problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Given point $(a,b)$ is outside the circle $x^{2}+y^{2}=r^{2}$, the position relationship between the line $ax+by=r^{2}$ and the circle is $( \ )$
A: Separate
B: Tangent
C: Intersecting and not passing through the center
D: Intersecting and passing through the center | C |
Two positive numbers $x$ and $y$ are in the ratio $a: b$ where $0 < a < b$. If $x+y = c$, then the smaller of $x$ and $y$ is
$\text{(A)} \ \frac{ac}{b} \qquad \text{(B)} \ \frac{bc-ac}{b} \qquad \text{(C)} \ \frac{ac}{a+b} \qquad \text{(D)}\ \frac{bc}{a+b}\qquad \text{(E)}\ \frac{ac}{b-a}$ | \frac{ac}{+b} |
Given that the function $f(x)=λ\sin x+\cos x$ has a symmetry axis with the equation $x=\frac{\pi}{6}$, determine the maximum value of this function. | \frac{2\sqrt{3}}{3} |
17. Given the set $S \subseteq\{1,2, \cdots, 12\},|S|=6$ ( $|S|$ represents the number of elements in the set $S$). If $a, b \in$ $S, a<b$, then $b$ is not a multiple of $a$. Then, the smallest element in the set $S$ is ( ).
(A) 2
(B) 3
(C) 4
(D) 5
(E) 7. | C |
The solution set of the inequality system $\left\{\begin{array}{l}{x-2≥-5①}\\{3x＜x+2②}\end{array}\right.$ is ______. | -3 \leq x < 1 |
5. Eight numbers $a_{1}, a_{2}, a_{3}, a_{4}$ and $b_{1}, b_{2}, b_{3}, b_{4}$ satisfy the relations
$$
\left\{\begin{aligned}
a_{1} b_{1}+a_{2} b_{3} & =1 \\
a_{1} b_{2}+a_{2} b_{4} & =0 \\
a_{3} b_{1}+a_{4} b_{3} & =0 \\
a_{3} b_{2}+a_{4} b_{4} & =1
\end{aligned}\right.
$$
It is known that $a_{2} b_{3}=7$. Find $a_... | -6 |
1. Compute the following expression: \\
$(\dfrac{4}{9})^{-\dfrac{3}{2}}+(\left(-2\right)^6)^{\dfrac{1}{2}}-\lg 0.4-2\lg 0.5-14\times\log_2 \sqrt{2}$ \\
2. Given a point $P(\sin \alpha,\cos \alpha)$ situated on the line $y=\dfrac{1}{2}x$, find the value of $\dfrac{\cos (\pi-\alpha)+\sin (\pi+\alpha)}{\cos (\dfrac{1}{... | -\dfrac{1}{5} |
4. Through the focus $F$ of the parabola $E: y^{2}=2 x$, two lines $l_{1}, l_{2}$ are drawn with the product of their slopes being $-\frac{1}{2}$. Line $l_{1}$ intersects $E$ at points $A, C$, and line $l_{2}$ intersects $E$ at points $B, D$. Then the minimum value of $|A C|+2|B D|$ is $\qquad$. | 8\sqrt{2}+6 |
Problem 2. A group of adventurers is showing off their loot. It is known that exactly 9 adventurers have rubies; exactly 8 have emeralds; exactly 2 have sapphires; exactly 11 have diamonds. Moreover, it is known that
- if an adventurer has diamonds, then they have either rubies or sapphires (but not both at the same t... | 17 |
19. $\tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}=\frac{r}{p}$. | \frac{r}{p} |
## Task A-2.7.
The rows of a $50 \times 50$ table are labeled with numbers $a_{1}, \ldots, a_{50}$, and the columns with numbers $b_{1}, \ldots, b_{50}$. These 100 numbers are all distinct, and exactly 50 of them are rational numbers. The table is filled such that for $i, j=1,2, \ldots, 50$, the number $a_{i}+b_{j}$ i... | 1250 |
2. In the interval $0 \leq x \leq \pi$ find the solutions to the equation
$$
\frac{1}{\sin x}-\frac{1}{\cos x}=2 \sqrt{2}
$$ | \frac{3\pi}{4},\frac{\pi}{12} |
How many two-digit numbers are there where the digit in the tens place is greater than the digit in the units place?
# | 45 |
Problem 9.7. The numbers $1, 2, 3, \ldots, 57$ are written on a board. What is the maximum number of these numbers that can be chosen so that no two chosen numbers differ exactly by a factor of 2.5? | 48 |
[ Investigation of a quadratic trinomial ] [ Methods for solving problems with parameters ]
For the quadratic trinomial $f(x)=a x^{2}-a x+1$, it is known that $|f(x)| \leq 1$ for $0 \leq x \leq 1$. Find the greatest possible value of $a$. | 8 |
If $2(z+\overline{z})+3(z-\overline{z})=4+6i$, then $z=\left(\ \ \right)$
A: $1-2i$
B: $1+2i$
C: $1+i$
D: $1-i$ | C |
8,9, |
In triangle $A B C \angle B=110^{\circ}, \angle C=50^{\circ}$. On side $A B$ a point $P$ is chosen such that $\angle P C B=30^{\circ}$, and on side $A C$ a point $Q$ is chosen such that
$\angle A B Q=40^{\circ}$. Find the angle $Q P C$. | 40 |
Example 3.3 Find the sum $\sum_{k=0}^{n} k(k-1)(k+2)$.
| \frac{1}{4}n^{4}+\frac{5}{6}n^{3}-\frac{1}{4}n^{2}-\frac{5}{6}n |
From 10 basketballs, one is randomly selected to test its quality. The sampling method that should be used is ____. | \text{Simple random sampling} |
In the Cartesian coordinate system $xOy$, the following definitions are given: the smaller value of the distances from point $A$ to the $x$-axis and the $y$-axis is called the "short distance" of point $A". When the "short distance" of point $P$ is equal to the "short distance" of point $Q$, points $P$ and $Q$ are call... | k = 1 \text{ or } 2 |
## Task 1.
Determine all pairs $(a, b)$ of real numbers for which
$$
a+\frac{a+8 b}{a^{2}+b^{2}}=2 \quad \text { and } \quad b+\frac{8 a-b}{a^{2}+b^{2}}=0
$$ | (,b)\in{(-1,2),(3,-2)} |
In how many ways can four married couples sit around a round table so that no man sits next to his own wife? | 1488 |
6. (3 points) On a $3 \times 3$ chessboard, there are knights, who always tell the truth, and liars, who always lie. Each of them said: "Among my neighbors, there are exactly three liars." How many liars are on the board?
Neighbors are considered to be people on cells that share a common side. | 5 |
Given $f(x,y)=(x-y)^{2}+(4+ \sqrt {1-x^{2}}+ \sqrt {1- \frac {y^{2}}{9}})^{2}$, find the maximum value of $f(x,y)$. | 28+6 \sqrt {3} |
A2. Which equation represents a line?
(A) $x y=5$
(B) $\frac{2 x}{3}-1=3 y$
(C) $y=x^{2}-3$
(D) $3 x-y^{2}-1=0$
(E) $y=x^{\frac{1}{2}}$ | \frac{2x}{3}-1=3y |
Given the function $f(x)=mx^{2}-2x+m$ with a range of $[0,+∞)$, find the value of the real number $m$. | 1 |
16. (5 points)
The Reuleaux triangle is a famous shape. By using the vertices of an equilateral triangle as the centers and its side length as the radius to draw three circular arcs, a Reuleaux triangle can be formed. In the figure below, the perimeter of the Reuleaux triangle is $314 \mathrm{~cm}$, then the side lengt... | 100 |
Given that $i$ is the imaginary unit, find the value of $\frac{1-\sqrt{3}i}{(\sqrt{3}+i)^2}=(\ \ \ \ \ )$
A: $\frac{1}{4}+\frac{\sqrt{3}}{4}i$
B: $\frac{1}{2}+\frac{\sqrt{3}}{2}i$
C: $-\frac{1}{2}-\frac{\sqrt{3}}{2}i$
D: $-\frac{1}{4}-\frac{\sqrt{3}}{4}i$ | D |
Given that the area of $\triangle ABC$ is $S$, and the sides opposite to angles $A$, $B$, and $C$ are respectively $a$, $b$, and $c$:
1. If $S=(a+b)^2-c^2$, and $a+b=4$, find the value of $\sin C$.
2. Prove that: $$\frac{a^2-b^2}{c^2} = \frac{\sin(A-B)}{\sin C}$$.
3. Compare the magnitude of $a^2+b^2+c^2$ and $4\sqrt{3... | a^2 + b^2 + c^2 \geq 4\sqrt{3}S |
6. Given an increasing sequence composed of powers of 3 or the sum of several different powers of 3: $1,3,4,9,10,12,13, \cdots$. Then the 100th term of this sequence is $\qquad$ | 981 |
Denote by $d(n)$ be the biggest prime divisor of $|n|>1$. Find all polynomials with integer coefficients satisfy;
$$P(n+d(n))=n+d(P(n)) $$
for the all $|n|>1$ integers such that $P(n)>1$ and $d(P(n))$ can be defined. | P(n) = n |
4. Divide 1996 into the sum of 96 positive integers, the maximum value of the product of these 96 positive integers is | 20^{20} \cdot 21^{76} |
(The full score of this question is 12 points) In a box, there are three cards labeled 1, 2, and 3, respectively. Now, two cards are drawn from this box with replacement in succession, and their labels are denoted as $x$ and $y$, respectively. Let $\xi = |x-2| + |y-x|$.
(1) Find the range of the random variable $\xi$... | \frac{2}{9} |
5. Given $[x]$ denotes the greatest integer not exceeding $x$, $a, b, c \in$ $R^{+}, a+b+c=1$, let $M=\sqrt{3 a+1}+\sqrt{3 b+1}+$ $\sqrt{3 c+1}$. Then the value of $[M]$ is $($. | 4 |
5. Let triangle $A B C$ be such that $A B=A C=22$ and $B C=11$. Point $D$ is chosen in the interior of the triangle such that $A D=19$ and $\angle A B D+\angle A C D=90^{\circ}$. The value of $B D^{2}+C D^{2}$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100 a+b$... | 36104 |
Given that a number $x$ is randomly chosen from the interval $[0,2π]$, find the probability that $2\sin x > 1$. | \frac{1}{3} |
Simplify first, then evaluate: $(a\sqrt{\frac{1}{a}}+\sqrt{4b})-(\frac{\sqrt{a}}{2}-b\sqrt{\frac{1}{b}})$, where $a=4$ and $b=3$. | 1 + 3\sqrt{3} |
Given a function $f(x) = ax^3 + bx^2 + cx$ where $a \neq 0$, and $6a + b = 0$, $f(1) = 4a$.
$(1)$ Discuss the monotonicity of $f(x)$.
$(2)$ If $x \in [0,3]$, the function $F(x) = f(x) - xe^{-x}$ has three zeros $x_1$, $x_2$, $x_3$ with $x_1 < x_2 < x_3$. Compare $x_1 + x_2 + x_3$ with $2$ and explain the reason. | x_1 + x_2 + x_3 < 2 |
If the ordinate of a point is three times its abscissa, then this point is called a "three times point." For example, $A(1,3)$, $B(-2,-6)$, $C(0,0)$ are all "three times points." In the range of $-3 \lt x \lt 1$, if the graph of the quadratic function $y=-x^{2}-x+c$ has at least one "three times point," what is the ran... | D |
1. Can the number 8 be represented as the sum of eight integers such that the product of these numbers is also equal to 8? | Yes |
Example 8. Find the region of convergence of the series
$$
\sum_{n=1}^{\infty} \frac{(1+i)^{n+1}}{z^{n}}
$$ | |z|>\sqrt{2} |
Problem 1. In an orthogonal coordinate system $x O y$ a set consisiting of 2000 points $M_{i}\left(x_{i}, y_{i}\right)$, is called "good" if $0 \leq x_{i} \leq 83,0 \leq$ $y_{i} \leq 1 \quad i=1,2, \ldots, 2000$ and $x_{i} \neq x_{j}$ for $i \neq j$. Find all natural numbers $n$ with the following properties: : a) For... | 25 |
5.2. Find the sum of all integer values of the argument $x$ for which the corresponding values of the function
$$
y=x^{2}+x\left(\log _{2} 20-\log _{5} 8\right)-\log _{2} 5-9 \log _{5} 2
$$
do not exceed 6. | -12 |
3. Let $O$ be the origin, $A$ be a moving point on the parabola $x=\frac{1}{4} y^{2}+1$, and $B$ be a moving point on the parabola $y=x^{2}+4$. Then the minimum value of the area of $\triangle O A B$ is $\qquad$ . | 2 |
Given an ellipse $C:\frac{{{x^2}}}{{{a^2}}}+\frac{{{y^2}}}{{{b^2}}}=1(a＞b＞0)$ passing through the point $({\sqrt{3},\frac{1}{2}})$, with its right focus at $F({\sqrt{3},0})$.$(1)$ Find the standard equation of the ellipse $C$;$(2)$ Let the right vertex of the ellipse $C$ be $A$. If points $P$ and $Q$ lie on the ellipse... | \frac{5}{3} |
6. Given in $\triangle A B C$, $\angle B A C=\frac{2 \pi}{3}, A B=2, A C=1, D$ is a point on side $B C$, $D C=2 B D$, then $\overrightarrow{A D} \cdot \overrightarrow{B C}=$ $\qquad$ - | -\frac{8}{3} |
Given that $S\_n=\\cos \\dfrac {π}{8}+\\cos \\dfrac {2π}{8}+…+\\cos \\dfrac {nπ}{8}(n∈N^{+})$, determine the number of positive terms in the sequence $S\_1$, $S\_2$, …, $S\_{2015}$.
A: $882$
B: $756$
C: $750$
D: $378$ | B |
Given that the angle between the plane vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $60^{\circ}$, $|\overrightarrow{a}| = 1$, and $\overrightarrow{b} = (3, 0)$, find $|2\overrightarrow{a} + \overrightarrow{b}|$. | \sqrt{19} |
Given that {a\_n} is a geometric sequence with all positive terms, if a\_4 * a\_8 = 4, then find the value of a\_5 * a\_6 * a\_7. | 8 |
In $\triangle ABC$, it is known that $A=\frac{\pi}{3}$, $b=1$, and the radius of the circumcircle of $\triangle ABC$ is $1$. Find the area of $\triangle ABC$. | \frac{\sqrt{3}}{2} |
4B. Determine the set of values of the expression $(x-y)(y-z)(z-x)$ if $x, y$ and $z$ are real numbers for which
$$
\sqrt{x-y+z}=\sqrt{x}-\sqrt{y}+\sqrt{z}
$$ | 0 |
Selecting $2$ out of $6$ experts including A and B to visit a certain place for inspection, the probability that at least one of A and B will be selected is ( )
A: $\frac{4}{5}$
B: $\frac{2}{3}$
C: $\frac{2}{5}$
D: $\frac{3}{5}$ | D |
Given that the focus of the parabola $y=x^{2}$ is $F$, a line passing through point $F$ intersects the parabola at points $A$ and $B$. If $|AB|=4$, find the distance from the midpoint of chord $AB$ to the $x$-axis. | \frac{7}{4} |
2. (8 points) From 1999 to 2010, over a period of 12 years, the price increase was $150 \%$ (i.e., items that could be bought with 100 yuan in 1999 would cost 150 yuan more in 2010). If the wages of frontline employees in a certain company have not changed over these 12 years, in terms of purchasing power, this is equi... | 60 |
Problem 4.3. Zhenya drew a square with a side of 3 cm, and then erased one of these sides. A figure in the shape of the letter "P" was obtained. The teacher asked Zhenya to place dots along this letter "P", starting from the edge, so that the next dot was 1 cm away from the previous one, as shown in the picture, and th... | 31 |
10.5. In a row, $n$ integers are written such that the sum of any three consecutive numbers is positive, while the sum of any five consecutive numbers is negative. For what largest $n$ is this possible | 6 |
2. If $a, b, c > 0, \frac{1}{a}+\frac{2}{b}+\frac{3}{c}=1$, then the minimum value of $a+2b+3c$ is . $\qquad$ | 36 |
A 3x3x3 cube is made of $27$ normal dice. Each die's opposite sides sum to $7$. What is the smallest possible sum of all of the values visible on the $6$ faces of the large cube?
$\text{(A)}\ 60 \qquad \text{(B)}\ 72 \qquad \text{(C)}\ 84 \qquad \text{(D)}\ 90 \qquad \text{(E)} 96$ | 90 |
8 The three sides of a triangle are $t^{2}+t+1$, $t^{2}-1$, and $2 t+1$, then the largest interior angle of the triangle is $\qquad$ . | 120 |
[ Examples and counterexamples. Constructions ]
It is known about numbers $a$ and $b$ that $a=b+1$. Can it happen that $a^{4}=b^{4}$? | =1/2,b=-1/2 |
274. $\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$.
Translate the above text into English, keeping the original text's line breaks and format, and output the translation result directly.
274. $\int_{0}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}$.
The above text is already in a mathematical format and does not require transl... | \frac{\pi}{6} |
7. A sequence begins $2023,2022,1, \ldots$. After the first two terms, each term is the positive difference between the previous two terms.
What is the value of the $25^{\text {th }}$ term?
A 2010
B 2009
C 2008
D 2007
E 2006 | 2007 |
A quadrilateral has two adjacent sides of lengths 2 and 1 units, respectively, and the included angle between them is $60^{\circ}$. The quadrilateral is both a cyclic and a tangential quadrilateral. What are the lengths of the other two sides of the quadrilateral? | \frac{\sqrt{33}+3}{6},\frac{\sqrt{33}-3}{6} |
The sum of all three-digit numbers that leave a remainder of 1 when divided by both 8 and 9 is ( )
A: 6492
B: 6565
C: 7501
D: 7574 | A |
Let \\(\triangle ABC\\) have internal angles \\(A\\), \\(B\\), and \\(C\\) opposite to sides \\(a\\), \\(b\\), and \\(c\\) respectively, satisfying \\((a+b+c)(a-b+c)=ac\\).
\\((\\)Ⅰ\\()\\) Find \\(B\\).
\\((\\)Ⅱ\\()\\) If \\(\sin A\sin C= \dfrac { \sqrt {3}-1}{4}\\), find \\(C\\). | C=15^{\circ} \text{ or } C=45^{\circ} |
## Problem Statement
Calculate the definite integral:
$$
\int_{1}^{e} \frac{x^{2}+\ln x^{2}}{x} d x
$$ | \frac{e^{2}+1}{2} |
In the diagram, $A B$ and $C D$ intersect at $E$. If $\triangle B C E$ is equilateral and $\triangle A D E$ is a right-angled triangle, what is the value of $x$ ?
(A) 90
(B) 60
(C) 25
(D) 45
(E) 30
Economist Sasha has a 2016-sided die. The probability of each face landing is the same. Sasha rolls this die 2016 times and counts the sum of the points that come up, but unfortu... | 2033138 |
4. Given real numbers $x, y$ satisfy
$\frac{x^{2}}{3}+y^{2}=1$, then the range of $P=|2 x+y-4|+|4-x-2 y|$ is $\qquad$ | [2,14] |
Given the function $f(x)= \frac {a-x}{x+1}e^{2x}(x>-1)$.
(I) Discuss the monotonicity of the function $f(x)$;
(II) Prove that when $m\in[0,1)$, the function $g(x)= \frac {2mx+m-e^{2x}}{x^{2}}(x>0)$ has a maximum value. Let the maximum value of $g(x)$ be $h(m)$, find the range of the function $h(m)$. | [-e^2,-2) |
Problem 4. Positive numbers $a, b, c, d$ are greater than 1. Find the smallest possible value of the expression
$$
\log _{a}\left(a b^{2}\right)+\log _{b}\left(b^{2} c^{3}\right)+\log _{c}\left(c^{5} d^{6}\right)+\log _{d}\left(d^{35} a^{36}\right)
$$ | 67 |
Let $\{a_n\}$ be an arithmetic sequence with a non-zero common difference. Given $a_1 = 4$ and that $a_1, a_3, a_6$ form a geometric sequence, find the sum of the first $n$ terms of $\{a_n\}$, denoted as $S_n$. | S_n = \frac{n^2 + 7n}{2} |
8. In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks.
How many cows will eat all the grass on six ... | 5 |
Let $\omega_1, \omega_2, \omega_3, \ldots, \omega_{2020!}$ be the distinct roots of $x^{2020!} - 1$. Suppose that $n$ is the largest integer such that $2^n$ divides the value $$\sum_{k=1}^{2020!} \frac{2^{2019!}-1}{\omega_{k}^{2020}+2}.$$ Then $n$ can be written as $a! + b$, where $a$ and $b$ are positive integers, and... | 31 |
Example 5 In $\triangle ABC$, $AB=BC>AC$, $AH$ and $AM$ are the altitude and median from vertex $A$ to side $BC$, respectively, and $\frac{S_{\triangle MHH}}{S_{\triangle ABC}}=\frac{3}{8}$. Determine the value of $\cos \angle BAC$. [s]
$(2009$, Beijing High School Mathematics Competition Preliminary (High $(-))$ | \frac{1}{4} |
2. A store sells a certain product and makes a profit of $m$ yuan per item, with a profit margin of $20\%$ (profit margin $=\frac{\text{selling price} - \text{cost price}}{\text{cost price}}$). If the cost price of this product increases by $25\%$, and the store raises the selling price so that it still makes a profit ... | C |
12. (2005 Czech-Polish-Slovak Mathematical Competition) Find all integer pairs $(x, y)$ that satisfy the equation $y(x+y)=x^{3}-7 x^{2}+11 x-3$. | (6,3),(6,-9),(1,1),(1,-2),(2,-1) |
If $\left(\frac{1}{2+3}\right)\left(\frac{1}{3+4}\right)=\frac{1}{x+5}$, the value of $x$ is
(A) 4
(B) 7
(C) 30
(D) 37
(E) 67 | 30 |
$13 \cdot 31$ The maximum value of the function $y=-x-\frac{9}{x}+18(x>0)$ is
(A) 24 .
(B) 18 .
(C) 12 .
(D) 2 .
(China Guangzhou, Wuhan, Fuzhou, etc. five cities junior high school mathematics league, 1991) | 12 |
Given an array $(a\_1, a\_2, a\_3, a\_4, a\_5)$ consisting of a permutation of the numbers $1$, $2$, $3$, $4$, $5$, such that each permutation corresponds to only one array, and in each array, there is exactly one element $a\_i$ such that $a\_i = i$ ($i=1,2,3,4,5$). The sum of all the numbers in all distinct arrays is ... | 675 |
15. In Bethany's class of 30 students, twice as many people played basketball as played football. Twice as many played football as played neither.
Which of the following options could have been the number of people who played both?
A 19
B 14
C 9
D 5
E 0 | 5 |
The minimum value of the function $f(x)=x^{3}+ \frac {3}{x}$ on the interval $(0,+\infty)$ is ______. | 4 |
Given the function $f(x)=|x-1|+|x-a|$ where $a\in \mathbb{R}$.
$(1)$ If $a=4$, find the solution set of the inequality $f(x) \geqslant 5$;
$(2)$ If there exists $x \in \mathbb{R}$ such that $f(x) \leqslant 4$ holds, find the range of values for $a$. | -3 \leqslant a \leqslant 5 |
Let's calculate the area of the triangle if we know the lengths of the medians $\left(s_{a}, s_{b}, s_{c}\right)$. | t_{\triangleABC}=\frac{1}{3}\sqrt{(s_a+s_b+s_c)(s_b+s_c-s_a)(s_a+s_c-s_b)(s_a+s_b-s_c)} |
5. Given that $x, y, z$ are 3 non-negative real numbers, and satisfy $3x + 2y + z = 5, x + y - z = 2$. If $s = 2x + y - z$, then the sum of the maximum and minimum values of $s$ is ( ).
(A) 5
(B) $\frac{23}{4}$
(C) $\frac{27}{4}$
(D) $\frac{35}{4}$ | A |
Find the sum of all positive integers $1\leq k\leq 99$ such that there exist positive integers $a$ and $b$ with the property that \[x^{100}-ax^k+b=(x^2-2x+1)P(x)\] for some polynomial $P$ with integer coefficients.
[i]Proposed by David Altizio[/i] | 117 |
3. Vasya's grandmother loves to pick berries in the forest and then make jam from them. Every weekend she goes by train, spending 200 rubles on tickets and collects 5 kilograms of berries. On the market, you can buy a kilogram of berries for 150 rubles, and sugar for 54 rubles per kilogram. From one kilogram of berries... | 1 |
4. Let square $A B C D$ have a side length of 2 and center $O$. Denote by $E$ the midpoint of side $A B$. Calculate the radius of the circumcircle of triangle $E O C$. | \frac{\sqrt{10}}{2} |
## Task $9 / 89$
Determine all non-negative real numbers $x$ that satisfy the equation:
$$
\sqrt[3]{x}+\sqrt[4]{x}=6 \sqrt[6]{x}
$$ | x_1=0,x_2=4096 |
11. (20 points) Let $x_{i} \in \mathbf{R}_{+}(i=1,2, \cdots, n, n \geqslant 2)$, $m \in \mathbf{N}, m \geqslant 2$, and $\sum_{i=1}^{n} x_{i}=T$, where $k$, $a$, $T$, and $m$ are given positive real numbers, and let $W=\sum_{i=1}^{n} \sqrt[m]{k x_{i}+a}$. Find the range of $W$. | \left[\sqrt[m]{k T+a}+(n-1) \sqrt[m]{a}, n \sqrt[m]{\frac{k T+n a}{n}}\right] |
Given $\cos \left(\frac{\pi}{4}+x\right)=\frac{3}{5}, \frac{17 \pi}{12}<x<\frac{7 \pi}{4}$, find $\frac{\sin 2 x+2 \sin ^{2} x}{1-\tan x}$. | \frac{2 \times \left(-\frac{7 \sqrt{2}}{10}\right) \times \left(-\frac{\sqrt{2}}{10}\right) + 2 \times \left(-\frac{7 \sqrt{2}}{10}\right)^2}{1 - 7} |
3. Color the points on a plane in the following way:
(1) Choose any positive integer $m$, let $K_{1}, K_{2}, \cdots, K_{m}$ be different circles with non-zero radii, and $K_{i} \subset K_{j}$ or $K_{j} \subset$ $\mathrm{K}_{i}(i \neq j)$;
(2) The color of any point outside a chosen circle is different from the color of... | 2019 |
4. Calculate $\sqrt{4+\sqrt{7}}-\sqrt{4-\sqrt{7}}$. | \sqrt{2} |
1. Today is Thursday. What day will it be in 2023 days' time?
A Tuesday
B Wednesday
C Thursday
D Friday
E Saturday | C |
5. In $\triangle A B C$, it is known that $\angle A B C=44^{\circ}, D$ is a point on side $B C$, satisfying $D C=2 A B, \angle B A D=$ $24^{\circ}$. Then the size of $\angle A C B$ is $\qquad$ | 22^{\circ} |
2.1. Gavriila found out that the front tires of the car last for 20000 km, while the rear tires last for 30000 km. Therefore, he decided to swap them at some point to maximize the distance the car can travel. Find this distance (in km). | 24000 |
Given an infinite geometric sequence $\{b_{n}\}$ with the first term $2$ and a common ratio of $\frac{1}{3}$, then $\sum_{i=1}^{+\infty}{{b}_{i}=}$______. | 3 |
## Task A-2.1.
Let $a, b$ and $c$ be real numbers such that $a+2b+c>0$ and $a-2b+c>0$ and $a > c$. | b^{2}> |
64. Rearrange the digits of a three-digit number, the difference between the largest and smallest three-digit numbers obtained is exactly equal to the original three-digit number. This three-digit number is $\qquad$ . | 495 |
Tímár Mihály found himself in a difficult situation because the red crescent fell off the bag where the treasure was hidden. All he knows is that the treasure is hidden in the heaviest of the four bags, buried in the wheat. It was revealed during three measurements that the first bag together with the second is lighter... | D |
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