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stringlengths 18
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|---|---|---|
The sum of the non-negative numbers \(a\), \(b\), and \(c\) is equal to 3. Find the maximum value of the expression \(ab + bc + 2ca\).
|
\frac{9}{2}
| 0.375 |
Given complex numbers \( z_{1}, z_{2}, z_{3} \) such that \( \left|z_{1}\right| \leq 1 \), \( \left|z_{2}\right| \leq 1 \), and \( \left|2 z_{3}-\left(z_{1}+z_{2}\right)\right| \leq \left|z_{1}-z_{2}\right| \). What is the maximum value of \( \left|z_{3}\right| \)?
|
\sqrt{2}
| 0.75 |
What is the maximum number of non-empty subsets that can be chosen from a set of 100 elements such that any two chosen subsets are either disjoint or one contains the other?
|
199
| 0.375 |
Let \( N \) denote the number of subsets of \(\{1,2,3, \ldots, 100\}\) that contain more prime numbers than multiples of 4. Compute the largest integer \( k \) such that \( 2^{k} \) divides \( N \).
|
52
| 0.75 |
Let \(a\), \(b\), and \(c\) be three positive real numbers such that \(abc \geq 1\). Show that:
\[
\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c} \leq 1
\]
Determine the case of equality.
|
a = b = c = 1
| 0.25 |
If \( a + x^2 = 2015 \), \( b + x^2 = 2016 \), \( c + x^2 = 2017 \), and \( abc = 24 \), find the value of \( \frac{a}{bc} + \frac{b}{ac} + \frac{c}{ab} - \frac{1}{a} - \frac{1}{b} - \frac{1}{c} \).
|
\frac{1}{8}
| 0.75 |
Samson writes down the number 123456789 on a piece of paper. He can insert multiplication signs between any two adjacent digits, any number of times at different places, or none at all. By reading the digits between the multiplication signs as individual numbers, he creates an expression made up of the products of these numbers. For example, 1234$\cdot$56$\cdot$789. What is the maximum possible value of the resulting number?
|
123456789
| 0.125 |
Five friends - Kristina, Nadya, Marina, Liza, and Galya - gather in the park every day after buying ice cream from the shop around the corner. One day, they had a conversation.
Kristina: There were five people in front of me.
Marina: I was the first in line!
Liza: There was no one after me.
Nadya: I was standing next to Marina.
Galya: There was only one person after me.
The girls are friends, so they do not lie to each other. How many people were between Kristina and Nadya?
|
3
| 0.375 |
There are 60 people sitting at a large round table; each person is either a knight or a liar. Each person made the statement: "Of the five people sitting consecutively to my right, at least two are liars." How many knights can be sitting at this table?
|
40
| 0.75 |
a) In how many ways can 9 people arrange themselves I) on a bench II) around a circular table?
b) In how many ways can 5 men and 4 women arrange themselves on a bench such that I) no two people of the same gender sit next to each other? II) the men and women sit in separate groups (only 1 man and 1 woman sit next to each other)?
|
5760
| 0.5 |
There are three batches of parts, each containing 20 parts. The number of standard parts in the first, second, and third batches is 20, 15, and 10 respectively. A part is randomly selected from one of these batches and is found to be standard. After returning the part to the batch, another part is randomly selected from the same batch and is also found to be standard. Find the probability that the parts were selected from the third batch.
|
\frac{4}{29}
| 0.875 |
a) The line segments meeting at the vertex form a number of angles, one between each adjacent pair of segments. Suppose we color half the angles white and half the angles black such that no two angles next to each other are the same color. Show that the sum of the black angles is $180^{\circ}$.
b) Show that the number of mountain folds and the number of valley folds meeting at the vertex differ by exactly 2.
|
2
| 0.375 |
Given \( n+1 \) distinct values of \( x \) such that the polynomials
$$
\begin{array}{l}
f(x)=a_{n} x^{n}+a_{n-1} x^{n-1}+\cdots+a_{1} x+a_{0} \\
g(x)=b_{n} x^{n}+b_{n-1} x^{n-1}+\cdots+b_{1} x+b_{0}
\end{array}
$$
have equal values, then \( f(x) = g(x) \).
|
f(x) = g(x)
| 0.875 |
On a circle, 15 white points and 1 black point are marked. Consider all possible (convex) polygons with their vertices at these points.
We will classify them into two types:
- Type 1: those with only white vertices.
- Type 2: those with the black point as one of the vertices.
Are there more polygons of Type 1 or Type 2? How many more are there?
|
105
| 0.875 |
In $\triangle ABC$, $AB > AC$, $\angle BAC = 45^\circ$. Point $E$ is the intersection of the external angle bisector of $\angle BAC$ with the circumcircle of $\triangle ABC$. Point $F$ is on $AB$ such that $EF \perp AB$. Given $AF = 1$ and $BF = 5$, find the area of $\triangle ABC$.
|
6 \sqrt{2}
| 0.25 |
In an acute triangle \( \mathrm{ABC} \), the distance between the centers of the circumcircles of triangles \( C_2HA_1 \) and \( A_1HA_2 \) is equal to \( AC \).
|
AC
| 0.625 |
The side of an equilateral triangle is $a$. Find the radius of the escribed circle.
|
\frac{a\sqrt{3}}{2}
| 0.625 |
In triangle $ABC$, the median $CM$ and the angle bisector $BL$ were drawn. Then, all segments and points except for the points $A(2, 8)$, $M(4, 11)$, and $L(6, 6)$ were erased from the diagram. What are the coordinates of point $C$?
|
(14, 2)
| 0.875 |
Consider all 7-digit numbers formed by all possible permutations of the digits in the number 1234567. How many of these numbers leave a remainder of 5 when divided by 7? The answer is \(6!\).
|
720
| 0.375 |
In triangle ABC, AC=1, AB=2, and O is the point of intersection of the angle bisectors. A segment passing through point O parallel to side BC intersects sides AC and AB at points K and M, respectively. Find the perimeter of triangle AKM.
|
3
| 0.75 |
Determine the largest positive integer $N$ such that there exists a $6 \times N$ table $T$ that satisfies:
(1) Each column is a permutation of $1, 2, \cdots, 6$.
(2) For any two columns $i \neq j$, there exists a row $r (r \in \{1,2, \cdots, 6\})$ such that $t_{ri}=t_{rj}$.
(3) For any two columns $i \neq j$, there exists a row $s (s \in \{1,2, \cdots, 6\})$ such that $t_{si} \neq t_{sj}$.
|
120
| 0.375 |
A store has two types of toys, Big White and Little Yellow, with a total of 60 toys. It is known that the price ratio of Big White to Little Yellow is 6:5 (both prices are in integer yuan). Selling all of them results in a total of 2016 yuan. How many Big Whites are there?
|
36
| 0.75 |
A function \( f \) satisfies the following conditions for all nonnegative integers \( x \) and \( y \):
- \( f(0, x) = f(x, 0) = x \)
- If \( x \geq y \geq 0 \), \( f(x, y) = f(x - y, y) + 1 \)
- If \( y \geq x \geq 0 \), \( f(x, y) = f(x, y - x) + 1 \)
Find the maximum value of \( f \) over \( 0 \leq x, y \leq 100 \).
|
101
| 0.25 |
A positive integer is said to be a "palindrome" if it reads the same from left to right as from right to left. For example, 2002 is a palindrome. Find the sum of all 4-digit palindromes.
|
495000
| 0.375 |
Find the largest interval over which \( f(x) = \sqrt{x - 1} + \sqrt{x + 24 - 10\sqrt{x - 1}} \) is real and constant.
|
[1, 26]
| 0.75 |
Calculate the limit of the function:
$$
\lim _{x \rightarrow 1}\left(1+e^{x}\right)^{\frac{\sin \pi x}{1-x}}
$$
|
(1 + e)^\pi
| 0.5 |
Given quadratic polynomials \( f_{1}(x), f_{2}(x), \ldots, f_{100}(x) \) with identical coefficients for \( x^{2} \) and \( x \), but differing constant terms; each polynomial has two roots. For each polynomial \( f_{i}(x) \), one root is chosen and denoted by \( x_{i} \). What values can the sum \( f_{2}\left(x_{1}\right) + f_{3}\left(x_{2}\right) + \ldots + f_{100}\left(x_{99}\right) + f_{1}\left(x_{100}\right) \) take?
|
0
| 0.625 |
Let $R$ be the set of all real numbers. Find all functions $f: R \rightarrow R$ such that for all $x, y \in R$, the following equation holds:
$$
f\left(x^{2}+f(y)\right)=y+(f(x))^{2}.
$$
|
f(x) = x
| 0.875 |
Find the remainder when \( 30! - 1 \) is divided by 930.
|
29
| 0.625 |
There are 36 criminal gangs operating in Chicago, some of which are hostile to each other. Each gangster belongs to several gangs, and any two gangsters are members of different sets of gangs. It is known that no gangster belongs to two gangs that are hostile to each other. Furthermore, if a gangster does not belong to a particular gang, that gang is hostile to one of the gangs the gangster is a member of. What is the maximum number of gangsters that can be in Chicago?
|
531441
| 0.375 |
An ordered pair of non-negative integers $(m, n)$ is called "simple" if, when performing the addition $m+n$, no carrying is required. The sum $m+n$ is called the sum of the ordered pair $(m, n)$. Find the number of "simple" ordered pairs of non-negative integers whose sum is 1492.
|
300
| 0.25 |
Given that $\alpha, \beta, \gamma$ are all acute angles and $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$, find the minimum value of $\tan \alpha \cdot \tan \beta \cdot \tan \gamma$.
|
2\sqrt{2}
| 0.875 |
An archipelago consists of \( N \geq 7 \) islands. Any two islands are connected by at most one bridge. It is known that there are no more than 5 bridges leading from each island, and among any 7 islands, there are always two islands connected by a bridge. What is the maximum possible value of \( N \)?
|
36
| 0.625 |
Let the lengths of the three sides of a triangle be integers \( l, m, n \) with \( l > m > n \). It is given that \(\left\{\frac{3^{l}}{10^{4}}\right\}= \left\{\frac{3^{m}}{10^{4}}\right\}=\left\{\frac{3^{n}}{10^{4}}\right\}\), where \(\{x\}=x-[x]\) and \([x]\) denotes the greatest integer less than or equal to \(x\). Find the minimum perimeter of such a triangle.
|
3003
| 0.75 |
The graphs of the functions \( y = ax^2 \), \( y = bx \), and \( y = c \) intersect at a point located above the x-axis. Determine how many roots the equation \( ax^2 + bx + c = 0 \) can have.
|
0
| 0.5 |
Given some triangles with side lengths \(a \,\text{cm}, 2 \,\text{cm}\) and \(b \,\text{cm}\), where \(a\) and \(b\) are integers and \(a \leq 2 \leq b\). If there are \(q\) non-congruent classes of triangles satisfying the above conditions, find the value of \(q\).
|
3
| 0.75 |
Solve the equation for real numbers:
$$
\frac{1}{\log _{\frac{1}{2}} x}+\frac{1}{\log _{\frac{2}{3}} x}+\cdots+\frac{1}{\log _{\frac{9}{10}} x}=1
$$
|
\frac{1}{10}
| 0.875 |
Let \(\log_{14} 16\) be equal to \(a\). Express \(\log_{8} 14\) in terms of \(a\).
|
\frac{4}{3a}
| 0.875 |
The edges of \( K_{2017} \) are each labeled with 1, 2, or 3 such that any triangle has a sum of labels of at least 5. Determine the minimum possible average of all \( \binom{2017}{2} \) labels.
|
2 - \frac{1}{2017}
| 0.125 |
Given point \( A \) is a point inside a unit circle centered at \( O \), satisfying \( |\overrightarrow{OA}| = \frac{1}{2} \). Points \( B \) and \( C \) are any two points on the unit circle \( O \). Determine the range of values for \( \overrightarrow{AC} \cdot \overrightarrow{BC} \).
|
[-\frac{1}{8}, 3]
| 0.125 |
There are four different passwords, $A$, $B$, $C$, and $D$, used by an intelligence station. Each week, one of the passwords is used, and each week it is randomly chosen with equal probability from the three passwords not used in the previous week. Given that the password used in the first week is $A$, find the probability that the password used in the seventh week is also $A$ (expressed as a simplified fraction).
|
\frac{61}{243}
| 0.5 |
Given a natural number \( n \geq 5 \), find:
1. The minimum number of distinct values produced by \( a_i + a_j \) for \( \{a_1, a_2, \cdots, a_n\} \) where \( 1 \leq i < j \leq n \).
2. Identify all \( n \)-element sets that achieve this minimum value.
|
2n-3
| 0.625 |
Solve the equation \(\frac{15}{x\left(\sqrt[3]{35-8 x^{3}}\right)}=2x+\sqrt[3]{35-8 x^{3}}\). Write the sum of all obtained solutions as the answer.
|
2.5
| 0.375 |
Show that the sum of the cubes of three consecutive numbers is always divisible by 9.
|
9
| 0.5 |
Let \(a, b, c > 0\) such that \(abc = 1\). Show that
\[ \sum_{cyc} \frac{ab}{ab + a^5 + b^5} \leq 1. \]
|
1
| 0.625 |
Let the set \( A = \left\{\frac{1}{2}, \frac{1}{7}, \frac{1}{11}, \frac{1}{13}, \frac{1}{15}, \frac{1}{32}\right\} \) have non-empty subsets \( A_{1}, A_{2}, \cdots, A_{63} \). Denote the product of all elements in the subset \( A_{i} \) (where \( i=1,2,\cdots,63 \)) as \( p_{i} \) (the product of elements in a single-element subset is the element itself). Then,
\[
p_{1} + p_{2} + \cdots + p_{63} =
\]
|
\frac{79}{65}
| 0.125 |
Katya placed a square with a perimeter of 40 cm next to a square with a perimeter of 100 cm as shown in the figure. What is the perimeter of the resulting figure in centimeters?
|
120 \text{ cm}
| 0.375 |
A parallelepiped $A B C D A_1 B_1 C_1 D_1$ is given. Points $M, L$, and $K$ are taken on the edges $A D, A_1 D_1$, and $B_1 C_1$ respectively, such that $B_1 K = \frac{1}{3} A_1 L$ and $A M = \frac{1}{2} A_1 L$. It is known that $K L = 2$. Find the length of the segment by which the plane $K L M$ intersects the parallelogram $A B C D$.
|
\frac{3}{2}
| 0.625 |
Nine integers from 1 to 5 are written on a board. It is known that seven of them are at least 2, six are greater than 2, three are at least 4, and one is at least 5. Find the sum of all the numbers.
|
26
| 0.875 |
A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?
The perimeter of a shape is the sum of its side lengths.
|
52
| 0.5 |
Show that \( x^2 + y^2 + 1 > x(y + 1) \) for all reals \( x, y \). Find the largest \( k \) such that \( x^2 + y^2 + 1 \geq kx(y + 1) \) for all reals \( x, y \). Find the largest \( k \) such that \( m^2 + n^2 + 1 \geq km(n + 1) \) for all integers \( m, n \).
|
\frac{3}{2}
| 0.75 |
Solve the inequality
$$
8 \cdot \frac{|x+1|-|x-7|}{|2x-3|-|2x-9|} + 3 \cdot \frac{|x+1|+|x-7|}{|2x-3|+|2x-9|} \leq 8
$$
Record the sum of its integer solutions that satisfy the condition $|x| < 120$.
|
6
| 0.375 |
To the eight-digit number 20222023, append one digit to the left and one digit to the right so that the resulting ten-digit number is divisible by 72. Determine all possible solutions.
|
3202220232
| 0.75 |
Determine the value of \(a\) such that in the equation
$$
x^{2}+a x+2 a=0
$$
one root is four times the other root.
|
\frac{25}{2}
| 0.25 |
Let function \( f(x) = 1 - |1 - 2x| \) and \( g(x) = x^2 - 2x + 1 \) for \( x \in [0,1] \), and define
\[
F(x) =
\begin{cases}
f(x) & \text{if } f(x) \geq g(x), \\
g(x) & \text{if } f(x) < g(x).
\end{cases}
\]
Determine the number of real roots of the equation \( F(x) \cdot 2^x = 1 \).
|
3
| 0.75 |
The function \( f(x) \) is defined on the set of real numbers, and satisfies the equations \( f(2+x) = f(2-x) \) and \( f(7+x) = f(7-x) \) for all real numbers \( x \). Let \( x = 0 \) be a root of \( f(x) = 0 \). Denote the number of roots of \( f(x) = 0 \) in the interval \(-1000 \leq x \leq 1000 \) by \( N \). Find the minimum value of \( N \).
|
401
| 0.5 |
Given a geometric sequence $\left\{a_{n}\right\}$ with a common ratio $q \in (1,2)$, and that $a_{n}$ is a positive integer for $1 \leq n \leq 6$, find the minimum value of $a_{6}$.
|
243
| 0.75 |
Calculate the limit of the function:
\[ \lim_{x \to 10} \frac{\lg x - 1}{\sqrt{x - 9} - 1} \]
|
\frac{1}{5 \ln 10}
| 0.875 |
From any \( n \)-digit \((n>1)\) number \( a \), we can obtain a \( 2n \)-digit number \( b \) by writing two copies of \( a \) one after the other. If \( \frac{b}{a^{2}} \) is an integer, find the value of this integer.
|
7
| 0.375 |
In a game, \( N \) people are in a room. Each of them simultaneously writes down an integer between 0 and 100 inclusive. A person wins the game if their number is exactly two-thirds of the average of all the numbers written down. There can be multiple winners or no winners in this game. Let \( m \) be the maximum possible number such that it is possible to win the game by writing down \( m \). Find the smallest possible value of \( N \) for which it is possible to win the game by writing down \( m \) in a room of \( N \) people.
|
34
| 0.625 |
It is given that \(a, b, c\) are three real numbers such that the roots of the equation \(x^{2} + 3x - 1 = 0\) also satisfy the equation \(x^{4} + a x^{2} + b x + c = 0\). Find the value of \(a + b + 4c + 100\).
|
93
| 0.875 |
The parabola with equation \( y = ax^2 + bx + c \) passes through the points \( (4, 0) \), \( \left(\frac{t}{3}, 0\right) \), and \( (0, 60) \). What is the value of \( a \)?
|
3
| 0.125 |
Over the course of three years, Marina did not invest the funds in her Individual Investment Account (IIA) into financial instruments and therefore did not receive any income from it. However, she gained the right to an investment tax deduction for depositing her own money into the IIA.
The tax deduction is provided for the amount of money deposited into the IIA during the tax period, but not more than 400,000 rubles in total per year.
Marina is entitled to receive 13% of the amount deposited into the IIA as a refund of the Personal Income Tax (PIT) that she paid on her income.
The PIT amount deducted from Marina's annual salary is equal to \( 30,000 \text{ rubles} \times 12 \text{ months} \times 0.13 = 46,800 \text{ rubles} \).
The tax deduction for the first year is \( 100,000 \text{ rubles} \times 0.13 = 13,000 \text{ rubles} \). This amount does not exceed the PIT deducted from Marina's annual salary.
The tax deduction for the second year is \( 400,000 \text{ rubles} \times 0.13 = 52,000 \text{ rubles} \). This amount exceeds the PIT deducted from Marina's annual salary. Therefore, the tax deduction will be limited to the PIT paid for the second year, or 46,800 rubles.
The tax deduction for the third year is \( 400,000 \text{ rubles} \times 0.13 = 52,000 \text{ rubles} \). This amount also exceeds the PIT deducted from Marina's annual salary. Therefore, the tax deduction will be limited to the PIT paid for the third year, or 46,800 rubles.
The total amount of tax deduction for 3 years is \( 13,000 + 46,800 + 46,800 = 106,600 \text{ rubles} \).
The return on Marina's transactions over 3 years is \( \frac{106,600}{1,000,000} \times 100 \% = 10.66 \% \).
The annual return on Marina's transactions is \( \frac{10.66 \%}{3} = 3.55 \% \).
|
3.55\%
| 0.625 |
Let \( a \) and \( b \) be positive integers where \( a \) is less than \( b \). Calculate the sum of the irreducible fractions between \( a \) and \( b \) with denominator 7.
|
3(b^2 - a^2)
| 0.5 |
In a certain kingdom, the workforce consists only of a clan of dwarves and a clan of elves. Historically, in this kingdom, dwarves and elves have always worked separately, and no enterprise has ever allowed itself to hire both at the same time. The aggregate labor supply of dwarves is given by the function \( w_{\text{dwarves}}^{S} = 1 + \frac{L}{3} \), and the aggregate labor supply of elves is \( w_{\text{elves}}^{S} = 3 + L \). The inverse function of aggregate labor demand for dwarves is \( w_{\text{dwarves}}^{D} = 10 - \frac{2L}{3} \), and the inverse function of aggregate labor demand for elves is \( w_{\text{elves}}^{D} = 18 - 2L \). Recently, a newly enthroned king became very concerned that the wage rates of his subjects are different, so he issued a law stating that the wages of elves and dwarves must be equal and that workers should not be discriminated against based on their clan affiliation. The king believes that regulatory intervention in wage rates will negatively impact the kingdom's economy overall and mandates that all his subjects behave in a perfectly competitive manner. By how many times will the wage of the group of workers whose wage was lower before the king's intervention increase if the firms in the kingdom are indifferent to hiring elves or dwarves?
|
1.25 \text{ times}
| 0.875 |
There are 10 cards, each with two different numbers from the set {1, 2, 3, 4, 5}, and no two cards have the exact same pair of numbers. These 10 cards are placed in five boxes labeled 1, 2, 3, 4, and 5. A card labeled with numbers i and j can only be placed in box i or box j. A placement is called "good" if and only if the number of cards in box 1 is greater than the number of cards in each of the other boxes. Find the total number of good placements.
|
120
| 0.25 |
Does there exist a natural number that, when divided by the sum of its digits, gives 2014 as both the quotient and the remainder? If there is more than one such number, write their sum as the answer. If no such number exists, write 0 as the answer.
|
0
| 0.5 |
Suppose that for the positive numbers \( x \), \( y \), and \( z \),
\[ x^2 + xy + y^2 = 9, \quad y^2 + yz + z^2 = 16, \quad z^2 + zx + x^2 = 25. \]
Determine the value of \( xy + yz + zx \).
|
8 \sqrt{3}
| 0.125 |
A square is inscribed in an isosceles right triangle such that one vertex of the square is located on the hypotenuse, the opposite vertex coincides with the right angle vertex of the triangle, and the other vertices lie on the legs. Find the side of the square if the leg of the triangle is $a$.
|
\frac{a}{2}
| 0.875 |
Two squares, $ABCD$ and $BEFG$, with side lengths of $8 \text{ cm}$ and $6 \text{ cm}$ respectively, are placed next to each other as shown in the diagram. Line $DE$ intersects $BG$ at point $P$. What is the area of the shaded region $APEG$ in the diagram?
|
18 \text{ cm}^2
| 0.375 |
There are 31 ones written on a board. Each minute, Karlson erases any two numbers and writes their sum on the board, then eats an amount of candies equal to the product of the two erased numbers. What is the maximum number of candies he could eat in 31 minutes?
|
465
| 0.875 |
Given the equation \(x^{2} - 2ax + 64 =0\):
1. If the equation has real roots, find the range of the real number \(a\).
2. If \(a\) is a positive integer and the equation has integer roots, find the maximum value of the positive integer \(a\).
|
17
| 0.875 |
Calculate the lengths of the arcs of the curves given by the equations in the rectangular coordinate system.
$$
y=2+\operatorname{ch} x, 0 \leq x \leq 1
$$
|
\sinh(1)
| 0.625 |
If \( a + b = 2 \) and \( a^{2} + b^{2} = 2 \), what is the value of \( a^{3} + b^{3} \)? And \( a^{4} + b^{4} \)?
|
a^4 + b^4 = 2
| 0.75 |
Given a positive integer \( n \), let \( P(x) \) and \( Q(x) \) be polynomials with real coefficients of degree no greater than \( n \). They satisfy the equation \( x^{n+1} P(x) + (x+1)^{n+1} Q(x) = 1 \). Find \( Q(x) \), and determine the value of \( Q\left(-\frac{1}{2}\right) \).
|
2^n
| 0.625 |
In the coordinate plane, a point is called an integer point if both its x-coordinate and y-coordinate are integers. For any natural number \( n \), the point \( O \) (the origin) is connected to the point \( A_n(n, n+3) \). Let \( f(n) \) denote the number of integer points on the line segment \( OA_n \) excluding the endpoints. Find the value of \( f(1) + f(2) + \cdots + f(1990) \).
|
1326
| 0.5 |
The bases \( AB \) and \( CD \) of the trapezoid \( ABCD \) are 155 and 13 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \( \overrightarrow{AD} \) and \( \overrightarrow{BC} \).
|
2015
| 0.375 |
At 1:00 PM, two identical recreational boats set off in opposite directions from a pier on a river. At the same time, a raft also departed from the pier. An hour later, one of the boats turned around and started moving back. The other boat did the same at 3:00 PM. What is the speed of the current if, at the moment the boats met, the raft had drifted 7.5 km from the pier?
|
2.5 \text{ km/h}
| 0.875 |
In a regular polygon with 67 sides, all segments joining two vertices, including the sides of the polygon, are drawn. We choose $n$ of these segments and assign each one a color from 10 possible colors. Find the minimum value of $n$ that guarantees, regardless of which $n$ segments are chosen and how the colors are assigned, that there will always be a vertex of the polygon that belongs to 7 segments of the same color.
|
2011
| 0.875 |
Find the smallest positive period of the function \( f(x) = \cos(\sqrt{2} x) + \sin\left(\frac{3}{8} \sqrt{2} x\right) \).
|
8 \sqrt{2} \pi
| 0.375 |
Each cell of a \(50 \times 50\) square contains a number equal to the count of \(1 \times 16\) rectangles (both vertical and horizontal) for which this cell is an endpoint. How many cells contain numbers that are greater than or equal to 3?
|
1600
| 0.125 |
As shown in Figure 1.4.23, in the isosceles triangle \( \triangle ABC \), \( AB = AC \) and \( \angle A = 120^\circ \). Point \( D \) is on side \( BC \), and \( BD = 1 \), \( DC = 2 \). Find the length of \( AD \).
|
1
| 0.625 |
Let \( f(x) \) be a function defined on \( \mathbf{R} \). If there exist two distinct real numbers \( x_{1}, x_{2} \in \mathbf{R} \) such that \( f\left(\frac{x_{1}+x_{2}}{2}\right) = \frac{f\left(x_{1}\right) + f\left(x_{2}\right)}{2} \), then the function \( f(x) \) is said to have property \( \mathrm{P} \). Determine which of the following functions does not have property \( \mathrm{P} \):
\[
1. \quad f(x) =
\begin{cases}
\frac{1}{x} & (x \neq 0) \\
0 & (x = 0)
\end{cases}
\]
\[
2. \quad f(x) = x^2
\]
\[
3. \quad f(x) = \left| x^2 - 1 \right|
\]
\[
4. \quad f(x) = x^3
\]
A: 1
B: 2
C: 3
D: 4
|
B
| 0.75 |
Find \( p \) if each of the numbers \( p \), \( 2p + 1 \), and \( 4p + 1 \) is prime.
|
3
| 0.5 |
Let \( k, \alpha \) and \( 10k - \alpha \) be positive integers. What is the remainder when the following number is divided by 11?
\[
8^{10k + \alpha} + 6^{10k - \alpha} - 7^{10k - \alpha} - 2^{10k + \alpha}
\]
|
0
| 0.875 |
Given 6 different thin rods with lengths $a, b, c, d, e, f$, any three of which can form a triangle. How many distinct tetrahedral edge frameworks can be assembled from the rods, where the frameworks are not equivalent by rotation or reflection?
|
30
| 0.875 |
How many times does 24 divide into 100 factorial (100!)?
|
32
| 0.875 |
Among the positive integers less than 10,000, if we exchange the digit in the highest place with the digit in the lowest place, we obtain a new number that is 1.2 times the original number. What is the sum of all numbers that satisfy this condition?
|
5535
| 0.375 |
On an island, there live three tribes: knights, who always tell the truth; liars, who always lie; and tricksters, who sometimes tell the truth and sometimes lie. At a round table sit 100 representatives of these tribes.
Each person at the table said two sentences: 1) "To my left sits a liar"; 2) "To my right sits a trickster". How many knights and liars are at the table if half of those present are tricksters?
|
25
| 0.625 |
As shown in the figure, the square $DEOF$ is inscribed in a quarter circle. If the radius of the circle is 1 centimeter, then the area of the shaded region is $\qquad$ square centimeters. (Use $\pi = 3.14$.)
|
0.285
| 0.625 |
The product of all natural numbers from 1 to \( n \) is denoted as \( n! \) (read as "n-factorial"). Which number is greater, \( 200! \) or \( 100^{200} \)?
|
100^{200}
| 0.75 |
Find the lateral surface area of a regular triangular pyramid if its height is 4 and its slant height is 8.
|
288
| 0.625 |
Xiaoming is riding a bicycle, while Xiaoming's father is walking. They start from locations $A$ and $B$ respectively, moving towards each other. After meeting, Xiaoming continues for another 18 minutes to reach $B$. It is known that Xiaoming's cycling speed is 4 times that of his father's walking speed, and it takes Xiaoming's father a certain number of minutes to walk from the meeting point to $A$. How long does Xiaoming's father need to walk from the meeting point to $A$?
|
288 \text{ minutes}
| 0.75 |
Given the sets \( M=\{x, xy, \lg(xy)\} \) and \( N=\{0, |x|, y\} \), and that \( M=N \), find the value of \( \left(x+\frac{1}{y}\right)+\left(x^{2}+\frac{1}{y^{2}}\right)+\left(x^{3}+\frac{1}{y^{3}}\right)+\cdots+\left(x^{2001}+\frac{1}{y^{2001}}\right) \).
|
-2
| 0.625 |
The number 2019 is expressed as a sum of different odd natural numbers. What is the maximum possible number of terms in this sum?
|
43
| 0.25 |
Three volleyballs with a radius of 18 lie on a horizontal floor, each pair touching one another. A tennis ball with a radius of 6 is placed on top of them, touching all three volleyballs. Find the distance from the top of the tennis ball to the floor. (All balls are spherical in shape.)
|
36
| 0.75 |
Given that the polynomial \( f \) satisfies the equation \( f(x^{2} + 1) - f(x^{2} - 1) = 4x^{2} + 6 \), determine the polynomial \( f(x^{2} + 1) - f(x^{2}) \).
|
2x^2 + 4
| 0.75 |
Determine all real values of \(a\) for which there exist five non-negative real numbers \(x_{1}, \ldots, x_{5}\) satisfying the following equations:
\[
\sum_{k=1}^{5} k x_{k} = a
\]
\[
\sum_{k=1}^{5} k^{3} x_{k} = a^{2}
\]
\[
\sum_{k=1}^{5} k^{5} x_{k} = a^{3}
\]
|
0, 1, 4, 9, 16, 25
| 0.875 |
In a math competition, the possible scores for each problem for each participating team are 0 points, 3 points, or 5 points. By the end of the competition, the sum of the total scores of three teams is 32 points. If the total score of any single team can reach 32 points, how many different possible combinations are there for the total scores of these three teams?
|
255
| 0.125 |
Pensioners on one of the planets in Alpha Centauri enjoy painting the cells of $2016 \times 2016$ boards with gold and silver paints. One day, it turned out that for all the boards painted that day, each $3 \times 3$ square contained exactly $A$ gold cells, and each $2 \times 4$ or $4 \times 2$ rectangle contained exactly $Z$ gold cells. For which values of $A$ and $Z$ is this possible?
|
A = 9, Z = 8
| 0.5 |
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