
olympiads
| 0.0625
|
Four homeowners: You see in the picture a square plot of land with four houses, four trees, a well (W) in the center, and fences with four gates (G).
Can you divide this plot so that each homeowner gets an equal amount of land, one tree, one gate, an equal length of fence, and unrestricted access to the well that doesn't cross the neighbor's plot?
|
The plot can indeed be divided as shown in the reference diagram to meet all specified conditions.
|
olympiads
| 0.1875
|
What is the geometric representation of the following equation:
$$
x^{2}+2 y^{2}-10 x+12 y+43=0
$$
|
(5, -3)
|
olympiads
| 0.453125
|
Find all continuous functions defined for all \( x \) that satisfy the equation \( f(x) = a^{x} f(x / 2) \), where \( a \) is a fixed positive number.
|
f(x) = C a^{2x}
|
olympiads
| 0.109375
|
Every third student in the sixth grade is a member of the math club, every fourth student is a member of the history club, and every sixth student is a member of the chemistry club. The rest of the students are members of the literature club. How many people are in the chemistry club if the number of members in the math club exceeds the number of members in the literature club by three?
|
6
|
olympiads
| 0.3125
|
Calculate the limit of the function:
\[
\lim _{x \rightarrow \frac{\pi}{2}}(\cos x+1)^{\sin x}
\]
|
1
|
olympiads
| 0.421875
|
Find all real values of \( x \) for which the fraction
$$
\frac{x^{2}+2x-3}{x^{2}+1}
$$
takes on integer values.
|
\{-3, -2, -1, 0, 1\}
|
olympiads
| 0.0625
|
Among all the positive integers that are multiples of 20, what is the sum of those that do not exceed 2014 and are also multiples of 14?
|
14700
|
olympiads
| 0.59375
|
Natural numbers \( a, b, c \) are chosen such that \( a < b < c \). It is also known that the system of equations \( 2x + y = 2033 \) and \( y = |x-a| + |x-b| + |x-c| \) has exactly one solution. Find the minimum possible value of \( c \).
|
1017
|
olympiads
| 0.078125
|
Santa Claus has many identical dials in the form of regular 12-sided polygons, with numbers from 1 to 12 printed on them. He stacks these dials one on top of the other (face-up). The vertices of the dials are aligned, but the numbers at the aligned vertices do not necessarily match. The Christmas tree will light up as soon as the sums of the numbers in all 12 columns have the same remainder when divided by 12. How many dials might be in the stack at that moment?
|
12
|
olympiads
| 0.53125
|
Find the polynomials with integer coefficients \( P \) such that:
$$
\forall n,\; n \text{ divides } P\left(2^{n}\right)
$$
|
P(x) = 0
|
olympiads
| 0.265625
|
Let $x_i\ (i = 1, 2, \cdots 22)$ be reals such that $x_i \in [2^{i-1},2^i]$ . Find the maximum possible value of $$ (x_1+x_2+\cdots +x_{22})(\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_{22}}) $$
|
2^{24}
|
aops_forum
| 0.15625
|
Let \( f(x) = |x - a| + |x - 15| + |x - a - 15| \), where \( a \leq x \leq 15 \) and \( 0 < a < 15 \). If \( Q \) is the smallest value of \( f(x) \), find the value of \( Q \).
|
15
|
olympiads
| 0.28125
|
Consider all possible 100-digit natural numbers, in which only the digits $1, 2, 3$ are used. How many of them are divisible by 3 exactly?
|
3^{99}
|
olympiads
| 0.078125
|
Ahmed is going to the store. One quarter of the way to the store, he stops to talk with Kee. He then continues for $12 \mathrm{~km}$ and reaches the store. How many kilometres does he travel altogether?
|
16
|
olympiads
| 0.546875
|
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