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math
|
Let $a_1, a_2, \ldots$ and $b_1, b_2, \ldots$ be sequences such that $a_ib_i - a_i - b_i = 0$ and $a_{i+1} = \frac{2-a_ib_i}{1-b_i}$ for all $i \ge 1$. If $a_1 = 1 + \frac{1}{\sqrt[4]{2}}$, then what is $b_{6}$?
[i]Proposed by Andrew Wu[/i]
|
257
| 118
| 3
|
math
|
## Problem Statement
Write the equation of the plane passing through point $A$ and perpendicular to vector $\overrightarrow{B C}$.
$A(-1 ; 2 ; -2)$
$B(13 ; 14 ; 1)$
$C(14 ; 15 ; 2)$
|
x+y+z+1=0
| 67
| 7
|
math
|
51. Calculate: $\quad\left(5 \frac{1}{10}\right)^{2}$.
|
26.01
| 25
| 5
|
math
|
8. (3 points) There are 10 pencil cases, among which 5 contain pencils, 4 contain pens, and 2 contain both pencils and pens. The number of empty pencil cases is $\qquad$.
|
3
| 46
| 1
|
math
|
1. Solve the inequality $2^{\log _{2}^{2} x}-12 \cdot x^{\log _{0.5} x}<3-\log _{3-x}\left(x^{2}-6 x+9\right)$.
|
x\in(2^{-\sqrt{2}},2)\cup(2,2^{\sqrt{2}})
| 56
| 25
|
math
|
The Torricelli experiment is to be performed with ether instead of mercury. The room temperature in one case is $t_{1}=$ $30^{\circ}$, and in the other case $t_{2}=-10^{\circ}$, the air pressure in both cases is equal to $750 \mathrm{~mm}$ of mercury pressure. The density of ether is $\sigma_{1}=0.715$, that of mercury is $\sigma_{2}=13.6$; the vapor pressure of ether is $30^{\circ}$, $634.8 \mathrm{~mm}$ of mercury column pressure at $10^{\circ}$, and $114.72 \mathrm{~mm}$ of mercury column pressure at $-10^{\circ}$. How high is the ether column at the two temperatures?
|
2191
| 186
| 4
|
math
|
2. $\prod_{i=1}^{2018} \sin \frac{i \pi}{2019}=$ $\qquad$
|
\frac{2019}{2^{2018}}
| 33
| 15
|
math
|
[ Mathematical logic (other).]
Several natives met (each one is either a liar or a knight), and each one declared to all the others: “You are all liars.” How many knights were among them?
#
|
1
| 45
| 1
|
math
|
(given to François Caddet and Philippe Cloarec). We play on a chessboard with a side of 100. A move consists of choosing a rectangle formed by squares of the chessboard and inverting their colors. What is the smallest number of moves that allows the entire chessboard to be turned black?
|
100
| 66
| 3
|
math
|
5. Given the number $500 \ldots 005$ (80 zeros). It is required to replace some two zeros with non-zero digits so that after the replacement, the resulting number is divisible by 165. In how many ways can this be done?
|
17280
| 60
| 5
|
math
|
13. In $\triangle A B C, A B=8, B C=13$ and $C A=15$. Let $H, I, O$ be the orthocentre, incentre and circumcentre of $\triangle A B C$ respectively. Find $\sin \angle H I O$.
(2 marks)
在 $\triangle A B C$ 中, $A B=8 、 B C=13 、 C A=15$ 。設 $H 、 I 、 O$ 分別為 $\triangle A B C$ 的垂
心、內心和外心。求 $\sin \angle H I O$ 。
(2 分)
|
\frac{7\sqrt{3}}{26}
| 146
| 13
|
math
|
Example 5 Let $0<\theta<\pi$, find the maximum value of $y=\sin \frac{\theta}{2}(1+\cos \theta)$.
|
\frac{4 \sqrt{3}}{9}
| 36
| 12
|
math
|
A school with 862 students will participate in a scavenger hunt, whose rules are:
I) The number of registrants must be a number between $\frac{2}{3}$ and $\frac{7}{9}$ of the total number of students in the school.
II) Since the students will be divided into groups of 11, the number of registrants must be a multiple of 11.
How many are the possible numbers of registrants for this school?
#
|
8
| 100
| 1
|
math
|
Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\).
|
100
| 81
| 3
|
math
|
Volume $A$ equals one fourth of the sum of the volumes $B$ and $C$, while volume $B$ equals one sixth of the sum of the volumes $A$ and $C$. There are relatively prime positive integers $m$ and $n$ so that the ratio of volume $C$ to the sum of the other two volumes is $\frac{m}{n}$. Find $m+n$.
|
35
| 85
| 2
|
math
|
1. Positive real numbers $a, b, c$ satisfy $\ln (a b), \ln (a c), \ln (b c)$ form an arithmetic sequence, and $4(a+c)=17 b$, then all possible values of $\frac{c}{a}$ are $\qquad$ .
|
16
| 64
| 2
|
math
|
Example 12 The sports meet lasted for $n$ days $(n>1)$, and a total of $m$ medals were awarded. On the first day, 1 medal was awarded, and then $\frac{1}{7}$ of the remaining $m-1$ medals were awarded. On the second day, 2 medals were awarded, and then $\frac{1}{7}$ of the remaining medals were awarded, and so on, until on the $n$-th day, exactly $n$ medals were awarded, and all were distributed. How many days did the sports meet last? How many medals were awarded in total?
(IMO - 9 problem)
|
6
| 140
| 1
|
math
|
11th Chinese 1996 Problem B1 Eight singers sing songs at a festival. Each song is sung once by a group of four singers. Every pair of singers sing the same number of songs together. Find the smallest possible number of songs. Solution
|
14
| 53
| 2
|
math
|
1. Restore an example of dividing two numbers if it is known that the quotient is five times smaller than the dividend and seven times larger than the divisor.
#
|
175:5=35
| 32
| 8
|
math
|
6.17 Suppose there are $n$ tennis players who are to play doubles matches with the rule that each player must play exactly one match against every other player as an opponent. For which natural numbers $n$ can such a tournament be organized?
|
n\equiv1(\bmod8)
| 51
| 9
|
math
|
Define the sequence $x_1, x_2, ...$ inductively by $x_1 = \sqrt{5}$ and $x_{n+1} = x_n^2 - 2$ for each $n \geq 1$. Compute
$\lim_{n \to \infty} \frac{x_1 \cdot x_2 \cdot x_3 \cdot ... \cdot x_n}{x_{n+1}}$.
|
1
| 97
| 1
|
math
|
Gapochnik A.I.
How many integers from 1 to 1997 have a sum of digits that is divisible by 5?
|
399
| 31
| 3
|
math
|
Find all positive integers $n$ for which $n^{n-1} - 1$ is divisible by $2^{2015}$, but not by $2^{2016}$.
|
n = 2^{2014}k - 1
| 44
| 15
|
math
|
9-5-1. The numbers 1 and 7 are written on the board. In one move, the greatest common divisor of both numbers on the board is added to each of them. For example, if at some point the numbers 20 and 50 are on the board, they will be replaced by the numbers 30 and 60.
What numbers will be on the board after 100 moves? Provide the answers in any order. Answer. 588 and 594.
|
588594
| 109
| 6
|
math
|
4. In the sequence $\left\{a_{n}\right\}$, $a_{1}=2, a_{2}=10$, for all positive integers $n$ we have $a_{n+2}=a_{n+1}-a_{n}$. Then $a_{2015}=$
|
-10
| 68
| 3
|
math
|
Let $R$ be the rectangle in the Cartesian plane with vertices at $(0,0)$, $(2,0)$, $(2,1)$, and $(0,1)$. $R$ can be divided into two unit squares, as shown. [asy]size(120); defaultpen(linewidth(0.7));
draw(origin--(2,0)--(2,1)--(0,1)--cycle^^(1,0)--(1,1));[/asy] Pro selects a point $P$ at random in the interior of $R$. Find the probability that the line through $P$ with slope $\frac{1}{2}$ will pass through both unit squares.
|
\frac{3}{4}
| 148
| 7
|
math
|
Given that the line $y=mx+k$ intersects the parabola $y=ax^2+bx+c$ at two points, compute the product of the two $x$-coordinates of these points in terms of $a$, $b$, $c$, $k$, and $m$.
|
\frac{c - k}{a}
| 61
| 9
|
math
|
14. Definition: If Team A beats Team B, Team B beats Team C, and Team C beats Team A, then Teams A, B, and C are called a "friendly group".
(1) If 19 teams participate in a round-robin tournament, find the maximum number of friendly groups;
(2) If 20 teams participate in a round-robin tournament, find the maximum number of friendly groups.
|
285
| 88
| 3
|
math
|
1. A mathematician left point A for point B. After some time, a physicist also left point A for point B. Catching up with the mathematician after 20 km, the physicist, without stopping, continued to point B and turned back. They met again 20 km from B. Then each continued in their respective directions. Upon reaching points A and B respectively, they turned and walked towards each other again. How many kilometers from point A will they meet for the third time, if the distance between points A and B is 100 km?
|
45
| 117
| 2
|
math
|
3.260. $\frac{4 \sin ^{4}\left(\alpha-\frac{3}{2} \pi\right)}{\sin ^{4}\left(\alpha-\frac{5 \pi}{2}\right)+\cos ^{4}\left(\alpha+\frac{5 \pi}{2}\right)-1}$.
|
-2\operatorname{ctg}^{2}\alpha
| 74
| 13
|
math
|
2. Let $n$ be a natural number such that the numbers $n, n+2$, and $n+4$ are prime. Determine the smallest natural number which, when divided by $n$ gives a remainder of 1, when divided by $n+2$ gives a remainder of 2, and when divided by $n+4$ gives a remainder of 6.
|
97
| 81
| 2
|
math
|
8. A frog starts climbing out of a 12-meter deep well at 8 o'clock. It climbs up 3 meters and then slides down 1 meter due to the slippery well walls. The time it takes to slide down 1 meter is one-third of the time it takes to climb up 3 meters. At 8:17, the frog reaches 3 meters below the well's mouth for the second time. How many minutes does it take for the frog to climb from the bottom of the well to the mouth of the well?
|
22
| 113
| 2
|
math
|
13.093. An alloy of copper and silver contains 1845 g more silver than copper. If a certain amount of pure silver, equal to $1 / 3$ of the mass of pure silver originally contained in the alloy, were added to it, then a new alloy would be obtained, containing $83.5\%$ silver. What is the mass of the alloy and what was the original percentage of silver in it?
|
3165;\approx79.1
| 95
| 10
|
math
|
2. (6 points) Given $16.2 \times\left[\left(4 \frac{1}{7}-\square \times 700\right) \div 1 \frac{2}{7}\right]=8.1$, then $\square=$
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly.
|
0.005
| 85
| 5
|
math
|
Two planes, $P$ and $Q$, intersect along the line $p$. The point $A$ is given in the plane $P$, and the point $C$ in the plane $Q$; neither of these points lies on the straight line $p$. Construct an isosceles trapezoid $ABCD$ (with $AB \parallel CD$) in which a circle can be inscribed, and with vertices $B$ and $D$ lying in planes $P$ and $Q$ respectively.
|
ABCD
| 109
| 3
|
math
|
4A. Determine all real functions $f$ defined for every real number $x$, such that $f(x) + x f(1-x) = x^2 - 1$.
|
f(x)=\frac{-x^{3}+3x^{2}-1}{x^{2}-x+1}
| 39
| 26
|
math
|
2. Let planar vectors $\boldsymbol{a}, \boldsymbol{b}$ satisfy: $|\boldsymbol{a}|=1,|\boldsymbol{b}|=2, \boldsymbol{a} \perp \boldsymbol{b}$. Points $O, A, B$ are three points on the plane, satisfying $\overrightarrow{O A}=$ $2 \boldsymbol{a}+\boldsymbol{b}, \overrightarrow{O B}=-3 \boldsymbol{a}+2 \boldsymbol{b}$, then the area of $\triangle A O B$ is
|
7
| 128
| 1
|
math
|
1. (1997 Shanghai High School Mathematics Competition) Let $S=\{1,2,3,4\}$, and the sequence $a_{1}, a_{2}, \cdots, a_{n}$ has the following property: for any non-empty subset $B$ of $S$, there are adjacent $|B|$ terms in the sequence that exactly form the set $B$. Find the minimum value of $n$.
|
8
| 92
| 1
|
math
|
4. In a company of 8 people, each person is acquainted with exactly 6 others. In how many ways can four people be chosen such that any two of them are acquainted? (We assume that if A is acquainted with B, then B is also acquainted with A, and that a person is not acquainted with themselves, as the concept of acquaintance applies to two different people.)
|
16
| 78
| 2
|
math
|
# Task 6. (14 points)
Determine the sign of the number
$$
A=\frac{1}{1}-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}+\ldots+\frac{1}{2012}+\frac{1}{2013}-\frac{1}{2014}-\frac{1}{2015}+\frac{1}{2016}
$$
The signs are arranged as follows: “+" before the first fraction, then two “-” and two “+" alternately. The last fraction has a "+" before it.
|
A>0
| 160
| 3
|
math
|
Find all integers $n$ such that
$$
n^{4}+6 n^{3}+11 n^{2}+3 n+31
$$
is a perfect square.
(Xu Wandang)
|
n=10
| 47
| 4
|
math
|
Example 4 Find the number of integer points that satisfy the system of inequalities: $\left\{\begin{array}{l}y \leqslant 3 x \\ y \geqslant \frac{1}{3} x \\ x+y \leqslant 100\end{array}\right.$
|
2551
| 69
| 4
|
math
|
Example 1 Find the number of positive integer solutions to the indeterminate equation
$$7 x+19 y=2012$$
|
15
| 29
| 2
|
math
|
For each positive integer $k$ greater than $1$, find the largest real number $t$ such that the following hold:
Given $n$ distinct points $a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k)$, $\ldots$, $a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k)$ in $\mathbb{R}^k$, we define the score of the tuple $a^{(i)}$ as
\[\prod_{j=1}^{k}\#\{1\leq i'\leq n\textup{ such that }\pi_j(a^{(i')})=\pi_j(a^{(i)})\}\]
where $\#S$ is the number of elements in set $S$, and $\pi_j$ is the projection $\mathbb{R}^k\to \mathbb{R}^{k-1}$ omitting the $j$-th coordinate. Then the $t$-th power mean of the scores of all $a^{(i)}$'s is at most $n$.
Note: The $t$-th power mean of positive real numbers $x_1,\ldots,x_n$ is defined as
\[\left(\frac{x_1^t+\cdots+x_n^t}{n}\right)^{1/t}\]
when $t\neq 0$, and it is $\sqrt[n]{x_1\cdots x_n}$ when $t=0$.
[i]Proposed by Cheng-Ying Chang and usjl[/i]
|
t = \frac{1}{k-1}
| 349
| 12
|
math
|
14th ASU 1980 Problem 5 Are there any solutions in positive integers to a 4 = b 3 + c 2 ?
|
6^4=28^2+8^3
| 33
| 12
|
math
|
9. Let $n$ be an integer greater than 2, and let $a_{n}$ be the largest $n$-digit number that is neither the sum of two perfect squares nor the difference of two perfect squares.
(1) Find $a_{n}$ (expressed as a function of $n$);
(2) Find the smallest value of $n$ such that the sum of the squares of the digits of $n$ is a perfect square.
|
a_{n}=10^{n}-2,\;n_{\}=66
| 97
| 18
|
math
|
11.1. The ore contains $21 \%$ copper, enriched - $45 \%$ copper. It is known that during the enrichment process, $60 \%$ of the mined ore goes to waste. Determine the percentage content of ore in the waste.
|
5
| 56
| 1
|
math
|
6. Given $f(x)=(\sin x+4 \sin \theta+4)^{2}+(\cos x$ $-5 \cos \theta)^{2}$, the minimum value of $f(x)$ is $g(\theta)$. Then the maximum value of $g(\theta)$ is
|
49
| 64
| 2
|
math
|
For a triangle, we know that $a b + b c + c a = 12$ (where $a, b$, and $c$ are the sides of the triangle.) What are the limits within which the perimeter of the triangle can fall?
|
6\leqk\leq4\sqrt{3}
| 53
| 14
|
math
|
2. Several brothers divided a certain amount of gold coins. The first brother took 100 gold coins and $\frac{1}{6}$ of the remainder. The second brother took 200 gold coins and $\frac{1}{6}$ of the new remainder, and so on. The last brother took whatever was left. It turned out that each brother took an equal number of gold coins. How many brothers participated in the division?
|
5
| 91
| 1
|
math
|
Example 10. Calculate the integral $I=\oint_{L}\left(x^{2}-y^{2}\right) d x+2 x y d y$, where $L-$ is the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$.
|
0
| 68
| 1
|
math
|
8. On each face of a cube, randomly fill in one of the numbers 1, 2, $\cdots$, 6 (the numbers on different faces are distinct). Then number the eight vertices such that the number assigned to each vertex is the product of the numbers on the three adjacent faces. The maximum value of the sum of the numbers assigned to the eight vertices is $\qquad$
|
343
| 81
| 3
|
math
|
94. A system of five linear equations. Solve the following system of equations:
$$
\left\{\begin{array}{l}
x+y+z+u=5 \\
y+z+u+v=1 \\
z+u+v+x=2 \\
u+v+x+y=0 \\
v+x+y+z=4
\end{array}\right.
$$
|
v=-2,2,1,3,u=-1
| 75
| 12
|
math
|
Example 1 Find all integers $x$ such that $1+5 \times 2^{x}$ is the square of a rational number. ${ }^{[1]}$
(2008, Croatia National Training (Grade 2))
|
x=-2 \text{ or } 4
| 51
| 10
|
math
|
Consider the following function:
$$
y=\frac{2 x^{2}-5 a x+8}{x-2 a}
$$
$1^{0}$. For which values of $a$ will this function be strictly increasing?
$2^{0}$. For which values of $a$ will this function have a maximum and a minimum?
$3^{0}$. The values of $a$ that separate the conditions in $1^{0}$. and $2^{0}$. are certain $a_{1}$ and $a_{2}$. What happens to the function for these values of $a$?
$4^{0}$. Imagine all the curves $C$ that correspond to the function for all possible values of $a$, plotted on the same coordinate system. What geometric locus is described by the intersection points of the asymptotes of these $C$ curves? Show that all $C$ curves pass through two fixed points!
|
P_{1}(4,10),\quadP_{2}(-4,-10)
| 195
| 21
|
math
|
11. (20 points) If the equation about $x$
$$
x^{2}+k x-12=0 \text { and } 3 x^{2}-8 x-3 k=0
$$
have a common root, find all possible values of the real number $k$.
|
1
| 66
| 1
|
math
|
2.283. $\frac{\left(p q^{-1}+1\right)^{2}}{p q^{-1}-p^{-1} q} \cdot \frac{p^{3} q^{-3}-1}{p^{2} q^{-2}+p q^{-1}+1}: \frac{p^{3} q^{-3}+1}{p q^{-1}+p^{-1} q-1}$.
|
1
| 96
| 1
|
math
|
10. Given the set $\Omega=\left\{(x, y) \mid x^{2}+y^{2} \leqslant 2008\right\}$. If points $P(x, y)$ and $P^{\prime}\left(x^{\prime}, y^{\prime}\right)$ satisfy $x \leqslant x^{\prime}$ and $y \geqslant y^{\prime}$, then point $P$ is said to dominate $P^{\prime}$. If a point $Q$ in the set $\Omega$ satisfies: there does not exist any other point in $\Omega$ that dominates $Q$, then the set of all such points $Q$ is
|
\left\{(x, y) \mid x^{2}+y^{2}=2008, x \leqslant 0\right. \text{ and } \left.y \geqslant 0\right\}
| 154
| 54
|
math
|
$$
\begin{array}{l}
\text { 9. For } \frac{1}{2} \leqslant x \leqslant 1 \text {, when } \\
(1+x)^{5}(1-x)(1-2 x)^{2}
\end{array}
$$
reaches its maximum value, $x=$ $\qquad$ .
|
x=\frac{7}{8}
| 83
| 8
|
math
|
3. Last year, the number of students (girls and boys) in a certain school was 850. This year, the number of boys decreased by $4 \%$, and the number of girls increased by $3 \%$, after which the number of students in the school is 844. What is the number of girls in the school this year?
|
412
| 75
| 3
|
math
|
5. Problem: A sequence of $A$ s and $B \mathrm{~s}$ is called antipalindromic if writing it backwards, then turning all the $A$ s into $B \mathrm{~s}$ and vice versa, produces the original sequence. For example $A B B A A B$ is antipalindromic. For any sequence of $A \mathrm{~s}$ and $B \mathrm{~s}$ we define the cost of the sequence to be the product of the positions of the $A \mathrm{~s}$. For example, the string $A B B A A B$ has cost $1 \cdot 4 \cdot 5=20$. Find the sum of the costs of all antipalindromic sequences of length 2020 .
|
2021^{1010}
| 175
| 10
|
math
|
Give the prime factorization of 2020.
Give the prime factorization of 2021.
|
2021=43^{1}\times47^{1}
| 24
| 16
|
math
|
6、$F_{1}, F_{2}$ are the two foci of the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, and $P$ is a point on the ellipse. If the area of $\Delta P F_{1} F_{2}$ is $1$, $\tan \angle P F_{1} F_{2}=\frac{1}{2}$, and $\tan \angle P F_{2} F_{1}=-2$. Then $a=\frac{\sqrt{15}}{2}$.
|
\frac{\sqrt{15}}{2}
| 133
| 11
|
math
|
55. How many different letter combinations can be obtained by rearranging the letters in the word a) NONNA; b) MATHEMATICA?
|
20
| 31
| 2
|
math
|
8th Irish 1995 Problem A2 Find all integers n for which x 2 + nxy + y 2 = 1 has infinitely many distinct integer solutions x, y.
|
all\n\except\-1,0,1
| 40
| 10
|
math
|
1. Determine for which pairs of positive integers $m$ and $n$ the following inequality holds:
$$
\sqrt{m^{2}-4}<2 \sqrt{n}-m<\sqrt{m^{2}-2} .
$$
|
(,n)=(,^{2}-2),where\geqq2
| 50
| 15
|
math
|
Three, (50 points) Try to find the smallest positive number $a$, such that there exists a positive number $b$, for which when $x \in [0,1]$, the inequality $\sqrt{1-x}+\sqrt{1+x} \leqslant 2 - b x^{a}$ always holds; for the obtained $a$, determine the largest positive number $b$ that satisfies the above inequality.
|
=2,b=\frac{1}{4}
| 90
| 10
|
math
|
[ The Law of Cosines [Isosceles, Inscribed, and Circumscribed Trapezoids]
The bases of the trapezoid are 3 cm and 5 cm. One of the diagonals of the trapezoid is 8 cm, and the angle between the diagonals is $60^{\circ}$. Find the perimeter of the trapezoid.
#
|
22
| 83
| 2
|
math
|
Problem 10.4. Consider the sequence
$$
a_{n}=\cos (\underbrace{100 \ldots 0^{\circ}}_{n-1})
$$
For example, $a_{1}=\cos 1^{\circ}, a_{6}=\cos 100000^{\circ}$.
How many of the numbers $a_{1}, a_{2}, \ldots, a_{100}$ are positive?
|
99
| 103
| 2
|
math
|
23. (CZS 4) Find all complex numbers $m$ such that polynomial
$$
x^{3}+y^{3}+z^{3}+\text { mxyz }
$$
can be represented as the product of three linear trinomials.
|
-3,-3\omega,-3\omega^2
| 59
| 12
|
math
|
8. Define the sequence $\left\{a_{n}\right\}: a_{n}=n^{3}+4, n \in \mathbf{N}_{+}$, and let $d_{n}=\left(a_{n}, a_{n+1}\right)$, i.e., $d_{n}$ is the greatest common divisor of $a_{n}$ and $a_{n+1}$, then the maximum value of $d_{n}$ is $\qquad$.
|
433
| 104
| 3
|
math
|
$ABCDEF$ is a hexagon inscribed in a circle such that the measure of $\angle{ACE}$ is $90^{\circ}$. What is the average of the measures, in degrees, of $\angle{ABC}$ and $\angle{CDE}$?
[i]2018 CCA Math Bonanza Lightning Round #1.3[/i]
|
45^\circ
| 77
| 4
|
math
|
5. A cube with side length 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}
| 53
| 7
|
math
|
4. Given an acute triangle $\triangle A B C$ with three interior angles satisfying $\angle A>\angle B>\angle C$, let $\alpha$ represent the minimum of $\angle A-\angle B$, $\angle B-\angle C$, and $90^{\circ}-\angle A$. Then the maximum value of $\alpha$ is $\qquad$
|
15^{\circ}
| 74
| 6
|
math
|
Example 2 The equation $2 x^{2}+5 x y+2 y^{2}=2007$ has $\qquad$ different integer solutions.
(2007, National Junior High School Mathematics League Sichuan Preliminary Competition)
|
4
| 54
| 1
|
math
|
30. Think of a single-digit number: double it, add 3, multiply by 5, add 7, using the last digit of the resulting number, write down a single-digit number, add 18 to it, and divide the result by 5. What number did you get? No matter what number you thought of, you will always get the same number in the final result. Explain why.
|
4
| 86
| 1
|
math
|
Find all functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that for all $x, y \in \mathbb{R}$,
$$
f\left(x^{666}+y\right)=f\left(x^{2023}+2 y\right)+f\left(x^{42}\right)
$$
|
f(x)=0
| 81
| 4
|
math
|
17th Chinese 2002 Problem B1 Given four distinct points in the plane such that the distance between any two is at least 1, what is the minimum possible value for the sum of the six distances between them?
|
6+(\sqrt{3}-1)
| 48
| 8
|
math
|
Given the numbers $1,2,3, \ldots, 1000$. Find the largest number $m$, such that: no matter which $m$ of these numbers are erased, among the remaining $1000-m$ numbers, there will be two such that one divides the other.
|
499
| 65
| 3
|
math
|
9. A coffee shop launched a "50% off coffee" promotion, stipulating: the first cup is at the original price, the second cup is at half price, and the third cup only costs 3 yuan. Xiao Zhou drank 3 cups of coffee that day, with an average cost of 19 yuan per cup. What is the original price of one cup of coffee? $(\quad)$ yuan.
|
36
| 86
| 2
|
math
|
234. $x^{2}-x y+y^{2}-x+3 y-7=0$, if it is known that the solution is: $x=3 ; y=1$.
|
(3,1),(-1,1),(3,-1),(-3,-1),(-1,-5)
| 42
| 24
|
math
|
3. Find the natural numbers of the form $\overline{ab}$ which, when divided by 36, give a remainder that is a perfect square.
(7 points)
## E:14178 from GM 11/2011
|
16,25,36,37,40,45,52,61,72,73,76,81,88,97
| 55
| 41
|
math
|
1. Define the operation " $\odot$ " as $\mathrm{a} \odot \mathrm{b}=\mathrm{a} \times \mathrm{b}-(\mathrm{a}+\mathrm{b})$, please calculate: $6 \odot(5 \odot 4)=$
|
49
| 66
| 2
|
math
|
Find all positive integers $(x, y, n, k)$ such that $x$ and $y$ are coprime and such that
$$
3^{n}=x^{k}+y^{k}
$$
|
(x,y,k)=(2,1,3)
| 46
| 10
|
math
|
## Task 6
Peter has 12 pennies, Annett gives him enough so that he has 20 pennies.
How much money does Annett give? Write an equation.
|
8
| 40
| 1
|
math
|
1. 1.1. An online electric scooter rental service charges a fixed amount for a ride and a certain amount for each minute of rental. Katya paid 78 rubles for a ride lasting 3 minutes. Lena paid 108 rubles for a ride lasting 8 minutes. It takes Kolya 5 minutes to get from home to work. How much will he pay for this ride?
|
90
| 86
| 2
|
math
|
9. A large rectangle is divided into four identical smaller rectangles by slicing parallel to one of its sides. The perimeter of the large rectangle is 18 metres more than the perimeter of each of the smaller rectangles. The area of the large rectangle is $18 \mathrm{~m}^{2}$ more than the area of each of the smaller rectangles. What is the perimeter in metres of the large rectangle?
|
28
| 84
| 2
|
math
|
833. Find all two-digit numbers that are equal to triple the product of their digits.
|
15,24
| 20
| 5
|
math
|
Find all pairs of non-negative integers $x$ and one-digit natural numbers $y$ for which
$$
\frac{x}{y}+1=x, \overline{y}
$$
The notation on the right side of the equation denotes a repeating decimal.
(K. Pazourek)
Hint. Is there any connection between the decimal expansions of the numbers $\frac{x}{y}+1$ and $\frac{1}{y}$?
|
1,30,9
| 92
| 6
|
math
|
8. Given that $A$ and $B$ are points on $C_{1}: x^{2}-y+1=0$ and $C_{2}: y^{2}-x+1=0$ respectively. Then the minimum value of $|A B|$ is $\qquad$
|
\frac{3\sqrt{2}}{4}
| 62
| 12
|
math
|
A [i]Beaver-number[/i] is a positive 5 digit integer whose digit sum is divisible by 17. Call a pair of [i]Beaver-numbers[/i] differing by exactly $1$ a [i]Beaver-pair[/i]. The smaller number in a [i]Beaver-pair[/i] is called an [i]MIT Beaver[/i], while the larger number is called a [i]CIT Beaver[/i]. Find the positive difference between the largest and smallest [i]CIT Beavers[/i] (over all [i]Beaver-pairs[/i]).
|
79200
| 132
| 5
|
math
|
Jennifer wants to do origami, and she has a square of side length $ 1$. However, she would prefer to use a regular octagon for her origami, so she decides to cut the four corners of the square to get a regular octagon. Once she does so, what will be the side length of the octagon Jennifer obtains?
|
1 - \frac{\sqrt{2}}{2}
| 71
| 12
|
math
|
B4. For the second round of the Math Olympiad, 999 students are invited. Melanie prepares invitation letters in the order of participant numbers: $1,2,3, \ldots$ For some values of $n \geqslant 100$, she notices the following: the number of participant numbers from $1$ to $n$ that end in a 5 is exactly equal to the number formed by the last two digits of $n$.
For how many values of $n$ (with $100 \leqslant n<1000$) does this hold?
|
9
| 131
| 1
|
math
|
4. A, B, and C went to the orchard to pick oranges, and they picked a total of 108 oranges. A first gave half of his oranges to B, then B gave $\frac{1}{5}$ of his oranges at that time to C, and finally C gave $\frac{1}{7}$ of his oranges at that time to A. If at this point the three of them had the same number of oranges, how many oranges did each of them initially pick?
|
60,15,33
| 103
| 8
|
math
|
9. Let $n$ be a natural number, for any real numbers $x, y, z$ there is always $\left(x^{2}+y^{2}+z^{2}\right)^{2} \leqslant n\left(x^{4}+y^{4}+z^{4}\right)$, then the minimum value of $n$ is $\qquad$
|
3
| 84
| 1
|
math
|
[Sum of interior and exterior angles of a polygon] [Sum of angles in a triangle. Theorem of the exterior angle.]
Find the sum of the interior angles:
a) of a quadrilateral;
b) of a convex pentagon
c) of a convex $n$-gon
#
|
)360;b)540;)180(n-2)
| 61
| 17
|
math
|
4B. Find all five-digit numbers $\overline{a b c d e}$ such that $\overline{a b}$, $\overline{b c}$, $\overline{c d}$, and $\overline{d e}$ are perfect squares.
|
81649
| 56
| 5
|
math
|
385*. Solve the system of equations:
$$
\left\{\begin{array}{l}
x=\frac{2 y^{2}}{1+z^{2}} \\
y=\frac{2 z^{2}}{1+x^{2}} \\
z=\frac{2 x^{2}}{1+y^{2}}
\end{array}\right.
$$
|
(0;0;0),(1;1;1)
| 77
| 13
|
math
|
4. Let $f(x)=(x+a)(x+b)$ (where $a, b$ are given positive real numbers), and $n \geqslant 2$ be a given integer. For non-negative real numbers $x_{1}, x_{2}, \cdots, x_{n}$ satisfying
$$
x_{1}+x_{2}+\cdots+x_{n}=1
$$
find the maximum value of
$$
F=\sum_{1 \leq i<j \leq n} \min \left\{f\left(x_{i}\right), f\left(x_{j}\right)\right\}
$$
(Leng Gangsong)
|
\frac{n-1}{2}\left(\frac{1}{n}+a+b+n a b\right)
| 145
| 25
|
math
|
Exercise 4. Determine all triplets of positive integers $(\mathrm{a}, \mathrm{b}, \mathrm{n})$ satisfying:
$$
a!+b!=2^{n}
$$
|
(1,1,1),(2,2,2),(2,3,3),(3,2,3)
| 41
| 25
|
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