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In a trapezoid, the lengths of the bases are 5 and 15, and the lengths of the diagonals are 12 and 16. Find the area of the trapezoid. | 96 | 0.625 |
Given two points on a plane and a line parallel to the line segment connecting the two points, along with the angle $\alpha$. Construct the segment on the parallel line that appears at an angle $\alpha$ from both points! | CD | 0.25 |
Let \( R_{n}=\frac{1}{2}\left(a^{n}+b^{n}\right) \) where \( a=3+2 \sqrt{2} \), \( b=3-2 \sqrt{2} \), and \( n=1,2, \cdots \). What is the units digit of \( R_{12445} \)? | 3 | 0.75 |
Find all values of \( a \) for which the equation \( x^{2} + 2ax = 8a \) has two distinct integer roots. Record the product of all such \( a \), rounding to the nearest hundredth if necessary. | 506.25 | 0.25 |
A battery of three guns fired a volley, and two shells hit the target. Find the probability that the first gun hit the target, given that the probabilities of hitting the target by the first, second, and third guns are $p_{1}=0,4$, $p_{2}=0,3$, and $p_{3}=0,5$, respectively. | \frac{20}{29} | 0.625 |
In a chess-playing club, some of the players take lessons from other players. It is possible (but not necessary) for two players both to take lessons from each other. It so happens that for any three distinct members of the club, $A, B$, and $C$, exactly one of the following three statements is true: $A$ takes lessons from $B$; $B$ takes lessons from $C$; $C$ takes lessons from $A$. What is the largest number of players there can be? | 4 | 0.375 |
Extend each side of the quadrilateral $ABCD$ in the same direction beyond itself, such that the new points are $A_{1}, B_{1}, C_{1}$, and $D_{1}$. Connect these newly formed points to each other. Show that the area of the quadrilateral $A_{1}B_{1}C_{1}D_{1}$ is five times the area of the quadrilateral $ABCD$. | 5 | 0.625 |
A metro network has at least 4 stations on each line, with no more than three transfer stations per line. No transfer station has more than two lines crossing. What is the maximum number of lines such a network can have if it is possible to travel from any station to any other station with no more than two transfers? | 10 | 0.375 |
Given the sequence $\left\{a_{n}\right\}$ that satisfies $a_{1}=1$ and $S_{n+1}=2 S_{n}-\frac{n(n+1)}{2}+1$, where $S_{n}=a_{1}+a_{2}+\cdots+a_{n}$ $(n=1,2, \cdots)$. If $\Delta a_{n}=a_{n+1}-a_{n}$, find the number of elements in the set $S=\left\{n \in \mathbf{N}^{*} \mid \Delta\left(\Delta a_{n}\right) \geqslant-2015\right\}$. | 11 | 0.625 |
Consider equations of the form \( ax^{2} + bx + c = 0 \), where \( a, b, c \) are all single-digit prime numbers. How many of these equations have at least one solution for \( x \) that is an integer? | 7 | 0.875 |
Find the sum of all \( x \) such that \( 0 \leq x \leq 360 \) and \( \cos 12 x^{\circ} = 5 \sin 3 x^{\circ} + 9 \tan^2 x^{\circ} + \cot^2 x^{\circ} \). | 540 | 0.75 |
A and B play a number-changing game on a $5 \times 5$ grid: A starts and both take turns filling empty spaces, with A filling each space with the number 1 and B filling each space with the number 0. After the grid is completely filled, the sum of the numbers in each $3 \times 3$ square is calculated, and the maximum sum among these squares is denoted as $A$. A tries to maximize $A$, while B tries to minimize $A$. What is the maximum value of $A$ that A can achieve?
(The problem is from the 35th IMO selection test) | 6 | 0.25 |
Given \( \theta_{1}, \theta_{2}, \theta_{3}, \theta_{4} \in \mathbf{R}^{+} \) and \( \theta_{1} + \theta_{2} + \theta_{3} + \theta_{4} = \pi \), find the minimum value of \( \left(2 \sin^{2} \theta_{1} + \frac{1}{\sin^{2} \theta_{1}}\right)\left(2 \sin^{2} \theta_{2} + \frac{1}{\sin^{2} \theta_{2}}\right)\left(2 \sin^{2} \theta_{3} + \frac{1}{\sin^{2} \theta_{3}}\right)\left(2 \sin^{2} \theta_{4} + \frac{1}{\sin^{2} \theta_{1}}\right) \). | 81 | 0.75 |
If a number \( N \) is chosen randomly from the set of positive integers, the probability of the units digit of \( N^4 \) being unity is \( \frac{P}{10} \). Find the value of \( P \). | 4 | 0.875 |
The numbers \(2^{2000}\) and \(5^{2000}\) are written consecutively. How many digits are written in total? | 2001 | 0.125 |
A person is walking parallel to railway tracks at a constant speed. A train also passes by this person at a constant speed. The person noticed that depending on the direction of the train's movement, it takes either $t_{1}=1$ minute or $t_{2}=2$ minutes to pass by him. Determine how long it would take for the person to walk from one end of the train to the other. | 4 \text{ minutes} | 0.75 |
Simplify the expression given by \(\frac{a^{-1} - b^{-1}}{a^{-3} + b^{-3}} : \frac{a^{2} b^{2}}{(a+b)^{2} - 3ab} \cdot \left(\frac{a^{2} - b^{2}}{ab}\right)^{-1}\) for \( a = 1 - \sqrt{2} \) and \( b = 1 + \sqrt{2} \). | \frac{1}{4} | 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 |
Find the maximum value of the function \( f(x) = 8 \sin x + 15 \cos x \). | 17 | 0.875 |
A jar contains 97 marbles that are either red, green, or blue. Neil draws two marbles from the jar without replacement and notes that the probability that they would be the same color is \( \frac{5}{12} \). After Neil puts his marbles back, Jerry draws two marbles from the jar with replacement. Compute the probability that the marbles that Jerry draws are the same color. | \frac{41}{97} | 0.875 |
Given the function
$$
f(x)=\left\{\begin{array}{ll}
\ln x, & \text{if } x>1 ; \\
\frac{1}{2} x + \frac{1}{2}, & \text{if } x \leq 1 .
\end{array}\right.
$$
If \( m < n \) and \( f(m) = f(n) \), find the minimum value of \( n - m \). | 3 - 2 \ln 2 | 0.875 |
Initially, the numbers $1, 2, \ldots, n$ were written on the board. It is allowed to erase any two numbers on the board and write the absolute value of their difference instead. What is the smallest number that can be left on the board after $(n-1)$ such operations a) for $n=111$; b) for $n=110$? | 1 | 0.75 |
In triangle \( ABC \), an incircle is inscribed with center \( I \) and points of tangency \( P, Q, R \) with sides \( BC, CA, AB \) respectively. Using only a ruler, construct the point \( K \) where the circle passing through vertices \( B \) and \( C \) is internally tangent to the incircle. | K | 0.875 |
The area of two parallel plane sections of a sphere are $9 \pi$ and $16 \pi$. The distance between the planes is given. What is the surface area of the sphere? | 100\pi | 0.5 |
Given that \(a, b, c\) are non-zero rational numbers and satisfy \(a b^{2}=\frac{c}{a}-b\), then \[\left(\frac{a^{2} b^{2}}{c^{2}}-\frac{2}{c}+\frac{1}{a^{2} b^{2}}+\frac{2 a b}{c^{2}}-\frac{2}{a b c}\right) \div\left(\frac{2}{a b}-\frac{2 a b}{c}\right) \div \frac{101}{c}=\] | -\frac{1}{202} | 0.875 |
Given a hyperbola \( H: x^{2}-y^{2}=1 \) with a point \( M \) in the first quadrant, and a line \( l \) tangent to the hyperbola \( H \) at point \( M \), intersecting the asymptotes of \( H \) at points \( P \) and \( Q \) (where \( P \) is in the first quadrant). If point \( R \) is on the same asymptote as \( Q \), then the minimum value of \( \overrightarrow{R P} \cdot \overrightarrow{R Q} \) is ______. | -\frac{1}{2} | 0.875 |
There are 1000 lights and 1000 switches. Each switch simultaneously controls all lights whose numbers are multiples of the switch's number. Initially, all lights are on. Now, if switches numbered 2, 3, and 5 are pulled, how many lights will remain on? | 499 | 0.375 |
The product of five different integers is 12. What is the largest of the integers? | 3 | 0.625 |
On a table, there are 100 identical-looking coins, of which 85 are counterfeit and 15 are real. You have a miracle tester at your disposal, in which you can place two coins and get one of three results - "both coins are real," "both coins are counterfeit," and "the coins are different." Can you find all the counterfeit coins in 64 such tests? (K. Knop) | Yes | 0.375 |
Let the function
$$
f(x, y) = \sqrt{x^{2}+y^{2}-6y+9} + \sqrt{x^{2}+y^{2}+2\sqrt{3}x+3} + \sqrt{x^{2}+y^{2}-2\sqrt{3}x+3}
$$
Find the minimum value of \( f(x, y) \). | 6 | 0.875 |
Let \( n \) be a natural number. Determine the number of subsets \( A \subset \{1, 2, \ldots, 2n\} \) such that \( x + y = 2n + 1 \) does not hold for any two elements \( x, y \in A \). | 3^n | 0.875 |
Let $a$ be a natural number. Define $M$ as the set of all integers $x$ that satisfy $|x-a| < a + \frac{1}{2}$, and $N$ as the set of all integers $x$ that satisfy $|x| < 2a$. What is the sum of all the integers belonging to $M \cap N$? | a(2a-1) | 0.625 |
Calculate the circulation of the vector field:
a) $\vec{A}=x^{2} y^{2} \vec{i}+\vec{j}+z \vec{k}$ along the circle $x^{2}+y^{2}=a^{2}, z=0$;
b) $\dot{A}=(x-2 z) \dot{i}+(x+3 y+z) \dot{j}+(5 x+y) \vec{k}$ along the perimeter of the triangle $A B C$ with vertices $A(1,0,0), B(0,1,0), C(0,0,1)$. | -3 | 0.875 |
Calculate the sum \(2\left[h\left(\frac{1}{2009}\right)+h\left(\frac{2}{2009}\right)+\ldots+h\left(\frac{2008}{2009}\right)\right]\), where
\[
h(t)=\frac{5}{5+25^{t}}, \quad t \in \mathbb{R}
\] | 2008 | 0.75 |
Through the points \( R \) and \( E \), located on the sides \( AB \) and \( AD \) of the parallelogram \( ABCD \), such that \( AR = \frac{2}{3} AB \) and \( AE = \frac{1}{3} AD \), a line is drawn. Find the ratio of the area of the parallelogram to the area of the resulting triangle. | 9 | 0.875 |
The functions \( f(x) \) and \( g(x) \) are defined for all \( x > 0 \). The function \( f(x) \) is equal to the greater of the numbers \( x \) and \( \frac{1}{x} \), and \( g(x) \) is equal to the smaller of the numbers \( x \) and \( \frac{1}{x} \). Solve the equation \( f(5x) \cdot g(8x) \cdot g(25x)=1 \). In the answer, specify the solution if it is unique, or the sum of the solutions if there are several. Round the answer to the nearest hundredth if necessary. | 0.09 | 0.75 |
Let \( M = \{1, 2, \ldots, 10\} \), and let \( A_1, A_2, \ldots, A_n \) be distinct non-empty subsets of \( M \). For \( i \neq j \), the intersection \( A_i \cap A_j \) contains at most two elements. Find the maximum value of \( n \). | 175 | 0.375 |
Given the function \( f(x) = \log_{2}(x + 1) \), the graph of \( y = f(x) \) is shifted 1 unit to the left and then the y-coordinates of all points on the graph are stretched to twice their original values (the x-coordinates remain unchanged), resulting in the graph of the function \( y = g(x) \). What is the maximum value of the function \( F(x) = f(x) - g(x) \)? | -2 | 0.625 |
For a positive integer \( n \), let \( x_n \) be the real root of the equation \( n x^{3} + 2 x - n = 0 \). Define \( a_n = \left[ (n+1) x_n \right] \) (where \( [x] \) denotes the greatest integer less than or equal to \( x \)) for \( n = 2, 3, \ldots \). Then find \( \frac{1}{1005} \left( a_2 + a_3 + a_4 + \cdots + a_{2011} \right) \). | 2013 | 0.875 |
A circle with radius 1 is circumscribed around triangle \( A P K \). The extension of side \( A P \) beyond vertex \( P \) cuts off a segment \( B K \) from the tangent to the circle through vertex \( K \), where \( B K \) is equal to 7. Find the area of triangle \( A P K \), given that angle \( A B K \) is equal to \( \arctan \frac{2}{7} \). | \frac{28}{53} | 0.125 |
In how many ways can all natural numbers from 1 to 200 be painted in red and blue so that no sum of two different numbers of the same color equals a power of two? | 256 | 0.25 |
Natural numbers \(a\) and \(b\) are such that \(a^{a}\) is divisible by \(b^{b}\), but \(a\) is not divisible by \(b\). Find the smallest possible value of \(a + b\), given that \(b\) is coprime with 210. | 374 | 0.625 |
A provincial TV station has $n$ ad inserts in one day, broadcasting a total of $m$ ads. The first time, they broadcast one ad and $\frac{1}{8}$ of the remaining $(m-1)$ ads. The second time, they broadcast two ads and $\frac{1}{8}$ of the remaining ads. Following this pattern, during the $n$-th insert, they broadcast the remaining $n$ ads (with $n > 1$). How many times did they broadcast ads that day? Find the total number of ads $m$. | n = 7, m = 49 | 0.125 |
Determine the following number:
\[
\frac{12346 \cdot 24689 \cdot 37033 + 12347 \cdot 37034}{12345^{2}}
\] | 74072 | 0.75 |
For \(x, y, z \geq 1\), find the minimum value of the expression
$$
A = \frac{\sqrt{3 x^{4} + y} + \sqrt{3 y^{4} + z} + \sqrt{3 z^{4} + x} - 3}{x y + y z + z x}
$$ | 1 | 0.875 |
Solve the following system of equations:
$$
\begin{aligned}
& \frac{6}{3x + 4y} + \frac{4}{5x - 4z} = \frac{7}{12} \\
& \frac{9}{4y + 3z} - \frac{4}{3x + 4y} = \frac{1}{3} \\
& \frac{2}{5x - 4z} + \frac{6}{4y + 3z} = \frac{1}{2}
\end{aligned}
$$ | x = 4, y = 3, z = 2 | 0.125 |
A sequence of integers \( a_1, a_2, a_3, \ldots \) is defined by
\[
\begin{array}{c}
a_1 = k, \\
a_{n+1} = a_n + 8n \text{ for all integers } n \geq 1.
\end{array}
\]
Find all values of \( k \) such that every term in the sequence is a square. | 1 | 0.75 |
Given that the integer part of the real number \(\frac{2+\sqrt{2}}{2-\sqrt{2}}\) is \(a\), and the decimal part is \(1 - b\), find the value of \(\frac{(b-1) \cdot (5-b)}{\sqrt{a^{2}-3^{2}}}\). | -1 | 0.875 |
How many natural numbers \( \mathrm{N} \) greater than 300 exist such that exactly two out of the numbers \( 4 \mathrm{~N}, \mathrm{~N} - 300, \mathrm{N} + 45, 2\mathrm{N} \) are four-digit numbers? | 5410 | 0.5 |
Calculate the double integral
$$
\iint_{D} \frac{x}{y^{5}} \, dx \, dy
$$
where the region $D$ is defined by the inequalities
$$
1 \leq \frac{x^{2}}{16}+y^{2} \leq 3, \quad y \geq \frac{x}{4}, \quad x \geq 0
$$ | 4 | 0.625 |
For a natural number \( N \), if at least seven out of the nine natural numbers from 1 to 9 are factors of \( N \), \( N \) is called a "seven-star number." What is the smallest "seven-star number" greater than 2000? | 2016 | 0.625 |
In triangle \( \triangle ABC \), it is known that \(AB = 2\), \(BC = 4\), \(BD = \sqrt{6}\) is the angle bisector of \(\angle B\). Find the length of the median \(BE\) on side \(AC\). | \frac{\sqrt{31}}{2} | 0.75 |
Let \( ABC \) be an isosceles triangle with \( AB = AC \). Let \( D \) and \( E \) be the midpoints of segments \( AB \) and \( AC \), respectively. Suppose that there exists a point \( F \) on ray \( \overrightarrow{DE} \) outside of \( ABC \) such that triangle \( BFA \) is similar to triangle \( ABC \). Compute \( \frac{AB}{BC} \). | \sqrt{2} | 0.5 |
Given the complex numbers \( z_{1}, z_{2}, z_{3} \) satisfying:
\[
\begin{array}{l}
\left|z_{1}\right| \leq 1, \left|z_{2}\right| \leq 2, \\
\left|2z_{3} - z_{1} - z_{2}\right| \leq \left|z_{1} - z_{2}\right|.
\end{array}
\]
What is the maximum value of \( \left|z_{3}\right| \)? | \sqrt{5} | 0.625 |
Count the number of functions \( f: \mathbb{Z} \rightarrow \{\text{'green','blue'}\} \) such that \( f(x) = f(x+22) \) for all integers \( x \) and there does not exist an integer \( y \) with \( f(y) = f(y+2) = \text{'green'} \). | 39601 | 0.125 |
What is the measure, in degrees, of the smallest positive angle \( x \) for which \( 4^{\sin ^{2} x} \cdot 2^{\cos ^{2} x} = 2 \sqrt[4]{8} \)? | 60^\circ | 0.875 |
How many different ways (up to rotation) are there of labeling the faces of a cube with the numbers 1, 2, ..., 6? | 30 | 0.875 |
Can the number, formed by writing out the integers from 1 to \( n \) (where \( n > 1 \)) in a row, be the same when read from left to right and from right to left? | \text{No} | 0.625 |
Given that point \( P \) is inside \( \triangle ABC \), and satisfies
$$
\overrightarrow{AP}=\frac{1}{3} \overrightarrow{AB}+\frac{1}{4} \overrightarrow{AC} ,
$$
and let the areas of \( \triangle PBC \), \( \triangle PCA \), and \( \triangle PAB \) be \( S_{1}, S_{2}, S_{3} \) respectively. Determine the ratio \( S_{1}: S_{2}: S_{3} \). | 5: 4: 3 | 0.875 |
Find the number of triples of natural numbers \(a, b,\) and \(c\) not exceeding 2017 such that the polynomial \(x^{11} + a x^7 + b x^3 + c\) has a rational root. | 2031120 | 0.375 |
In a paper, a $4 \times 6$ grid was drawn, and then the diagonal from $A$ to $B$ was traced.
Observe that the diagonal $AB$ intersects the grid at 9 points.
If the grid were of size $12 \times 17$, at how many points would the diagonal $AB$ intersect the grid? | 29 | 0.75 |
Given that the positive rational numbers \(a\) and \(b\) satisfy:
\[ a+b=a^{2}+b^{2}=s \]
If \(s\) is not an integer, then denote:
\[ s=\frac{m}{n} \text{ where } (m, n) \text{ are positive integers with } \gcd(m, n)=1. \]
Let \(p\) be the smallest prime factor of \(n\). Find the smallest possible value of \(p\). | 5 | 0.75 |
Write in a row five numbers such that the sum of each pair of adjacent numbers is negative, and the sum of all the numbers is positive. | 3, -4, 3, -4, 3 | 0.125 |
13 children sat at a round table and agreed that boys would lie to girls and tell the truth to each other, while girls would lie to boys and tell the truth to each other. One of the children said to his right neighbor: "Most of us are boys." That neighbor said to his right neighbor: "Most of us are girls," and he said to his right neighbor: "Most of us are boys," and so on, until the last child said to the first: "Most of us are boys." How many boys were at the table? | 7 | 0.5 |
Find the coefficient of \( x^{6} \) in the expansion of
$$
(x+1)^{6} \cdot \sum_{i=0}^{6} x^{i}
$$ | 64 | 0.375 |
Masha wrote a three-digit number on the board, and Vera wrote the same number next to it but swapped the last two digits. After that, Polina added the resulting numbers and got a four-digit sum, the first three digits of which are 195. What is the last digit of this sum? (The answer needs to be justified.) | 4 | 0.5 |
As shown in Figure 2, in the quadrilateral pyramid $P-ABCD$, $PA \perp$ the base $ABCD$, $BC=CD=2$, $AC=4$, $\angle ACB=\angle ACD=\frac{\pi}{3}$, and $F$ is the midpoint of $PC$. Given that $AF \perp PB$, find:
1. The length of $PA$;
2. The sine value of the dihedral angle $B-AF-D$. | \frac{3\sqrt{7}}{8} | 0.375 |
Anya did not tell Misha how old she is but mentioned that on each of her birthdays, her mother puts as many coins into a piggy bank as Anya's age. Misha estimated that there are no less than 110 and no more than 130 coins in the piggy bank. How old is Anya? | 15 | 0.875 |
A roulette can land on any number from 0 to 2007 with equal probability. The roulette is spun repeatedly. Let $P_{k}$ be the probability that at some point the sum of the numbers that have appeared in all spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008}$? | P_{2007} | 0.375 |
Let the set \( A = \{0, 1, 2, \ldots, 9\} \). The collection \( \{B_1, B_2, \ldots, B_k\} \) is a family of non-empty subsets of \( A \). When \( i \neq j \), the intersection \( B_i \cap B_j \) has at most two elements. Find the maximum value of \( k \). | 175 | 0.75 |
If \( y \) is the greatest value of \(\frac{14}{5+3 \sin \theta}\), find \( y \).
In the figure, \(100 \cos \alpha = k\). Find \( k \).
When \(3 x^{2}+4 x + a\) is divided by \(x+2\), the remainder is 5. Find \(a\).
The solution for \(3 t^{2}-5 t-2<0\) is \(-\frac{1}{3}<t<m\). Find \(m\). | 2 | 0.875 |
Find the sum of the squares of the natural divisors of the number 1800. (For example, the sum of the squares of the natural divisors of the number 4 is \(1^{2} + 2^{2} + 4^{2} = 21\)). | 5035485 | 0.375 |
If the domains of the functions \( f(x) \) and \( g(x) \) are both the set of non-negative real numbers, and for any \( x \geq 0 \), \( f(x) \cdot g(x) = \min \{ f(x), g(x) \} \), given \( f(x) = 3 - x \) and \( g(x) = \sqrt{2x + 5} \), then the maximum value of \( f(x) \cdot g(x) \) is ______ . | 2 \sqrt{3} - 1 | 0.375 |
A locomotive approaching with a speed of $20 \mathrm{~m/s}$ sounded its whistle, and it was heard by a person standing at the railway crossing 4 seconds before the train arrived. How far was the locomotive when it started whistling? (The speed of sound is $340 \mathrm{~m/s}$.) | 85 \ \text{m} | 0.75 |
$[a]$ denotes the greatest integer less than or equal to $a$. Given that $\left(\left[\frac{1}{7}\right]+1\right) \times\left(\left[\frac{2}{7}\right]+1\right) \times\left(\left[\frac{3}{7}\right]+1\right) \times \cdots \times$ $\left(\left[\frac{\mathrm{k}}{7}\right]+1\right)$ leaves a remainder of 7 when divided by 13, find the largest positive integer $k$ not exceeding 48. | 45 | 0.125 |
Find the natural values of \( x \) that satisfy the system of inequalities
\[
\begin{cases}
\log _{\sqrt{2}}(x-1) < 4 \\
\frac{x}{x-3} + \frac{x-5}{x} < \frac{2x}{3-x}
\end{cases}
\] | x = 2 | 0.625 |
From the numbers $1, 2, 3, \cdots, 2003$, select $k$ numbers such that among the selected $k$ numbers, it is always possible to find three numbers that can form the side lengths of an acute-angled triangle. Find the minimum value of $k$ that satisfies the above condition. | 29 | 0.125 |
In triangle \( \triangle ABC \), the sides opposite to angles \( \angle A \), \( \angle B \), and \( \angle C \) are \( a \), \( b \), and \( c \) respectively. Given that:
\[ a^2 + 2(b^2 + c^2) = 2\sqrt{2} \]
find the maximum value of the area of triangle \( \triangle ABC \). | \frac{1}{4} | 0.625 |
Find the smallest natural decimal number \(n\) whose square starts with the digits 19 and ends with the digits 89. | 1383 | 0.375 |
Let $[x]$ denote the greatest integer less than or equal to $x$. When $0 \leqslant x \leqslant 10$, find the number of distinct integers represented by the function $f(x) = [x] + [2x] + [3x] + [4x]$. | 61 | 0.25 |
Seryozha and Lena have several chocolate bars, each weighing no more than 100 grams. No matter how they divide these chocolate bars, the total weight of the chocolate bars for one of them will not exceed 100 grams. What is the maximum possible total weight of all the chocolate bars? | 300 \text{ grams} | 0.125 |
Anton writes down all positive integers divisible by 2 in sequence. Berta writes down all positive integers divisible by 3 in sequence. Clara writes down all positive integers divisible by 4 in sequence. The orderly Dora notes down the numbers written by the others, arranges them in ascending order, and does not write down any number more than once. What is the 2017th number in her list? | 3026 | 0.5 |
The function \( f \) is such that \( f(2x - 3y) - f(x + y) = -2x + 8y \) for all \( x \) and \( y \). Find all possible values of the expression \(\frac{f(5t) - f(t)}{f(4t) - f(3t)}\). | 4 | 0.75 |
Find the largest value of \( x \) such that \(\sqrt[3]{x} + \sqrt[3]{10 - x} = 1\). | 5 + 2\sqrt{13} | 0.75 |
Given the function \( f(x) = x^2 \cos \frac{\pi x}{2} \), and the sequence \(\left\{a_n\right\}\) in which \( a_n = f(n) + f(n+1) \) where \( n \in \mathbf{Z}_{+} \). Find the sum of the first 100 terms of the sequence \(\left\{a_n\right\}\), denoted as \( S_{100} \). | 10200 | 0.375 |
Point \( C \) is located 12 km downstream from point \( B \). A fisherman set out in a rowboat from point \( A \), which is upstream from point \( B \), and reached \( C \) in 4 hours. For the return trip, he took 6 hours. On another occasion, the fisherman used a motorized boat, tripling his relative speed in the water, and traveled from \( A \) to \( B \) in 45 minutes. Determine the speed of the current, assuming it is constant. | 1 \text{ km/h} | 0.625 |
Calculate the limit of the numerical sequence:
$$
\lim _{n \rightarrow x}\left(\sin \sqrt{n^{2}+1} \cdot \operatorname{arctg} \frac{n}{n^{2}+1}\right)
$$ | 0 | 0.875 |
Seven dwarfs stood at the corners of their garden, each at one corner, and stretched a rope around the entire garden. Snow White started from Doc and walked along the rope. First, she walked four meters to the east where she met Prof. From there, she continued two meters north before reaching Grumpy. From Grumpy, she walked west and after two meters met Bashful. Continuing three meters north, she reached Happy. She then walked west and after four meters met Sneezy, from where she had three meters south to Sleepy. Finally, she followed the rope by the shortest path back to Doc, thus walking around the entire garden.
How many square meters is the entire garden?
Hint: Draw the shape of the garden, preferably on graph paper. | 22 | 0.625 |
A fly and $\mathrm{k}$ spiders are moving on a $2019 \times 2019$ grid. On its turn, the fly can move by 1 square and the $k$ spiders can each move by 1 square. What is the minimal $k$ for which the spiders are guaranteed to catch the fly? | 2 | 0.125 |
When the number "POTOP" was added together 99,999 times, the resulting number had the last three digits of 285. What number is represented by the word "POTOP"? (Identical letters represent identical digits.) | 51715 | 0.75 |
In the rectangular parallelepiped $A B C D A_{1} B_{1} C_{1} D_{1}$, the lengths of the edges are given as $A B=54, A D=90, A A_{1}=60$. A point $E$ is marked at the midpoint of edge $A_{1} B_{1}$, and a point $F$ is marked at the midpoint of edge $B_{1} C_{1}$. Find the distance between lines $A E$ and $B F$. | 43.2 | 0.25 |
a) The difference between two consecutive but not adjacent angles is $100^{\circ}$. Determine the angle formed by their bisectors.
b) In the drawing below, $D A$ is the bisector of angle $\angle C A B$. Determine the value of angle $\angle D A E$ knowing that $\angle C A B + \angle E A B = 120^{\circ}$ and $\angle C A B - \angle E A B = 80^{\circ}$. | 30^\circ | 0.125 |
In parallelogram \(ABCD\), points \(A_{1}, A_{2}, A_{3}, A_{4}\) and \(C_{1}, C_{2}, C_{3}, C_{4}\) are respectively the quintisection points of \(AB\) and \(CD\). Points \(B_{1}, B_{2}\) and \(D_{1}, D_{2}\) are respectively the trisection points of \(BC\) and \(DA\). Given that the area of quadrilateral \(A_{4} B_{2} C_{4} D_{2}\) is 1, find the area of parallelogram \(ABCD\).
| \frac{5}{3} | 0.625 |
Solve the equation for all values of the parameter \( a \):
$$
3 x^{2}+2 a x-a^{2}=\ln \frac{x-a}{2 x}
$$ | x = -a | 0.875 |
Among 30 people with different ages, select two groups: the first group with 12 people and the second group with 15 people, such that the oldest person in the first group is younger than the youngest person in the second group. Determine the number of ways to make this selection. | 4060 | 0.375 |
What is the largest integer divisible by all positive integers less than its cube root? | 420 | 0.625 |
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 |
Let \( x, y, z, u, v \in \mathbf{R}_{+} \). The maximum value of
\[
f = \frac{x y + y z + z u + u v}{2 x^{2} + y^{2} + 2 z^{2} + u^{2} + 2 v^{2}}
\]
is $\qquad$ . | \frac{\sqrt{6}}{4} | 0.375 |
A case contains 20 cassette tapes with disco music and 10 with techno music. A DJ randomly picks two tapes one after the other. What is the probability that
1) The first tape contains disco music;
2) The second tape also contains disco music.
Consider two scenarios:
a) The DJ returns the first tape to the case before picking the second tape;
b) The DJ does not return the first tape to the case. | \frac{19}{29} | 0.125 |
Let the function \( f(x) = \cos x \cos (x - \theta) - \frac{1}{2} \cos \theta \), where \( x \in \mathbb{R} \) and \( 0 < \theta < \pi \). It is known that \( f(x) \) attains its maximum value at \( x = \frac{\pi}{3} \).
(1) Find the value of \( \theta \).
(2) Let \( g(x) = 2 f\left(\frac{3}{2} x\right) \). Determine the minimum value of the function \( g(x) \) on the interval \(\left[0, \frac{\pi}{3}\right] \). | -\frac{1}{2} | 0.875 |
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