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stringlengths 18
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|---|---|---|
Find the minimum value of the expression \((\sqrt{2(1+\cos 2x)} - \sqrt{3-\sqrt{2}} \sin x + 1) \cdot (3 + 2\sqrt{7-\sqrt{2}} \cos y - \cos 2y)\). If the answer is not an integer, round it to the nearest whole number.
|
-9
| 0.375 |
\(\left(1-\operatorname{ctg}^{2}\left(\frac{3}{2} \pi - 2 \alpha\right)\right) \sin^{2}\left(\frac{\pi}{2} + 2 \alpha\right) \operatorname{tg}\left(\frac{5}{4} \pi - 2 \alpha\right) + \cos\left(4 \alpha - \frac{\pi}{2}\right)\).
|
1
| 0.875 |
90 students arrived at the camp. It is known that among any 10 students, there will always be two friends. A group of students is said to form a friendship chain if the children in the group can be numbered from 1 to \(k\) such that all the students can be divided into no more than 9 groups, each of which forms a friendship chain. (A group of one student also forms a friendship chain.)
|
9
| 0.625 |
Let \( S \) be the sum of the digits of the number \( 11^{2017} \). Find the remainder when \( S \) is divided by 9.
|
2
| 0.75 |
In the tetrahedron \(ABCD\), \(CD \perp BC\), \(AB \perp BC\), \(CD = AC\), \(AB = BC = 1\). The dihedral angle between the planes \(BCD\) and \(ABC\) is \(45^\circ\). Find the distance from point \(B\) to the plane \(ACD\).
|
\frac{\sqrt{3}}{3}
| 0.75 |
Find all three-digit numbers $\overline{МГУ}$, comprised of different digits $M$, $\Gamma$, and $Y$, for which the equality $\overline{\text{МГУ}} = (M + \Gamma + Y) \times (M + \Gamma + Y - 2)$ holds.
|
195
| 0.125 |
The ring road is divided by kilometer posts, and it is known that the number of posts is even. One of the posts is painted yellow, another is painted blue, and the rest are painted white. The distance between posts is defined as the length of the shortest arc connecting them. Find the distance from the blue post to the yellow post if the sum of the distances from the blue post to the white posts is 2008 km.
|
17 \text{ km}
| 0.625 |
Find the largest value of \( c \), if \( c = 2 - x + 2 \sqrt{x - 1} \) and \( x > 1 \).
|
2
| 0.875 |
Given 5 distinct real numbers, any two of which are summed to yield 10 sums. Among these sums, the smallest three are 32, 36, and 37, and the largest two are 48 and 51. What is the largest of these 5 numbers?
|
27.5
| 0.5 |
Let \( A B C \) be a triangle. The midpoints of the sides \( B C, A C \), and \( A B \) are denoted by \( D, E \), and \( F \) respectively.
The two medians \( A D \) and \( B E \) are perpendicular to each other and have lengths \( \overline{A D} = 18 \) and \( \overline{B E} = 13.5 \).
Calculate the length of the third median \( CF \) of this triangle.
|
22.5
| 0.625 |
Oleg has four cards, each with a natural number on each side (a total of 8 numbers). He considers all possible sets of four numbers where the first number is written on the first card, the second number on the second card, the third number on the third card, and the fourth number on the fourth card. For each set of four numbers, he writes the product of the numbers in his notebook. What is the sum of the eight numbers on the cards if the sum of the sixteen numbers in Oleg’s notebook is $330?
|
21
| 0.5 |
The numbers $1,2, \ldots, 2016$ are grouped into pairs in such a way that the product of the numbers in each pair does not exceed a certain natural number $N$. What is the smallest possible value of $N$ for which this is possible?
|
1017072
| 0.125 |
The tetrahedron $A B C D$ has edge lengths of $7, 13, 18, 27, 36, 41$, with $A B = 41$. What is the length of $C D$?
|
13
| 0.625 |
A car was traveling at a speed of \( V \). Upon entering a city, the driver reduced the speed by \( x \% \), and upon leaving the city, increased it by \( 0.5 x \% \). It turned out that this new speed is \( 0.6 x \% \) less than the speed \( V \). Find the value of \( x \).
|
20
| 0.875 |
Find all pairs \((a, b)\) of positive integers such that
\[
1 + 5^a = 6^b .
\]
|
(1,1)
| 0.75 |
At a club, twenty gentlemen met. Some of them were wearing hats, and some were without hats. From time to time, one of the gentlemen took off his hat and put it on one of those who did not have a hat at that moment. In the end, ten gentlemen counted that each of them had given away a hat more times than they had received one. How many gentlemen came to the club wearing hats?
|
10
| 0.875 |
Given an integer \( n \geq 2 \), for any pairwise coprime positive integers \( a_1, a_2, \ldots, a_n \), let \( A = a_1 + a_2 + \ldots + a_n \). Denote by \( d_i \) the greatest common divisor (gcd) of \( A \) and \( a_i \) for \( i = 1, 2, \ldots, n \). Denote by \( D_i \) the gcd of the remaining \( n-1 \) numbers after removing \( a_i \). Find the minimum value of \( \prod_{i=1}^{n} \frac{A - a_i}{d_i D_i} \).
|
(n-1)^n
| 0.75 |
Given a positive integer \( n \), find the smallest real number \( \lambda \) such that there exist real numbers \( a_1, a_2, \ldots, a_n \) within the interval \([0, 1]\), for any real numbers \( x_1, x_2, \ldots, x_n \) satisfying \( 0 \leq x_1 \leq x_2 \leq \cdots \leq x_n \leq 1 \), we have
$$
\min_{1 \leq i \leq n} \left|x_i - a_i\right| \leq \lambda
$$
|
\frac{1}{2n}
| 0.375 |
\( \Delta ABC \) is an isosceles triangle with \( AB = 2 \) and \( \angle ABC = 90^{\circ} \). Point \( D \) is the midpoint of \( BC \) and point \( E \) is on \( AC \) such that the area of quadrilateral \( AEDB \) is twice the area of triangle \( ECD \). Find the length of \( DE \).
|
\frac{\sqrt{17}}{3}
| 0.875 |
As shown in the figure, Leilei uses 12 small wooden sticks to form a $3 \times 3$ square. Fanfan uses 9 small wooden sticks to cut it into 3 small $1 \times 2$ rectangles and 3 small $1 \times 1$ squares. If Leilei uses 40 small wooden sticks to form an $8 \times 12$ rectangle, then Fanfan needs to use $\qquad$ more small wooden sticks to cut it into 40 small rectangles, such that each small rectangle is either $1 \times 2$ or $1 \times 3$.
|
116
| 0.25 |
Two distinct numbers \( x \) and \( y \) (not necessarily integers) satisfy the equation \( x^2 - 2000x = y^2 - 2000y \). Find the sum of the numbers \( x \) and \( y \).
|
2000
| 0.875 |
Let \( v(X) \) be the sum of elements of a nonempty finite set \( X \), where \( X \) is a set of numbers. Calculate the sum of all numbers \( v(X) \) where \( X \) ranges over all nonempty subsets of the set \(\{1,2,3, \ldots, 16\}\).
|
4456448
| 0.875 |
Anton, Vasya, Sasha, and Dima were driving from city A to city B, each taking turns at the wheel. The entire journey was made at a constant speed.
Anton drove the car for half the time Vasya did, and Sasha drove for as long as Anton and Dima together. Dima was at the wheel for only one-tenth of the distance. What fraction of the distance did Vasya drive? Provide your answer as a decimal.
|
0.4
| 0.625 |
Calculate: $\left(1+\frac{1}{2}\right) \times \left(1-\frac{1}{2}\right) \times \left(1+\frac{1}{3}\right) \times \left(1-\frac{1}{3}\right) \times \cdots \times \left(1+\frac{1}{10}\right) \times \left(1-\frac{1}{10}\right) = \ ?$
|
\frac{11}{20}
| 0.625 |
In a cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ with a side length of 1, points $E$ and $F$ are located on $A A_{1}$ and $C C_{1}$ respectively, such that $A E = C_{1} F$. Determine the minimum area of the quadrilateral $E B F D_{1}$.
|
\frac{\sqrt{6}}{2}
| 0.875 |
Find the last digit of the number $1 \cdot 2 + 2 \cdot 3 + \dots + 999 \cdot 1000$.
|
0
| 0.5 |
Find \(\lim _{x \rightarrow \frac{\pi}{2}} \frac{\tan x}{\tan 3x}\).
|
3
| 0.5 |
Roman numerals I, V, X, L, C, D, M represent 1, 5, 10, 50, 100, 500, 1000 respectively. Calculate \(2 \times M + 5 \times L + 7 \times X + 9 \times I\).
|
2329
| 0.875 |
During training shooting, each of the soldiers shot 10 times. One of them successfully completed the task and scored 90 points. How many times did he score 9 points if he scored four 10s, and the results of the other shots were 7s, 8s, and 9s, with no misses?
|
3
| 0.125 |
Let $ABC$ be an isosceles right triangle at $C$, and $D$ and $E$ be two points on $[CA]$ and $[CB]$ such that $CD = CE$. Then let $U$ and $V$ be two points on $[AB]$ such that $DU$ and $CV$ are perpendicular to $(AE)$. Show that $UV = VB$.
|
UV = VB
| 0.875 |
If there are 5 medical boxes, each containing a specific medicine common to exactly 2 of the boxes, and each type of medicine appears in exactly 2 of the boxes, how many types of medicine are there?
|
10
| 0.75 |
Find the smallest natural number that is divisible by 11, which after being increased by 1 is divisible by 13.
|
77
| 0.75 |
Each rational number is painted with one of two colors, white or red. A coloring is called "sanferminera" if for any two rational numbers \( x \) and \( y \) with \( x \neq y \), the following conditions are satisfied:
a) \( xy = 1 \),
b) \( x + y = 0 \),
c) \( x + y = 1 \),
then \( x \) and \( y \) are painted different colors. How many "sanferminera" colorings are there?
|
2
| 0.75 |
Liza found the total of the interior angles of a convex polygon. She missed one of the angles and obtained the result $2017^\circ$. Which of the following was the angle she missed?
A $37^\circ$
B $53^\circ$
C $97^\circ$
D $127^\circ$
E $143^\circ$
|
143^\circ
| 0.875 |
Given an isosceles triangle \( A B C \) where \( A B = A C \) and \( \angle A B C = 53^\circ \). Point \( K \) is such that \( C \) is the midpoint of \( A K \). Point \( M \) is chosen such that:
- \( B \) and \( M \) are on the same side of the line \( A C \);
- \( K M = A B \);
- the angle \( \angle M A K \) is the largest possible.
What is the measure of the angle \( \angle B A M \) in degrees?
|
44^\circ
| 0.875 |
\[
\frac{\left(\left(4.625 - \frac{13}{18} \cdot \frac{9}{26}\right) : \frac{9}{4} + 2.5 : 1.25 : 6.75\right) : 1 \frac{53}{68}}{\left(\frac{1}{2} - 0.375\right) : 0.125 + \left(\frac{5}{6} - \frac{7}{12}\right) : (0.358 - 1.4796 : 13.7)}
\]
|
\frac{17}{27}
| 0.875 |
Let's divide a sequence of natural numbers into groups:
\((1), (2,3), (4,5,6), (7,8,9,10), \ldots\)
Let \( S_{n} \) denote the sum of the \( n \)-th group of numbers. Find \( S_{16} - S_{4} - S_{1} \).
|
2021
| 0.75 |
Calculate the limit of the function:
\[
\lim _{x \rightarrow 0} \frac{e^{2 x}-e^{3 x}}{\operatorname{arctg} x-x^{2}}
\]
|
-1
| 0.875 |
Let $\left\{a_{n}\right\}$ be a sequence of positive terms with the first term being 1, and satisfying the recurrence relation $(n+1) a_{n+1}^{2} - n a_{n}^{2} + a_{n} a_{n+1} = 0$ for $n=1, 2, 3, \ldots$. Find the general term $a_{n}$.
|
\frac{1}{n}
| 0.625 |
Find the maximum and minimum values of the function:
1) \( z = x^{2} - y^{2} + 2a^{2} \) in the circle \( x^{2} + y^{2} \leqslant a^{2} \);
2) \( v = 2x^{3} + 4x^{2} + y^{2} - 2xy \) in the closed region bounded by the lines \( y = x^{2} \) and \( y = 4 \).
|
0 \text{ and } 32
| 0.5 |
On a computer keyboard, the key for the digit 1 is not working. For example, if you try to type the number 1231234, only the number 23234 will actually print.
Sasha tried to type an 8-digit number, but only 202020 was printed. How many 8-digit numbers satisfy this condition?
|
28
| 0.25 |
On each side of a square, a point is taken. It turns out that these points are the vertices of a rectangle whose sides are parallel to the diagonals of the square. Find the perimeter of the rectangle if the diagonal of the square is 6.
|
12
| 0.625 |
Calculate as simply as possible the value of the expression
$$
a^{3}+b^{3}+3\left(a^{3} b+a b^{3}\right)+6\left(a^{3} b^{2}+a^{2} b^{3}\right)
$$
if $a+b=1$.
|
1
| 0.875 |
The positive integers \( x \) and \( y \), for which \( \gcd(x, y) = 3 \), are the coordinates of the vertex of a square centered at the origin with an area of \( 20 \cdot \operatorname{lcm}(x, y) \). Find the perimeter of the square.
|
24\sqrt{5}
| 0.875 |
In the language of the Ancient Tribe, the alphabet consists of only two letters: M and O. Two words are synonyms if one can be obtained from the other by
a) Removing the letter combinations МО or ООММ,
b) Adding the letter combination ОМ in any place
Are the words ОММ and МОО synonyms in the language of the Ancient Tribe?
|
\text{No}
| 0.625 |
Given that \(a\) is a digit from 1 to 9, and the repeating decimal \(0.1a = \frac{1}{a}\), find the value of \(a\).
|
6
| 0.875 |
At the "Economics and Law" congress, a "Best of the Best" tournament was held, in which more than 220 but fewer than 254 delegates—economists and lawyers—participated. During one match, participants had to ask each other questions within a limited time and record correct answers. Each participant played with each other participant exactly once. A match winner got one point, the loser got none, and in case of a draw, both participants received half a point each. By the end of the tournament, it turned out that in matches against economists, each participant gained half of all their points. How many lawyers participated in the tournament? Provide the smallest possible number as the answer.
|
105
| 0.5 |
Let the set \( \mathrm{S} = \{1, 2, 3, \ldots, 10\} \). The subset \( \mathrm{A} \) of \( \mathrm{S} \) satisfies \( \mathrm{A} \cap \{1, 2, 3\} \neq \emptyset \) and \( \mathrm{A} \cup \{4, 5, 6\} \neq \mathrm{S} \). Find the number of such subsets \( \mathrm{A} \).
|
888
| 0.375 |
Arrange the leftmost numbers of Pascal's triangle in the following manner:
Diagonals are a sequence of numbers connected by arrows in a slanted direction. For example, the first diagonal is " $1 \leftarrow 1$", the second diagonal is " $1 \longleftrightarrow \rightarrow 2$", and the third diagonal is " $1 \longleftrightarrow 3 \longleftrightarrow \rightarrow 1$".
Question: What is the sum of all the numbers in the 13th diagonal?
|
610
| 0.125 |
Let \( p \) be a function associated with a permutation. The order of this permutation is defined as the smallest integer \( k \) such that \( p^{(k)} = \text{Id} \).
What is the largest order for a permutation of size 11?
|
30
| 0.5 |
An exchange point conducts two types of operations:
1) Give 2 euros - receive 3 dollars and a candy as a gift;
2) Give 5 dollars - receive 3 euros and a candy as a gift.
When the wealthy Buratino came to the exchange point, he had only dollars. When he left, he had fewer dollars, he did not acquire any euros, but he received 50 candies. How many dollars did this "gift" cost Buratino?
|
10 \text{ dollars}
| 0.75 |
Inside the tetrahedron \( ABCD \), points \( X \) and \( Y \) are given. The distances from point \( X \) to the faces \( ABC, ABD, ACD, BCD \) are \( 14, 11, 29, 8 \) respectively. The distances from point \( Y \) to the faces \( ABC, ABD, ACD, BCD \) are \( 15, 13, 25, 11 \) respectively. Find the radius of the inscribed sphere of tetrahedron \( ABCD \).
|
17
| 0.75 |
In a country with 15 cities, some of which are connected by airlines belonging to three different companies, it is known that even if any one of the airlines ceases operations, it will still be possible to travel between any two cities (possibly with transfers) using the remaining two companies' flights. What is the minimum number of airline routes in the country?
|
21
| 0.375 |
Find the limits:
1) $\lim _{x \rightarrow 4} \frac{x^{2}-16}{x^{2}-5 x+4}$
2) $\lim _{x \rightarrow 1} \frac{x^{3}-3 x+2}{x^{3}-x^{2}-x+1}$
3) $\lim _{x \rightarrow 0} \frac{x^{3}-6 x+6 \sin x}{x^{5}}$
4) $\lim _{x \rightarrow 0} \frac{\operatorname{tg} x-x}{x-\sin x}$
5) $\lim _{x \rightarrow \infty} \frac{a x^{2}+b}{c x^{2}+d}$
6) $\lim _{x \rightarrow 0} \frac{\ln \sin 5 x}{\ln \sin 2 x}$
|
1
| 0.5 |
In the parallelogram \(ABCD\), the longer side \(AD\) is 5. The angle bisectors of angles \(A\) and \(B\) intersect at point \(M\).
Find the area of the parallelogram, given that \(BM = 2\) and \(\cos \angle BAM = \frac{4}{5}\).
|
16
| 0.5 |
It is known that the sequence of numbers \(a_{1}, a_{2}, \ldots\) is an arithmetic progression, and the sequence of products \(a_{1}a_{2}, a_{2}a_{3}, a_{3}a_{4}, \ldots\) is a geometric progression. It is given that \(a_{1} = 1\). Find \(a_{2017}\).
|
1
| 0.375 |
A five-digit number $\overline{abcde}$ is given. When any two digits from this number are selected to form a two-digit number keeping the order as in the original five-digit number, there are 10 two-digit numbers: $33, 37, 37, 37, 38, 73, 77, 78, 83, 87$. What is the five-digit number $\overline{abcde}$?
|
37837
| 0.75 |
What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is palindromic if the sequence of decimal digits is the same when read backward.)
|
21
| 0.5 |
Find \(\lim _{x \rightarrow 0}[(x-\sin x) \ln x]\).
|
0
| 0.625 |
Calculate the areas of the figures enclosed by the lines given in polar coordinates.
$$
r=\sin \phi, \quad r=2 \sin \phi
$$
|
\frac{3\pi}{4}
| 0.75 |
For every positive integer \( N \), determine the smallest real number \( b_{N} \) such that, for all real \( x \),
\[
\sqrt[N]{\frac{x^{2N}+1}{2}} \leqslant b_{N}(x-1)^2 + x.
\]
|
\frac{N}{2}
| 0.375 |
Let \(ABC\) be a right triangle with \(\angle A = 90^\circ\). A circle \(\omega\) centered on \(BC\) is tangent to \(AB\) at \(D\) and \(AC\) at \(E\). Let \(F\) and \(G\) be the intersections of \(\omega\) and \(BC\) so that \(F\) lies between \(B\) and \(G\). If lines \(DG\) and \(EF\) intersect at \(X\), show that \(AX = AD\).
|
AX = AD
| 0.875 |
You are allowed to cut out any 18 squares from a $20 \times 20$ chessboard, and after that, place several rooks on the remaining squares such that no two rooks attack each other. Rooks attack each other if they are in the same row or column of the board and there are no cut-out squares between them. What is the maximum number of rooks that can be placed under these conditions?
|
38
| 0.125 |
Find the limits:
1) \(\lim_{x \to 3}\left(\frac{1}{x-3}-\frac{6}{x^2-9}\right)\)
2) \(\lim_{x \to \infty}\left(\sqrt{x^2 + 1}-x\right)\)
3) \(\lim_{n \to \infty} 2^n \sin \frac{x}{2^n}\)
4) \(\lim_{x \to 1}(1-x) \tan \frac{\pi x}{2}\).
|
\frac{2}{\pi}
| 0.5 |
The archipelago consists of $N \geqslant 7$ islands. Any two islands are connected by no more than one bridge. It is known that from each island, there are no more than 5 bridges, and among any 7 islands, there are always two islands connected by a bridge. What is the largest possible value of $N$?
|
36
| 0.375 |
Find all positive integers \( n \) such that \( n+1 \) divides \( 2n^2 + 5n \). Verify that the found \( n \) are solutions.
|
2
| 0.625 |
The sum of all numbers in the first row is equal to the sum of all numbers in the second row. What number should be placed in the position of the "?" in the second row?
\[
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline 1 & 3 & 5 & 7 & 9 & 11 & 13 & 15 & 17 & 19 & 200 \\
\hline
\end{tabular}
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|}
\hline 1 & 4 & 7 & 10 & 13 & 16 & 19 & 22 & 25 & 28 & $?$ \\
\hline
\end{tabular}
\]
|
155
| 0.875 |
A locomotive is moving on a horizontal section of track at a speed of 72 km/h. In how much time and at what distance will it be stopped by the brake if the resistance to motion after the start of braking is equal to 0.2 of its weight?
|
102 \ \text{meters}
| 0.625 |
Indiana Jones reached an abandoned temple in the jungle and entered the treasury. There were 5 chests, with the knowledge that only one contains the treasure, and the others trigger a stone slab to fall on the head of anyone who tries to open them. The chests are numbered from left to right.
- The first, fourth, and fifth chests are made of cedar.
- The second and third chests are made of sandalwood.
- The inscriptions on the chests are as follows:
- On the first chest: "The treasure is in me or in the 4th chest."
- On the second chest: "The treasure is in the chest to the left of me."
- On the third chest: "The treasure is in me or in the chest at the far right."
- On the fourth chest: "There is no treasure in the chests to the left of me."
- On the fifth chest: "All the inscriptions on other chests are false."
The last guardian of the temple, before dying, revealed to Indiana a secret: an equal number of false statements are written on both cedar and sandalwood chests.
In which chest is the treasure?
|
2
| 0.875 |
A natural number \( n \) was multiplied by the sum of the digits of the number \( 3n \), and the resulting number was then multiplied by 2. The result was 2022. Find \( n \).
|
337
| 0.875 |
Find an integer \( x \) such that \(\left(1+\frac{1}{x}\right)^{x+1}=\left(1+\frac{1}{2003}\right)^{2003}\).
|
-2004
| 0.375 |
Find the smallest positive integer \( n \) such that there exists a complex number \( z \), with positive real and imaginary part, satisfying \( z^{n} = (\bar{z})^{n} \).
|
3
| 0.75 |
The sequence \(\left\{x_{n}\right\}\) is defined as follows: \(x_{1}=\frac{1}{2}, x_{k+1}=x_{k}^{2}+x_{k}\). Calculate the integer part of the sum \(\frac{1}{x_{1}+1}+\frac{1}{x_{2}+1}+\cdots+\frac{1}{x_{100}+1}\).
|
1
| 0.875 |
Let $[x]$ denote the greatest integer less than or equal to the real number $x$. The equation $\lg ^{2} x-[\lg x]-2=0$ has how many real roots?
|
3
| 0.875 |
A villager A and a motorcycle with a passenger, villager B, set off at the same time from the village to the station along a single road. Before reaching the station, the motorcyclist dropped off villager B and immediately went back towards the village, while villager B walked to the station. Upon meeting villager A, the motorcyclist picked him up and took him to the station. As a result, both villagers arrived at the station simultaneously. What fraction of the journey from the village to the station did villager A travel by motorcycle, considering that both villagers walked at the same speed, which was 9 times slower than the speed of the motorcycle?
|
\frac{5}{6}
| 0.5 |
For a triangle with side lengths \(a\), \(b\), and \(c\), an area of \(\frac{1}{4}\), and a circumradius of 1, compare the magnitude of \(q = \sqrt{a} + \sqrt{b} + \sqrt{c}\) to \(t = \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\).
|
q < t
| 0.125 |
Bob Barker decided to raise the intellectual level of The Price is Right by having contestants guess how many objects exist of a certain type, without going over. The number of points you will get is the percentage of the correct answer, divided by 10, with no points for going over (i.e., a maximum of 10 points).
The first object for our contestants is a table of shape \((5,4,3,2,1)\). This table is an arrangement of the integers 1 through 15 with five numbers in the top row, four in the next, three in the next, two in the next, and one in the last, such that each row and each column is increasing (from left to right, and top to bottom, respectively). For instance:
```
6
13 14
15
```
is one such table. How many tables are there?
|
292864
| 0.125 |
Solve the equation
$$
\begin{aligned}
& x(x+1)+(x+1)(x+2)+(x+2)(x+3)+(x+3)(x+4)+\ldots \\
& \ldots+(x+8)(x+9)+(x+9)(x+10)=1 \cdot 2+2 \cdot 3+\ldots+8 \cdot 9+9 \cdot 10
\end{aligned}
$$
|
x = 0, x = -10
| 0.375 |
Find all integers \( k \geq 1 \) so that the sequence \( k, k+1, k+2, \ldots, k+99 \) contains the maximum number of prime numbers.
|
k=2
| 0.5 |
Given that \(a, b, c, d\) are integers with \(ad \neq bc\), show that \(\frac{1}{(ax+b)(cx+d)}\) can be written in the form \(\frac{r}{ax+b} + \frac{s}{cx+d}\). Then, find the sum \(\frac{1}{1 \cdot 4} + \frac{1}{4 \cdot 7} + \frac{1}{7 \cdot 10} + \ldots + \frac{1}{2998 \cdot 3001}\).
|
\frac{1000}{3001}
| 0.875 |
Let \(\triangle ABC\) be a triangle with \(AB=7\), \(BC=1\), and \(CA=4\sqrt{3}\). The angle trisectors of \(C\) intersect \(\overline{AB}\) at \(D\) and \(E\), and lines \(\overline{AC}\) and \(\overline{BC}\) intersect the circumcircle of \(\triangle CDE\) again at \(X\) and \(Y\), respectively. Find the length of \(XY\).
|
\frac{112}{65}
| 0.875 |
There are some natural numbers such that when each of them is multiplied by 7, the last four digits of the product are 2012. The smallest of these natural numbers is $\qquad$ .
|
1716
| 0.75 |
A polynomial \( P(x) \) of degree 10 with a leading coefficient of 1 is given. The graph of \( y = P(x) \) lies entirely above the x-axis. The polynomial \( -P(x) \) was factored into irreducible factors (i.e., polynomials that cannot be represented as the product of two non-constant polynomials). It is known that at \( x = 2020 \), all the resulting irreducible polynomials take the value -3. Find \( P(2020) \).
|
243
| 0.875 |
Mathematician Wiener, the founder of cybernetics, was asked about his age during his Ph.D. awarding ceremony at Harvard University because he looked very young. Wiener's interesting response was: "The cube of my age is a four-digit number, and the fourth power of my age is a six-digit number. These two numbers together use all the digits from 0 to 9 exactly once, with no repetition or omission." What is Wiener's age that year? (Note: The cube of a number \(a\) is equal to \(a \times a \times a\), and the fourth power of a number \(a\) is equal to \(a \times a \times a \times a\)).
|
18
| 0.75 |
A grid sheet of size \(5 \times 7\) was cut into \(2 \times 2\) squares, L-shaped pieces covering 3 cells, and \(1 \times 3\) strips. How many \(2 \times 2\) squares could be obtained?
|
5
| 0.375 |
The lengths of the edges of a regular tetrahedron \(ABCD\) are 1. \(G\) is the center of the base \(ABC\). Point \(M\) is on line segment \(DG\) such that \(\angle AMB = 90^\circ\). Find the length of \(DM\).
|
\frac{\sqrt{6}}{6}
| 0.875 |
In square \( A B C D \), \( P \) and \( Q \) are points on sides \( C D \) and \( B C \), respectively, such that \( \angle A P Q = 90^\circ \). If \( A P = 4 \) and \( P Q = 3 \), find the area of \( A B C D \).
|
\frac{256}{17}
| 0.375 |
Fifty children went to the zoo, with 36 of them seeing the pandas, 28 seeing the giraffes, and 15 seeing the pandas but not the giraffes. How many children saw the giraffes but not the pandas?
|
7
| 0.875 |
Find the smallest prime number that cannot be expressed in the form $\left|3^{a}-2^{b}\right|$, where $a$ and $b$ are non-negative integers.
|
41
| 0.25 |
There are \( n \) people, and it is known that any two of them can call each other at most once. Among them, the total number of calls between any group of \( n-2 \) people is the same and is \( 3^{k} \) times, where \( k \) is a natural number. Find all possible values of \( n \).
|
5
| 0.5 |
Along a straight alley, there are 400 streetlights placed at equal intervals, numbered consecutively from 1 to 400. Alla and Boris start walking towards each other from opposite ends of the alley at the same time but with different constant speeds (Alla from the first streetlight and Boris from the four-hundredth streetlight). When Alla is at the 55th streetlight, Boris is at the 321st streetlight. At which streetlight will they meet? If the meeting occurs between two streetlights, indicate the smaller number of the two in the answer.
|
163
| 0.75 |
Given two positive integers \(x\) and \(y\), \(xy - (x + y) = \operatorname{HCF}(x, y) + \operatorname{LCM}(x, y)\), where \(\operatorname{HCF}(x, y)\) and \(\operatorname{LCM}(x, y)\) are respectively the greatest common divisor and the least common multiple of \(x\) and \(y\). If \(c\) is the maximum possible value of \(x + y\), find \(c\).
|
10
| 0.625 |
Find all values of \( a \) such that the roots \( x_1, x_2, x_3 \) of the polynomial
\[ x^3 - 6x^2 + ax + a \]
satisfy
\[ \left(x_1 - 3\right)^3 + \left(x_2 - 3\right)^3 + \left(x_3 - 3\right)^3 = 0. \]
|
-9
| 0.5 |
In \(\triangle ABC\), \(\sin A = \frac{3}{5}\), \(\cos B = \frac{5}{13}\). What is the value of \(\cos C\)?
|
\frac{16}{65}
| 0.75 |
In the coordinate plane, a point whose x-coordinate and y-coordinate are both integers is called a lattice point. For any positive integer \( n \), connect the origin \( O \) with the point \( A_{n}(n, n+3) \). Let \( f(n) \) denote the number of lattice points on the line segment \( OA_{n} \) excluding the endpoints. Determine the value of \( f(1)+f(2)+\cdots+f(1990) \).
|
1326
| 0.75 |
The first two digits of a natural four-digit number are either both less than 5 or both greater than 5. The same condition applies to the last two digits. How many such numbers are there in total? Justify your answer.
|
1476
| 0.75 |
How many integers from 1 to 300 can be divided by 7 and also be divisible by either 2 or 5?
|
25
| 0.75 |
If we replace some of the " × " signs with " ÷ " signs in the expression $9 \times 8 \times 7 \times 6 \times 5 \times 4 \times 3 \times 2 \times 1$ such that the final result is still a natural number, denoted as $N$, what is the minimum value of $N$?
|
70
| 0.25 |
A polynomial of degree 10 has three distinct roots. What is the maximum number of zero coefficients it can have?
|
9
| 0.5 |
On an island, there are knights who always tell the truth and liars who always lie. Before a friendly match, 30 islanders gathered wearing shirts with numbers on them—random natural numbers. Each of them said, "I have a shirt with an odd number." After that, they exchanged shirts, and each said, "I have a shirt with an even number." How many knights participated in the exchange?
|
15
| 0.875 |
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