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stringlengths 33
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|---|---|---|---|
Let \( p \) be an arbitrary prime number. Find all non-negative integer solutions to the equation
$$
\sqrt{x}+\sqrt{y}=\sqrt{p}.
$$
|
(0, p) \quad \text{or} \quad (p, 0)
|
olympiads
| 0.125 |
Let \( \triangle ABC \) be a triangle with \( \angle BAC = 60^\circ \). Let \( E \) be the point on the side \( BC \) such that \( 2 \angle BAE = \angle ACB \). Let \( D \) be the second point of intersection of \( AB \) and the circumcircle of the triangle \( \triangle AEC \), and let \( P \) be the second point of intersection of \( CD \) and the circumcircle of the triangle \( \triangle DBE \). Calculate the angle \( \angle BAP \).
|
30^ extcirc
|
olympiads
| 0.203125 |
Find all triplets $ (x,y,z) $ of real numbers such that
\[ 2x^3 + 1 = 3zx \]\[ 2y^3 + 1 = 3xy \]\[ 2z^3 + 1 = 3yz \]
|
(1, 1, 1)
|
aops_forum
| 0.15625 |
How many positive integers divide at least one of \( 10^{40} \) and \( 20^{30} \)?
|
2301
|
olympiads
| 0.296875 |
In the acute-angled triangle $ABC$ , the altitudes $BP$ and $CQ$ were drawn, and the point $T$ is the intersection point of the altitudes of $\Delta PAQ$ . It turned out that $\angle CTB = 90 {} ^ \circ$ . Find the measure of $\angle BAC$ .
|
45^ ext{\circ}
|
aops_forum
| 0.25 |
Rectangle \(ABCD\) is congruent to rectangle \(ABEF\), and \(D-AB-E\) forms a dihedral angle. \(M\) is the midpoint of \(AB\). \(FM\) makes an angle \(\theta\) with \(BD\), where \(\sin \theta = \frac{\sqrt{78}}{9}\). Find the value of \(\frac{AB}{BC}\).
|
\frac{\sqrt{2}}{2}
|
olympiads
| 0.078125 |
To represent integers or decimal fractions, a special type of "stem-and-leaf" diagram is often used. In such diagrams, it is convenient to depict the age of people. Suppose that in the studied group there are 5 people aged 19, 34, 37, 42, and 48 years. For this group, the diagram will look as shown in Figure 2. The left column is the "stem", and to the right of it are the "leaves".
While studying a group of patients, on December 1st, the doctor created a diagram of their ages (Figure 3a). Figure 3b shows the new diagram of their ages, which was also created on December 1st after several years. During these years, the composition of the group remained the same — everyone who was there stayed, and no new members joined the group. However, the numbers on the new diagram are not visible — instead, there are asterisks. Determine how many years have passed and restore the diagram.
$$
\begin{array}{l|llllll}
0 & & & & & & \\
1 & 0 & 0 & 1 & 2 & 2 & 3 \\
2 & 1 & 5 & 6 & & & \\
3 & 0 & 2 & 4 & 6 & & \\
4 & 1 & 6 & & & &
\end{array}
$$
(Figure 3a)
|
6
|
olympiads
| 0.0625 |
Find all real numbers $a$ for which there exists a function $f: R \to R$ asuch that $x + f(y) =a(y + f(x))$ for all real numbers $x,y\in R$ .
I.Voronovich
|
a = \pm 1
|
aops_forum
| 0.140625 |
Petya takes black and red cards out of a bag and stacks them into two piles. Placing a card on another card of the same color is not allowed. The tenth and the eleventh cards placed by Petya are red, and the twenty-fifth card is black. What is the color of the twenty-sixth card placed?
|
ext{Red}
|
olympiads
| 0.546875 |
Determine the work (in joules) done in lifting a satellite from the Earth's surface to a height of $H$ km. The satellite's mass is $m$ tons, and the Earth's radius $R_{3}$ is $6380$ km. Assume the acceleration due to gravity $g$ at the Earth's surface is $10 \text{ m/s}^2$.
$$
m = 3.0 \text{ tons}, \; H = 650 \text{ km}
$$
|
1.77 imes 10^{10} \, \text{J}
|
olympiads
| 0.140625 |
A sphere is inscribed around a cylinder. The radius of the sphere is $2 \mathrm{~cm}$ larger than the radius of the base of the cylinder and $1 \mathrm{~cm}$ larger than the height of the cylinder. What is the volume of the sphere?
|
6550 \frac{2}{3} \pi \, \text{cm}^3
|
olympiads
| 0.109375 |
On the board, there are natural numbers from 1 to 1000, each written once. Vasya can erase any two numbers and write one of the following in their place: their greatest common divisor or their least common multiple. After 999 such operations, one number remains on the board, which is equal to a natural power of ten. What is the maximum value it can take?
|
10000
|
olympiads
| 0.0625 |
Find all real solutions of the system $$ \begin{cases} x_1 +x_2 +...+x_{2000} = 2000 x_1^4 +x_2^4 +...+x_{2000}^4= x_1^3 +x_2^3 +...+x_{2000}^3\end{cases} $$
|
x_1 = x_2 = \cdots = x_{2000} = 1
|
aops_forum
| 0.359375 |
Calculate the value of the expression \(\sin \frac{b \pi}{36}\), where \(b\) is the sum of all distinct numbers obtained from the number \(a = 987654321\) by cyclic permutations of its digits (in a cyclic permutation, all the digits of the number, except the last one, are shifted one place to the right, and the last digit moves to the first place).
|
\frac{\sqrt{2}}{2}
|
olympiads
| 0.140625 |
What number corresponds to point $\mathrm{P}$ on the scale below?
|
12.50
|
olympiads
| 0.0625 |
How many multiples of 3 are there between 1 and 2015 whose units digit in the decimal representation is 2?
|
67
|
olympiads
| 0.09375 |
An equilateral triangle is inscribed in a circle of radius \( R \). The altitudes of the triangle are extended until they intersect with the circle. These points of intersection are connected, forming a new triangle. Calculate the portion of the area of the circle that lies outside these triangles.
|
R^2 (\pi - \sqrt{3})
|
olympiads
| 0.203125 |
Ana has $22$ coins. She can take from her friends either $6$ coins or $18$ coins, or she can give $12$ coins to her friends. She can do these operations many times she wants. Find the least number of coins Ana can have.
|
4
|
aops_forum
| 0.296875 |
Mona has 12 match sticks of length 1, and she needs to use them to make regular polygons, with each match being a side or a fraction of a side of a polygon, and no two matches overlapping or crossing each other. What is the smallest total area of the polygons Mona can make?
|
\sqrt{3}
|
olympiads
| 0.09375 |
There is a point source of light in an empty universe. What is the minimum number of solid balls (of any size) one must place in space so that any light ray emanating from the light source intersects at least one ball?
|
4
|
aops_forum
| 0.28125 |
Calculate the volumes of the solids formed by the rotation of figures bounded by the function graphs. The axis of rotation is the $x$-axis.
$$
y = x e^x, \quad y = 0, \quad x = 1
$$
|
\frac{\pi (e^2 - 1)}{4}
|
olympiads
| 0.40625 |
Given that $\alpha$ and $\beta$ are acute angles, and $\alpha - \beta = \frac{\pi}{3}$, find the value of $\sin^{2} \alpha + \cos^{2} \beta - \sqrt{3} \sin \alpha \cdot \cos \beta$.
|
\frac{1}{4}
|
olympiads
| 0.0625 |
Given that \(\cos \alpha + \cos \beta + \cos \gamma = \sqrt{\frac{1}{5}}\) and \(\sin \alpha + \sin \beta + \sin \gamma = \sqrt{\frac{4}{5}}\), find \(\cos (\alpha - \beta) + \cos (\beta - \gamma) + \cos (\gamma - \alpha)\).
|
-1
|
olympiads
| 0.078125 |
Each of the lateral edges of the pyramid is equal to \( b \). Its base is a right triangle with legs that are in the ratio \( m: n \), and the hypotenuse is equal to \( c \). Calculate the volume of the pyramid.
|
\frac{m n c^2 \sqrt{4 b^2 - c^2}}{12(m^2 + n^2)}
|
olympiads
| 0.0625 |
The notation \( |x| \) is used to denote the absolute value of a number, regardless of sign. For example, \( |7| = |-7| = 7 \). The graphs \( y = |2x| - 3 \) and \( y = |x| \) are drawn on the same set of axes. What is the area enclosed by them?
|
9
|
olympiads
| 0.21875 |
The figure below represents routes of a postman. Starting at the post office, the postman walks through all the 12 points and finally returns to the post office. If he takes 10 minutes from a point to another adjacent point by walk and $K$ is the number of hours required for the postman to finish the routes, find the smallest possible value of $K$.
|
4
|
olympiads
| 0.0625 |
A total of 20 birds – 8 starlings, 7 wagtails, and 5 woodpeckers – fly into a photo studio. Each time the photographer clicks the camera shutter, one bird flies away (permanently). How many photos can the photographer take to ensure that at least four birds of one species and at least three birds of another species remain?
|
7
|
olympiads
| 0.078125 |
How many solutions does the equation \(\left|\left| |x-1| - 1 \right| - 1 \right| = 1\) have? The modulus function \( |x| \) evaluates the absolute value of a number; for example \( |6| = |-6| = 6 \).
|
4
|
olympiads
| 0.234375 |
If \( m \in \mathbf{R} \), then find the range of values for \( m \) in the set \( \left\{m, m^{2}+3m\right\} \).
|
\{ m \mid m \neq 0 \text{, and } m \neq -2 \}
|
olympiads
| 0.140625 |
Find all such \( a \) and \( b \) that \( |a| + |b| \geq \frac{2}{\sqrt{3}} \) and the inequality \( |a \sin x + b \sin 2x| \leq 1 \) holds for all \( x \).
|
\left( \pm \frac{4}{3\sqrt{3}}, \pm \frac{2}{3\sqrt{3}} \right)
|
olympiads
| 0.0625 |
In the middle of a river (the main channel), the water flow speed is 10 km/h, and along the riverbank, the water flow speed is 8 km/h. A boat travels downstream in the middle of the river for 10 hours and covers 360 km. How many hours will it take for the boat to return to its original place along the riverbank?
|
20
|
olympiads
| 0.390625 |
Let \( m \) be a natural number greater than 1. The sequence \( x_{0}, x_{1}, x_{2}, \ldots \) is defined by
\[ x_{i} = \begin{cases}2^{i}, & \text{for } 0 \leq i \leq m-1 \\ \sum_{j=1}^{m} x_{i-j}, & \text{for } i \geq m\end{cases} \]
Find the largest \( k \) such that there are \( k \) consecutive sequential elements that are all divisible by \( m \).
|
m-1
|
olympiads
| 0.0625 |
Let \(\left(a_{n}\right)\) be defined by \(a_{1}=3\), \(a_{2}=2\), and for \(n \geqslant 1\), \(a_{n+2}\) is the remainder of the Euclidean division of \(a_{n}+a_{n+1}\) by 100. Compute the remainder of the Euclidean division of:
\[ a_{1}^{2}+a_{2}^{2}+\cdots+a_{2007}^{2} \]
by 8.
|
1
|
olympiads
| 0.109375 |
Anna has five circular discs, each of a different size. She decides to build a tower using three of her discs so that each disc in her tower is smaller than the disc below it. How many different towers could Anna construct?
|
10
|
olympiads
| 0.25 |
Find all bounded sequences $(a_n)_{n=1}^\infty$ of natural numbers such that for all $n \ge 3$ , \[ a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}. \]
|
a_i = 2 \text{ for all } i
|
aops_forum
| 0.078125 |
How many roots of the equation
$$
z^{4}-5z+1=0
$$
are in the annulus $1<|z|<2$?
|
3
|
olympiads
| 0.1875 |
As shown in the figure, quadrilateral \(ABCD\) is a square, \(ABGF\) and \(FGCD\) are rectangles, point \(E\) is on \(AB\), and \(EC\) intersects \(FG\) at point \(M\). If \(AB = 6\) and the area of \(\triangle ECF\) is 12, find the area of \(\triangle BCM\).
|
6
|
olympiads
| 0.140625 |
Solve the equation
\[ x^3 - 7x^2 + 36 = 0 \]
given that the product of two of its roots is 18.
|
-2, 3, 6
|
olympiads
| 0.28125 |
$OABC$ is a rectangle on the Cartesian plane, with sides parallel to the coordinate axes. Point $O$ is the origin, and point $B$ has coordinates $(11, 8)$. Inside the rectangle, there is a point $X$ with integer coordinates. What is the minimum possible area of the triangle $OBX$?
|
\frac{1}{2}
|
olympiads
| 0.0625 |
Cut a square into 4 parts, from which you can form two squares.
|
Rearranging the four parts can form two squares.
|
olympiads
| 0.0625 |
Compute the minimum value of $cos(a-b) + cos(b-c) + cos(c-a)$ as $a,b,c$ ranges over the real numbers.
|
-\frac{3}{2}
|
aops_forum
| 0.171875 |
Four married couples dined together. After dessert, Diana smoked three cigarettes, Elizabeth smoked two, Nicole smoked four, and Maud smoked one. Simon smoked the same amount as his wife, Pierre smoked twice as much as his wife, Louis smoked three times as much as his wife, and Christian smoked four times as much as his wife.
If all attendees smoked a total of 32 cigarettes, can you determine the name of Louis' wife?
|
Maud
|
olympiads
| 0.234375 |
Let \(P\) and \(P+2\) be both prime numbers satisfying \(P(P+2) \leq 2007\). If \(S\) represents the sum of such possible values of \(P\), find the value of \(S\).
|
106
|
olympiads
| 0.3125 |
Determine all functions \( f: \mathbb{R}_{+} \rightarrow \mathbb{R}_{+} \) such that for all \( x \geqslant 0 \), we have \( f(f(x)) + f(x) = 6x \).
|
f(x) = 2x
|
olympiads
| 0.109375 |
Using a compass and straightedge, construct a triangle given two angles \( A \), \( B \), and the perimeter \( P \).
|
A'B'C'
|
olympiads
| 0.0625 |
\( AB \) and \( AC \) are two chords forming an angle \( BAC \) equal to \( 70^\circ \). Tangents are drawn through points \( B \) and \( C \) until they intersect at point \( M \). Find \(\angle BMC\).
|
40^{\circ}
|
olympiads
| 0.09375 |
How many boards, each 6 arshins long and 6 vershoks wide, are needed to cover the floor of a square room with a side of 12 arshins? The answer is 64 boards. Determine from this data how many vershoks are in an arshin.
|
16
|
olympiads
| 0.125 |
There is a rectangular table. Two players take turns placing one euro coin on it in such a way that the coins do not overlap each other. The player who can't make a move loses. Who will win with perfect play?
|
Player 1 wins with optimal play.
|
olympiads
| 0.25 |
In trapezoid \(ABCD\) with the shorter base \(BC\), a line is drawn through point \(B\) parallel to \(CD\) and intersects diagonal \(AC\) at point \(E\). Compare the areas of triangles \(ABC\) and \(DEC\).
|
The areas of the triangles are equal.
|
olympiads
| 0.171875 |
Find the smallest possible side of a square in which five circles of radius $1$ can be placed, so that no two of them have a common interior point.
|
2 + 2\sqrt{2}
|
aops_forum
| 0.09375 |
Yura, Ira, Olya, Sasha, and Kolya are standing in line for ice cream. Yura is ahead of Ira, but behind Kolya. Olya and Kolya are not standing next to each other, and Sasha is not next to Kolya, Yura, or Olya. In what order are they standing?
|
\text{Kolya}, \text{Yura}, \text{Olya}, \text{Ira}, \text{Sasha}
|
olympiads
| 0.15625 |
Masha has two-ruble and five-ruble coins. If she takes all her two-ruble coins, she will be 60 rubles short of buying four pies. If she takes all her five-ruble coins, she will be 60 rubles short of buying five pies. She is also 60 rubles short of buying six pies overall. How much does one pie cost?
|
20 \text{ rubles}
|
olympiads
| 0.5625 |
Calculate the volume of a regular tetrahedron if the radius of the circumscribed circle around one of its faces is $R$.
|
\frac{R^3 \sqrt{6}}{4}
|
olympiads
| 0.109375 |
Find the smallest positive integer \( k \) which can be represented in the form \( k = 19^n - 5^m \) for some positive integers \( m \) and \( n \).
|
14
|
olympiads
| 0.28125 |
Calculate the product \(\frac{2^{3}-1}{2^{3}+1} \cdot \frac{3^{3}-1}{3^{3}+1} \cdot \ldots \cdot \frac{n^{3}-1}{n^{3}+1}\).
|
\frac{2}{3}\left(1+\frac{1}{n(n+1)}\right)
|
olympiads
| 0.0625 |
The perimeter of a square inscribed in a circle is \( p \). What is the area of the square that circumscribes the circle?
|
\frac{p^2}{8}
|
olympiads
| 0.328125 |
A sequence of positive integers \(a_{n}\) begins with \(a_{1}=a\) and \(a_{2}=b\) for positive integers \(a\) and \(b\). Subsequent terms in the sequence satisfy the following two rules for all positive integers \(n\):
\[a_{2 n+1}=a_{2 n} a_{2 n-1}, \quad a_{2 n+2}=a_{2 n+1}+4 .\]
Exactly \(m\) of the numbers \(a_{1}, a_{2}, a_{3}, \ldots, a_{2022}\) are square numbers. What is the maximum possible value of \(m\)? Note that \(m\) depends on \(a\) and \(b\), so the maximum is over all possible choices of \(a\) and \(b\).
|
1012
|
olympiads
| 0.0625 |
Find the specific solution of the system
$$
\left\{\begin{array}{l}
\frac{d x}{d t}=1-\frac{1}{y} \\
\frac{d y}{d t}=\frac{1}{x-t}
\end{array}\right.
$$
satisfying the initial conditions \( x(0)=1 \) and \( y(0)=1 \).
|
x = t + e^{-t}, \quad y = e^t
|
olympiads
| 0.0625 |
Li Gang read a book. On the first day, he read $\frac{1}{5}$ of the book. On the second day, he read 24 pages. On the third day, he read $150\%$ of the total number of pages from the first two days. At this point, there was still $\frac{1}{4}$ of the book left unread. How many pages does the book have in total?
|
240
|
olympiads
| 0.359375 |
At the base of the pyramid lies a triangle with sides 3, 4, and 5. The lateral faces are inclined to the plane of the base at an angle of $45^{\circ}$. What can the height of the pyramid be?
|
1 \text{ or } 2 \text{ or } 3 \text{ or } 6
|
olympiads
| 0.09375 |
A car traveled half of the distance at a speed of 60 km/h, then one-third of the remaining distance at a speed of 120 km/h, and the rest of the distance at a speed of 80 km/h.
Find the average speed of the car for this trip. Provide the answer in km/h.
|
72
|
olympiads
| 0.546875 |
There are two alloys of copper and zinc. In the first alloy, there is twice as much copper as zinc, and in the second alloy, there is five times less copper than zinc. In what proportion should these alloys be mixed to get a new alloy in which there is twice as much zinc as copper?
|
1 : 2
|
olympiads
| 0.09375 |
Points \( A_{1}, B_{1}, C_{1} \) are the midpoints of the sides \( BC, AC, \) and \( AB \) of triangle \( ABC \), respectively. It is known that \( A_{1}A \) and \( B_{1}B \) are the angle bisectors of angles of triangle \( A_{1}B_{1}C_{1} \). Find the angles of triangle \( ABC \).
|
60^ullet, 60^ullet, 60^ullet
|
olympiads
| 0.234375 |
What is the maximum number of points at which 4 circles can intersect?
|
12
|
olympiads
| 0.140625 |
In an arm wrestling tournament, there are $2^{n}$ athletes, where $n$ is a natural number greater than 7. For each win, an athlete receives 1 point; for a loss, 0 points. Before each round, pairs are randomly formed from participants who have an equal number of points (those who cannot be paired receive a point automatically). After the seventh round, it turned out that exactly 42 participants had scored 5 points. What is the value of $n$?
|
8
|
olympiads
| 0.078125 |
Find the smallest real number \( m \) such that for any positive integers \( a, b, \) and \( c \) satisfying \( a + b + c = 1 \), the inequality \( m\left(a^{3}+b^{3}+c^{3}\right) \geqslant 6\left(a^{2}+b^{2}+c^{2}\right)+1 \) holds.
|
27
|
olympiads
| 0.09375 |
If several elementary school students go to buy cakes and if each student buys $\mathrm{K}$ cakes, the cake shop has 6 cakes left. If each student buys 8 cakes, the last student can only buy 1 cake. How many cakes are there in total in the cake shop?
|
97
|
olympiads
| 0.09375 |
Petya and Vasya are playing a game where they take turns naming non-zero decimal digits. It is forbidden to name a digit that is a divisor of any already named digit. The player who cannot make a move loses. Petya starts. Who wins if both play optimally?
|
Petya wins with optimal play.
|
olympiads
| 0.109375 |
For which values of $p$ and $q$ does the equation $x^{2}+px+q=0$ have two distinct solutions $2p$ and $p+q$?
|
p = \frac{2}{3}, q = -\frac{8}{3}
|
olympiads
| 0.21875 |
Find the real number $t$ , such that the following system of equations has a unique real solution $(x, y, z, v)$ :
\[ \left\{\begin{array}{cc}x+y+z+v=0 (xy + yz +zv)+t(xz+xv+yv)=0\end{array}\right. \]
|
t \in \left( \frac{3 - \sqrt{5}}{2}, \frac{3 + \sqrt{5}}{2} \right)
|
aops_forum
| 0.0625 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow 1} \frac{e^{x}-e}{\sin \left(x^{2}-1\right)}
\]
|
\frac{e}{2}
|
olympiads
| 0.578125 |
Compute
$$
\int_{1}^{2} \frac{9x+4}{x^{5}+3x^{2}+x} \, dx.
|
\ln \frac{80}{23}
|
olympiads
| 0.09375 |
Find the maximum possible area of a quadrilateral where the product of any two adjacent sides is equal to 1.
|
1
|
olympiads
| 0.328125 |
Given two skew lines $a$ and $b$ that form an angle $\theta$, with their common perpendicular line $A^{\prime} A$ having a length of $d$. Points $E$ and $F$ are taken on lines $a$ and $b$ respectively. Let $A^{\prime} E = m$ and $A F = n$. Find the distance $E F$. ($A^{\prime}$ lies on line $a$ perpendicular to $A$, and $A$ lies on line $b$ perpendicular to $A^{\prime}$).
|
\sqrt{d^2 + m^2 + n^2 \pm 2mn \cos \theta}
|
olympiads
| 0.0625 |
The points \( A=\left(4, \frac{1}{4}\right) \) and \( B=\left(-5, -\frac{1}{5}\right) \) lie on the hyperbola \( xy=1 \). The circle with diameter \( AB \) intersects this hyperbola again at points \( X \) and \( Y \). Compute \( XY \).
|
\sqrt{\frac{401}{5}}
|
olympiads
| 0.15625 |
Find the distance from point $M_0$ to the plane passing through the three points $M_1$, $M_2$, $M_3$.
$M_1(1, 5, -7)$
$M_2(-3, 6, 3)$
$M_3(-2, 7, 3)$
$M_0(1, -1, 2)$
|
7
|
olympiads
| 0.15625 |
Xiaoliang's family bought 72 eggs, and they also have a hen that lays one egg every day. If Xiaoliang's family eats 4 eggs every day, how many days will these eggs last them?
|
24
|
olympiads
| 0.59375 |
At the end of the term, Vovochka wrote down his current singing grades in a row and placed multiplication signs between some of them. The product of the resulting numbers turned out to be 2007. What grade does Vovochka have for the term in singing? (The singing teacher does not give "kol" grades.)
|
3
|
olympiads
| 0.21875 |
Let $\mathbb{R}^*$ be the set of all real numbers, except $1$ . Find all functions $f:\mathbb{R}^* \rightarrow \mathbb{R}$ that satisfy the functional equation $$ x+f(x)+2f\left(\frac{x+2009}{x-1}\right)=2010 $$ .
|
f(x) = \frac{1}{3}\left(x + 2010 - 2\frac{x+2009}{x-1}\right)
|
aops_forum
| 0.109375 |
The lengths of the sides of a rectangle are all integers. Four times its perimeter is numerically equal to one less than its area. Find the largest possible perimeter of such a rectangle.
|
164
|
olympiads
| 0.140625 |
Given the inequality \( x^2 - a x - 6a < 0 \) with respect to \( x \), determine the range of values of \( a \) such that the length of the solution interval does not exceed 5.
|
-25 \leq a <-24 \text{ or } 0 < a \leq 1
|
olympiads
| 0.0625 |
If \( x \in \mathbf{C} \) and \( x^{10} = 1 \), then find the value of \( 1 + x + x^2 + x^3 + \cdots + x^{2009} + x^{2010} \).
|
1
|
olympiads
| 0.203125 |
If \( R \) is the unit digit of the value of \( 8^{Q} + 7^{10Q} + 6^{100Q} + 5^{1000Q} \), find the value of \( R \).
|
8
|
olympiads
| 0.15625 |
Construct the $ \triangle ABC$ , given $ h_a$ , $ h_b$ (the altitudes from $ A$ and $ B$ ) and $ m_a$ , the median from the vertex $ A$ .
|
\triangle ABC
|
aops_forum
| 0.4375 |
On a plane, two vectors $\overrightarrow{OA}$ and $\overrightarrow{OB}$ satisfy $|\overrightarrow{OA}| = a$ and $|\overrightarrow{OB}| = b$, with $a^2 + b^2 = 4$ and $\overrightarrow{OA} \cdot \overrightarrow{OB} = 0$. Given the vector $\overrightarrow{OC} = \lambda \overrightarrow{OA} + \mu \overrightarrow{OB}$ ($\lambda, \mu \in \mathbf{R}$), and the condition $\left(\lambda - \frac{1}{2}\right)^2 a^2 + \left(\mu - \frac{1}{2}\right)^2 b^2 = 1$, determine the maximum value of $|\overrightarrow{OC}|$.
|
2
|
olympiads
| 0.171875 |
In how many ways can you place 8 rooks on a chessboard so that they do not attack each other?
A rook moves horizontally and vertically. For example, a rook on d3 threatens all the squares in column d and row 3.
|
40320
|
olympiads
| 0.4375 |
Two circles with radii \(R\) and \(r\) (\(R > r\)) touch each other externally. Find the radii of the circles that touch both of the given circles and their common external tangent.
|
\frac{Rr}{(\sqrt{R} \pm \sqrt{r})^2}
|
olympiads
| 0.078125 |
There is a quadrilateral drawn on a sheet of transparent paper. Specify a way to fold this sheet (possibly multiple times) to determine whether the original quadrilateral is a rhombus.
|
ABCD is a rhombus
|
olympiads
| 0.0625 |
After a football match, the coach lined up the team as shown in the picture and commanded: "Run to the locker room if your number is less than either of your neighbors' numbers." After several players ran off, he repeated his command. The coach continued until only one player was left. What is Igor's number if it is known that after he ran off, 3 people remained in the line? (After each command, one or more players ran off, and the line closed up so that there were no empty spots between the remaining players.)
|
5
|
olympiads
| 0.140625 |
Determine the functions \( f:(0,1) \rightarrow \mathbf{R} \) for which:
\[ f(x \cdot y) = x \cdot f(x) + y \cdot f(y) \]
|
f(x) = 0
|
olympiads
| 0.203125 |
Let $f(x)=cx(x-1)$ , where $c$ is a positive real number. We use $f^n(x)$ to denote the polynomial obtained by composing $f$ with itself $n$ times. For every positive integer $n$ , all the roots of $f^n(x)$ are real. What is the smallest possible value of $c$ ?
|
4
|
aops_forum
| 0.078125 |
As shown in the diagram, the area of trapezoid $\mathrm{ABCD}$ is 117 square centimeters. $\mathrm{EF}$ is 13 centimeters, $\mathrm{MN}$ is 4 centimeters, and it is also known that there is point $\mathrm{O}$. What is the total area of the shaded region in square centimeters?
|
65 \text{ square cm}
|
olympiads
| 0.09375 |
The distance of a point light source from a sphere is equal to three times the radius of the sphere. How does the illuminated area of the sphere compare to the lateral surface area of the cone of light?
|
\frac{2}{5}
|
olympiads
| 0.0625 |
Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\]
|
5
|
aops_forum
| 0.109375 |
Let $n$ be a positive integer. Determine, in terms of $n$ , the greatest integer which divides
every number of the form $p + 1$ , where $p \equiv 2$ mod $3$ is a prime number which does not divide $n$ .
|
6
|
aops_forum
| 0.1875 |
If complex numbers \(z_{1}, z_{2}, z_{3}\) satisfy \(\frac{z_{3}-z_{1}}{z_{2}-z_{1}}=a i \ (a \in \mathbf{R}, a \neq 0)\), then the angle between the vectors \(\overrightarrow{Z_{1} Z_{2}}\) and \(\overrightarrow{Z_{1} Z_{3}}\) is \(\qquad\).
|
\frac{\pi}{2}
|
olympiads
| 0.296875 |
A certain number is written in the base-12 numeral system. For which divisor \( m \) is the following divisibility rule valid: if the sum of the digits of the number is divisible by \( m \), then the number itself is divisible by \( m \)?
|
11
|
olympiads
| 0.09375 |
Seller reduced price of one shirt for $20\%$ ,and they raised it by $10\%$ . Does he needs to reduce or raise the price and how many, so that price of shirt will be reduced by $10\%$ from the original price
|
2 \text{ dollars}
|
aops_forum
| 0.125 |
Calculate the limit of the function:
$$\lim_{x \rightarrow \pi} (x + \sin x)^{\sin x + x}$$
|
\pi^{\pi}
|
olympiads
| 0.078125 |
Given \( 0 < x < 1 \) and \( a, b \) are both positive constants, the minimum value of \( \frac{a^{2}}{x}+\frac{b^{2}}{1-x} \) is ______.
|
(a + b)^2
|
olympiads
| 0.09375 |
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