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stringlengths 33
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0.59
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|---|---|---|---|
Find any seven consecutive natural numbers, each of which can be changed (increased or decreased) by 1 in such a way that the product of the seven resulting numbers equals the product of the seven original numbers.
|
2, \ 5, \ 6, \ 6, \ 7, \ 9, \ 8
|
olympiads
| 0.09375
|
Natural numbers from 1 to 2021 are written in a row in some order. It turned out that any number has neighbors with different parity. What number can be in the first place?
|
Any odd number
|
olympiads
| 0.1875
|
Determine the trinomial whose cube is:
$$
x^{6}-6 x^{5}+15 x^{4}-20 x^{3}+15 x^{2}-6 x+1
$$
|
x^{2} - 2x + 1
|
olympiads
| 0.140625
|
In each cell of $5 \times 5$ table there is one number from $1$ to $5$ such that every number occurs exactly once in every row and in every column. Number in one column is *good positioned* if following holds:
- In every row, every number which is left from *good positoned* number is smaller than him, and every number which is right to him is greater than him, or vice versa.
- In every column, every number which is above from *good positoned* number is smaller than him, and every number which is below to him is greater than him, or vice versa.
What is maximal number of good positioned numbers that can occur in this table?
|
5
|
aops_forum
| 0.265625
|
What simpler property characterizes the sequences in which the ratio of any two adjacent terms is equal to the ratio of the corresponding (same index) terms in the sequence of differences?
|
Geometric Sequence
|
olympiads
| 0.234375
|
Anya is arranging pebbles on the sand. First, she placed one pebble, then added pebbles to form a pentagon, then made an outer larger pentagon with the pebbles, and continued this process to make further outer pentagons, as shown in the illustration. The number of pebbles she had arranged in the first four images is: 1, 5, 12, and 22. Continuing to create such images, how many pebbles will be in the 10th image?
|
145
|
olympiads
| 0.25
|
The legs of a right triangle are 9 cm and 12 cm. Find the distance between the incenter (the point of intersection of its angle bisectors) and the centroid (the point of intersection of its medians).
|
1 \ ext{cm}
|
olympiads
| 0.265625
|
Points $\boldsymbol{A}$ and $\boldsymbol{B}$ are located on a straight highway running from west to east. Point B is 9 km east of A. A car departs from point A heading east at a speed of 40 km/h. Simultaneously, from point B, a motorcycle starts traveling in the same direction with a constant acceleration of 32 km/h². Determine the greatest distance that can be between the car and the motorcycle during the first two hours of their movement.
|
16 \text{ km}
|
olympiads
| 0.234375
|
We call a number antitriangular if it can be expressed in the form \(\frac{2}{n(n+1)}\) for some natural number \(n\). For how many numbers \(k\) (where \(1000 \leq k \leq 2000\)) can the number 1 be expressed as the sum of \(k\) antitriangular numbers (not necessarily distinct)?
|
1001
|
olympiads
| 0.125
|
For what value of \( d \) does a continuous function \( f: [0, 1] \to \mathbb{R} \) with \( f(0) = f(1) \) always have a horizontal chord of length \( d \)?
|
d \in \left\{1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \ldots \right\}
|
olympiads
| 0.21875
|
The sum of three positive integers is 15, and the sum of their reciprocals is $\frac{71}{105}$. Determine the numbers!
|
3, 5, 7
|
olympiads
| 0.078125
|
To celebrate the $20$ th LMT, the LHSMath Team bakes a cake. Each of the $n$ bakers places $20$ candles on the cake. When they count, they realize that there are $(n -1)!$ total candles on the cake. Find $n$ .
|
n = 6
|
aops_forum
| 0.359375
|
A spider has 8 identical socks and 8 identical shoes. In how many different orders can it put them on, given that, obviously, on each leg, it must put on the shoe after the sock?
|
81729648000
|
olympiads
| 0.078125
|
The product of 3 integers is -13. What values can their sum take? Be sure to specify all options.
|
-11 \text{ or } 13
|
olympiads
| 0.078125
|
On side $BC$ of triangle $ABC$, a point $P$ is selected such that $PC = 2BP$. Find $\angle ACB$ if $\angle ABC = 45^\circ$ and $\angle APC = 60^\circ$.
|
\angle ACB = 75^\circ
|
olympiads
| 0.078125
|
Given that \(0 \leq x_{0} \leq \frac{\pi}{2}\) and \(x_{0}\) satisfies the equation \(\sqrt{\sin x + 1} - \sqrt{1 - \sin x} = \sin \frac{x}{2}\). If \(d = \tan x_{0}\), find the value of \(d\).
|
0
|
olympiads
| 0.3125
|
Given a regular pentagon \(ABCDE\). Point \(K\) is marked on side \(AE\), and point \(L\) is marked on side \(CD\). It is known that \(\angle LAE + \angle KCD = 108^\circ\) and \(AK: KE = 3:7\). Find \(CL: AB\).
A regular pentagon is a pentagon where all sides and all angles are equal.
|
0.7
|
olympiads
| 0.140625
|
On a table, there are two spheres with radii 4 and 1, with centers \(O_{1}\) and \(O_{2}\), respectively, touching each other externally. A cone touches the side surface of the table and both spheres (externally). The apex \(C\) of the cone is located on the segment connecting the points where the spheres touch the table. It is known that the rays \(CO_{1}\) and \(CO_{2}\) form equal angles with the table. Find the angle at the apex of the cone. (The angle at the apex of the cone is defined as the angle between its generatrices in the axial section.)
|
2\arctg\left( \frac{2}{5} \right)
|
olympiads
| 0.15625
|
William is thinking of an integer between 1 and 50, inclusive. Victor can choose a positive integer $m$ and ask William: "does $m$ divide your number?", to which William must answer truthfully. Victor continues asking these questions until he determines William's number. What is the minimum number of questions that Victor needs to guarantee this?
|
15
|
aops_forum
| 0.078125
|
How many pairs of positive integers \((x, y)\) have the property that the ratio \(x: 4\) equals the ratio \(9: y\)?
|
9
|
olympiads
| 0.203125
|
In a certain year, the month of October has 5 Saturdays and 4 Sundays. What day of the week is October 1st in that year?
|
Thursday
|
olympiads
| 0.265625
|
In the quadrilateral pyramid \( P-ABCD \), it is given that \( AB \parallel CD \), \( AB \perp AD \), \( AB = 4 \), \( AD = 2\sqrt{2} \), \( CD = 2 \), and \( PA \perp \) plane \( ABCD \) with \( PA = 4 \). Let \( Q \) be a point on the segment \( PB \), and the sine of the angle between the line \( QC \) and the plane \( PAC \) is \( \frac{\sqrt{3}}{3} \). Find the value of \( \frac{PQ}{PB} \).
|
\frac{1}{2}
|
olympiads
| 0.078125
|
Determine all functions \( f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+} \) such that for all \( x, y > 0 \):
\[
f(x y) \leq \frac{x f(y) + y f(x)}{2}
\]
|
f(x) = ax, \, a > 0
|
olympiads
| 0.078125
|
How many roots does the following equation have:
$$
10^{\lg \cos x}=\frac{1}{7} x-1 ?
$$
|
3
|
olympiads
| 0.15625
|
Malcolm writes a positive integer on a piece of paper. Malcolm doubles this integer and subtracts 1, writing this second result on the same piece of paper. Malcolm then doubles the second integer and adds 1, writing this third integer on the paper. If all of the numbers Malcolm writes down are prime, determine all possible values for the first integer.
|
2 \text{ and } 3
|
aops_forum
| 0.171875
|
For any integer \( n \geq 0 \), let \( S(n) = n - m^2 \), where \( m \) is the largest integer satisfying \( m^2 \leq n \). The sequence \(\{a_k\}_{k=0}^{\infty}\) is defined as follows:
\[
a_0 = A, \quad a_{k+1} = a_k + S(a_k), \quad k \geq 0.
\]
Determine for which positive integers \( A \), this sequence eventually becomes constant.
(The 52nd Putnam Mathematical Competition, 1991)
|
A is a perfect square
|
olympiads
| 0.140625
|
Given \( k \) is a positive real number, solve the system of equations:
\[ x_1 |x_1| = x_2 |x_2| + (x_1 - k) |x_1 - k| \]
\[ x_2 |x_2| = x_3 |x_3| + (x_2 - k) |x_2 - k| \]
\[ \vdots \]
\[ x_n |x_n| = x_1 |x_1| + (x_n - k) |x_n - k| \]
|
x_i = k \ \text{for all} \ i
|
olympiads
| 0.3125
|
The centers of three spheres, with radii 3, 4, and 6, are located at the vertices of an equilateral triangle with a side length of 11. How many planes exist that are tangent to all of these spheres simultaneously?
|
3
|
olympiads
| 0.125
|
In a triangle, the lengths of two sides \( a \) and \( b \) and the angle \( \alpha \) between them are given. Find the length of the height drawn to the third side.
|
\frac{a b \, \sin \alpha}{\sqrt{a^2 + b^2 - 2 a b \cos \alpha}}
|
olympiads
| 0.21875
|
Is there a right triangle in which the legs \(a\) and \(b\) and the altitude \(m\) to the hypotenuse satisfy the following relationship:
\[ m = \frac{1}{5} \sqrt{9b^{2} - 16a^{2}} \]
|
2a = b
|
olympiads
| 0.09375
|
The product of two two-digit numbers is 777. Find these numbers.
|
21 \text{ and } 37
|
olympiads
| 0.40625
|
The length of the common external tangent of two externally tangent circles is $20 \mathrm{~cm}$. What are the radii of these circles if the radius of the circle that is tangent to both of these circles and their common tangent, placed between them, is as large as possible?
|
R = 10 \text{ cm}, r = 10 \text{ cm}
|
olympiads
| 0.109375
|
For which values of the parameter \( p \) does the equation
\[ x+1=\sqrt{p x} \]
have exactly one real root?
|
p = 4 \text{ or } p \leq 0
|
olympiads
| 0.40625
|
A random variable $X$ in the interval $(-c, c)$ is given by the probability density function $f(x) = \frac{1}{\pi \sqrt{c^2 - x^2}}$; outside this interval, $f(x) = 0$. Find the expected value of the variable $X$.
|
0
|
olympiads
| 0.59375
|
The diagram shows a quadrilateral composed of 18 identical small equilateral triangles. In the diagram, some adjacent small equilateral triangles can form several larger equilateral triangles. How many large and small equilateral triangles marked with "**" are there in total in the diagram?
|
6
|
olympiads
| 0.078125
|
There are four inequalities: $\sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2}}}}<2$, $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}<2$, $\sqrt{3 \sqrt{3 \sqrt{3 \sqrt{3}}}}<3$, $\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3}}}}<3$. The number of incorrect inequalities is ______.
|
0
|
olympiads
| 0.078125
|
A snail crawls on a plane, turning $90^{\circ}$ after each meter of travel. What is the maximum distance it could be from its starting point after crawling 300 meters, if it made a total of 99 left turns and 200 right turns?
|
100\sqrt{2}
|
olympiads
| 0.078125
|
At a physical education class, 27 seventh graders attended, some of whom brought one ball each. Occasionally during the class, a seventh grader would give their ball to another seventh grader who did not have a ball.
At the end of the class, \( N \) seventh graders said: "I received balls less often than I gave them away!" Find the maximum possible value of \( N \) given that nobody lied.
|
13
|
olympiads
| 0.34375
|
In the Cartesian coordinate system, a fixed point \( B \) is on the negative half of the y-axis. A variable line \( l \) passing through \( B \) intersects the ellipse \(\frac{x^2}{2} + y^2 = 1\) at points \( C \) and \( D \). The circle with diameter \( CD \) always passes through a fixed point \( A \) located above the x-axis. Determine the coordinates of \( A \).
|
A(0, 1)
|
olympiads
| 0.078125
|
From a right prism with a square base with side length \( L_{1} \) and height \( H \), we extract a frustum of a pyramid, not necessarily right, with square bases, side lengths \( L_{1} \) (for the lower base) and \( L_{2} \) (for the upper base), and height \( H \). The two pieces obtained are shown in the following image.
If the volume of the frustum of the pyramid is \(\frac{2}{3}\) of the total volume of the prism, what is the value of \( \frac{L_{1}}{L_{2}} \)?
|
\frac{L_1}{L_2} = \frac{1 + \sqrt{5}}{2}
|
olympiads
| 0.140625
|
A large rectangle is divided into four identical smaller rectangles by slicing parallel to one of its sides. The perimeter of the large rectangle is 18 metres more than the perimeter of each of the smaller rectangles. The area of the large rectangle is 18 square metres more than the area of each of the smaller rectangles. What is the perimeter in metres of the large rectangle?
|
28
|
olympiads
| 0.078125
|
Let $AD$ and $BC$ be the parallel sides of a trapezium $ABCD$ . Let $P$ and $Q$ be the midpoints of the diagonals $AC$ and $BD$ . If $AD = 16$ and $BC = 20$ , what is the length of $PQ$ ?
|
2
|
aops_forum
| 0.0625
|
Find \(\log _{n}\left(\frac{1}{2}\right) \log _{n-1}\left(\frac{1}{3}\right) \cdots \log _{2}\left(\frac{1}{n}\right)\) in terms of \(n\).
|
(-1)^{n-1}
|
olympiads
| 0.09375
|
Find all pairs of positive integers \((a, b)\) that satisfy the equation:
$$
2a^2 = 3b^3.
$$
|
(18d^3, 6d^2)
|
olympiads
| 0.0625
|
Compute the limit of the function:
\[ \lim _{x \rightarrow 0}\left(\frac{11 x+8}{12 x+1}\right)^{\cos ^{2} x} \]
|
8
|
olympiads
| 0.203125
|
Find the integral \(\int \frac{dx}{\sqrt{3x^2 - 6x + 9}}\).
|
\frac{1}{\sqrt{3}} \ln \left| x - 1 + \sqrt{x^2 - 2x + 3} \right| + C
|
olympiads
| 0.171875
|
Find the equation of the tangent line to the given curve at the point with the x-coordinate \( x_{0} \).
\[ y = \frac{x^{2} - 2x - 3}{4}, \quad x_{0} = 4 \]
|
y = \frac{3}{2} x - \frac{19}{4}
|
olympiads
| 0.34375
|
A bag contains red and yellow balls. When 60 balls are taken out, exactly 56 of them are red. Thereafter, every time 18 balls are taken out, 14 of them are always red, until the last batch of 18 balls is taken out. If the total number of red balls in the bag is exactly four-fifths of the total number of balls, how many red balls are in the bag?
|
336
|
olympiads
| 0.171875
|
Let \( D \) be the midpoint of the hypotenuse \( BC \) of the right triangle \( ABC \). On the leg \( AC \), a point \( M \) is chosen such that \(\angle AMB = \angle CMD\). Find the ratio \(\frac{AM}{MC}\).
|
1:2
|
olympiads
| 0.21875
|
Find all integers \( b, n \in \mathbb{N} \) such that \( b^2 - 8 = 3^n \).
|
b = 3, n = 0
|
olympiads
| 0.25
|
Find the value of \( a \) such that the equation \( \sin 4x \cdot \sin 2x - \sin x \cdot \sin 3x = a \) has a unique solution in the interval \([0, \pi)\).
|
1
|
olympiads
| 0.078125
|
In the Karabas Barabas theater, a chess tournament took place among the actors. Each participant played exactly one match against each of the other participants. A win awarded one solido, a draw awarded half a solido, and a loss awarded nothing. It turned out that among every three participants, there was one chess player who earned exactly 1.5 solidos in the games against the other two. What is the maximum number of actors that could have participated in such a tournament?
|
5
|
olympiads
| 0.0625
|
Find the largest positive integer $n$ such that the following is true:
There exists $n$ distinct positive integers $x_1,~x_2,\dots,x_n$ such that whatever the numbers $a_1,~a_2,\dots,a_n\in\left\{-1,0,1\right\}$ are, not all null, the number $n^3$ do not divide $\sum_{k=1}^n a_kx_k$ .
|
9
|
aops_forum
| 0.0625
|
Sasha was descending the stairs from his apartment to his friend Kolya who lives on the first floor. When he had descended a few floors, it turned out that he had covered one-third of the way. After descending one more floor, he still had half of the way to go. On which floor does Sasha live?
|
7
|
olympiads
| 0.0625
|
If \( f(x) = \frac{25^x}{25^x + P} \) and \( Q = f\left(\frac{1}{25}\right) + f\left(\frac{2}{25}\right) + \cdots + f\left(\frac{24}{25}\right) \), find the value of \( Q \).
|
12
|
olympiads
| 0.265625
|
In figure 1, \(ABCD\) is a rectangle, and \(E\) and \(F\) are points on \(AD\) and \(DC\), respectively. Also, \(G\) is the intersection of \(AF\) and \(BE\), \(H\) is the intersection of \(AF\) and \(CE\), and \(I\) is the intersection of \(BF\) and \(CE\). If the areas of \(AGE\), \(DEHF\), and \(CIF\) are 2, 3, and 1, respectively, find the area of the grey region \(BGHI\).
|
6
|
olympiads
| 0.0625
|
A game for two participants consists of a rectangular field $1 \times 25$, divided into 25 square cells, and 25 tokens. The cells are numbered consecutively $1, 2, \ldots, 25$. In one move, a player either places a new token in one of the free cells or moves a previously placed token to the nearest free cell with a higher number. In the initial position, all cells are free. The game ends when all cells are occupied by tokens, and the winner is the player who makes the last move. Players take turns. Who will win with perfect play: the starting player or their opponent?
|
first player
|
olympiads
| 0.109375
|
In an isosceles trapezoid $ABCD$ with $AB = CD = 3$, base $AD = 7$, and $\angle BAD = 60^\circ$, a point $M$ is located on the diagonal $BD$ such that $BM : MD = 3 : 5$.
Which side of the trapezoid, $BC$ or $CD$, does the extension of the segment $AM$ intersect?
|
CD
|
olympiads
| 0.203125
|
Indicate a solution to the puzzle: \(2014 + \text{YEAR} = \text{SOCHI}\).
|
2014 + 789 = 2803, \quad 2014 + 896 = 2910
|
olympiads
| 0.09375
|
Given that \( x + y - 2 \) is a factor of the polynomial \( x^{2} + axy + by^{2} - 5x + y + 6 \) in terms of \( x \) and \( y \), determine the value of \( a - b \).
|
1
|
olympiads
| 0.1875
|
Two dice are thrown. What is the probability that the sum of the points on both dice will not exceed 5?
|
\frac{5}{18}
|
olympiads
| 0.40625
|
If \( x + k \) is a factor of \( 3x^2 + 14x + a \), find \( k \). (\( k \) is an integer.)
|
4
|
olympiads
| 0.078125
|
A segment \( AB \) of unit length, which is a chord of a sphere with radius 1, is positioned at an angle of \( \pi / 3 \) to the diameter \( CD \) of this sphere. The distance from the end \( C \) of the diameter to the nearest end \( A \) of the chord \( AB \) is \( \sqrt{2} \). Determine the length of segment \( BD \).
|
|BD| = \sqrt{3}
|
olympiads
| 0.109375
|
As shown in the figure, a polyhedron is obtained by cutting a regular quadrangular prism through the vertices of the base $B$ with the section $A_{1} B C_{1} D_{1}$. Given that $A A_{1}=O C_{1}$, the dihedral angle between the section $A_{1} B C_{1} D_{1}$ and the base $A B C D$ is $45^{\circ}$, and $A B=1$. Find the volume of this polyhedron.
|
\frac{\sqrt{2}}{2}
|
olympiads
| 0.109375
|
In the Land of Fools, there are \(N^2\) cities arranged in a square grid, with the distance between neighboring cities being 10 km. The cities are connected by a road system consisting of straight sections that are parallel to the sides of the square. What is the minimum length of such a road system if it is known that it is possible to travel from any city to any other city?
|
10(N^2 - 1)
|
olympiads
| 0.171875
|
Given the function \( f(x)=\left\{\begin{array}{ll}1 & x \geqslant 0 \\ -1 & x<0\end{array}\right. \), find the solution set for the inequality \( x + (x + 2) \cdot f(x + 2) \leqslant 5 \).
|
(-\infty, \frac{3}{2} ]
|
olympiads
| 0.171875
|
In the vertices of a unit square, perpendiculars are erected to its plane. On them, on one side of the plane of the square, points are taken at distances of 3, 4, 6, and 5 from this plane (in order of traversal). Find the volume of the polyhedron whose vertices are the specified points and the vertices of the square.
|
4.5
|
olympiads
| 0.0625
|
Given is a chessboard 8x8. We have to place $n$ black queens and $n$ white queens, so that no two queens attack. Find the maximal possible $n$ .
(Two queens attack each other when they have different colors. The queens of the same color don't attack each other)
|
8
|
aops_forum
| 0.078125
|
In Figure 6, a square-based pyramid is cut into two shapes by a cut running parallel to the base and made \(\frac{2}{3}\) of the way up. Let \(1: c\) be the ratio of the volume of the small pyramid to that of the truncated base. Find the value of \(c\).
|
\frac{19}{8}
|
olympiads
| 0.0625
|
Let \( x_{1}, \ldots, x_{100} \) be defined so that for each \( i, x_{i} \) is a (uniformly) random integer between 1 and 6 inclusive. Find the expected number of integers in the set \( \left\{ x_{1}, x_{1} + x_{2}, \ldots, x_{1} + x_{2} + \ldots + x_{100} \right\} \) that are multiples of 6.
|
\frac{50}{3}
|
olympiads
| 0.234375
|
Given the set \( S = \left\{ z \mid |z - 7 - 8i| = |z_1^4 + 1 - 2z_1^2| ; z, z_1 \in \mathbb{C}, |z_1| = 1 \right\} \), find the area of the region corresponding to \( S \) in the complex plane.
|
16\pi
|
olympiads
| 0.0625
|
Consider an isosceles triangle \( T \) with base 10 and height 12. Define a sequence \( \omega_1, \omega_2, \ldots \) of circles such that \( \omega_1 \) is the incircle of \( T \) and \( \omega_{i+1} \) is tangent to \( \omega_i \) and both legs of the isosceles triangle for \( i > 1 \).
Find the radius of \( \omega_1 \).
|
\frac{10}{3}
|
olympiads
| 0.34375
|
The blackboard displays the integer 1000, and there are 1000 matchsticks on the table. Two players, Player A and Player B, play a game where Player A goes first and they take turns. The player whose turn it is can either take up to 5 matchsticks from the pile or return up to 5 matchsticks to the pile (initially, neither player has any matchsticks, so they can only return matchsticks they've previously taken). The current count of matchsticks in the pile is then written on the blackboard. If a player writes down a number that has already been written on the blackboard, they lose. Assuming both players use the optimal strategy, who will win?
|
Player B wins
|
olympiads
| 0.125
|
Find the relationship between the coefficients of the equation \(a x^2 + b x + c = 0\), given that the sum of its roots is twice their difference.
|
3 b^2 = 16 a c
|
olympiads
| 0.296875
|
Calculate the area of the figures bounded by the lines given in polar coordinates.
$$
r=\sin 3 \phi
$$
|
\frac{\pi}{4}
|
olympiads
| 0.1875
|
We know that the segments of the altitudes of a triangle that fall on the inscribed circle are of equal length. Does it follow that the triangle is equilateral?
|
The triangle is equilateral.
|
olympiads
| 0.078125
|
For four pairwise distinct numbers \( x, y, s, t \), the equality \(\frac{x+s}{x+t} = \frac{y+t}{y+s}\) holds. Find the sum of all four numbers.
|
0
|
olympiads
| 0.21875
|
How many natural numbers are divisors of the number 1,000,000 and do not end in 0?
|
7
|
olympiads
| 0.25
|
Inside rectangle \(ABCD\), point \(M\) is chosen such that \(\angle BMC + \angle AMD = 180^\circ\). Find the measure of \(\angle BCM + \angle DAM\).
|
90^{\circ}
|
olympiads
| 0.21875
|
In triangle \(ABC\), the lengths of the sides are \(AB = 8\), \(BC = 6\), and the length of the angle bisector \(BD = 6\). Find the length of the median \(AE\).
|
\frac{\sqrt{190}}{2}
|
olympiads
| 0.0625
|
Cat and Claire are having a conversation about Cat’s favorite number. Cat says, “My favorite number is a two-digit perfect square!”
Claire asks, “If you picked a digit of your favorite number at random and revealed it to me without telling me which place it was in, is there any chance I’d know for certain what it is?”
Cat says, “Yes! Moreover, if I told you a number and identified it as the sum of the digits of my favorite number, or if I told you a number and identified it as the positive difference of the digits of my favorite number, you wouldn’t know my favorite number.”
Claire says, “Now I know your favorite number!” What is Cat’s favorite number?
|
25
|
aops_forum
| 0.171875
|
Find all integer sequences of the form $ x_i, 1 \le i \le 1997$ , that satisfy $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$ .
|
x_i = 0
|
aops_forum
| 0.09375
|
A packet of seeds was passed around the table. The first person took 1 seed, the second took 2 seeds, the third took 3 seeds, and so on: each subsequent person took one more seed than the previous one. It is known that on the second round, a total of 100 more seeds were taken than on the first round. How many people were sitting at the table?
|
10
|
olympiads
| 0.09375
|
Let \( f(x) = x^3 - 20x^2 + x - a \) and \( g(x) = x^4 + 3x^2 + 2 \). If \( h(x) \) is the highest common factor of \( f(x) \) and \( g(x) \), find \( b = h(1) \).
|
2
|
olympiads
| 0.109375
|
Two cones have a common vertex, and the slant height of the first cone is the height of the second cone. The angle at the vertex of the axial section of the first cone is $\arccos \frac{1}{3}$, and of the second cone is $120^{\circ}$. Find the angle between the sides where the lateral surfaces of the cones intersect.
|
60^{\circ}
|
olympiads
| 0.109375
|
It is known that an arithmetic progression includes the terms \( a_{2n} \) and \( a_{2m} \) such that \( a_{2n} / a_{2m} = -1 \). Is there a term in this progression that is equal to zero? If so, what is the index of this term?
|
n + m
|
olympiads
| 0.0625
|
The height of a cone is $h$. The development of the lateral surface of this cone is a sector with a central angle of $120^\circ$. Calculate the volume of the cone.
|
\frac{\pi h^3}{24}
|
olympiads
| 0.09375
|
In triangle \(ABC\), points \(M\) and \(N\) are the midpoints of sides \(AC\) and \(BC\) respectively. It is known that the intersection of the medians of triangle \(AMN\) is the orthocenter of triangle \(ABC\). Find the angle \(ABC\).
|
45^
\circ
|
olympiads
| 0.078125
|
Given that \(ABCD\) is a parallelogram, where \(\overrightarrow{AB} = \vec{a}\) and \(\overrightarrow{AC} = \vec{b}\), and \(E\) is the midpoint of \(CD\), find \(\overrightarrow{EB} = \quad\).
|
\frac{3}{2} \vec{a} - \vec{b}
|
olympiads
| 0.09375
|
A king traversed a $9 \times 9$ chessboard, visiting each square exactly once. The king's route is not a closed loop and may intersect itself. What is the maximum possible length of such a route if the length of a move diagonally is $\sqrt{2}$ and the length of a move vertically or horizontally is 1?
|
16 + 64 \sqrt{2}
|
olympiads
| 0.0625
|
What is the last digit of the sum $4.1 \times 1 + 2 \times 2 + 3 \times 3 + \ldots + 2011 \times 2011 + 2012 \times 2012$?
|
0
|
olympiads
| 0.21875
|
Let $P$ be a regular $2006$ -gon. A diagonal is called *good* if its endpoints divide the boundary of $P$ into two parts, each composed of an odd number of sides of $P$ . The sides of $P$ are also called *good*.
Suppose $P$ has been dissected into triangles by $2003$ diagonals, no two of which have a common point in the interior of $P$ . Find the maximum number of isosceles triangles having two good sides that could appear in such a configuration.
|
1003
|
aops_forum
| 0.234375
|
Find the inverse matrix of
$$
A=\left(\begin{array}{rr}
2 & -1 \\
4 & 3
\end{array}\right)
$$
|
\begin{pmatrix}
\frac{3}{10} & \frac{1}{10} \\
-\frac{2}{5} & \frac{1}{5}
\end{pmatrix}
|
olympiads
| 0.1875
|
Find all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ for which $f(0)=0$ and
\[f(x^2-y^2)=f(x)f(y) \]
for all $x,y\in\mathbb{N}_0$ with $x>y$ .
|
f(x) = 0
|
aops_forum
| 0.234375
|
How old is the eldest brother?
Determine this given that the age of the middle brother is the product of the ages of his two brothers, the sum of the ages of all three brothers is 35, and the sum of the decimal logarithms of their ages is 3.
|
10
|
olympiads
| 0.125
|
Find the integral \(\int \cos^{3} x \sin^{2} x \, dx\).
|
\frac{\sin^3 x}{3} - \frac{\sin^5 x}{5} + C
|
olympiads
| 0.140625
|
(**4**) Let $ a$ , $ b$ be constants such that $ \lim_{x\rightarrow1}\frac {(\ln(2 \minus{} x))^2}{x^2 \plus{} ax \plus{} b} \equal{} 1$ . Determine the pair $ (a,b)$ .
|
(a, b) = (-2, 1)
|
aops_forum
| 0.078125
|
The altitudes of an acute-angled triangle \(ABC\) drawn from points \(B\) and \(C\) are extended to intersect with the circumcircle at points \(B_1\) and \(C_1\). It turned out that the segment \(B_1C_1\) passes through the center of the circumcircle. Find the angle \(BAC\).
|
45^{\circ}
|
olympiads
| 0.234375
|
If the sum of interior angles of \( m \) regular \( n \)-sided polygons is divisible by 27, what is the minimum value of \( m + n \)?
|
6
|
olympiads
| 0.15625
|
As shown in Figure 4, car A starts from point A and car B starts from point B at the same time and travels towards each other. After the first meeting, car A continues to travel for 4 hours to reach point B, while car B takes only 1 hour to reach point A. What is the ratio of the speeds of car A and car B?
|
\frac{1}{2}
|
olympiads
| 0.078125
|
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