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|---|---|---|---|
Consider set \( S \) formed by the points inside and on the edges of a regular hexagon with side length 1. Find the minimum value of \( r \) such that there is a way to color the points in \( S \) with three colors such that the distance between any two points of the same color is less than \( r \).
|
\frac{3}{2}
|
olympiads
| 0.109375
|
\[
\sum_{k=1}^{70} \frac{k}{x-k} \geq \frac{5}{4}
\]
is a union of disjoint intervals the sum of whose lengths is 1988.
|
1988
|
olympiads
| 0.125
|
Find \(\lim _{x \rightarrow 0}\left(\frac{1}{4 \sin ^{2} x}-\frac{1}{\sin ^{2} 2 x}\right)\).
|
-\frac{1}{4}
|
olympiads
| 0.203125
|
\(\frac{\sin 20^{\circ} \cos 10^{\circ} + \cos 160^{\circ} \cos 100^{\circ}}{\sin 21^{\circ} \cos 9^{\circ} + \cos 159^{\circ} \cos 99^{\circ}} = 1\).
|
1
|
olympiads
| 0.328125
|
If $B$ is a sufficient condition for $A$, will $B$ be a necessary condition for $A$?
|
Negative
|
olympiads
| 0.125
|
Determine the area of a segment if its perimeter is $p$ and the arc contains $120^\circ$.
|
\frac{3p^2 \left(4\pi - 3\sqrt{3}\right)}{4 \left(2\pi + 3\sqrt{3}\right)^2}
|
olympiads
| 0.125
|
Three distinct diameters are drawn on a unit circle such that chords are drawn as shown. If the length of one chord is \(\sqrt{2}\) units and the other two chords are of equal lengths, what is the common length of these chords?
|
\sqrt{2-\sqrt{2}}
|
olympiads
| 0.140625
|
Let $ABCD$ be a convex quadrilateral with $\angle ABD = \angle BCD$ , $AD = 1000$ , $BD = 2000$ , $BC = 2001$ , and $DC = 1999$ . Point $E$ is chosen on segment $DB$ such that $\angle ABD = \angle ECD$ . Find $AE$ .
|
1000
|
aops_forum
| 0.0625
|
Given the function \( f(x) \) presented in Table 1, \( a_{k} (0 \leqslant k \leqslant 4) \) represents the number of times \( k \) appears among \( a_{0}, a_{1}, \cdots, a_{4} \). Then \( a_{0}+a_{1}+a_{2}+a_{3} = \qquad \).
|
5
|
olympiads
| 0.25
|
Find all $ f:N\rightarrow N$ , such that $\forall m,n\in N $ $ 2f(mn) \geq f(m^2+n^2)-f(m)^2-f(n)^2 \geq 2f(m)f(n) $
|
f(n) = n^2 \quad \forall n \in \mathbb{N}
|
aops_forum
| 0.109375
|
Find all functions \( f: \mathbb{Z}_{+} \rightarrow \mathbb{Z}_{+} \) that satisfy
\[ f(x + y f(x)) = x + f(x) f(y) \]
for all positive integers \( x \) and \( y \).
|
f(x) = x
|
olympiads
| 0.5625
|
For a numerical sequence \(\{x_{n}\}\), where all members starting from \(n \geq 2\) are distinct, the relation
\[ x_{n} = \frac{x_{n-1} + 398x_{n} + x_{n+1}}{400} \]
is satisfied. Find
\[
\sqrt{\frac{x_{2023} - x_{2}}{2021} \cdot \frac{2022}{x_{2023} - x_{1}}} + 2021.
\]
|
2022
|
olympiads
| 0.125
|
Find all polynomials $f(x)$ with integer coefficients such that $f(n)$ and $f(2^{n})$ are co-prime for all natural numbers $n$ .
|
f(x) \equiv 1 \text{ or } f(x) \equiv -1
|
aops_forum
| 0.078125
|
Three equal circles of radius $r$ each pass through the centres of the other two.
What is the area of intersection that is common to all the three circles?
|
r^2 \left( \frac{\pi - \sqrt{3}}{2} \right)
|
aops_forum
| 0.546875
|
Find all integers \( x, y \geq 1 \) such that \( 7^x = 3^y + 4 \).
|
(1, 1)
|
olympiads
| 0.234375
|
Let $A_1B_1C_1D_1A_2B_2C_2D_2$ be a unit cube, with $A_1B_1C_1D_1$ and $A_2B_2C_2D_2$ opposite square faces, and let $M$ be the center of face $A_2 B_2 C_2 D_2$ . Rectangular pyramid $MA_1B_1C_1D_1$ is cut out of the cube. If the surface area of the remaining solid can be expressed in the form $a + \sqrt{b}$ , where $a$ and $b$ are positive integers and $b$ is not divisible by the square of any prime, find $a+b$ .
|
11
|
aops_forum
| 0.0625
|
Robinson Crusoe was swimming in a circular lake when a cannibal appeared on its shore. Robinson knew that the cannibal could run around the shore four times as fast as he could swim in the water, but Robinson was much faster than the cannibal on land. Thus, if he could reach the shore without the cannibal being directly there, he could escape. Did he manage to escape?
|
Robinson successfully escaped!
|
olympiads
| 0.125
|
Solve the equation \(x^{3}+y^{3}+1=x^{2} y^{2}\) in the set of integers.
|
(3, 2), (2, 3), (1, -1), (-1, 1), (0, -1), (-1, 0)
|
olympiads
| 0.09375
|
Determine all prime numbers \( p \) such that
$$
5^{p} + 4 \cdot p^{4}
$$
is a perfect square, i.e., the square of an integer.
|
5
|
olympiads
| 0.078125
|
Calculate the limit of the function:
\[
\lim _{x \rightarrow 4} \frac{\sqrt{1+2 x}-3}{\sqrt{x}-2}
\]
|
\frac{4}{3}
|
olympiads
| 0.0625
|
A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$ , so that $CN/BN = AC/BC = 2/1$ . The segments $CM$ and $AN$ meet at $O$ . Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$ . The bisector of $\angle ABC$ meets the line through $K$ perpendicular to $AN$ at point $T$ .
Determine $\angle MTB$ .
|
\angle MTB = 90^ au
|
aops_forum
| 0.0625
|
A smooth sphere with a radius of 1 cm was dipped in blue paint and placed between two absolutely smooth concentric spheres with radii of 4 cm and 6 cm, respectively (the sphere was outside the smaller sphere but inside the larger one). When in contact with both spheres, the sphere leaves a blue trace. While moving, the sphere traveled along a closed path, resulting in a region on the smaller sphere outlined in blue with an area of 17 square centimeters. Find the area of the region outlined in blue on the larger sphere. Provide the answer in square centimeters, rounding to two decimal places if necessary.
|
38.25
|
olympiads
| 0.21875
|
In the complex plane, non-zero complex numbers \( z_{1} \) and \( z_{2} \) lie on the circle centered at \( \mathrm{i} \) with a radius of 1. The real part of \( \overline{z_{1}} \cdot z_{2} \) is zero, and the principal argument of \( z_{1} \) is \( \frac{\pi}{6} \). Find \( z_{2} \).
|
-\frac{\sqrt{3}}{2} + \frac{3}{2}i
|
olympiads
| 0.078125
|
Let \( \triangle ABC \) be a triangle with its incircle tangent to the perpendicular bisector of \( BC \). If \( BC = AE = 20 \), where \( E \) is the point where the \( A \)-excircle touches \( BC \), then compute the area of \( \triangle ABC \).
|
100 \sqrt{2}
|
olympiads
| 0.0625
|
Let $n$ points with integer coordinates be given in the $xy$ -plane. What is the minimum value of $n$ which will ensure that three of the points are the vertices of a triangel with integer (possibly, 0) area?
|
5
|
aops_forum
| 0.078125
|
The shape in the figure is made up of squares. Find the side length of the bottom left square if the side length of the smallest square is 1.
|
4
|
olympiads
| 0.09375
|
Two people agreed to meet at a specific location between 12 PM and 1 PM. The condition is that the first person to arrive will wait for the second person for 15 minutes and then leave. What is the probability that these two people will meet if each of them chooses their moment of arrival at the agreed location randomly within the interval between 12 PM and 1 PM?
|
\frac{7}{16}
|
olympiads
| 0.3125
|
The secret object is a rectangle measuring $200 \times 300$ meters. Outside the object, there is one guard stationed at each of its four corners. An intruder approached the perimeter of the secret object from the outside, and all the guards ran towards the intruder using the shortest paths along the outer perimeter (the intruder remained stationary). Three guards collectively ran a total of 850 meters to reach the intruder. How many meters did the fourth guard run to reach the intruder?
|
150 \text{ meters}
|
olympiads
| 0.25
|
Calculate the arc lengths of the curves given by the equations in polar coordinates.
$$
\rho=2 e^{4 \varphi / 3},-\frac{\pi}{2} \leq \varphi \leq \frac{\pi}{2}
$$
|
5 \, \sinh\left(\frac{2\pi}{3}\right)
|
olympiads
| 0.0625
|
Compare the numbers $\frac{\sin 2016^{\circ}}{\sin 2017^{\circ}}$ and $\frac{\sin 2018^{\circ}}{\sin 2019^{\circ}}$.
|
The second expression is larger.
|
olympiads
| 0.0625
|
Determine all pairs of polynomials $(P, Q)$ with real coefficients satisfying $$ P(x + Q(y)) = Q(x + P(y)) $$ for all real numbers $x$ and $y$ .
|
P(x) = Q(x) \text{ or } P(x) = x + a, Q(x) = x + b \text{ for some constants } a \text{ and } b
|
aops_forum
| 0.0625
|
Find the cosine of the angle at the base of an isosceles triangle if the point of intersection of its altitudes lies on the circle inscribed in the triangle.
|
\frac{1}{2}
|
olympiads
| 0.21875
|
In triangle \(ABC\), angle \(A\) is \(60^\circ\). The distances from vertices \(B\) and \(C\) to the incenter of triangle \(ABC\) are 3 and 4, respectively. Find the radius of the circumscribed circle (circumcircle) of triangle \(ABC\).
|
\sqrt{\frac{37}{3}}
|
olympiads
| 0.0625
|
If \(\tan \theta = \frac{-7}{24}, 90^{\circ} < \theta < 180^{\circ}\) and \(100 \cos \theta = r\), find \(r\).
|
-96
|
olympiads
| 0.4375
|
The chord \( AB \) divides the circle into two arcs, with the smaller arc being \( 130^{\circ} \). The larger arc is divided by chord \( AC \) in the ratio \( 31:15 \) from point \( A \). Find the angle \( BAC \).
|
37^{\circ} 30'
|
olympiads
| 0.0625
|
The unit square $ABCD$ has side $AB$ divided at its midpoint $E$ and side $DC$ divided at its midpoint $F$. The line $DE$ intersects the diagonal $AC$ at point $G$, and the line $BF$ intersects the diagonal $AC$ at point $H$. What is the length of the segment $GH$?
|
\frac{\sqrt{2}}{3}
|
olympiads
| 0.0625
|
Let the ellipse $\frac{x^{2}}{9}+\frac{y^{2}}{4}=1$ have a point $P(x, y)$ moving along it, and a fixed point $M(a, 0)$ where $0 < a < 3$. If the minimum distance $|PM|$ between $P$ and $M$ is 1, then determine the value of $a$.
|
a = \frac{\sqrt{15}}{2}
|
olympiads
| 0.09375
|
If \( f(x) \) is a linear function with \( f(k) = 4 \), \( f(f(k)) = 7 \), and \( f(f(f(k))) = 19 \), what is the value of \( k \)?
|
\frac{13}{4}
|
olympiads
| 0.5625
|
Find three prime numbers whose product is three times their sum.
|
2, 3, 5
|
olympiads
| 0.125
|
In a cube with side length 6, what is the volume of the tetrahedron formed by any vertex and the three vertices connected to that vertex by edges of the cube?
|
36
|
olympiads
| 0.15625
|
Suppose that \( x \) and \( y \) are positive real numbers such that
\[ x - y^2 = 3 \]
\[ x^2 + y^4 = 13 \]
Find \( x \).
|
\frac{3 + \sqrt{17}}{2}
|
olympiads
| 0.4375
|
\(5.29 \sin ^{3} z \cos z-\sin z \cos ^{3} z=\frac{\sqrt{2}}{8}\).
|
z = (-1)^{k+1} \frac{\pi}{16} + \frac{\pi k}{4}, \quad k \in \mathbb{Z}
|
olympiads
| 0.0625
|
On the board, two-digit numbers are written. Each number is composite, but any two numbers are relatively prime.
What is the maximum number of numbers that can be written?
|
4
|
olympiads
| 0.078125
|
The expression \(\frac{7n + 12}{2n + 3}\) takes integer values for certain integer values of \(n\).
What is the sum of all such integer values of the expression?
|
14
|
olympiads
| 0.078125
|
The numbers 100 and 90 were divided by the same number. In the first case, the remainder was 4, and in the second case, the remainder was 18. What number were they divided by?
|
24
|
olympiads
| 0.375
|
The larger base \(AD\) of trapezoid \(ABCD\) is \(a\), and the smaller base \(BC\) is \(b\). The diagonal \(AC\) is divided into three equal parts, and through the point of division \(M\), which is closest to \(A\), a line parallel to the bases is drawn. Find the segment of this line that is enclosed between the diagonals.
|
\frac{1}{3}(2a - b)
|
olympiads
| 0.0625
|
Arrange $n+1$ squares with a side length of 1 as shown in figure 4. Points $A, A_1, A_2, \cdots, A_{n+1}, M_0, M_1, M_2, \cdots, M_n$ are vertices of the squares. Connect $A M_i \ (i=1, 2, \cdots, n)$ and intersect with side $A_i M_{i-1}$ at point $N_i$. Let the area of quadrilateral $M_i N_i A_i A_{i+1}$ be $S_i \ (i=1,2, \cdots, n)$. What is $S_{2017}$?
|
\frac{4035}{4036}
|
olympiads
| 0.09375
|
Distinct positive integers \(a, b, c, d\) satisfy
\[
\begin{cases}
a \mid b^2 + c^2 + d^2 \\
b \mid a^2 + c^2 + d^2 \\
c \mid a^2 + b^2 + d^2 \\
d \mid a^2 + b^2 + c^2
\end{cases}
\]
and none of them is larger than the product of the three others. What is the largest possible number of primes among them?
|
3
|
olympiads
| 0.1875
|
\(\frac{b^{2}-3b-(b-1)\sqrt{b^{2}-4}+2}{b^{2}+3b-(b+1)\sqrt{b^{2}-4}+2} \cdot \sqrt{\frac{b+2}{b-2}} \quad ; \quad b > 2\)
|
\frac{1-b}{1+b}
|
olympiads
| 0.171875
|
In quadrilateral $ABCD$ , diagonals $AC$ and $BD$ intersect at $O$ . If the area of triangle $DOC$ is $4$ and the area of triangle $AOB$ is $36$ , compute the minimum possible value of the area of $ABCD$ .
|
80
|
aops_forum
| 0.09375
|
On the table, there are a banana, a pear, a melon, a tomato, and an apple. It is known that the melon weighs more than the pear but less than the tomato. Match the items with their weights in grams. Weights: 120, 140, 150, 170, 1500. Items: banana, pear, melon, tomato, and apple.
|
\begin{aligned}
&\text{Banana} = 140 \, \text{g} \\
&\text{Pear} = 120 \, \text{g} \\
&\text{Melon} = 1500 \, \text{g} \\
&\text{Tomato} = 150 \, \text{g} \\
&\text{Apple} = 170 \, \text{g}
\end{aligned}
|
olympiads
| 0.25
|
Find all ordered pairs \((m, n)\) of integers such that \(4^m - 4^n = 255\).
|
(4,0)
|
olympiads
| 0.125
|
Let $a=1+10^{-4}$ . Consider some $2023\times 2023$ matrix with each entry a real in $[1,a]$ . Let $x_i$ be the sum of the elements of the $i$ -th row and $y_i$ be the sum of the elements of the $i$ -th column for each integer $i\in [1,n]$ . Find the maximum possible value of $\dfrac{y_1y_2\cdots y_{2023}}{x_1x_2\cdots x_{2023}}$ (the answer may be expressed in terms of $a$ ).
|
1
|
aops_forum
| 0.140625
|
For which positive integers \( n \) and \( k \) do there exist integers \( a_{1}, a_{2}, \ldots, a_{n}, b_{1}, b_{2}, \ldots, b_{k} \) such that the products \( a_{i} b_{j} \) (for \( 1 \leq i \leq n, 1 \leq j \leq k \)) give pairwise distinct remainders when divided by \( n k \)?
|
n \text{ and } k \text{ are relatively prime
|
olympiads
| 0.515625
|
Given that \( a \) and \( b \) are positive integers, and \( b - a = 2013 \). If the equation \( x^{2} - a x + b = 0 \) has a positive integer solution, what is the smallest value of \( a \)?
|
93
|
olympiads
| 0.078125
|
Find the point of intersection of the line and the plane.
\(\frac{x+1}{-2}=\frac{y}{0}=\frac{z+1}{3}\)
\(x+4y+13z-23=0\)
|
(-3, 0, 2)
|
olympiads
| 0.453125
|
Determine how many tens of meters are contained in the length of the path that Balda rode on a mare. (1 verst = 500 sazhen, 1 sazhen = 3 arshins, 1 arshin = 71 cm).
|
106
|
olympiads
| 0.234375
|
In \( a : b = 5 : 4 \), \( b : c = 3 : x \), and \( a : c = y : 4 \), find \( y \).
|
y = 15
|
olympiads
| 0.140625
|
Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for any \( x, y \in \mathbf{R} \), the following equation holds:
\[ f(x f(y)) + x = x y + f(x). \]
|
f(x) = x \text{ or } f(x) = -x
|
olympiads
| 0.296875
|
Which sides does the Euler line intersect in an acute-angled and obtuse-angled triangle?
|
\text{In an acute-angled triangle: largest and smallest sides, in an obtuse-angled triangle: largest and median sides.}
|
olympiads
| 0.296875
|
For the celebration of Birthday Day in the 5th-grade parallelogram, several pizzas were ordered. A total of 10 pizzas were ordered for all the boys, with each boy getting an equal portion. Each girl received an equal portion, but it was half as much as what each boy received. How many pizzas were ordered if it is known that there are 11 girls in this parallelogram and more boys than girls? Pizzas can be divided into parts.
|
11
|
olympiads
| 0.09375
|
There are $N>2$ piles on a table with one nut in each pile. Two players take turns. On each turn, a player needs to choose two piles where the numbers of nuts are coprime and combine these piles into one. The player who makes the last move wins. For each $N$, determine which player can always win, regardless of the opponent's strategy.
|
Second player wins
|
olympiads
| 0.0625
|
A set of 8 problems was prepared for an examination. Each student was given 3 of them. No two students received more than one common problem. What is the largest possible number of students?
|
8
|
olympiads
| 0.078125
|
Calculate the integral \( I = \oint_{L} \left( x^2 - y^2 \right) dx + 2xy \, dy \), where \( L \) is the ellipse \(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\).
|
I = 0
|
olympiads
| 0.359375
|
Teacher Wang distributes bananas to children. If each child receives 3 bananas, there are 4 bananas left. If each child receives 4 bananas, 3 more bananas are needed. How many children are there in total?
|
7
|
olympiads
| 0.421875
|
With the help of a compass and a straightedge, construct a chord of a given circle that is equal in length and parallel to a given segment.
|
\text{Construct the chord that meets the problem's requirements, verifying through parallel translation and intersections.}
|
olympiads
| 0.125
|
Find all integer solutions to the equation \(21x + 48y = 6\).
|
(-2 + 16k, 1 - 7k)
|
olympiads
| 0.078125
|
In triangle \( \triangle ABC \), where \( AB \) is the longest side, given that \( \sin A \sin B = \frac{2 - \sqrt{3}}{4} \), find the maximum value of \( \cos A \cos B \).
|
\frac{2 + \\sqrt{3}}{4}
|
olympiads
| 0.0625
|
Find the sum \(1 \cdot 1! + 2 \cdot 2! + \cdot \cdot \cdot + n \cdot n!\).
|
(n+1)! - 1
|
olympiads
| 0.0625
|
Ludvík noticed in a certain division problem that when he doubles the dividend and increases the divisor by 12, he gets his favorite number as the result. He would get the same number if he reduced the original dividend by 42 and halved the original divisor. Determine Ludvík’s favorite number.
|
7
|
olympiads
| 0.140625
|
In an isosceles triangle, the side is divided by the point of tangency of the inscribed circle in the ratio 7:5 (starting from the vertex). Find the ratio of the side to the base.
|
\frac{6}{5}
|
olympiads
| 0.078125
|
For a positive integer $n$ determine all $n\times n$ real matrices $A$ which have only real eigenvalues and such that there exists an integer $k\geq n$ with $A + A^k = A^T$ .
|
A = 0
|
aops_forum
| 0.34375
|
Among the 2019 consecutive natural numbers starting from 1, how many numbers are neither multiples of 3 nor multiples of 4?
|
1010
|
olympiads
| 0.53125
|
In a rectangle $ABCD$, the sides $AB = 3$ and $BC = 4$ are given. Point $K$ is at distances $\sqrt{10}$, 2, and 3 from points $A$, $B$, and $C$, respectively. Find the angle between lines $CK$ and $BD$.
|
\arcsin \left(\frac{4}{5}\right)
|
olympiads
| 0.109375
|
Point \( O \), lying inside a convex quadrilateral with area \( S \), is reflected symmetrically with respect to the midpoints of its sides.
Find the area of the quadrilateral with vertices at the resulting points.
|
2S
|
olympiads
| 0.078125
|
Given the function \( f(x) = 2 + \log_{3} x \) for \( x \in [1, 9] \), find the range of \( y = [f(x)]^{2} + f(x^{2}) \).
|
[6, 22]
|
olympiads
| 0.359375
|
A family approached a bridge at night. The father can cross it in 1 minute, the mother in 2 minutes, the child in 5 minutes, and the grandmother in 10 minutes. They have one flashlight. The bridge can hold only two people at a time. How can they cross the bridge in 17 minutes? (If two people cross together, they move at the slower person's speed. Crossing the bridge without a flashlight is not allowed. Shining the flashlight from afar is not allowed. Carrying each other is not allowed.)
|
17 \text{ minutes}
|
olympiads
| 0.5
|
Calculate the limit of the function:
$$\lim _{x \rightarrow 0}(\cos \pi x)^{\frac{1}{x \cdot \sin \pi x}}$$
|
e^{- rac{ ext{π}}{2}}
|
olympiads
| 0.125
|
Solve the equation, where $x$ and $y$ are positive integers: $$ x^3-y^3=999 $$
|
(x, y) = (12, 9) and (x, y) = (10, 1)
|
aops_forum
| 0.078125
|
In the complex plane, non-zero complex numbers \( z_1 \) and \( z_2 \) lie on the circle with center \( i \) and radius 1. The real part of \( \bar{z_1} \cdot z_2 \) is zero, and the principal value of the argument of \( z_1 \) is \( \frac{\pi}{6} \). Find \( z_2 \).
|
z_{2} = -\frac{\sqrt{3}}{2} + \frac{3}{2}i
|
olympiads
| 0.0625
|
Which natural numbers \( m \) and \( n \) satisfy \( 2^n + 1 = m^2 \)?
|
m = 3 \, \text{and} \, n = 3
|
olympiads
| 0.171875
|
Calculate the limit of the function:
\[
\lim_{x \rightarrow a} \frac{a^{\left(x^{2}-a^{2}\right)}-1}{\operatorname{tg} \ln \left(\frac{x}{a}\right)}
\]
|
2a^2 \ln a
|
olympiads
| 0.078125
|
How many ways are there to use diagonals to divide a regular 6-sided polygon into triangles such that at least one side of each triangle is a side of the original polygon and that each vertex of each triangle is a vertex of the original polygon?
|
12
|
olympiads
| 0.0625
|
Car A and Car B are traveling in opposite directions on a road parallel to a railway. A 180-meter-long train is moving in the same direction as Car A at a speed of 60 km/h. The time from when the train catches up with Car A until it meets Car B is 5 minutes. If it takes the train 30 seconds to completely pass Car A and 6 seconds to completely pass Car B, after how many more minutes will Car A and Car B meet once Car B has passed the train?
|
1.25 \text{ minutes}
|
olympiads
| 0.0625
|
Determine the number of all positive integers which cannot be written in the form $80k + 3m$ , where $k,m \in N = \{0,1,2,...,\}$
|
79
|
aops_forum
| 0.0625
|
Find all the real roots of the system of equations: $$ \begin{cases} x^3+y^3=19 x^2+y^2+5x+5y+xy=12 \end{cases} $$
|
(3, -2) ext{ or } (-2, 3)
|
aops_forum
| 0.109375
|
Given a positive integer $n,$ what is the largest $k$ such that the numbers $1,2,\dots,n$ can be put into $k$ boxes so that the sum of the numbers in each box is the same?
[When $n=8,$ the example $\{1,2,3,6\},\{4,8\},\{5,7\}$ shows that the largest $k$ is *at least* 3.]
|
\max(k) =
\begin{cases}
\frac{n+1}{2}, & \text{if } n \equiv 1 \pmod{2} \\
\frac{n}{2}, & \text{if } n \equiv 0 \pmod{2}
\end{cases}
|
aops_forum
| 0.109375
|
There are \( N \) pieces of candy, packed into boxes, with each box containing 45 pieces. \( N \) is a non-zero perfect cube, and 45 is one of its factors. What is the least possible number of boxes that can be packed?
|
75
|
olympiads
| 0.28125
|
Let $n>1$ be a natural number. Find the real values of the parameter $a$ , for which the equation $\sqrt[n]{1+x}+\sqrt[n]{1-x}=a$ has a single real root.
|
a = 2
|
aops_forum
| 0.125
|
There are 2 pizzerias in a town, with 2010 pizzas each. Two scientists $A$ and $B$ are taking turns ( $A$ is first), where on each turn one can eat as many pizzas as he likes from one of the pizzerias or exactly one pizza from each of the two. The one that has eaten the last pizza is the winner. Which one of them is the winner, provided that they both use the best possible strategy?
|
B
|
aops_forum
| 0.25
|
There are exactly \( n \) values of \( \theta \) satisfying the equation \( \left(\sin ^{2} \theta-1\right)\left(2 \sin ^{2} \theta-1\right)=0 \), where \( 0^{\circ} \leq \theta \leq 360^{\circ} \). Find \( n \).
|
6
|
olympiads
| 0.515625
|
Yannick is playing a game with 100 rounds, starting with 1 coin. During each round, there is an \( n \% \) chance that he gains an extra coin, where \( n \) is the number of coins he has at the beginning of the round. What is the expected number of coins he will have at the end of the game?
|
1.01^{100}
|
olympiads
| 0.125
|
For what integers $ n\ge 3$ is it possible to accommodate, in some order, the numbers $ 1,2,\cdots, n$ in a circular form such that every number divides the sum of the next two numbers, in a clockwise direction?
|
3
|
aops_forum
| 0.171875
|
An analyst predicted the dollar exchange rate change for each of the next three months: by what percentage (a number greater than $0\%$ and less than $100\%$) the rate will change in July, by what percentage in August, and by what percentage in September. It turned out that for each month he correctly predicted the percentage change in the rate, but he was wrong about the direction of the change (i.e., if he predicted the rate would increase by \( x\%\), it decreased by \( x\%\), and vice versa). However, after three months, the actual rate matched the forecast. In which direction did the rate change in the end?
|
Decreased
|
olympiads
| 0.09375
|
Let \( A_{10} \) denote the answer to problem 10. Two circles lie in the plane; denote the lengths of the internal and external tangents between these two circles by \( x \) and \( y \), respectively. Given that the product of the radii of these two circles is \( \frac{15}{2} \), and that the distance between their centers is \( A_{10} \), determine \( y^{2} - x^{2} \).
|
30
|
olympiads
| 0.28125
|
Find all the extrema of the function \( y = \sin^2(3x) \) on the interval \( (0, 0.6) \).
|
x = \frac{\pi}{6}
|
olympiads
| 0.09375
|
The height and the median of a triangle, drawn from one of its vertices, are distinct and form equal angles with the sides emanating from the same vertex. Determine the radius of the circumscribed circle if the median is equal to \( m \).
|
m
|
olympiads
| 0.125
|
In the plane, a rhombus \(ABCD\) and two circles are constructed, circumscribed around triangles \(BCD\) and \(ABD\). Ray \(BA\) intersects the first circle at point \(P\) (different from \(B\)), and ray \(PD\) intersects the second circle at point \(Q\). It is known that \(PD = 1\) and \(DQ = 2 + \sqrt{3}\). Find the area of the rhombus.
|
4
|
olympiads
| 0.0625
|
In triangle \( \triangle ABC \), \( AB = BC = 2 \) and \( AC = 3 \). Let \( O \) be the incenter of \( \triangle ABC \). If \( \overrightarrow{AO} = p \overrightarrow{AB} + q \overrightarrow{AC} \), find the value of \( \frac{p}{q} \).
|
\frac{2}{3}
|
olympiads
| 0.234375
|
In the first pile, there were 7 nuts, and in the rest, there were 12 nuts each. How many piles were there in total?
|
5
|
olympiads
| 0.0625
|
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