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|---|---|---|---|
Let $M= \{ 1, 2, \cdots, 19 \}$ and $A = \{ a_{1}, a_{2}, \cdots, a_{k}\}\subseteq M$ . Find the least $k$ so that for any $b \in M$ , there exist $a_{i}, a_{j}\in A$ , satisfying $b=a_{i}$ or $b=a_{i}\pm a_{i}$ ( $a_{i}$ and $a_{j}$ do not have to be different) .
|
5
|
aops_forum
| 0.0625
|
Try to find all natural numbers that are 5 times greater than their last digit.
|
25
|
olympiads
| 0.109375
|
On one side of an acute angle, points $A$ and $B$ are taken. Find a point $C$ on the other side of the angle such that the angle $A C B$ is the largest possible. Construct the point $C$ using a compass and straightedge.
|
C
|
olympiads
| 0.515625
|
Given a line \( l \) with intercepts on the coordinate axes that are mutual opposites, and a point \( M(1, -1) \) with a distance of \( \sqrt{2} \) from \( l \). How many lines \( l \) meet these conditions?
|
1 \text{ line}
|
olympiads
| 0.25
|
Given a sequence $\left\{a_{n}\right\}$ with 100 terms, where $a_{1}=0$ and $a_{100}=475$, and $\left|a_{k+1}-a_{k}\right|=5$ for $k=1,2,\cdots, 99$, determine the number of different sequences that satisfy these conditions.
|
4851
|
olympiads
| 0.0625
|
\( 3 \cdot 4^{(x-2)} + 27 = a + a \cdot 4^{(x-2)} \). For which values of \( a \) does the equation have a solution?
|
a \in (3, 27)
|
olympiads
| 0.0625
|
A number \( A \) consisting of eight non-zero digits is added to a seven-digit number consisting of identical digits, resulting in an eight-digit number \( B \). It turns out that \( B \) can be obtained by permuting some of the digits of \( A \). What digit can \( A \) start with if the last digit of \( B \) is 5?
|
5
|
olympiads
| 0.15625
|
The distinct prime factors of 4446 are 2, 3, and 13. What is the sum of all of the distinct prime factors of \(4446\)?
|
37
|
olympiads
| 0.09375
|
In triangle $ABC$, point $K$ is chosen on side $AB$, and the angle bisector $KE$ of triangle $AKC$ and the altitude $KH$ of triangle $BKC$ are drawn. It turns out that $\angle EKH$ is a right angle. Find $BC$ if $HC=5$.
|
10
|
olympiads
| 0.171875
|
There is exactly one isosceles triangle that has a side of length 10 and a side of length 22. What is the perimeter of this triangle?
|
54
|
olympiads
| 0.140625
|
Calculate the limit of the numerical sequence:
\[
\lim _{n \rightarrow \infty} \frac{\sqrt{n^{7}+5}-\sqrt{n-5}}{\sqrt[7]{n^{7}+5}+\sqrt{n-5}}
\]
|
1
|
olympiads
| 0.265625
|
Consider the sequences \(\{a_{n}\}\) and \(\{b_{n}\}\), where \(a_{1} = p\) and \(b_{1} = q\). It is known that \(a_{n} = p a_{n-1}\) and \(b_{n} = q a_{n-1} + r b_{n-1}\) (with constants \(p, q, r\), and \(q > 0\), \(p > r > 0\), \(n \geq 2\)). Find the general term formula for the sequence \(\{b_{n}\}\): \(b_{n} = \_\).
|
\frac{q(p^n - r^n)}{p-r}
|
olympiads
| 0.28125
|
Given the set $\Omega=\left\{(x, y) \mid x^{2}+y^{2} \leqslant 2008\right\}$, if point $P(x, y)$ and point $P^{\prime}\left(x^{\prime}, y^{\prime}\right)$ satisfy $x \leqslant x^{\prime}$ and $y \geqslant y^{\prime}$, then point $P$ is said to be superior to point $P^{\prime}$. If a point $Q$ in set $\Omega$ satisfies that there is no other point in $\Omega$ superior to $Q$, then the set of all such points $Q$ is ___.
|
\{(x, y) \mid x^2 + y^2 = 2008, \ x \leq 0, \ y \geq 0 \}
|
olympiads
| 0.078125
|
Find all natural numbers $n$ for which $n + 195$ and $n - 274$ are perfect cubes.
|
2002
|
aops_forum
| 0.078125
|
Evaluate \(\frac{2016!^{2}}{2015! \cdot 2017!}\). Here \( n! \) denotes \( 1 \times 2 \times \cdots \times n \).
|
\frac{2016}{2017}
|
olympiads
| 0.46875
|
If $x > 10$ , what is the greatest possible value of the expression
\[
{( \log x )}^{\log \log \log x} - {(\log \log x)}^{\log \log x} ?
\]
All the logarithms are base 10.
|
0
|
aops_forum
| 0.171875
|
In a math competition, each correct answer for a mandatory question earns 3 points, and each incorrect answer deducts 2 points; each correct answer for an optional question earns 5 points, and each incorrect answer earns 0 points. Xiaoming answered all the questions and got 15 correct answers in total, with a score of 49 points. How many mandatory questions were there in this math competition?
|
13
|
olympiads
| 0.34375
|
Let $O$ be the origin, and $F$ be the right focus of the ellipse $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ where $a > b > 0$. The line $l$ passing through $F$ intersects the ellipse $C$ at points $A$ and $B$. Two points $P$ and $Q$ on the ellipse satisfy
$$
\overrightarrow{O P}+\overrightarrow{O A}+\overrightarrow{O B}=\overrightarrow{O P}+\overrightarrow{O Q}=0,
$$
and the points $P, A, Q,$ and $B$ are concyclic. Find the eccentricity of the ellipse $C$.
|
\frac{\sqrt{2}}{2}
|
olympiads
| 0.15625
|
Let \(x_{1}, x_{2}, \ldots, x_{200}\) be natural numbers greater than 2 (not necessarily distinct). In a \(200 \times 200\) table, the numbers are arranged as follows: at the intersection of the \(i\)-th row and the \(k\)-th column, the number \(\log _{x_{k}} \frac{x_{i}}{9}\) is written. Find the smallest possible value of the sum of all the numbers in the table.
|
-40000
|
olympiads
| 0.078125
|
13 children sat around a circular table and agreed that boys would lie to girls but tell the truth to each other, while girls would lie to boys but tell the truth to each other. One of the children told his right-hand neighbor, "Most of us are boys." The neighbor then told his right-hand neighbor, "Most of us are girls." This pattern continued in the same way, alternating statements, until the last child told the first, "Most of us are boys." How many boys are at the table?
|
7
|
olympiads
| 0.296875
|
On the coordinate plane \( Oxy \), the graph of the function \( y = x^2 \) was drawn. Then the coordinate axes were erased - only the parabola remained. How can you restore the coordinate axes and the unit of length using only a compass and a ruler?
|
Coordinate Axes and Unit Length Reconstructed
|
olympiads
| 0.09375
|
A banquet has invited 44 guests. There are 15 identical square tables, each of which can seat 1 person per side. By appropriately combining the square tables (to form rectangular or square tables), ensure that all guests are seated with no empty seats. What is the minimum number of tables in the final arrangement?
|
11
|
olympiads
| 0.265625
|
Suppose you are given that for some positive integer $n$ , $1! + 2! + \ldots + n!$ is a perfect square. Find the sum of all possible values of $n$ .
|
4
|
aops_forum
| 0.40625
|
In how many ways can a square with a side length of 5 units be placed on an $8 \times 8$ chessboard such that each vertex of the square falls on the center of a square? (Solutions that can be transformed into each other by rotation or reflection are not considered different.)
|
4
|
olympiads
| 0.109375
|
Find all functions \( f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} \) such that
\[ f(x + f(y + xy)) = (y + 1) f(x + 1) - 1 \]
for all \( x, y \in \mathbb{R}^+ \).
(Note: \(\mathbb{R}^+\) denotes the set of positive real numbers.)
|
f(x) = x \text{ for all } x \in \mathbb{R}^{+}
|
olympiads
| 0.171875
|
Find the residue of the function
$$
f(z)=z^{3} \cdot \sin \frac{1}{z^{2}}
$$
at its singular point.
|
0
|
olympiads
| 0.296875
|
Given that $a$ and $b$ are positive integers, and $a - b \sqrt{3} = (2 - \sqrt{3})^{100}$, find the unit digit of $a \cdot b$.
|
2
|
olympiads
| 0.109375
|
The diagonals of a convex quadrilateral are $d_{1}$ and $d_{2}$. What is the maximum possible value of its area?
|
\frac{1}{2} d_1 d_2
|
olympiads
| 0.453125
|
What number must be added to both terms of a fraction to obtain the reciprocal of that same fraction?
|
-(a + b)
|
olympiads
| 0.34375
|
Given \( a, b, c, d \in [0, \sqrt[4]{2}) \), such that \( a^{3} + b^{3} + c^{3} + d^{3} = 2 \).
|
2
|
olympiads
| 0.125
|
Calculate \(\sqrt{31 \times 30 \times 29 \times 28 + 1}\).
|
869
|
olympiads
| 0.09375
|
The circles \( S_1 \) and \( S_2 \) with centers \( O_1 \) and \( O_2 \) respectively touch each other externally; a line touches the circles at distinct points \( A \) and \( B \) respectively. It is known that the intersection point of the diagonals of quadrilateral \( O_1 A B O_2 \) lies on one of the circles. Find the ratio of the radii of the circles.
|
1:2
|
olympiads
| 0.15625
|
Find \( y'' \) if \( x = \ln t \) and \( y = \sin 2t \).
|
-4 t^2 \sin 2 t + 2 t \cos 2 t
|
olympiads
| 0.171875
|
Denis painted some faces of his cubes gray. Vova selected 10 cubes so that each one had a different coloring, and then arranged them as shown in the picture. How many total white faces does the top cube have?
|
5
|
olympiads
| 0.15625
|
The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\circ}$ and $ SB \equal{} SC \equal{} 1$ . Find the volume of the tetrahedron.
|
\frac{1}{8}
|
aops_forum
| 0.09375
|
A table consists of several rows of numbers. From the second row onwards, each number in a row is equal to the sum of the two numbers directly above it. The last row contains only one number. The first row is made up of the first 100 positive integers arranged in ascending order. What is the number in the last row? (You may use exponential notation to express the number).
|
101 \times 2^{98}
|
olympiads
| 0.15625
|
A pool has two drain taps, A and B. If both taps are opened simultaneously, it takes 30 minutes to drain the full pool. If both taps are opened for 10 minutes and then tap A is closed while tap B continues, it also takes 30 more minutes to drain the full pool. How many minutes will it take to completely drain the full pool with only tap B open?
|
45
|
olympiads
| 0.296875
|
Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$ , such that inequality holds $$ \frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}} $$
|
x \in (\min(a, b), \max(a, b))
|
aops_forum
| 0.0625
|
In a kindergarten, 5 children eat porridge every day, 7 children eat porridge every other day, and the rest never eat porridge. Yesterday, 9 children ate porridge. How many children will eat porridge today?
|
8
|
olympiads
| 0.203125
|
In the trapezoid \(ABCD\), if \(AB = 8\), \(DC = 10\), the area of \(\triangle AMD\) is 10, and the area of \(\triangle BCM\) is 15, then the area of trapezoid \(ABCD\) is \(\quad\).
|
45
|
olympiads
| 0.125
|
In an acute triangle \( K L M \) with \(\angle K L M = 68^\circ\), point \( V \) is the orthocenter, and point \( P \) is the foot of the altitude from \( K \) to the side \( L M \). The angle bisector of \(\angle P V M\) is parallel to side \( K M \).
Compare the sizes of angles \(\angle M K L\) and \(\angle L M K\).
|
\angle MKL = \angle LMK
|
olympiads
| 0.1875
|
Dmitry has socks in his closet: 6 blue pairs, 18 black pairs, and 12 white pairs. Dmitry bought several more pairs of black socks and found that now the black socks make up 3/5 of the total number of socks. How many pairs of black socks did Dmitry buy?
|
9
|
olympiads
| 0.515625
|
On the line \( y = -\frac{13}{6} \), find the point \( M \) through which pass two tangents to the graph of the function \( y = \frac{x^2}{2} \) that form an angle of \( 60^\circ \) between them.
|
M(\pm 2, -\frac{13}{6})
|
olympiads
| 0.0625
|
Petya bought himself football shorts in the store.
- If he had bought the shorts with a T-shirt, the cost of the purchase would have been twice as much.
- If he had bought the shorts with boots, the cost of the purchase would have been five times as much.
- If he had bought the shorts with shin guards, the cost of the purchase would have been three times as much.
By how many times more would the cost of the purchase be if Petya bought the shorts, T-shirt, boots, and shin guards?
|
8
|
olympiads
| 0.4375
|
The rectangle $3 \times 100$ consists of 300 squares $1 \times 1$. What is the largest number of diagonals that can be drawn in the squares so that no two diagonals share endpoints? (In one square, two diagonals can be drawn without sharing endpoints. Shared internal points are allowed.)
|
200
|
olympiads
| 0.09375
|
A circle is circumscribed around a square with side length $a$, and a regular hexagon is circumscribed around the circle. Determine the area of the hexagon.
|
\sqrt{3} a^2
|
olympiads
| 0.1875
|
The lateral surface area of a regular triangular pyramid is 5 times greater than the area of its base. Find the dihedral angle at the apex of the pyramid.
|
2 \arctan \left(\frac{\sqrt{3}}{5}\right)
|
olympiads
| 0.234375
|
C and C' are fixed circles. A is a fixed point on C, and A' is a fixed point on C'. B is a variable point on C. B' is the point on C' such that A'B' is parallel to AB. Find the locus of the midpoint of BB'.
|
The locus of the midpoint of BB' is a circle.
|
olympiads
| 0.0625
|
Buckets $A$ and $B$ have the same weight of water. If 2.5 kilograms of water is poured from bucket $A$ into bucket $B$, the weight of the water in bucket $B$ will be six times the weight of the water in bucket $A$. How many kilograms of water was originally in bucket $B$?
|
3.5 \text{ kilograms}
|
olympiads
| 0.515625
|
Points \( M, N, P, Q \) are taken on the diagonals \( D_1A, A_1B, B_1C, C_1D \) of the faces of cube \( ABCD A_1B_1C_1D_1 \) respectively, such that:
\[ D_1M: D_1A = BA_1: BN = B_1P: B_1C = DQ: DC_1 = \mu, \]
and the lines \( MN \) and \( PQ \) are mutually perpendicular. Find \( \mu \).
|
\frac{1}{\sqrt{2}}
|
olympiads
| 0.09375
|
Given the equation \(3 \sin x + 4 \cos x = a\) has exactly two equal real roots \(\alpha, \beta\) within the interval \((0, 2\pi)\). Find \(\alpha + \beta\).
|
\pi - 2 \arcsin \frac{4}{5} \text{ or } 3\pi - 2 \arcsin \frac{4}{5}
|
olympiads
| 0.109375
|
Compute the limit of the function:
$$
\lim _{x \rightarrow 2 \pi} \frac{(x-2 \pi)^{2}}{\tan(\cos x - 1)}
$$
|
-2
|
olympiads
| 0.125
|
Let point \( P \) be on the ellipse \( \frac{x^{2}}{5}+y^{2}=1 \), and let \( F_{1} \) and \( F_{2} \) be the two foci of the ellipse. If the area of \( \triangle F_{1} P F_{2} \) is \( \frac{\sqrt{3}}{3} \), find \( \angle F_{1} P F_{2} \).
|
60^
\circ
|
olympiads
| 0.203125
|
Write an \( n \)-digit number using the digits 1 and 2, such that no two consecutive digits are both 1. Denote the number of such \( n \)-digit numbers as \( f(n) \). Find \( f(10) \).
|
144
|
olympiads
| 0.546875
|
In a positive non-constant geometric progression, the arithmetic mean of the second, seventh, and ninth terms is equal to some term of this progression. What is the smallest possible index of this term?
|
3
|
olympiads
| 0.078125
|
Given that the complex numbers \( z_1 \) and \( z_2 \) satisfy \( \left|z_1\right| = 2 \) and \( \left|z_2\right| = 3 \). If the angle between the corresponding vectors is \( 60^\circ \), then compute \( \left|\frac{z_1 + z_2}{z_1 - z_2}\right| \).
|
\frac{\sqrt{133}}{7}
|
olympiads
| 0.0625
|
In a certain forest, there are \( n \geqslant 3 \) starling nests, and the distances between these nests are all distinct. Each nest contains one starling. At certain moments, some starlings fly from their nests and land in other nests. If among the flying starlings, the distance between any pair of starlings is less than the distance between another pair of starlings, then after landing, the distance between the first pair of starlings will always be greater than that between the second pair of starlings. For what value(s) of \( n \) can this be achieved?
|
3
|
olympiads
| 0.40625
|
There are 2019 numbers written on the board. One of them occurs more frequently than the others - 10 times. What is the minimum number of different numbers that could be written on the board?
|
225
|
olympiads
| 0.0625
|
Ninety-nine children are standing in a circle, each initially holding a ball. Every minute, each child with a ball throws their ball to one of their two neighbors. If two balls end up with the same child, one of these balls is irrevocably lost. What is the minimum time required for the children to have only one ball left?
|
98
|
olympiads
| 0.28125
|
Given the equation \(x^{2}+2xy+y^{2}+3x+y=0\) represents a parabola in the \(xOy\) plane, find the equation of its axis of symmetry.
|
x + y + 1 = 0
|
olympiads
| 0.125
|
Given the function \( y = a^{2x} + 2a^x - 1 \), find the value of \( a \) for which the maximum value of the function in the interval \([-1, 1]\) is 14.
|
a = \frac{1}{3} \text{ or } 3
|
olympiads
| 0.359375
|
Several cells of a \(14 \times 14\) board are marked. It is known that no two marked cells lie in the same row or the same column, and that a knight can start from one of the marked cells and visit all the marked cells exactly once by making several jumps. What is the maximum possible number of marked cells?
|
13
|
olympiads
| 0.390625
|
Adva van egy egyenes körkúp; tengelye merőleges az első képsíkra és egy $L$ pont; meghatározandó az $L$-en átmenő két érintősík $(\varphi)$ hajlásszöge.
|
\varphi
|
olympiads
| 0.296875
|
Place \( 4 \) points so that when measuring the pairwise distances between them, only two distinct numbers are obtained. Find all such arrangements.
|
The arrangement involves 4 points chosen from a regular pentagon's vertices.
|
olympiads
| 0.109375
|
The set $M=\{1,2,\ldots,2007\}$ has the following property: If $n$ is an element of $M$ , then all terms in the arithmetic progression with its first term $n$ and common difference $n+1$ , are in $M$ . Does there exist an integer $m$ such that all integers greater than $m$ are elements of $M$ ?
|
m = 2007
|
aops_forum
| 0.109375
|
How many 3-digit positive integers have exactly one even digit?
|
350
|
olympiads
| 0.1875
|
1. Baron Munchhausen was told that some polynomial $P(x)=a_{n} x^{n}+\ldots+a_{1} x+a_{0}$ is such that $P(x)+P(-x)$ has exactly 45 distinct real roots. Baron doesn't know the value of $n$ . Nevertheless he claims that he can determine one of the coefficients $a_{n}, \ldots, a_{1}, a_{0}$ (indicating its position and value). Isn't Baron mistaken?
Boris Frenkin
|
a_0 = 0
|
aops_forum
| 0.453125
|
It is known that the equation $ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$ .
|
1001000
|
aops_forum
| 0.15625
|
This year, the sum of the ages of person A and person B is 70 years. Some years ago, when person A's age was exactly the current age of person B, the age of person B was exactly half of person A's age at that time. How old is person A this year?
|
42
|
olympiads
| 0.28125
|
The pentagon $ABCDE$ is obtained by rotating the pentagon $HICBG$ around point $C$, and the pentagon $FGBAE$ is obtained by rotating the pentagon $ABCDE$ around point $E$, as shown in the sketch. The length of the segment $AB$ is $7 \text{ cm}$.
What is the total length of the 11 segments?
|
77 \, \text{cm}
|
olympiads
| 0.125
|
Define a sequence \(\left\{a_{n}\right\}\) by \(a_{1}=1\) and \(a_{n}=\left(a_{n-1}\right)!+1\) for every \(n>1\). Find the least \(n\) for which \(a_{n}>10^{10}\).
|
6
|
olympiads
| 0.59375
|
There are $n$ ellipses centered at the origin, with their axes being the coordinate axes, and their directrices are all $x=1$. If the eccentricity of the $k$-th $(k=1,2, \cdots, n)$ ellipse is $e_{k}=2^{-k}$, what is the sum of the lengths of the major axes of these $n$ ellipses?
|
2 - 2^{1-n}
|
olympiads
| 0.125
|
Fifty children went to the zoo, with 36 of them seeing the pandas, 28 seeing the giraffes, and 15 seeing the pandas but not the giraffes. How many children saw the giraffes but not the pandas?
|
7
|
olympiads
| 0.484375
|
Masha came up with the number \( A \), and Pasha came up with the number \( B \). It turned out that \( A + B = 2020 \), and the fraction \( \frac{A}{B} \) is less than \( \frac{1}{4} \). What is the maximum value that the fraction \( \frac{A}{B} \) can take?
|
\frac{403}{1617}
|
olympiads
| 0.28125
|
A cafeteria serves ham and cheese sandwiches, ham and tomato sandwiches, and tomato and cheese sandwiches. On a certain day, 80 customers had meals containing both ham and cheese; 90 had meals containing both ham and tomatoes; 100 had meals containing both tomatoes and cheese; and 20 customers had meals that included all three ingredients. How many customers were there?
|
230
|
olympiads
| 0.078125
|
While waiting for customers, a watermelon seller sequentially weighed 20 watermelons (weighing 1 kg, 2 kg, 3 kg, ..., 20 kg), balancing each watermelon on one side of the scale with one or two weights on the other side (possibly identical weights). The seller recorded on a piece of paper the weights of the scales he used. What is the smallest number of different weights that could appear in his records if the mass of each weight is an integer kilogram?
|
6
|
olympiads
| 0.078125
|
There are three defendants, and only one of them is guilty. Each defendant accused one of the other two, but it is not remembered who accused whom. It is known that some defendants told the truth while others lied, but the specific details of who lied and who told the truth are not remembered. The White Knight recalls an additional point: either there were two consecutive truths or two consecutive lies in the statements, but he does not remember which of these scenarios is accurate.
Can you determine who is guilty?
|
C
|
olympiads
| 0.265625
|
I have two coins that sum up to 15 kopecks. One of them is not a five-kopek coin. What are these coins?
|
5 kopecks and 10 kopecks
|
olympiads
| 0.28125
|
Given the real numbers \(\alpha\) and \(\beta\) that satisfy the system of equations
\[
\left\{\begin{array}{l}
\alpha^{3} - 3\alpha^{2} + 5\alpha - 17 = 0, \\
\beta^{3} - 3\beta^{2} + 5\beta + 11 = 0,
\end{array}\right.
\]
find \(\alpha + \beta\).
|
2
|
olympiads
| 0.109375
|
Let \( a_{n} = \frac{1}{3} + \frac{1}{12} + \frac{1}{30} + \frac{1}{60} + \cdots + \frac{2}{n(n-1)(n-2)} + \frac{2}{(n+1) n(n-1)} \), find \( \lim_{n \rightarrow \infty} a_{n} \).
|
\frac{1}{2}
|
olympiads
| 0.265625
|
Find all prime numbers $p$ and $q$ such that $2^2+p^2+q^2$ is also prime.
Please remember to hide your solution. (by using the hide tags of course.. I don't literally mean that you should hide it :ninja: )
|
(3, 2) \text{ and } (2, 3)
|
aops_forum
| 0.40625
|
Five people are standing in a line, each wearing a different hat numbered $1, 2, 3, 4, 5$. Each person can only see the hats of the people in front of them. Xiao Wang cannot see any hats; Xiao Kong can only see hat number 4; Xiao Tian cannot see hat number 3 but can see hat number 1; Xiao Yan sees three hats but does not see hat number 3; Xiao Wei sees hats numbered 3 and 2. What number hat is Xiao Tian wearing?
|
2
|
olympiads
| 0.15625
|
In the diagram below, the two circles have the same center. Point \( A \) is on the inner circle and point \( B \) is on the outer circle. Line segment \( AB \) has length 5 and is tangent to the inner circle at \( A \). What is the area of the shaded region?
|
25\pi
|
olympiads
| 0.296875
|
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x\in \mathbb{R} \ \ f(x) = max(2xy-f(y))$ where $y\in \mathbb{R}$ .
|
f(x) = x^2
|
aops_forum
| 0.234375
|
\(\triangle ABC\) is acute-angled with \(AB = 13\) and \(BC = 7\). \(D\) and \(E\) are points on \(AB\) and \(AC\) respectively such that \(BD = BC\) and \(\angle DEB = \angle CEB\). Find the product of all possible values of the length of \(AE\).
|
\frac{507}{10}
|
olympiads
| 0.171875
|
Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of $n$ between $1$ and $2013$ (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes.
|
1007
|
aops_forum
| 0.109375
|
In a row of compartments containing several balls, two players, A and B, take turns moving the balls. The rules for moving are: each person can choose any number of balls from any compartment and move them to the immediately adjacent compartment to the right. The game ends when all the balls are moved to the compartment on the far right, and the player who moves the last ball wins.
Given the situation illustrated in the figure (the numbers in each compartment represent the number of balls, and the numbers below represent the compartment numbers), if the first player has a winning strategy, to ensure victory, the first player should move $\qquad$ balls from compartment $\qquad$.
|
1
|
olympiads
| 0.234375
|
As shown in the figure, the area of the rectangle is $24 \mathrm{~cm}^{2}$, the total area of the two shaded triangles is $7.5 \mathrm{~cm}^{2}$, and the area of quadrilateral $ABCD$ is $\qquad \mathrm{cm}^{2}$.
|
16.5
|
olympiads
| 0.546875
|
8 singers are participating in a festival and are scheduled to perform \( m \) times, with 4 singers performing in each show. Design a plan to minimize the number of performances \( m \) so that any two singers perform together the same number of times.
|
14
|
olympiads
| 0.09375
|
A natural number $n$ is called *perfect* if it is equal to the sum of all its natural divisors other than $n$ . For example, the number $6$ is perfect because $6 = 1 + 2 + 3$ . Find all even perfect numbers that can be given as the sum of two cubes positive integers.
|
28
|
aops_forum
| 0.125
|
The sum of the digits of a ten-digit number is four. What can be the sum of the digits of the square of this number?
|
7 or 16
|
olympiads
| 0.171875
|
If \( p \) and \( q \) are positive integers and \(\frac{2008}{2009} < \frac{p}{q} < \frac{2009}{2010} \), what is the minimum value of \( p \)?
|
4017
|
olympiads
| 0.34375
|
A boy named Fred bought a banana and was asked by a friend how much he paid for it. Fred replied that the seller receives half as many sixpences for sixteen dozen dozen bananas as he gives bananas for a five-pound note. How much did Fred pay for the banana?
|
1.25 \text{ pence}
|
olympiads
| 0.0625
|
The sum of a two-digit number and its reversed number is a perfect square. Find all such numbers.
|
29, 38, 47, 56, 65, 74, 83, 92
|
olympiads
| 0.171875
|
Given positive real numbers \(a\) and \(b\) satisfying \(a(a+b) = 27\), find the maximum value of \(a^2 b\).
|
54
|
olympiads
| 0.140625
|
Find the sequences $\left(a_{n}\right)$ consisting of positive integers such that for every $i \neq j$, the greatest common divisor of $a_{i}$ and $a_{j}$ is equal to the greatest common divisor of $i$ and $j$.
|
a_n = n
|
olympiads
| 0.0625
|
Calculate the area of the parallelogram formed by the vectors \(a\) and \(b\).
Given:
\[ a = 6p - q \]
\[ b = p + q \]
\[ |p| = 3 \]
\[ |q| = 4 \]
\[ (\widehat{p, q}) = \frac{\pi}{4} \]
|
42\sqrt{2}
|
olympiads
| 0.1875
|
In the figure, the area of triangle $\triangle \mathrm{ABC}$ is 60. Points $\mathrm{E}$ and $\mathrm{F}$ are on $\mathrm{AB}$ and $\mathrm{AC}$, respectively, satisfying $\mathrm{AB}=3 \mathrm{AE}$ and $\mathrm{AC}=3 \mathrm{AF}$. Point $D$ is a moving point on segment $BC$. Let the area of $\triangle FBD$ be $S_{1}$, and the area of $\triangle EDC$ be $S_{2}$. Find the maximum value of $S_{1} \times S_{2}$.
|
400
|
olympiads
| 0.0625
|
Calculate the integral
$$
\int_{|z|=2} \frac{\cosh(iz)}{z^{2} + 4z + 3} \, dz
$$
|
\pi i \cos 1
|
olympiads
| 0.140625
|
Let point \( A(-2,0) \) and point \( B(2,0) \) be given, and let point \( P \) be on the unit circle. What is the maximum value of \( |PA| \cdot |PB| \)?
|
5
|
olympiads
| 0.171875
|
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