problem
stringlengths 33
2.6k
| answer
stringlengths 1
359
| source
stringclasses 2
values | llama8b_solve_rate
float64 0.06
0.59
|
|---|---|---|---|
The diagonals $AC$ and $CE$ of the regular hexagon $ABCDEF$ are divided by inner points $M$ and $N$ respectively, so that \[ {AM\over AC}={CN\over CE}=r. \] Determine $r$ if $B,M$ and $N$ are collinear.
|
r = \frac{\sqrt{3}}{3}
|
aops_forum
| 0.0625
|
If $\frac{2^{12} + 1}{2^{4} + 1} = 2^{a_{1}} + 2^{a_{2}} + \ldots + 2^{a_{k}}$ where $a_{1}, a_{2}, \ldots, a_{k}$ are natural numbers and $a_{1} < a_{2} < \cdots < a_{k}$, then $k =$ ______
|
5
|
olympiads
| 0.0625
|
Consider a sphere $S$ of radius $R$ tangent to a plane $\pi$ . Eight other spheres of the same radius $r$ are tangent to $\pi$ and tangent to $S$ , they form a "collar" in which each is tangent to its two neighbors. Express $r$ in terms of $R$ .
|
r = R(2 - \\sqrt{2})
|
aops_forum
| 0.0625
|
There were 115 tanks in total at the famous tank battle of "Lübeck" during World War II. The number of German tanks was 2 more than twice the number of Allied tanks, and the number of Allied tanks was 1 more than three times the number of "Sanlian" tanks. How many more tanks did the Germans have compared to "Sanlian"?
|
59
|
olympiads
| 0.265625
|
Determine the number of angles \(\theta\) between 0 and \(2\pi\), other than integer multiples of \(\pi / 2\), such that the quantities \(\sin \theta\), \(\cos \theta\), and \(\tan \theta\) form a geometric sequence in some order.
|
4
|
olympiads
| 0.0625
|
To make a very modern painting for his school, Roberto divides a square canvas into eight parts with four stripes of the same width and a diagonal, as shown in the figure. He paints the picture in blue and green, so that two adjacent parts always have different colors. In the end, he notices that he used more green than blue. What fraction of the picture was painted blue?
|
\frac{3}{8}
|
olympiads
| 0.171875
|
Find the largest positive integer \( n \) such that there exist \( n \) lattice points on the coordinate plane, and any 3 points form a triangle whose centroid is not a lattice point.
|
8
|
olympiads
| 0.125
|
Suppose we have a regular hexagon and draw all its sides and diagonals. Into how many regions do the segments divide the hexagon? (No proof is necessary.)
|
24
|
olympiads
| 0.09375
|
Given a regular tetrahedron with four vertices \(A, B, C, D\) and each edge of length 1 meter, a bug starts at point \(A\) and moves according to the following rule: at each vertex, it randomly chooses one of the three edges connected to that vertex with equal probability and crawls to the other end of the chosen edge. What is the probability that the bug will be exactly at vertex \(A\) after crawling a total of 7 meters?
|
\frac{182}{729}
|
olympiads
| 0.0625
|
From a common point, two tangents are drawn to a circle. The radius of the circle is 11, and the sum of the lengths of the tangents is 120.
Find the distance from the center of the circle to the common point of tangency.
|
61
|
olympiads
| 0.125
|
Let's define "addichiffrer" as the process of adding all the digits of a number. For example, if we addichiffrer 124, we get $1+2+4=7$.
What do we obtain when we addichiffrer $1998^{1998}$, then addichiffrer the result, and continue this process three times in total?
|
9
|
olympiads
| 0.203125
|
The operation $\ominus$ is defined by \(a \ominus b = a^b - b^a\). What is the value of \(2 \ominus(2 \odot 5)\)?
|
79
|
olympiads
| 0.203125
|
Construct the equation whose roots are the squares of the roots of the equation
$$
a b x^{2}-(a+b) x+1=0
$$
|
a^2 b^2 x^2 - (a^2 + b^2)x + 1 = 0
|
olympiads
| 0.078125
|
In the Cartesian coordinate plane $xOy$, the region determined by the system of inequalities
$$
\left\{\begin{array}{l}
|x| \leqslant 2, \\
|y| \leqslant 2, \\
|| x|-| y|| \leqslant 1
\end{array}\right.
$$
has an area of _______.
|
12
|
olympiads
| 0.15625
|
Suppose that $n$ is a positive integer and that $a$ is the integer equal to $\frac{10^{2n}-1}{3\left(10^n+1\right)}.$ If the sum of the digits of $a$ is 567, what is the value of $n$ ?
|
189
|
aops_forum
| 0.0625
|
A student was given a problem requiring them to divide the first given number by the second and then multiply the quotient by the number of which $\frac{2}{3}$ equals 4. However, the student instead subtracted the second given number from the first and added $\frac{2}{3}$ of 6 to the resulting difference. Despite this, the answer they obtained (10) was the same as it would have been with the correct solution. What were the two numbers given in the problem?
|
15 \text{ and } 9
|
olympiads
| 0.140625
|
Given that \(\alpha\), \(\beta\), and \(\gamma\) are acute angles such that \(\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1\), find the minimum value of \(\tan \alpha \cdot \tan \beta \cdot \tan \gamma\).
|
2\sqrt{2}
|
olympiads
| 0.09375
|
A firecracker was thrown vertically upwards with a speed of $20 \text{ m/s}$. One second after the start of the flight, it exploded into two fragments of equal mass. The first fragment flew horizontally at a speed of $48 \text{ m/s}$ immediately after the explosion. Find the magnitude of the speed of the second fragment (in m/s) immediately after the explosion. Assume the acceleration due to gravity is $10 \text{ m/s}^2$.
|
52
|
olympiads
| 0.125
|
Solve the equation
$$
2\left|\log _{1 / 2} x-1\right|-\left|\log _{4}\left(x^{2}\right)+2\right|=-\frac{1}{2} \log _{\sqrt{2}} x
$$
|
1
|
olympiads
| 0.203125
|
Each face of a $6 \times 6 \times 6$ cube is divided into $1 \times 1$ cells. The cube is covered with $2 \times 2$ squares such that each square covers exactly four cells, no squares overlap, and each cell is covered by the same number of squares. What is the maximum possible value of this identical number? (A square can be bent over an edge.)
|
3
|
olympiads
| 0.109375
|
The solution set of the inequality \(\log _{a}\left(a-\frac{x^{2}}{2}\right)>\log _{a}(a-x)\) is \(A\), and \(A \cap \mathbf{Z}=\{1\}\). What is the range of values for \(a\)?
|
(1, +\infty)
|
olympiads
| 0.1875
|
Determine \( m \) such that
\[ 2 x^{4} + 4 a x^{3} - 5 a^{2} x^{2} - 3 a^{3} x + m a^{4} \]
is divisible by \((x-a)\).
|
2
|
olympiads
| 0.25
|
The decimal representation of the natural number $N$ contains each digit from 0 to 9 exactly once. Let $A$ be the sum of the five two-digit numbers formed by the pairs (first and second, third and fourth,..., ninth and tenth) digits of $N$, and $B$ be the sum of the four two-digit numbers formed by the pairs (second and third, fourth and fifth,..., eighth and ninth) digits of $N$. It turns out that $A$ equals $B$. Can $N$ begin with an even digit?
|
Нет
|
olympiads
| 0.4375
|
We draw squares on the sides of a parallelogram. What type of quadrilateral is determined by the centers of these squares?
|
Square
|
olympiads
| 0.078125
|
Find the value of the expression \(\left(\sqrt[3]{x^{2}} \cdot x^{-0.5}\right):\left(\left(\sqrt[6]{x^{2}}\right)^{2} \cdot \sqrt{x}\right)\) for \(x = \frac{1}{2}\).
|
2
|
olympiads
| 0.40625
|
Origami. Mitya is going to fold a square sheet of paper $ABCD$. Mitya calls a fold beautiful if the side $AB$ intersects the side $CD$ and the four resulting right triangles are equal.
Before this, Vanya selects a random point $F$ on the sheet. Find the probability that Mitya can make a beautiful fold passing through the point $F$.
|
\frac{1}{2}
|
olympiads
| 0.21875
|
Given the lines \( l_{1}: A x + B y + C_{1} = 0 \) and \( l_{2}: A x + B y + C_{2} = 0 \) where \( C_{1} \neq C_{2} \). Also given that \( A - B + C_{1} + C_{2} = 0 \), find the equation of the line \( l \) that passes through the point \( H(-1,1) \) and has its midpoint \( M \), of the segment intercepted by the parallel lines \( l_{1} \) and \( l_{2} \), lying on the line \( x - y - 1 = 0\).
|
x + y = 0
|
olympiads
| 0.0625
|
Define the sequences $ a_n, b_n, c_n$ as follows. $ a_0 \equal{} k, b_0 \equal{} 4, c_0 \equal{} 1$ .
If $ a_n$ is even then $ a_{n \plus{} 1} \equal{} \frac {a_n}{2}$ , $ b_{n \plus{} 1} \equal{} 2b_n$ , $ c_{n \plus{} 1} \equal{} c_n$ .
If $ a_n$ is odd, then $ a_{n \plus{} 1} \equal{} a_n \minus{} \frac {b_n}{2} \minus{} c_n$ , $ b_{n \plus{} 1} \equal{} b_n$ , $ c_{n \plus{} 1} \equal{} b_n \plus{} c_n$ .
Find the number of positive integers $ k < 1995$ such that some $ a_n \equal{} 0$ .
|
u(2u + 1)
|
aops_forum
| 0.0625
|
Find all functions $f : Z^+ \to Z^+$ such that $n^3 - n^2 \le f(n) \cdot (f(f(n)))^2 \le n^3 + n^2$ for every $n$ in positive integers
|
f(n) \in \{n-1, n, n+1\}
|
aops_forum
| 0.0625
|
The product of three natural numbers is equal to 60. What is the largest possible value of their sum?
|
62
|
olympiads
| 0.140625
|
Find all positive integers $n$ such that $1! + 2! + \ldots + n!$ is a perfect square.
|
1 \text{ and } 3
|
olympiads
| 0.390625
|
A triangle has sides \(a, b, c\). Semicircles are constructed outwardly on each of its sides as diameters, forming a figure \(F\) composed of the triangle and the three semicircles. Find the diameter of \(F\) (the diameter of a set in the plane is the greatest distance between its points).
|
\frac{a+b+c}{2}
|
olympiads
| 0.0625
|
The triangle ABC has ∠B = 90°. The point D is taken on the ray AC, the other side of C from A, such that CD = AB. The ∠CBD = 30°. Find AC/CD.
|
2
|
olympiads
| 0.34375
|
Simplify the following expression:
$$
\frac{a}{x}+\sqrt{\frac{a^{2}}{x^{2}}+1}, \quad \text { where } \quad x=\frac{a^{2}-b^{2}}{2b}
$$
|
\frac{a+b}{a-b} \text{ or } \frac{b-a}{a+b}
|
olympiads
| 0.359375
|
Matěj and his friends went caroling. Each of the boys received oranges in addition to apples, nuts, and gingerbread. Jarda received one orange, Milan also received one. Radek, Patrik, Michal, and Dušan each received two oranges. Matěj received four oranges, which was the most among all the boys. The other boys received three oranges each. How many boys went caroling if they received a total of 23 oranges?
|
10
|
olympiads
| 0.109375
|
Anya is arranging stones on the sand. First, she placed one stone, then added stones to form a pentagon, then made a larger outer pentagon from the stones, then another larger outer pentagon, and so on, as shown in the illustration. The number of stones that were arranged in the first four images are 1, 5, 12, and 22. If we continue creating such images further, how many stones will there be in the 10th image?
|
145
|
olympiads
| 0.15625
|
Given the equation about \( x \)
$$
\sqrt{m-x^{2}}=\log _{2}(x+n)
$$
When there is one positive and one negative real root, the range of \( n \) is \( 3 \leqslant n<4 \). Then the value of the constant \( m \) is $\qquad$ .
|
4
|
olympiads
| 0.21875
|
The numbers \( a \) and \( b \) are such that the polynomial \( x^{4} + x^{3} + 2x^{2} + ax + b \) is the square of some other polynomial. Find \( b \).
|
\frac{49}{64}
|
olympiads
| 0.53125
|
Ashwin the frog is traveling on the $xy$-plane in a series of $2^{2017}-1$ steps, starting at the origin. At the $n^{\text{th}}$ step, if $n$ is odd, Ashwin jumps one unit to the right. If $n$ is even, Ashwin jumps $m$ units up, where $m$ is the greatest integer such that $2^{m}$ divides $n$. If Ashwin begins at the origin, what is the area of the polygon bounded by Ashwin's path, the line $x=2^{2016}$, and the $x$-axis?
|
2^{2015} \cdot(2^{2017} - 2018)
|
olympiads
| 0.078125
|
What is the smallest three-digit number \( K \) which can be written as \( K = a^b + b^a \), where both \( a \) and \( b \) are one-digit positive integers?
|
100
|
olympiads
| 0.125
|
In the pyramid \(ABCD\), the face \(ABC\) is an equilateral triangle with side \(a\), and \(AD = BD = CD = b\). Find the cosine of the angle formed by the lines \(AD\), \(BD\), and \(CD\) with the plane \(ABC\).
|
\frac{a}{b\sqrt{3}}
|
olympiads
| 0.140625
|
The area and the perimeter of the triangle with sides $10,8,6$ are equal. Find all the triangles with integral sides whose area and perimeter are equal.
|
\{6, 25, 29\}, \{7, 15, 20\}, \{9, 10, 17\}, \{5, 12, 13\}, \{6, 8, 10\}
|
aops_forum
| 0.09375
|
On side AC of triangle ABC with a 120-degree angle at vertex B, points D and E are marked such that AD = AB and CE = CB. A perpendicular DF is dropped from point D to line BE. Find the ratio BD / DF.
|
2
|
olympiads
| 0.21875
|
Let \( x, y, z \) be real numbers such that \( |x| \geq 2 \), \( |y| \geq 2 \), and \( |z| \geq 2 \). How small can \( |xyz + 2(x + y + z)| \) be?
|
4
|
olympiads
| 0.171875
|
For each positive integer \( n \), consider a cinema with \( n \) seats in a row, numbered left to right from 1 up to \( n \). There is a cup holder between any two adjacent seats, and there is a cup holder on the right of seat \( n \). So seat 1 is next to one cup holder, while every other seat is next to two cup holders. There are \( n \) people, each holding a drink, waiting in a line to sit down. In turn, each person chooses an available seat uniformly at random and carries out the following:
- If they sit next to two empty cup holders, then they place their drink in the left cup holder with probability \(\frac{1}{2}\) or in the right cup holder with probability \(\frac{1}{2}\).
- If they sit next to one empty cup holder, then they place their drink in that empty cup holder.
- If they sit next to zero empty cup holders, then they hold their drink in their hands.
Let \( p_{n} \) be the probability that all \( n \) people place their drink in a cup holder. Determine \( p_{1} + p_{2} + p_{3} + \cdots \).
|
\frac{2 \sqrt{e} - 2}{2 - \sqrt{e}}
|
olympiads
| 0.0625
|
Which is greater: \( (1.01)^{1000} \) or 1000?
|
(1.01)^{1000}
|
olympiads
| 0.375
|
Four boys - Alyosha, Borya, Vanya, and Grisha - competed in a race. The next day, when asked about their placements, they responded as follows:
Alyosha: I was neither first nor last.
Borya: I was not last.
Vanya: I was first.
Grisha: I was last.
It is known that three of these statements are true, and one is false. Who lied? Who was first?
|
Vanya is the one who lied; Borya was first
|
olympiads
| 0.09375
|
Let the function \( f(x) = \sqrt{x^2 + 1} - ax \), where \( a > 0 \). Determine the range of values for \( a \) such that the function \( f(x) \) is monotonic on \([0, +\infty)\).
|
a \geq 1
|
olympiads
| 0.125
|
Adam places down cards one at a time from a standard 52 card deck (without replacement) in a pile. Each time he places a card, he gets points equal to the number of cards in a row immediately before his current card that are all the same suit as the current card. For instance, if there are currently two hearts on the top of the pile (and the third card in the pile is not hearts), then placing a heart would be worth 2 points, and placing a card of any other suit would be worth 0 points. What is the expected number of points Adam will have after placing all 52 cards?
|
\frac{624}{41}
|
aops_forum
| 0.125
|
Given the function
\[ f(x) = \frac{x(1-x)}{x^3 - x + 1} \quad (0 < x < 1) \]
If the maximum value of \( f(x) \) occurs at \( f(x_0) \), then \( x_0 = \) $\qquad$ .
|
\frac{\sqrt{2} + 1 - \sqrt{2\sqrt{2} - 1}}{2}
|
olympiads
| 0.0625
|
Find the hyperbola with asymptotes $2x \pm 3y = 0$ that passes through the point $(1,2)$.
|
\frac{9}{32} y^2 - \frac{x^2}{8} = 1
|
olympiads
| 0.09375
|
Find all triples of positive numbers \( a, b, c \) that satisfy the conditions \( a + b + c = 3 \), \( a^2 - a \geq 1 - bc \), \( b^2 - b \geq 1 - ac \), and \( c^2 - c \geq 1 - ab \).
|
a = b = c = 1
|
olympiads
| 0.34375
|
The parabola \( y = ax^2 + bx + c \) has its vertex at \(\left( \frac{1}{4}, -\frac{9}{8} \right) \). If \( a > 0 \) and \( a + b + c \) is an integer, find the minimum possible value of \( a \).
|
\frac{2}{9}
|
olympiads
| 0.171875
|
Let $f_n(x,\ y)=\frac{n}{r\cos \pi r+n^2r^3}\ (r=\sqrt{x^2+y^2})$ , $I_n=\int\int_{r\leq 1} f_n(x,\ y)\ dxdy\ (n\geq 2).$ Find $\lim_{n\to\infty} I_n.$ *2009 Tokyo Institute of Technology, Master Course in Mathematics*
|
0
|
aops_forum
| 0.59375
|
Find any set of natural numbers \(x, y, z\) that satisfy the equation \(x^{3} + y^{4} = z^{5}\). Is the set of natural number solutions finite or infinite?
|
Infinite
|
olympiads
| 0.09375
|
Determine all functions \( f \) from the set of positive integers into the set of positive integers such that for all \( x \) and \( y \) there exists a non-degenerated triangle with sides of lengths
\[ x, \quad f(y) \text{ and } f(y+f(x)-1) \text{.} \]
|
f(x) = x
|
olympiads
| 0.234375
|
Find all quadratic polynomials \( f(x) \) for which the following equality holds: \( f(2x + 1) = 4x^2 + 14x + 7 \).
|
f(x) = x^2 + 5x + 1
|
olympiads
| 0.46875
|
Given the sine function $y = A \sin (\omega x + \varphi)$ with parameters $A > 0$, $\omega > 0$, and $0 < \varphi < 2 \pi$, where its graph intersects the y-axis at point $(0,1)$ and has a maximum point at $(2, \sqrt{2})$ within the same period, determine the expression of the function.
|
y = \sqrt{2} \sin \left( \frac{\pi}{8} x + \frac{\pi}{4} \right) \text{ or } y = \sqrt{2} \sin \left( \frac{7\pi}{8} x + \frac{3\pi}{4} \right)
|
olympiads
| 0.234375
|
Let the function \( \operatorname{lac}(x) \), defined on the set of real numbers, be given by the following rule:
$$
\operatorname{lac}(x) =
\begin{cases}
x, & \text{if } x \in [2n, 2n+1], \text{ where } n \in \mathbb{Z} \\
-x + 4n + 3, & \text{if } x \in (2n+1, 2n+2), \text{ where } n \in \mathbb{Z}
\end{cases}
$$
Solve the equation \( \operatorname{lac}(2x^2 + x + 4) = \operatorname{lac}(x^2 + 7x - 1) \) in the set of real numbers.
|
x = 1 \text{ and } x = 5
|
olympiads
| 0.265625
|
Which arithmetic sequence is it that, when summing any number of its terms, always results in the square of twice the number of terms?
|
4, 12, 20, 28, 36, \ldots
|
olympiads
| 0.140625
|
Five children \( A, B, C, D, \) and \( E \) sit in a circle. The teacher gives \( A, B, C, D, \) and \( E \) 2, 4, 6, 8, and 10 balls, respectively. Starting from \( A \), in a clockwise order, the game proceeds as follows: if a child's left neighbor has fewer balls than them, they give 2 balls to their left neighbor; if their left neighbor has the same or more balls, they don't give any balls. This process continues in sequence up to the fourth round. How many balls does each child have at the end of the fourth round?
|
6, 6, 6, 6, 6
|
olympiads
| 0.078125
|
A kindergarten teacher gave the children in the class 55 apples, 114 cookies, and 83 chocolates. After evenly distributing each item, 3 apples, 10 cookies, and 5 chocolates were left. What is the maximum number of children in the class?
|
26
|
olympiads
| 0.546875
|
Find all solutions in the set of positive real numbers for the system of equations:
$$
\left\{\begin{array}{l}
x(x+y+z) = 26 \\
y(x+y+z) = 27 \\
z(x+y+z) = 28
\end{array}\right.
$$
|
\left(\frac{26}{9}, 3, \frac{28}{9}\right)
|
olympiads
| 0.28125
|
A circle of radius \( 2 \) cm is inscribed in \( \triangle ABC \). Let \( D \) and \( E \) be the points of tangency of the circle with the sides \( AC \) and \( AB \), respectively. If \( \angle BAC = 45^{\circ} \), find the length of the minor arc \( DE \).
|
\pi \text{ cm}
|
olympiads
| 0.3125
|
Alice and Bob play a game together as a team on a $100 \times 100$ board with all unit squares initially white. Alice sets up the game by coloring exactly $k$ of the unit squares red at the beginning. After that, a legal move for Bob is to choose a row or column with at least $10$ red squares and color all of the remaining squares in it red. What is the
smallest $k$ such that Alice can set up a game in such a way that Bob can color the entire board red after finitely many moves?
Proposed by *Nikola Velov, Macedonia*
|
100
|
aops_forum
| 0.109375
|
King Midas spent $\frac{100}{x}\%$ of his gold deposit yesterday. He is set to earn gold today. What percentage
of the amount of gold King Midas currently has would he need to earn today to end up with as much
gold as he started?
|
\frac{100}{x-1}\%
|
aops_forum
| 0.140625
|
As shown in the figure, points $C$, $E$, and $B$ are collinear, $CB \perp AB$, $AE \parallel DC$, $AB = 8$, and $CE = 5$. What is the area of $\triangle AED$?
|
20
|
olympiads
| 0.140625
|
In the diagram below, circles \( C_{1} \) and \( C_{2} \) have centers \( O_{1} \) and \( O_{2} \), respectively. The radii of the circles are \( r_{1} \) and \( r_{2} \), with \( r_{1} = 3r_{2} \). Circle \( C_{2} \) is internally tangent to \( C_{1} \) at point \( P \). Chord \( XY \) of \( C_{1} \) has length 20, is tangent to \( C_{2} \) at point \( Q \), and is parallel to the line segment \( O_{2}O_{1} \). Determine the area of the shaded region, which is the region inside \( C_{1} \) but not \( C_{2} \).
|
160\pi
|
olympiads
| 0.109375
|
The graph of \((x^2 + y^2 - 2x)^2 = 2(x^2 + y^2)^2\) meets the \(x\)-axis in \(p\) different places and meets the \(y\)-axis in \(q\) different places. What is the value of \(100p + 100q\)?
|
400
|
olympiads
| 0.171875
|
There are $2^{2n+1}$ towns with $2n+1$ companies and each two towns are connected with airlines from one of the companies. What’s the greatest number $k$ with the following property:
We can close $k$ of the companies and their airlines in such way that we can still reach each town from any other (connected graph).
|
k = n
|
aops_forum
| 0.078125
|
Given positive integers \(a, b, c\) satisfying \(a<b<c\), find all triplets \((a, b, c)\) that satisfy the equation:
$$
\frac{(a, b)+(b, c)+(c, a)}{a+b+c} = \frac{1}{2}
$$
where \((x, y)\) denotes the greatest common divisor of the positive integers \(x\) and \(y\).
|
(d, 2d, 3d) \, \text{or} \, (d, 3d, 6d)
|
olympiads
| 0.09375
|
Given the set
$$
T=\left\{n \mid n=5^{a}+5^{b}, 0 \leqslant a \leqslant b \leqslant 30, a, b \in \mathbf{Z}\right\},
$$
if a number is randomly selected from set $T$, what is the probability that the number is a multiple of 9?
|
\frac{5}{31}
|
olympiads
| 0.0625
|
Determine all real-valued functions such that for all real numbers \( x \) and \( y \), the following equation holds:
$$
2 f(x) = f(x + y) + f(x + 2y)
$$
|
f(x) = c \text{ for some constant } c \in \mathbb{R}
|
olympiads
| 0.203125
|
In a deck of cards with four suits, each suit having 13 cards, how many cards must be drawn at a minimum to guarantee that there are 4 cards of the same suit?
|
13
|
olympiads
| 0.359375
|
Teams \(A\) and \(B\) are playing soccer until someone scores 29 goals. Throughout the game the score is shown on a board displaying two numbers - the number of goals scored by \(A\) and the number of goals scored by \(B\). A mathematical soccer fan noticed that several times throughout the game, the sum of all the digits displayed on the board was 10. (For example, a score of \(12: 7\) is one such possible occasion). What is the maximum number of times throughout the game that this could happen?
|
5
|
olympiads
| 0.0625
|
Given $f(x) = x^2 - 6x + 5$, find the region in the plane where the points $(x, y)$ satisfy $f(x) + f(y) \leq 0$ and $f(x) - f(y) \geq 0$. Provide a sketch of the region.
|
(x, y) \text{ in the shaded intersection region}
|
olympiads
| 0.078125
|
Compose the equation of the plane $\alpha$ that passes through the midpoint $M$ of segment $AD$ and is perpendicular to line $CB$, given $A(-1, 2, -3)$, $D(-5, 6, -1)$, $C(-3, 10, -5)$, and $B(3, 4, 1)$.
|
x - y + z + 9 = 0
|
olympiads
| 0.59375
|
Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble?
|
\frac{6480}{2401}
|
aops_forum
| 0.140625
|
Monday will be in five days after the day before yesterday. What day of the week will it be tomorrow?
|
Saturday
|
olympiads
| 0.15625
|
Write a twelve-digit number that is not a perfect cube.
|
100000000000
|
olympiads
| 0.0625
|
Given a sequence \( A = a_1, a_2, \cdots, a_{2005} \) with \( a_i \in \mathbf{N}^{+} \), let \( m(A) \) be the number of triplets \(\left(a_i, a_j, a_k\right)\) that satisfy \(a_j = a_i + 1\) and \(a_k = a_j + 1\) for \(1 \leq i < j < k \leq 2005\). Determine the maximum value of \( m(A) \) for all possible sequences \( A \).
|
668^2 \times 669
|
olympiads
| 0.15625
|
In this story, we have a caravan traveling through the Sahara Desert. One night, the caravan stopped to rest. There are three main characters, A, B, and C. A hated C and decided to kill him by poisoning his drinking water (the only water supply C had). Independently, another caravaner B also decided to kill C and, not knowing that the water had already been poisoned by A, made a small hole in the water container so that the water would slowly leak out. A few days later, C died of thirst. The question is, who is the killer? A or B?
Some people argue that B is the killer because C never had a chance to drink the poisoned water and would have died even if A had not poisoned it. Others argue that A is the killer because B's actions did not affect the outcome; since A poisoned the water, C was doomed and would have died even if B had not made the hole in the water container.
Whose reasoning is correct?
|
B
|
olympiads
| 0.171875
|
Let $ f, g: \mathbb {R} \to \mathbb {R} $ function such that $ f (x + g (y)) = - x + y + 1 $ for each pair of real numbers $ x $ e $ y $ . What is the value of $ g (x + f (y) $ ?
|
-x + y - 1
|
aops_forum
| 0.078125
|
Take the x-axis as horizontal and the y-axis as vertical. A gun at the origin can fire at any angle into the first quadrant (x, y ≥ 0) with a fixed muzzle velocity \( v \). Assuming the only force on the pellet after firing is gravity (acceleration \( g \)), which points in the first quadrant can the gun hit?
|
y \leq \frac{v^2}{2 g} - \frac{g x^2}{2 v^2}
|
olympiads
| 0.09375
|
Ten children play a "tangling" game: they stand in a circle and move toward the center with eyes closed and arms outstretched. Each child grabs the hand of someone else with both hands. Then, they open their eyes and start "untangling" themselves by moving under each other's arms, stepping over each other's hands, etc. (everyone is sufficiently flexible) without letting go of each other's hands. In what percentage of all cases is it true that if at one point two neighboring children let go of each other's hands, the ten children still holding hands form a single connected chain?
|
48\%
|
olympiads
| 0.109375
|
Given: $\sin \alpha=\cos \beta, \cos \alpha=\sin 2 \beta$, find $\sin ^{2} \beta+\cos ^{2} \alpha$.
|
\frac{3}{2}
|
olympiads
| 0.15625
|
If the real number \( x \) satisfies \( \log _{2} x = 1 + \cos \theta \) where \( \theta \in \left[ -\frac{\pi}{2}, 0 \right] \), then the maximum value of the function \( f(x) = |x-1| + 2|x-3| \) is ____________.
|
5
|
olympiads
| 0.328125
|
Petya places 500 kings on the cells of a $100 \times 50$ board in such a way that they do not attack each other. Vasya places 500 kings on the white cells (in a checkerboard pattern) of a $100 \times 100$ board in such a way that they do not attack each other. Who has more ways to do this?
|
Vasya has more
|
olympiads
| 0.078125
|
Find all pairs of real numbers \( (x ; y) \) that satisfy the inequality \( \sqrt{x+y-1} + x^{4} + y^{4} - \frac{1}{8} \leq 0 \). As an answer, write down the maximum value of the product \( x y \) for all the pairs \( (x ; y) \) found.
|
\frac{1}{4}
|
olympiads
| 0.21875
|
Person A departs from point $A$ to meet person B, traveling 80 kilometers to reach point $B$. At that moment, person B had already left point $B$ for point $C$ half an hour earlier. Person A had been traveling for 2 hours by the time they left point $A$, and then decides to proceed to point $C$ at twice their original speed. After 2 more hours, both A and B arrive at point $C$ simultaneously. What is the speed of person B in kilometers per hour?
|
64
|
olympiads
| 0.109375
|
Jenny and Kenny are walking in the same direction, Kenny at 3 feet per second and Jenny at 1 foot per second, on parallel paths that are 200 feet apart. A tall circular building 100 feet in diameter is centered midway between the paths. At the instant when the building first blocks the line of sight between Jenny and Kenny, they are 200 feet apart. Let $t$ be the amount of time, in seconds, before Jenny and Kenny can see each other again. If $t$ is written as a fraction in lowest terms, what is the sum of the numerator and denominator?
|
163
|
aops_forum
| 0.078125
|
Find the radius of the circle if an inscribed angle subtended by an arc of \(120^{\circ}\) has sides with lengths 1 and 2.
|
1
|
olympiads
| 0.09375
|
There are 300 non-zero integers written in a circle such that each number is greater than the product of the three numbers that follow it in the clockwise direction. What is the maximum number of positive numbers that can be among these 300 written numbers?
|
200
|
olympiads
| 0.078125
|
The angles of triangle \(A B C\) are such that \(\sin \angle A + \cos \angle B = \sqrt{2}\) and \(\cos \angle A + \sin \angle B = \sqrt{2}\). Find the measure of angle \(C\).
|
90^ ext{o}
|
olympiads
| 0.59375
|
Given a prime number \( p > 2 \) and \( p \nmid d \), the necessary and sufficient condition for \( d \) to be a quadratic residue modulo \( p \) is:
\[ d^{\frac{p-1}{2}} \equiv 1 \pmod{p} \]
The necessary and sufficient condition for \( d \) to be a non-quadratic residue modulo \( p \) is:
\[ d^{\frac{p-1}{2}} \equiv -1 \pmod{p} \]
|
The required conditions are proved as stated.
|
olympiads
| 0.1875
|
A dog has three trainers:
- The first trainer gives him a treat right away.
- The second trainer makes him jump five times, then gives him a treat.
- The third trainer makes him jump three times, then gives him no treat.
The dog will keep picking trainers with equal probability until he gets a treat. (The dog's memory isn't so good, so he might pick the third trainer repeatedly!) What is the expected number of times the dog will jump before getting a treat?
|
8
|
aops_forum
| 0.0625
|
What is the largest possible value of \(| |a_1 - a_2| - a_3| - \ldots - a_{1990}|\), where \(a_1, a_2, \ldots, a_{1990}\) is a permutation of \(1, 2, 3, \ldots, 1990\)?
|
1989
|
olympiads
| 0.09375
|
On the side $AC$ of triangle $ABC$, a circle is constructed such that it intersects sides $AB$ and $BC$ at points $D$ and $E$ respectively, and is constructed with $AC$ as its diameter. The angle $\angle EDC$ is equal to $30^\circ$, $AE = \sqrt{3}$, and the area of triangle $DBE$ is related to the area of triangle $ABC$ as \(1:2\). Find the length of segment $BO$, where $O$ is the point of intersection of segments $AE$ and $CD$.
|
2
|
olympiads
| 0.09375
|
Determine if the number $\underbrace{11 \ldots 1}_{2016} 2 \underbrace{11 \ldots 1}_{2016}$ is a prime number or a composite number.
|
The number is composed
|
olympiads
| 0.0625
|
The law of motion for the first tourist is given by $S=\sqrt{1+6 t}-1$, and for the second tourist, it is $-S=6\left(t-\frac{1}{6}\right)$ for $t \geq \frac{1}{6}$; $S=0$ for $t<\frac{1}{6}$.
The required condition is met when both tourists are on the same side of the sign, and not met when the sign is between them. The first tourist reaches the sign at time $t_{1}$ where $\sqrt{1+6 t_{1}}-1=2$, resulting in $t_{1}=\frac{4}{3}$. The second tourist reaches the sign at time $t_{2}$ where $6\left(t_{2}-\frac{1}{6}\right)=2$, resulting in $t_{2}=\frac{1}{2}$. Therefore, the suitable times are $t \in\left[0, \frac{1}{2}\right] \cup\left[\frac{4}{3},+\infty\right]$.
|
t \in \left[ 0, \frac{1}{2} \right] \cup \left[ \frac{4}{3}, +\infty \right]
|
olympiads
| 0.5625
|
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.