problem
stringlengths 33
2.6k
| answer
stringlengths 1
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values | llama8b_solve_rate
float64 0.06
0.59
|
|---|---|---|---|
Given \(\cos^2 \alpha - \cos^2 \beta = m\), find \(\sin (\alpha + \beta) \sin (\alpha - \beta)\).
|
-m
|
olympiads
| 0.15625
|
The plane is divided into equilateral triangles of side length 1. Consider an equilateral triangle of side length \( n \) whose sides lie on the grid lines. There is a tile on each grid point on the edge and inside this triangle. In one turn, a unit triangle is selected which has a checker on exactly 2 corners. The two stones are removed and a new stone is placed on the third corner. For which \( n \) is it possible that only one piece remains after a finite number of moves?
|
n \text{ is divisible by 3}
|
olympiads
| 0.140625
|
Let $\alpha$ and $\beta$ be the roots of the equations $\log _{2} x + x + 2 = 0$ and $2^{x} + x + 2 = 0$ respectively. What is the value of $\alpha + \beta$?
|
-2
|
olympiads
| 0.0625
|
Find all numbers \( n \) with the following property: there is exactly one set of 8 different positive integers whose sum is \( n \).
|
36, 37
|
olympiads
| 0.109375
|
As shown in the figure, in the right triangle \( \triangle ABC \), squares \( ACDE \) and \( CBF G \) are constructed on the legs \( AC \) and \( BC \) respectively. Connect \( DG \) and \( AF \), intersecting \( BC \) at \( W \). Then, connect \( GW \). If \( AC = 14 \) and \( BC = 28 \), find the area of \( \triangle AGW \).
|
(196)
|
olympiads
| 0.09375
|
In the arithmetic sequence \(\{a_{n}\}\), \(a_{20}=\frac{1}{a}, a_{201}=\frac{1}{b}, a_{2012}=\frac{1}{c}\). Find the value of \(1992 a c - 1811 b c - 181 a b\).
( Note: Given constants \(a\) and \(b\) satisfy \(a, b > 0, a \neq 1\), and points \(P(a, b)\) and \(Q(b, a)\) are on the curve \(y=\cos(x+c)\), where \(c\) is a constant. Find the value of \(\log _{a} b\).
|
0
|
olympiads
| 0.15625
|
The 97 numbers \( \frac{49}{1}, \frac{49}{2}, \frac{49}{3}, \ldots, \frac{49}{97} \) are written on a blackboard. We repeatedly pick two numbers \( a, b \) on the board and replace them by \( 2ab - a - b + 1 \) until only one number remains. What are the possible values of the final number?
|
1
|
olympiads
| 0.28125
|
Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation
\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]
|
f(x) = C
|
aops_forum
| 0.203125
|
Provide an example of integers \( a \) and \( b \) such that \( a b(2 a + b) = 2015 \).
|
a = 13, b = 5
|
olympiads
| 0.140625
|
What is the maximum possible value of $k$ for which $2013$ can be written as a sum of $k$ consecutive positive integers?
|
61
|
aops_forum
| 0.140625
|
Find all possible rectangles with integer sides for which the perimeter numerically equals the area.
|
(3, 6) \text{ and } (4, 4)
|
olympiads
| 0.078125
|
In a Turkish village, there are many long-living inhabitants, including old Ihsan, who is surrounded by his children, grandchildren, great-grandchildren, and great-great-grandchildren. Altogether, there are 2801 people, including Ihsan himself. Each of them has the same number of children, except for the great-great-grandchildren, who do not yet have children, and all the children are alive. Can you tell how many children old Ihsan has?
|
7
|
olympiads
| 0.15625
|
Two triangles intersect to form seven finite disjoint regions, six of which are triangles with area 1. The last region is a hexagon with area \(A\). Compute the minimum possible value of \(A\).
|
6
|
aops_forum
| 0.0625
|
On the graph of the function \( y = \frac{1}{x} \), Misha marked consecutive points with abscissas \( 1, 2, 3, \ldots \) until he got tired. Then Masha came and shaded all rectangles with one vertex at a marked point, another vertex at the origin, and the other two on the axes (the illustration shows the rectangle Masha would have shaded for the marked point P). The teacher then asked the students to calculate the area of the figure consisting of all points that were shaded exactly once. What is the resulting area?
|
1
|
olympiads
| 0.296875
|
Find all positive integers $n$ for which the number $1!+2!+3!+\cdots+n!$ is a perfect power of an integer.
|
n = 1 and n = 3
|
aops_forum
| 0.171875
|
The period of small oscillations $T$ of a simple pendulum is calculated using the formula $T=2 \pi \sqrt{\frac{l}{g}}$, where $l$ is the length of the pendulum and $g$ is the acceleration due to gravity. Which of the quantities in this formula are constants, and which are variables?
|
2, \pi \text{ (constants)}, g \text{ (constant)}, l, T \text{ (variables)}
|
olympiads
| 0.234375
|
Five points lie on the same line. When we list the 10 distances between two of these points, from smallest to largest, we find $2, 4, 5, 7, 8, k, 13, 15, 17,$ and 19. What is the value of $k$?
|
k = 12
|
olympiads
| 0.125
|
Twenty-five people who always tell the truth or always lie are standing in a queue. The man at the front of the queue says that everyone behind him always lies. Everyone else says that the person immediately in front of them always lies. How many people in the queue always lie?
|
13
|
olympiads
| 0.140625
|
It is known that $\log_{b} a = m$ and $\log_{c} b = n$. Find $\log_{bc}(ab)$.
|
\frac{n(m+1)}{n+1}
|
olympiads
| 0.171875
|
Three different numbers are chosen at random from the list \(1, 3, 5, 7, 9, 11, 13, 15, 17, 19\). The probability that one of them is the mean of the other two is \(p\). What is the value of \(\frac{120}{p}\) ?
|
720
|
olympiads
| 0.0625
|
Two players, A and B, take turns removing a square (along the grid lines) from a $4000 \times 2019$ grid, ensuring that after each move, the remaining grid squares are connected (i.e., it is possible to travel from any square to any other square along shared edges). If player A goes first, who has a winning strategy?
|
Player A has a winning strategy
|
olympiads
| 0.40625
|
Vernonia High School has 85 seniors, each of whom plays on at least one of the school's three varsity sports teams: football, baseball, and lacrosse. It so happens that 74 are on the football team; 26 are on the baseball team; 17 are on both the football and lacrosse teams; 18 are on both the baseball and football teams; and 13 are on both the baseball and lacrosse teams. Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team.
|
11
|
olympiads
| 0.34375
|
At the first stop, 18 passengers boarded an empty bus. At each subsequent stop, 4 people got off and 6 people got on. How many passengers were on the bus between the fourth and fifth stops?
|
24
|
olympiads
| 0.3125
|
The planes are given by the equations:
\[
\begin{aligned}
& 3x + 4y - 2z = 5 \\
& 2x + 3y - z = 3
\end{aligned}
\]
Write in standard form the system of equations describing the line of intersection of these planes.
|
\frac{x-3}{2} = \frac{y+1}{-1} = \frac{z}{1}
|
olympiads
| 0.203125
|
Find all natural numbers \( n \) for which prime numbers \( p \) and \( q \) exist such that
\[ p(p+1) + q(q+1) = n(n+1) \]
|
n=3 \text{ or } n=6
|
olympiads
| 0.0625
|
Given the sequence: \( y_{1}=y_{2}=1 \), \( y_{n+2}=(4 k-5) y_{n+1}-y_{n}+4-2 k \) for \( n \geq 1 \). Find all integers \( k \) such that every term in the sequence is a perfect square.
|
1 \text{ and } 3
|
olympiads
| 0.078125
|
Each of $2k+1$ distinct 7-element subsets of the 20 element set intersects with exactly $k$ of them. Find the maximum possible value of $k$ .
|
2
|
aops_forum
| 0.078125
|
The side of each square in a grid paper is 1 unit. A rectangle with sides \( m \) and \( n \) is constructed along the grid lines. Is it possible to draw a closed polygonal chain (broken line) along the grid lines within the rectangle such that it passes exactly once through each vertex of the grid located inside or on the boundary of the rectangle? If possible, what is its length?
|
(n+1)(m+1)
|
olympiads
| 0.0625
|
Find all positive integer number $n$ such that $[\sqrt{n}]-2$ divides $n-4$ and $[\sqrt{n}]+2$ divides $n+4$ . Note: $[r]$ denotes the integer part of $r$ .
|
n = 2, 4, 11, 20, 31, 36, 44 , and n = a^2 + 2a - 4 for a \in \mathbb{N} \setminus \{1, 2\}.
|
aops_forum
| 0.0625
|
Let ${ a\uparrow\uparrow b = {{{{{a^{a}}^a}^{\dots}}}^{a}}^{a}} $ , where there are $ b $ a's in total. That is $ a\uparrow\uparrow b $ is given by the recurrence \[ a\uparrow\uparrow b = \begin{cases} a & b=1 a^{a\uparrow\uparrow (b-1)} & b\ge2\end{cases} \] What is the remainder of $ 3\uparrow\uparrow( 3\uparrow\uparrow ( 3\uparrow\uparrow 3)) $ when divided by $ 60 $ ?
|
27
|
aops_forum
| 0.484375
|
Find the integer values of \( x \) for which the inequality holds:
\[ 8.58 \log _{4} x+\log _{2}(\sqrt{x}-1)<\log _{2} \log _{\sqrt{5}} 5 \]
|
2 \text{ and } 3
|
olympiads
| 0.078125
|
Compute the limit of the function:
\[
\lim _{x \rightarrow 0}\left(\frac{1 + x \cdot 2^{x}}{1 + x \cdot 3^{x}}\right)^{\frac{1}{x^{2}}}
\]
|
\frac{2}{3}
|
olympiads
| 0.171875
|
For any subset \( U \) of the real numbers \( \mathbf{R} \), we define the function
\[ f_{U}(x) = \begin{cases}
1, & \text{if } x \in U \\
0, & \text{if } x \notin U
\end{cases}. \]
If \( A \) and \( B \) are two subsets of the real numbers \( \mathbf{R} \), find the necessary and sufficient condition for \( f_{A}(x) + f_{B}(x) \equiv 1 \).
|
A \cup B = \mathbf{R}, \; \text{and} \; A \cap B = \varnothing
|
olympiads
| 0.4375
|
Seongcheol has $3$ red shirts and $2$ green shirts, such that he cannot tell the difference between his three red shirts and he similarly cannot tell the difference between his two green shirts. In how many ways can he hang them in a row in his closet, given that he does not want the two green shirts next to each other?
|
6
|
aops_forum
| 0.234375
|
Solve the equation \( c \frac{(x-a)(x-b)}{(c-a)(c-b)}+b \frac{(x-a)(x-c)}{(b-a)(b-c)}+a \frac{(x-b)(x-c)}{(a-b)(a-c)}=x \).
|
x \text{ is any real number}
|
olympiads
| 0.109375
|
Vasya plays the strings of a 6-string guitar from 1 to 6 and then back. Each subsequent hit lands on an adjacent string. On which string number will the 2000th hit land? (The order of string hits is: $1-2-3-4-5-6-5-4-3-2-1-2-\ldots)$
|
2
|
olympiads
| 0.15625
|
For which digits $a$ do exist integers $n \geq 4$ such that each digit of $\frac{n(n+1)}{2}$ equals $a \ ?$
|
5 \text{ and } 6
|
aops_forum
| 0.125
|
In the picture, arrows indicate the entrance and exit of the labyrinth. You can move through it such that you only move to the right, down, or up (you cannot turn back). How many different paths can you take through this labyrinth?
Answer: 16.
|
16
|
olympiads
| 0.375
|
If \( a b c + c b a = 1069 \), then there are \(\qquad\) such \(\underline{a b c}\).
|
8
|
olympiads
| 0.0625
|
In a square \(ABCD\) with a side length of 2, \(E\) is the midpoint of \(AB\). The figure is folded along segments \(EC\) and \(ED\) so that segments \(EA\) and \(EB\) coincide. The resulting tetrahedron is \(CDEA\) (with point \(B\) coinciding with point \(A\)). What is the volume of this tetrahedron?
|
\frac{\sqrt{3}}{3}
|
olympiads
| 0.078125
|
4. $\log _{b} N=\frac{\log _{a} N}{\log _{a} b}(N>0, a>0, b>0, a \neq 1, b \neq 1)$.
|
\log_{b}(N) = \frac{\log_{a}(N)}{\log_{a}(b)}
|
olympiads
| 0.40625
|
Calculate the limit of the function:
\[
\lim _{x \rightarrow \frac{\pi}{4}} \frac{1-\sin 2 x}{(\pi-4 x)^{2}}
\]
|
\frac{1}{8}
|
olympiads
| 0.171875
|
I could ask any guest in the castle a question that would inevitably make them answer "ball".
|
\textcolor{green!50!black}{True}
|
olympiads
| 0.515625
|
Solve the inequality \( n^{3} - n < n! \) for positive integers \( n \). (Here, \( n! \) denotes the factorial of \( n \), which is the product of all positive integers from 1 to \( n \)).
|
n=1 \text{ or } n \geq 6
|
olympiads
| 0.0625
|
Let \( p(x) = a_{n} x^{n} + a_{n-1} x^{n-1} + \ldots + a_{0} \), where each \( a_{i} \) is either 1 or -1. Let \( r \) be a root of \( p \). If \( |r| > \frac{15}{8} \), what is the minimum possible value of \( n \)?
|
4
|
olympiads
| 0.21875
|
Given a function \( f: \mathbf{R} \rightarrow \mathbf{R} \) that satisfies, for all \( x, y \in \mathbf{R} \),
$$
f\left[x^{2}+f(y)\right]=y+[f(x)]^{2}.
$$
Find \( f(x) \).
|
f(x) = x
|
olympiads
| 0.3125
|
A young man allocates his monthly salary in the following way: half of his salary is deposited in the bank, half of the remaining amount minus 300 yuan is used to pay off the mortgage, half of the remaining amount plus 300 yuan is used for meal expenses, and he is left with 800 yuan. What is his monthly salary?
|
7600
|
olympiads
| 0.171875
|
On a number line, points with coordinates $0, 1, 2, 3, 5, 8, 2016$ are marked. Consider the set of lengths of segments with endpoints at these points. How many elements does this set contain?
|
14
|
olympiads
| 0.140625
|
Given a triangle $ABC$, using a double-sided ruler, draw no more than eight lines to construct a point $D$ on side $AB$ such that $AD: BD = BC: AC$.
|
D
|
olympiads
| 0.140625
|
Let the probabilities of heads and tails be $p$ and $q=1-p$, respectively. Form the equation
$$
C_{10}^{7} p^{7} q^{3}=C_{10}^{6} p^{6} q^{4}
$$
from which it follows: $120 p=210 q ; \frac{p}{q}=\frac{7}{4} ; p=\frac{7}{11}$.
|
\frac{7}{11}
|
olympiads
| 0.546875
|
Calculate the area of the figure bounded by the curves \( y^2 = 4x \) and \( x^2 = 4y \).
|
\frac{16}{3} \text{ square units}
|
olympiads
| 0.25
|
Five friends approached a river and found a boat on the shore that can accommodate all five of them. They decided to take a ride on the boat. Each time, a group of one or more people crosses from one bank to the other. The friends want to organize the rides so that each possible group crosses the river exactly once. Is it possible for them to do this?
|
It is not possible to organize the rides as desired.
|
olympiads
| 0.0625
|
What is the largest number of natural numbers less than 50 that can be chosen so that any two of them are relatively prime?
|
16
|
olympiads
| 0.0625
|
Express \(\frac{\sin 10^\circ + \sin 20^\circ + \sin 30^\circ + \sin 40^\circ + \sin 50^\circ + \sin 60^\circ + \sin 70^\circ + \sin 80^\circ}{\cos 5^\circ \cos 10^\circ \cos 20^\circ}\) without using trigonometric functions.
|
4\sqrt{2}
|
olympiads
| 0.15625
|
A deck of 54 cards is divided into several piles by a magician. An audience member writes a natural number on each card equal to the number of cards in that pile. The magician then shuffles the cards in a special way and redistributes them into several piles. The audience member again writes a natural number on each card equal to the number of cards in the new pile. This process continues. What is the minimum number of times this process needs to be performed so that the (unordered) array of numbers written on the cards becomes unique for each card?
|
3
|
olympiads
| 0.109375
|
For which values of \( n \) is the number \( 3\left(n^{2} + n\right) + 7 \) divisible by 5?
|
n = 5k + 2
|
olympiads
| 0.109375
|
Assume \( f_{1}(x)=\sqrt{x^{2}+32} \) and \( f_{n+1}(x)=\sqrt{x^{2}+\frac{16}{3} f_{n}(x)} \) for \( n=1,2, \cdots \). For each \( n \), find the real solution of \( f_{n}(x)=3 x \).
|
2
|
olympiads
| 0.28125
|
There are 19 cards. Is it possible to write a non-zero digit on each of these cards so that they can be arranged to form exactly one 19-digit number that is divisible by 11?
|
Possible
|
olympiads
| 0.0625
|
Express the numerical expression \( 2 \cdot 2009^{2} + 2 \cdot 2010^{2} \) as a sum of the squares of two natural numbers.
|
4019^{2} + 1^{2} \text{ or } 2911^{2} + 2771^{2}
|
olympiads
| 0.1875
|
Let the function \( f(x)=\frac{a x}{2 x+3} \). If \( f[f(x)] = x \) always holds, find the value of the real number \( a \).
|
-3
|
olympiads
| 0.515625
|
Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$ has exactly $ n$ nonzero coefficients.
|
0, 50, 100, 101
|
aops_forum
| 0.40625
|
The polynomial $x^3-kx^2+20x-15$ has $3$ roots, one of which is known to be $3$ . Compute the greatest possible sum of the other two roots.
|
5
|
aops_forum
| 0.21875
|
There are 200 computers in a computer center, with some of them connected by a total of 345 cables. A "cluster" is defined as a set of computers such that a signal can travel from any computer in the set to any other computer in the set, possibly through intermediate computers. Originally, all the computers formed one cluster. However, one night an evil hacker cut several cables, resulting in 8 clusters. Find the maximum possible number of cables that were cut.
|
153
|
olympiads
| 0.140625
|
A bag contains \( d \) balls of which \( x \) are black, \( x+1 \) are red, and \( x+2 \) are white. If the probability of drawing a black ball randomly from the bag is less than \(\frac{1}{6}\), find the value of \( d \).
|
3
|
olympiads
| 0.453125
|
The diagonals of a quadrilateral are equal, and the lengths of its midline segments are \( p \) and \( q \). Find the area of the quadrilateral.
|
pq
|
olympiads
| 0.140625
|
Find all triples $m$ ; $n$ ; $l$ of natural numbers such that $m + n = gcd(m; n)^2$ ; $m + l = gcd(m; l)^2$ ; $n + l = gcd(n; l)^2$ :
|
(2, 2, 2)
|
aops_forum
| 0.109375
|
In a classroom, there is a teacher and several students. It is known that the teacher's age is 24 years greater than the average age of the students and 20 years greater than the average age of everyone present in the classroom. How many students are in the classroom?
|
5
|
olympiads
| 0.3125
|
Calculate \(\cos (\alpha - \beta)\) if \(\cos \alpha + \cos \beta = -\frac{4}{5}\) and \(\sin \alpha + \sin \beta = \frac{1}{3}\).
|
-\frac{28}{225}
|
olympiads
| 0.234375
|
On a plane, an equilateral triangle is drawn in black. There are nine triangular tiles of the same size and shape. The tiles need to be placed on the plane such that they do not overlap and each tile covers at least part of the black triangle (at least one point inside it). How can this be done?
|
The configuration is correct and satisfies all the problem constraints.
|
olympiads
| 0.171875
|
In how many ways can two identical pencils be distributed among five people?
|
15
|
olympiads
| 0.5
|
A bamboo pole of 10 feet in height is broken. If the top part is bent to the ground, the tip of the pole will be 3 feet away from the base. What is the length of the broken part?
|
5 \frac{9}{20}
|
olympiads
| 0.171875
|
Starting with a positive integer \( m \), Alicia creates a sequence by applying the following algorithm:
- Step 1: Alicia writes down the number \( m \) as the first term of the sequence.
- Step 2: If \( m \) is even, Alicia sets \( n = \frac{1}{2} m \). If \( m \) is odd, Alicia sets \( n = m + 1 \).
- Step 3: Alicia writes down the number \( m + n + 1 \) as the next term of the sequence.
- Step 4: Alicia sets \( m \) equal to the value of the term that she just wrote down in Step 3.
- Step 5: Alicia repeats Steps 2, 3, 4 until she has five terms, at which point she stops.
For example, starting with \( m = 1 \), Alicia's sequence would be \( 1, 4, 7, 16, 25 \).
Alicia starts a sequence with \( m = 3 \). What is the fifth term of her sequence?
|
43
|
olympiads
| 0.25
|
Wang Qiang walked to the park and took a car back, spending one and a half hours in total for the round trip. If he walked both ways, it would take two and a half hours. How many minutes would it take if he took a car both ways?
|
30
|
olympiads
| 0.359375
|
Using 4 different colors to paint the 4 faces of a regular tetrahedron (each face is an identical equilateral triangle) so that different faces have different colors, how many different ways are there to paint it? (Coloring methods that remain different even after any rotation of the tetrahedron are considered different.)
|
2
|
olympiads
| 0.09375
|
Point $I$ is the center of the inscribed circle of triangle $ABC$, $M$ is the midpoint of the side $AC$, and $W$ is the midpoint of the arc $AB$ of the circumcircle, not containing $C$. It turns out that $\angle AIM = 90^\circ$. In what ratio does point $I$ divide the segment $CW$?
|
2 : 1
|
olympiads
| 0.0625
|
The bases of a trapezoid have lengths 10 and 21, and the legs have lengths $\sqrt{34}$ and $3 \sqrt{5}$ . What is the area of the trapezoid?
|
\frac{93}{2}
|
aops_forum
| 0.1875
|
The lengths of the three sides $a, b, c$ with $a \le b \le c$ , of a right triangle is an integer. Find all the sequences $(a, b, c)$ so that the values of perimeter and area of the triangle are the same.
|
(5, 12, 13)
|
aops_forum
| 0.078125
|
The triangle $K_2$ has as its vertices the feet of the altitudes of a non-right triangle $K_1$ . Find all possibilities for the sizes of the angles of $K_1$ for which the triangles $K_1$ and $K_2$ are similar.
|
A = 60^
\circ, B = 60^
\circ, C = 60^
\circ
|
aops_forum
| 0.234375
|
In the picture, arrows mark the entrance and exit of the maze. You can move through it by only moving down, left, or right (you cannot turn around). How many different ways are there to traverse this maze?
|
16
|
olympiads
| 0.078125
|
In the Cartesian coordinate system, the set of points
$$
M=\left\{(x, y) \left\lvert\,\left\{\begin{array}{l}
x=\sin \alpha+\cos \beta, \\
y=\cos \alpha+\sin \beta,
\end{array}, \beta \in \mathbf{R}\right\}\right.,\right.
$$
The area of the plane figure covered by the set $M$ is .
|
4 \pi
|
olympiads
| 0.171875
|
Let \( b \) be the maximum of the function \( y = \left|x^{2} - 4\right| - 6x \) (where \( -2 \leq x \leq 5 \) ). Find the value of \( b \).
|
12
|
olympiads
| 0.265625
|
Given that \(a, b, c,\) and \(d\) are all integers, and \(m = a^2 + b^2\) and \(n = c^2 + d^2\), express \(mn\) as a sum of squares of two integers. The form is: \(mn =\) \[\_\_\_\]
|
(ac - bd)^2 + (ad + bc)^2 \text{ or } (ac + bd)^2 + (ad - bc)^2
|
olympiads
| 0.5
|
Find all sequences $\left\{a_{1}, a_{2}, \cdots\right\}$ such that:
$a_{1}=1$ and $\left|a_{n}-a_{m}\right| \leq \frac{2mn}{m^{2}+n^{2}}$ (for all positive integers $m, n$).
|
a_n = 1 \text{ for all } n \in \mathbb{N}
|
olympiads
| 0.578125
|
Baba Yaga told her 33 students, "It's too early for you to see this," and ordered, "Close your eyes!" All boys and one-third of the girls closed their right eye. All girls and one-third of the boys closed their left eye. How many students still saw what they were not supposed to see?
|
22
|
olympiads
| 0.265625
|
Write the decomposition of the vector $x$ in terms of the vectors $p, q, r$:
$x=\{8 ; 9 ; 4\}$
$p=\{1 ; 0 ; 1\}$
$q=\{0 ; -2 ; 1\}$
$r=\{1 ; 3 ; 0\}$
|
\mathbf{x} = 7\mathbf{p} - 3\mathbf{q} + \mathbf{r}
|
olympiads
| 0.15625
|
Given real numbers \( a \) and \( b \) that satisfy the equations
\[ a + \lg a = 10 \]
\[ b + 10^b = 10 \]
Find \(\lg (a + b)\).
|
1
|
olympiads
| 0.0625
|
There are two alloys consisting of zinc, copper, and tin. It is known that the first alloy contains 40% tin, and the second contains 26% copper. The percentage of zinc in the first and second alloys is the same. By melting 150 kg of the first alloy and 250 kg of the second, a new alloy was obtained which contains 30% zinc. How much tin is contained in the new alloy?
|
170 \text{ kg}
|
olympiads
| 0.109375
|
Find the coordinates of point $A$, which is equidistant from points $B$ and $C$.
$A(x ; 0 ; 0)$
$B(4 ; 0 ; 5)$
$C(5 ; 4 ; 2)$
|
A(2, 0, 0)
|
olympiads
| 0.265625
|
Let \(a < b < c < d\) be real numbers. Arrange the sums \(ab + cd\), \(ac + bd\), and \(ad + bc\) in increasing order.
|
ad + bc, ac + bd, ab + cd
|
olympiads
| 0.375
|
Find the natural integer pairs \((x, y)\) such that \((x+y)^{2}+3x+y+1\) is a perfect square.
|
(x, x)
|
olympiads
| 0.109375
|
A circle with radius \( R \) touches the base \( AC \) of an isosceles triangle \( ABC \) at its midpoint and intersects side \( AB \) at points \( P \) and \( Q \), and side \( CB \) at points \( S \) and \( T \). The circumcircles of triangles \( SQB \) and \( PTB \) intersect at points \( B \) and \( X \). Find the distance from point \( X \) to the base of the triangle \( ABC \).
|
R
|
olympiads
| 0.3125
|
If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$ , determine the product of all possible values of $ab$ .
|
1
|
aops_forum
| 0.25
|
Four men are each given a unique number from $1$ to $4$ , and four women are each given a unique number from $1$ to $4$ . How many ways are there to arrange the men and women in a circle such that no two men are next to each other, no two women are next to each other, and no two people with the same number are next to each other? Note that two configurations are considered to be the same if one can be rotated to obtain the other one.
|
12
|
aops_forum
| 0.0625
|
In a castle, there are 9 identical square rooms forming a $3 \times 3$ grid. Each of these rooms is occupied by one of 9 people: either liars or knights (liars always lie, knights always tell the truth). Each of these 9 people said: "At least one of my neighboring rooms has a liar." Rooms are considered neighbors if they share a wall. What is the maximum number of knights that could be among these 9 people?
|
6
|
olympiads
| 0.109375
|
One day in a room there were several inhabitants of an island where only truth-tellers and liars live. Three of them made the following statements:
- There are no more than three of us here. We are all liars.
- There are no more than four of us here. Not all of us are liars.
- There are five of us here. Three of us are liars.
How many people are in the room and how many of them are liars?
|
4 \text{ people}, 2 \text{ liars}
|
olympiads
| 0.078125
|
After the World Hockey Championship, three journalists wrote articles about the German team - each for their own newspaper.
- The first wrote: "The German team scored more than 10 but less than 17 goals throughout the championship."
- The second wrote: "The German team scored more than 11 but less than 18 goals throughout the championship."
- The third wrote: "The German team scored an odd number of goals throughout the championship."
It turned out that only two of the journalists were correct. How many goals could the German team have scored in the championship? List all possible options.
|
11, 12, 14, 16, 17
|
olympiads
| 0.078125
|
Suppose you are one of the inhabitants of the island of knights and liars. You love a girl and want to marry her. However, your chosen one has strange preferences: for some unknown reason, she does not want to marry a knight and prefers to marry only a liar. Moreover, she wants a rich liar (for convenience, let's assume that all liars on the island are either rich or poor). Suppose you are a rich liar. You are allowed to say only one phrase to your chosen one. How can you convince your beloved with just one phrase that you are a rich liar?
|
I am a poor liar
|
olympiads
| 0.125
|
In a chess tournament, each pair of players plays exactly one game. The winner of each game receives 2 points, the loser receives 0 points, and in case of a draw, both players receive 1 point each. Four scorers have recorded the total score of the tournament, but due to negligence, the scores recorded by each are different: 1979, 1980, 1984, and 1985. After verification, it is found that one of the scorers has recorded the correct total score. How many players participated in the tournament?
|
45
|
olympiads
| 0.125
|
In an arithmetic sequence $\left\{a_{n}\right\}$, if $a_{3}+a_{4}+a_{10}+a_{11}=2002$, find the value of $a_{1}+a_{5}+a_{7}+a_{9}+a_{13}$.
|
2502.5
|
olympiads
| 0.234375
|
Calculate the limit of the function:
$$
\lim _{x \rightarrow 0} \frac{1-\cos ^{3} x}{4x^{2}}
$$
|
\frac{3}{8}
|
olympiads
| 0.203125
|
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