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Calculate
$$
\left|\begin{array}{rrr}
2 & -1 & 1 \\
3 & 2 & 2 \\
1 & -2 & 1
\end{array}\right|
$$
| 5 | 0.375 |
A researcher receives a container for analysis holding approximately 150 samples of oil. Each sample has specific characteristics regarding sulfur content - either low-sulfur or high-sulfur - and density - either light or heavy. The relative frequency (statistical probability) that a randomly chosen sample is heavy oil is $\frac{2}{11}$. The relative frequency that a randomly chosen sample is light oil with low sulfur content is $\frac{7}{13}$. How many samples of high-sulfur oil are there in the container if there are no low-sulfur samples among the heavy oil samples? | 66 | 0.25 |
Consider the parabola consisting of the points \((x, y)\) in the real plane satisfying
\[
(y + x) = (y - x)^2 + 3(y - x) + 3.
\]
Find the minimum possible value of \(y\). | -\frac{1}{2} | 0.875 |
Let \( a, b, c > 0 \) such that \( a + b + c = 1 \). Show that
\[ \left(a+\frac{1}{a}\right)^{2}+\left(b+\frac{1}{b}\right)^{2}+\left(c+\frac{1}{c}\right)^{2} \geq \frac{100}{3} \] | \frac{100}{3} | 0.875 |
What is the greatest possible value of the ratio of a three-digit number to the sum of its digits? | 100 | 0.75 |
Fill integers $1, 2, 3, \cdots, 2016^{2}$ into a $2016 \times 2016$ grid, with each cell containing one number, and all numbers being different. Connect the centers of any two cells with a vector, with the direction from the smaller number to the bigger number. If the sum of numbers in each row and each column of the grid is equal, find the sum of all such vectors. | 0 | 0.5 |
In a company, employees have a combined monthly salary of $10,000. A kind manager proposes to triple the salaries of those earning up to $500, and to increase the salaries of others by $1,000, resulting in a total salary of $24,000. A strict manager proposes to reduce the salaries of those earning more than $500 to $500, while leaving others' salaries unchanged. What will the total salary be in this case? | 7000 | 0.5 |
A natural number is considered "cool" when each of its digits is greater than any of the other digits to its left. For example, 3479 is cool, while 2231 is not. How many cool numbers are there between 3000 and 8000? | 35 | 0.625 |
Schoolboy Alexey told his parents that he is already an adult and can manage his finances independently. His mother suggested using a duplicate bank card from her account. For participation in a charitable Christmas program, Alexey wants to buy 40 "Joy" chocolate bars and donate them to an orphanage. However, the bank, where Alexey's parents are clients, has implemented a new system to protect against unauthorized card payments. The protection system analyzes the root mean square (RMS) value of expenses for the last 3 purchases (S) using the formula \(S=\sqrt{\frac{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}{3}}\), where \(x_{1}, x_{2}\), and \(x_{3}\) are the costs of the previous purchases, and compares the value of \(S\) with the cost of the current purchase. If the cost of the current payment exceeds the value \(S\) by 3 times, the bank blocks the payment and requires additional verification (e.g., a call from mom to the call center). In the last month, payments made on the card were only for cellphone bills in the amount of 300 rubles each. Into how many minimum number of receipts should Alexey split the purchase so that he can buy all 40 "Joy" chocolate bars at a cost of 50 rubles each? | 2 | 0.75 |
I am thinking of a four-digit number, where each digit is different. When I omit the last two digits of this number, I get a prime number. Similarly, I get a prime number if I omit the second and fourth digits, and also if I omit the middle two digits. However, the number I am thinking of is not a prime number itself - it can be divided by three without a remainder. There are several numbers with these properties, but mine is the largest of them. Which number am I thinking of? | 4731 | 0.5 |
Given the real number \( x \), \([x] \) denotes the integer part that does not exceed \( x \). Find the positive integer \( n \) that satisfies:
\[
\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right] = 1994
\] | 312 | 0.375 |
In a regular tetrahedron \(ABCD\), \(AO \perp\) plane \(BCD\) at the foot \(O\). Let \(M\) be a point on segment \(AO\) such that \(\angle BMC = 90^\circ\). Then, find \(\frac{AM}{MO} = \quad\). | 1 | 0.625 |
How many nondecreasing sequences \(a_{1}, a_{2}, \ldots, a_{10}\) are composed entirely of at most three distinct numbers from the set \(\{1,2, \ldots, 9\}\)? For example, the sequences \(1,1,1,2,2,2,3,3,3,3\) and \(2,2,2,2,5,5,5,5,5,5\) are both allowed. | 3357 | 0.25 |
Determine the number of ordered pairs of integers \((m, n)\) for which \(m n \geq 0\) and \(m^{3}+n^{3}+99 m n=33^{3}\). | 35 | 0.75 |
Let \( a, b, c, d, e \) be positive integers whose sum is 2018. Let \( M = \max (a+b, b+c, c+d, d+e) \). Find the smallest possible value of \( M \). | 673 | 0.5 |
Find the minimum value of the function \( f(x, y) = \frac{2015(x+y)}{\sqrt{2015 x^{2} + 2015 y^{2}}} \) and specify all pairs \((x, y)\) where it is attained. | -\sqrt{4030} | 0.875 |
In $\triangle ABC$, \(AB = 5\), \(AC = 4\), and \(\overrightarrow{AB} \cdot \overrightarrow{AC} = 12\). Let \(P\) be a point on the plane of \(\triangle ABC\). Find the minimum value of \(\overrightarrow{PA} \cdot (\overrightarrow{PB} + \overrightarrow{PC})\). | -\frac{65}{8} | 0.75 |
Calculate the area of one petal of the curve $\rho = \sin^2 \varphi$. | \frac{3\pi}{16} | 0.25 |
An arbitrary point \( E \) inside the square \( ABCD \) with side length 1 is connected by line segments to its vertices. Points \( P, Q, F, \) and \( T \) are the points of intersection of the medians of triangles \( BCE, CDE, DAE, \) and \( ABE \) respectively. Find the area of the quadrilateral \( PQFT \). | \frac{2}{9} | 0.875 |
In triangle $A B C$, the sides are given as: $A B=4, A C=3, B C=\sqrt{37}$. Point $P$ is the midpoint of side $A B$, and point $Q$ is on side $A C$ at a distance of 1 from point $C$. Find $P Q$. | 2\sqrt{3} | 0.75 |
A pedestrian departed from point \( A \) to point \( B \). After walking 8 km, a second pedestrian left point \( A \) following the first pedestrian. When the second pedestrian had walked 15 km, the first pedestrian was halfway to point \( B \), and both pedestrians arrived at point \( B \) simultaneously. What is the distance between points \( A \) and \( B \)? | 40 | 0.625 |
Given \( f_{1}(x)=|1-2 x| \) for \( x \in [0,1] \) and \( f_{n}(x)=f_{1}(f_{n-1}(x)) \), determine the number of solutions to the equation \( f_{2005}(x)=\frac{1}{2} x \). | 2^{2005} | 0.625 |
Find the smallest natural number that is greater than the sum of its digits by 1755. | 1770 | 0.875 |
Find all pairs of two-digit natural numbers whose arithmetic mean is $25/24$ times greater than their geometric mean. In the answer, indicate the largest of the arithmetic means for all such pairs. | 75 | 0.5 |
Natural numbers \(a\) and \(b\) are such that \(5 \times \text{LCM}(a, b) + 2 \times \text{GCD}(a, b) = 120\). Find the greatest possible value of \(a\). | 20 | 0.625 |
In how many ways can we place two bishops of the same color on a chessboard such that they are on different rows, columns, and squares of distinct colors? | 768 | 0.25 |
Given the cryptarithm: REKA + KARE = ABVAD. Each identical letter corresponds to the same digit, and different letters correspond to different digits. Find the value of the letter B. | 2 | 0.375 |
Does there exist a two-variable polynomial \(P(x, y)\) with real number coefficients such that \(P(x, y)\) is positive exactly when \(x\) and \(y\) are both positive? | \text{No} | 0.625 |
In the regular tetrahedron \(ABCD\), the midpoint \(M\) is taken on the altitude \(AH\), connecting \(BM\) and \(CM\), then \(\angle BMC =\) | 90^{\circ} | 0.875 |
How many triangles exist such that the lengths of the sides are integers not greater than 10? | 125 | 0.125 |
The sequence \(a_1, a_2, a_3, \ldots\) of positive integers is determined by its first two members and the rule \( a_{n+2} = \frac{a_{n+1} + a_n}{\text{gcd}(a_n, a_{n+1})} \). For which values of \(a_1\) and \(a_2\) is this sequence bounded? | a_1 = a_2 = 2 | 0.875 |
A right-angled triangle has sides of integer length. One of its sides has length 20. List all the different possible hypotenuses of such triangles and find the sum of all the numbers in this list. | 227 | 0.375 |
Choose 4 different numbers from $1, 2, 3, 4, 5$ and fill them into the 4 squares in the expression $\square+\square > \square+\square$. There are $\qquad$ different ways to do this to make the expression true. (Hint: $1+5>2+3$ and $5+1>2+3$ are considered different ways.) | 48 | 0.5 |
Find all functions \( f: \mathbf{R} \rightarrow \mathbf{R} \) such that for all \( x, y \in \mathbf{R} \), the following holds:
\[ f(1 + xy) - f(x + y) = f(x) f(y), \]
and \( f(-1) \neq 0 \). | f(x) = x - 1 | 0.5 |
Find all integer values that the expression
$$
\frac{p q + p^{p} + q^{q}}{p + q}
$$
can take, where \( p \) and \( q \) are prime numbers. | 3 | 0.625 |
On the ground, there are three points: \( A \), \( B \), and \( C \). A frog is at point \( P \) on the ground, which is 0.27 meters away from point \( C \). The frog's first jump is a "symmetric jump" from \( P \) to the symmetric point \( P_{1} \) with respect to point \( A \). The second jump is a symmetric jump from \( P_{1} \) with respect to point \( B \) to reach \( P_{2} \). The third jump is a symmetric jump from \( P_{2} \) with respect to point \( C \) to reach \( P_{3} \). The fourth jump is a symmetric jump from \( P_{3} \) with respect to point \( A \) to reach \( P_{4} \), and so on, jumping in this way continuously. After the 2009th symmetric jump, the frog reaches \( P_{2009} \). What is the distance between \( P_{2009} \) and the starting point \( P \) in centimeters? | 54 | 0.875 |
Calculator ACH-2016 can perform two operations: taking the cube root and taking the tangent. Initially, the number \(2^{-243}\) was entered into the calculator. What is the minimum number of operations required to obtain a number greater than 1? | 7 | 0.5 |
Solve the equation
\[ x^3 = 3y^3 + 9z^3 \]
in non-negative integers \(x\), \(y\), and \(z\). | (0, 0, 0) | 0.875 |
On a line, we have four points $A, B, C,$ and $D$ arranged in that order such that $AB = CD$. E is a point outside the line such that $CE = DE$. Show that $\angle CED = 2 \angle AEB$ if and only if $AC = EC$. | AC = EC | 0.875 |
A cube with a side length of 2 is inscribed in a sphere. A second cube, with faces parallel to the first, is inscribed between the sphere and one face of the first cube. What is the length of a side of the smaller cube? | \frac{2}{3} | 0.25 |
Three workers are digging a hole. They take turns working, with each one working as long as it takes the other two to dig half the hole. Working in this way, they finished digging the hole. How many times faster would they have finished the job if they had worked simultaneously? | 3 | 0.125 |
A real number \( x \) is chosen uniformly at random from the interval \( (0, 10) \). Compute the probability that \( \sqrt{x}, \sqrt{x+7} \), and \( \sqrt{10-x} \) are the side lengths of a non-degenerate triangle. | \frac{22}{25} | 0.625 |
Given the sequence of positive integers \(\left\{a_{n}\right\}\) defined by \(a_{0}=m\) and \(a_{n+1}=a_{n}^{5}+487\) for \(n \geqslant 0\), find the value of \(m\) such that the number of perfect squares in the sequence \(\left\{a_{n}\right\}\) is maximized. | 9 | 0.75 |
Find the smallest positive integer \( n \) such that a cube with side length \( n \) can be divided into 1996 smaller cubes, each with side length a positive integer. | 13 | 0.375 |
In the interval \([0, \pi]\), the number of solutions to the trigonometric equation \(\cos 7x = \cos 5x\) is $\qquad$. | 7 | 0.875 |
$A A_1$ is the altitude of the acute-angled triangle $ABC$, $H$ is the orthocenter, and $O$ is the circumcenter of the triangle $ABC$. Find $OH$, given that $AH=3$, $A_1H=2$, and the radius of the circumcircle is 4. | 2 | 0.375 |
In triangle \(ABC\), a median \(AM\) is drawn. Circle \(\alpha\) passes through point \(A\), touches line \(BC\) at point \(M\), and intersects sides \(AB\) and \(AC\) at points \(D\) and \(E\) respectively. On arc \(AD\) that does not contain point \(E\), a point \(F\) is chosen such that \(\angle BFE = 72^\circ\). It is found that \(\angle DEF = \angle ABC\). Find \(\angle CME\). | 36^\circ | 0.625 |
The numbers \(1, 2, \ldots, 2016\) are divided into pairs such that the product of the numbers in each pair does not exceed a certain natural number \(N\). What is the smallest \(N\) for which this is possible? | 1017072 | 0.5 |
Let the function \( f(x) \) be defined on \( \mathbb{R} \), and for any \( x \), the condition \( f(x+2) + f(x) = x \) holds. It is also known that \( f(x) = x^3 \) on the interval \( (-2, 0] \). Find \( f(2012) \). | 1006 | 0.5 |
In the figure, the area of triangle $\triangle \mathrm{ABC}$ is 60. Points $\mathrm{E}$ and $\mathrm{F}$ are on $\mathrm{AB}$ and $\mathrm{AC}$, respectively, satisfying $\mathrm{AB}=3 \mathrm{AE}$ and $\mathrm{AC}=3 \mathrm{AF}$. Point $D$ is a moving point on segment $BC$. Let the area of $\triangle FBD$ be $S_{1}$, and the area of $\triangle EDC$ be $S_{2}$. Find the maximum value of $S_{1} \times S_{2}$. | 400 | 0.75 |
Scientists found a fragment of an ancient mechanics manuscript. It was a piece of a book where the first page was numbered 435, and the last page was numbered with the same digits, but in some different order. How many sheets did this fragment contain? | 50 | 0.5 |
As shown in the figure, the right triangle \( \triangle ABC \) has all three vertices on the given parabola \( x^{2}=2py \) (where \( p > 0 \)), and the hypotenuse \( AB \) is parallel to the \( x \)-axis. Find the height \( |CD| \) from the vertex \( C \) to the hypotenuse. | 2p | 0.75 |
The sequence of functions is defined by the formulas
\[ f_{0}(x)=3 \sin x, \quad f_{n+1}(x)=\frac{9}{3-f_{n}(x)} \]
for any integer \( n \geq 0 \). Find \( f_{2023}\left(\frac{\pi}{6}\right) \). | 6 | 0.75 |
Given \( x, y, z \in \mathbf{Z} \) such that \(\left\{\begin{array}{l}x+y+z=3 \\ x^{3}+y^{3}+z^{3}=3\end{array}\right.\), find the set of all possible values of \( x^{2}+y^{2}+z^{2} \). | \{3, 57\} | 0.75 |
Gavrila found out that the front tires of a car last for 24,000 km, and the rear tires last for 36,000 km. Therefore, he decided to swap them at some point to maximize the total distance the car can travel. Find this maximum possible distance (in km). | 28800 | 0.875 |
If \( P \) is an arbitrary point in the interior of the equilateral triangle \( ABC \), find the probability that the area of \( \triangle ABP \) is greater than each of the areas of \( \triangle ACP \) and \( \triangle BCP \). | \frac{1}{3} | 0.75 |
The numbers from 1 to 8 are placed at the vertices of a cube such that the sum of the numbers at any three vertices lying on the same face is at least 10. What is the minimum possible sum of the numbers at the vertices of one face? | 16 | 0.375 |
On the segment $OA$ of length $L$ on the number line $Ox$, two points, $B(x)$ and $C(y)$, are randomly placed. Find the probability that from the three resulting segments a triangle can be formed. | \frac{1}{4} | 0.75 |
A math competition problem: The probabilities that A, B, and C solve the problem independently are $\frac{1}{a}$, $\frac{1}{b}$, and $\frac{1}{c}$ respectively, where $a$, $b$, and $c$ are all single-digit numbers. If A, B, and C attempt the problem independently and the probability that exactly one of them solves the problem is $\frac{7}{15}$, then the probability that none of them solves the problem is $\qquad$. | \frac{4}{15} | 0.25 |
Maria received an enormous chocolate bar as an Easter gift. She decides to divide it into pieces to eat it gradually. On the first day, she divides it into 10 pieces and eats only one of them. On the second day, she divides one of the pieces left from the previous day into 10 pieces and eats only one of them. On the third day, she does the same, i.e., divides one of the pieces left from the previous day into 10 others and eats only one of them. She continues repeating this procedure until the next year's Easter.
a) How many pieces will she have at the end of the third day?
b) Is it possible for her to have exactly 2014 pieces on any given day? | \text{No} | 0.5 |
Let \(ABC\) be a triangle with integral side lengths such that \(\angle A = 3 \angle B\). Find the minimum value of its perimeter. | 21 | 0.125 |
In a certain logarithm system, $\log 450$ is 1.1639 more than $\log 40$. What is the base of the system and what are the logarithms of the two numbers in this system? | 8 | 0.25 |
Katya correctly solves a problem with a probability of $4 / 5$, and the magic pen solves a problem correctly without Katya's help with a probability of $1 / 2$. In a test containing 20 problems, solving 13 correctly is enough to get a "good" grade. How many problems does Katya need to solve on her own and how many should she leave to the magic pen to ensure that the expected number of correct answers is at least 13? | 10 | 0.625 |
Several points, including points \( A \) and \( B \), are marked on a line. We consider all possible segments with endpoints at the marked points. Vasya counted that point \( A \) is inside 40 of these segments, and point \( B \) is inside 42 segments. How many points were marked? (The endpoints of a segment are not considered its internal points.) | 14 | 0.75 |
The intervals on which the function \( y = \left\{
\begin{array}{cc}
2x + 3, & x \leq 0, \\
x + 3, & 0 < x \leq 1, \\
-x + 5, & x > 1
\end{array}
\right. \) is monotonically increasing. | (-\infty, 1] | 0.875 |
The sequence $\left\{a_{n}\right\}$ satisfies $a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+a_{n}, n \in \mathbf{N}^{*}, b_{n}=\frac{1}{1+a_{n}}$.
Given:
$$
S_{n}=b_{1}+b_{2}+\cdots+b_{n}, P_{n}=b_{1} b_{2} \cdots b_{n},
$$
find the value of $2 P_{n}+S_{n}$. | 2 | 0.125 |
The sum of two natural numbers is 2013. If the last two digits of one of them are crossed out, 1 is added to the resulting number, and the result is then multiplied by 5, you get the other number. Find these numbers. Enter the largest of them in the provided field. | 1913 | 0.75 |
For all real numbers \( x \) and \( y \) satisfying \( |x| + |y| \leq 1 \), the inequality \(\left|2x - 3y + \frac{3}{2}\right| + |y - 1| + |2y - x - 3| \leq a\) always holds. What is the minimum value of the real number \( a \)? | \frac{23}{2} | 0.5 |
Triangle \(ABC\) has side lengths \(AB=231\), \(BC=160\), and \(AC=281\). Point \(D\) is constructed on the opposite side of line \(AC\) as point \(B\) such that \(AD=178\) and \(CD=153\). Compute the distance from \(B\) to the midpoint of segment \(AD\). | 208 | 0.75 |
A scatterbrained scientist in his laboratory has developed a unicellular organism, which has a probability of 0.6 of dividing into two identical organisms and a probability of 0.4 of dying without leaving any offspring. Find the probability that after some time the scatterbrained scientist will have no such organisms left. | \frac{2}{3} | 0.875 |
In the rectangular prism \(A B C D - A_{1} B_{1} C_{1} D_{1}\), \(A B = A A_{1} = 2\), \(A D = 2 \sqrt{3}\). Point \(M\) lies within plane \(B A_{1} C_{1}\). Find the minimum value of \(\overrightarrow{M A} \cdot \overrightarrow{M C}\). | -\frac{16}{7} | 0.875 |
We consider 2021 lines in the plane, no two of which are parallel and no three of which are concurrent. Let E be the set of their intersection points. We want to assign a color to each point in E such that any two points on the same line, whose connecting segment contains no other point of E, have different colors. What is the minimum number of colors needed to achieve such a coloring? | 3 | 0.25 |
Find the angle $\Theta$ between the gradients of the functions
$$
u=\sqrt{x^{2}+y^{2}} \text{ and } v=x+y+2\sqrt{xy}
$$
at the point $M_{0}(1, I)$. | 0 | 0.875 |
At the points of intersection of the graph of the function \( y = \frac{20x^{2} - 16x + 1}{5x - 2} \) with the \( O_x \)-axis, tangents to this graph were drawn. Find the angles of inclination of these lines to the \( O_x \)-axis. | \arctan(8) | 0.25 |
In triangle \( \triangle ABC \), the angle bisectors of \( \angle C \) and \( \angle A \) intersect at point \( O \). Given that \( AC + AO = BC \) and \( \angle B = 25^\circ \), find the measure of \( \angle ACB \). | 105^\circ | 0.25 |
Two circles touch externally. A tangent to the first circle passes through the center of the second circle. The distance from the point of tangency to the center of the second circle is three times the radius of the second circle. By what factor is the circumference of the first circle greater than the circumference of the second circle? | 4 | 0.875 |
Find the smallest natural number \( n \) such that the sum of the digits of each of the numbers \( n \) and \( n+1 \) is divisible by 17. | 8899 | 0.25 |
Viewers rate a movie with an integer score from 0 to 10. At any given moment, the rating of the movie is computed as the sum of all scores divided by their quantity. At a certain moment \( T \), the rating was an integer, and then with each subsequent voter, the rating decreased by one unit. What is the maximum number of viewers who could have voted after moment \( T \)? | 5 | 0.375 |
Let the sets \( M = \{u \mid u = 12m + 8n + 4l, \; m, n, l \in \mathbf{Z}\} \) and \( N = \{u \mid u = 20p + 16q + 12r, \; p, q, r \in \mathbf{Z}\} \). Determine the relationship between \( M \) and \( N \). | M = N | 0.75 |
There is a magical tree with 123 fruits. On the first day, 1 fruit falls from the tree. From the second day onwards, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number of fruits that should fall on a given day, the falling process restarts from 1 fruit on that day and a new cycle begins. Following this pattern, on which day will all the fruits have fallen from the tree? | 17 \text{ days} | 0.25 |
An acute angle of $60^{\circ}$ contains two circles that are externally tangent to each other. The radius of the smaller circle is $r$. Find the radius of the larger circle. | 3r | 0.875 |
In a big box, there is a medium box, and inside the medium box, there is a small box. 100 balls are placed in the boxes. Let $n$ be the number of balls in the big box but not in the medium box, and $m$ be the number of balls in the medium box but not in the small box. Using $m$ and $n$, determine the number of balls in the small box. | 100 - n - m | 0.75 |
Let the following system of equations be satisfied for positive numbers \(x, y, z\):
\[
\left\{
\begin{array}{l}
x^{2}+x y+y^{2}=147 \\
y^{2}+y z+z^{2}=16 \\
z^{2}+x z+x^{2}=163
\end{array}\right.
\]
Find the value of the expression \( x y+y z+x z \). | 56 | 0.5 |
Find the value of the parameter \( p \) for which the equation \( p x^{2} = |x-1| \) has exactly three solutions. | \frac{1}{4} | 0.625 |
Let $\alpha$ and $\beta$ be a pair of complex conjugates. If $|\alpha - \beta| = 2 \sqrt{3}$ and $\frac{\alpha}{\beta^{2}}$ is a real number, then $|\alpha| = \qquad$. | 2 | 0.875 |
In an isosceles triangle \(ABC\) with base \(AC\), point \(D\) divides side \(BC\) in the ratio \(3:1\) starting from vertex \(B\), and point \(E\) is the midpoint of segment \(AD\). It is known that \(BE = \sqrt{7}\) and \(CE = 3\). Find the radius of the circumcircle of triangle \(ABC\). | \frac{8}{3} | 0.625 |
Determine the remainder when
\[
\sum_{i=0}^{2015}\left\lfloor\frac{2^{i}}{25}\right\rfloor
\]
is divided by 100, where \(\lfloor x\rfloor\) denotes the largest integer not greater than \(x\). | 14 | 0.375 |
Let \( a, b \), and \( c \) be real numbers. Consider the system of simultaneous equations in variables \( x \) and \( y \):
\[
\begin{aligned}
a x + b y &= c - 1 \\
(a+5) x + (b+3) y &= c + 1
\end{aligned}
\]
Determine the value(s) of \( c \) in terms of \( a \) such that the system always has a solution for any \( a \) and \( b \). | \frac{2a + 5}{5} | 0.5 |
There are three piles of stones. Each time A moves one stone from one pile to another, A earns a reward from B. The reward is equal to the difference in the number of stones between the pile to which A moves the stone and the pile from which A takes the stone. If this difference is negative, A must pay B that amount instead (if A doesn’t have enough money, A can owe it temporarily). At a certain moment, all the stones are in the piles where they initially started. Determine the maximum possible amount of money A can have by that moment. | 0 | 0.5 |
What is the minimum force required to press a cube with a volume of $10 \mathrm{~cm}^{3}$, floating in water, so that it is completely submerged? The density of the cube's material is $700 \mathrm{~kg/m}^{3}$, and the density of water is $1000 \mathrm{~kg/m}^{3}$. Give the answer in SI units. Take the acceleration due to gravity as $10 \mathrm{~m/s}^{2}$. | 0.03 \, \text{N} | 0.625 |
There are 10 numbers written on a circle, and their sum equals 100. It is known that the sum of any three consecutive numbers is at least 29.
What is the smallest number \( A \) such that in any such set of numbers, each number does not exceed \( A \)? | 13 | 0.5 |
On an island, there are only 50 knights, who always tell the truth, and 15 civilians, who can either tell the truth or lie. A forgetful professor, who came to the island to give a lecture, forgot what color hat he is wearing. What is the minimum number of local inhabitants that the professor needs to ask about the color of his hat in order to be certain about its color? | 31 | 0.875 |
Given that \( f(x) \) is an odd function on \( \mathbf{R} \) and \( f(1) = 1 \), with the condition that for any \( x < 0 \), \( f\left(\frac{x}{x-1}\right) = x f(x) \). Find the value of \( f(1) f\left(\frac{1}{100}\right) + f\left(\frac{1}{2}\right) f\left(\frac{1}{99}\right) + f\left(\frac{1}{3}\right) f\left(\frac{1}{98}\right) + \cdots + f\left(\frac{1}{50}\right) f\left(\frac{1}{51}\right) \). | \frac{2^{98}}{99!} | 0.25 |
Given the linear function \( y = ax + b \) and the hyperbolic function \( y = \frac{k}{x} \) (where \( k > 0 \)) intersect at points \( A \) and \( B \), with \( O \) being the origin. If the triangle \( \triangle OAB \) is an equilateral triangle with an area of \( \frac{2\sqrt{3}}{3} \), find the value of \( k \). | \frac{2}{3} | 0.25 |
The dwarfs went to work, and Snow White is bored. She placed a pile of 36 stones on the table. Every minute, Snow White splits one of the existing piles into two and then adds a new stone to one of them. After a while, she has seven piles, each with an equal number of stones. How many stones are there in each pile? | 6 | 0.875 |
In Vegetable Village it costs 75 pence to buy 2 potatoes, 3 carrots, and 2 leeks at the Outdoor Market, whereas it costs 88 pence to buy 2 potatoes, 3 carrots, and 2 leeks at the Indoor Market. To buy a potato, a carrot, and a leek costs 6 pence more at the Indoor Market than it does at the Outdoor Market. What is the difference, in pence, between the cost of buying a carrot at the Outdoor Market and the cost of buying a carrot at the Indoor Market? | 1 | 0.75 |
A laboratory has flasks of two different sizes (volume $V$ and volume $V/3$) with a total of 100 flasks, with at least 2 flasks of each size. A technician randomly selects two flasks in sequence, filling the first one with a 70% salt solution and the second with a 40% salt solution. The contents of these two flasks are then mixed in one dish, and the percentage of salt in the mixture is determined. What is the minimum number of large flasks $N$ needed to ensure that the event "the percentage of salt in the dish is between 50% and 60% inclusive" occurs less frequently than the event "when two fair coins are flipped, one shows heads and the other shows tails (in any order)"? Justify your answer. | 46 | 0.5 |
Given a prime number \( p \) and a positive integer \( n \) (\( p \geq n \geq 3 \)), the set \( A \) consists of different sequences of length \( n \) formed from the elements of the set \(\{1, 2, \cdots, p\}\) (such that not all elements in a sequence are the same). If, for any two sequences \((x_{1}, x_{2}, \cdots, x_{n})\) and \((y_{1}, y_{2}, \cdots, y_{n})\) in the set \( A \), there exist three different positive integers \( k, l, m \) such that \( x_{k} \neq y_{k}, x_{l} \neq y_{l}, x_{m} \neq y_{m} \), find the maximum number of elements in the set \( A \). | p^{n-2} | 0.25 |
Gustave has 15 steel bars of masses \(1 \mathrm{~kg}, 2 \mathrm{~kg}, 3 \mathrm{~kg}, \ldots, 14 \mathrm{~kg}, 15 \mathrm{~kg}\). He also has 3 bags labelled \(A, B, C\). He places two steel bars in each bag so that the total mass in each bag is equal to \(M \mathrm{~kg}\). How many different values of \(M\) are possible? | 19 | 0.625 |
Let $s$ be the set of all rational numbers $r$ that satisfy the following conditions: $0<r<1$, and $r$ can be represented as a repeating decimal of the form $0.abcabcabc\cdots = 0.\dot{a}b\dot{c}$, where the digits $a, b, c$ do not necessarily have to be distinct. How many different numerators are there when the elements of $s$ are written as reduced fractions? | 660 | 0.25 |
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