problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Find the equation of the hyperbola that passes through point $A(3,-1)$ and has its axes of symmetry along the coordinate axes. | \dfrac{x^2}{8} - \dfrac{y^2}{8} = 1 |
Example 6 Let $n$ be a positive integer. Calculate
$$
P=\prod_{k=0}^{n-1} \sin \left(\theta+\frac{k \pi}{n}\right) .
$$ | \frac{\sinn\theta}{2^{n-1}} |
Given the function f(x) = 2^x + 2^(-x), find the value of f(log₂3). | \frac{10}{3} |
Given 95 numbers $a_{1}, a_{2}, \cdots, a_{95}$, each of which can only take one of the two values +1 or -1, then the sum of their pairwise products
$$
a_{1} a_{2}+a_{1} a_{3}+\cdots+a_{94} a_{95}
$$
the smallest positive value is $\qquad$. | 13 |
Galperin G.A.
For each pair of numbers $x$ and $y$, a certain number $x^{*} y$ is assigned. Find 1993*1935, given that for any three numbers $x, y, z$ the identities $x^{*} x=0$ and $x^{*}\left(y^{*} z\right)=\left(x^{*} y\right)+z$ hold. | 58 |
## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(-1 ; 2 ; 4)$
$M_{2}(-1 ;-2 ;-4)$
$M_{3}(3 ; 0 ;-1)$
$M_{0}(-2 ; 3 ; 5)$ | \frac{5}{9} |
1. Let $a_{1}, a_{2}, \ldots, a_{n}$ be integers $(n>1)$ satisfying $a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=2005$. Find the smallest possible value of $n$.
(1 mark)
Let $a_{1}, a_{2}, \ldots, a_{n}$ be integers $(n>1)$, such that $a_{1}+a_{2}+\cdots+a_{n}=a_{1} a_{2} \cdots a_{n}=2005$. Find the smallest pos... | 5 |
We denote by gcd (...) the greatest common divisor of the numbers in (...). (For example, gcd$(4, 6, 8)=2$ and gcd $(12, 15)=3$.) Suppose that positive integers $a, b, c$ satisfy the following four conditions:
$\bullet$ gcd $(a, b, c)=1$,
$\bullet$ gcd $(a, b + c)>1$,
$\bullet$ gcd $(b, c + a)>1$,
$\bullet$ gcd $(c, a... | 30 |
Let $f : \mathbb{N} \to \mathbb{N}$ be a function such that the following conditions hold:
$\qquad\ (1) \; f(1) = 1.$
$\qquad\ (2) \; \dfrac{(x + y)}{2} < f(x + y) \le f(x) + f(y) \; \forall \; x, y \in \mathbb{N}.$
$\qquad\ (3) \; f(4n + 1) < 2f(2n + 1) \; \forall \; n \ge 0.$
$\qquad\ (4) \; f(4n + 3) \le 2f(2n +... | 1012 |
Let point $P$ be an intersection point of an ellipse and a hyperbola with common foci $F\_1$ and $F\_2$, and $\cos \angle F\_1PF\_2 = \frac{3}{5}$. The eccentricity of the ellipse is $e\_1$, and the eccentricity of the hyperbola is $e\_2$. If $e\_2 = 2e\_1$, then $e\_1 = (\ \ \ )$
A: $\frac{\sqrt{10}}{4}$
B: $\frac{\s... | D |
In the Nanjing area, the weather in July and August is relatively hot. Xiaohua collected the highest temperature for ten consecutive days, obtaining the following set of data in sequence: 34, 35, 36, 34, 36, 37, 37, 36, 37, 37 (unit: ℃). The mode of this set of data is ▲, and the median is ▲. | 36 |
The interval of monotonic increase for the function $y = \log_{\frac{1}{2}}(x^{2} + 2x - 3)$ is $(\ )$
A: $(-\infty, -3)$
B: $(-\infty, -1)$
C: $(-1, +\infty)$
D: $(1, +\infty)$ | \text{A} |
9. In $\triangle A B C$, let $B C=a, C A=b, A B=c$. If $9 a^{2}+9 b^{2}-19 c^{2}=0$, then $\frac{\cot C}{\cot A+\cot B}=$ $\qquad$ . | \frac{5}{9} |
Given $$\sin \frac{\theta}{2} = \frac{3}{5}, \cos \frac{\theta}{2} = -\frac{4}{5}$$, then $\theta$ is an angle in the ( ) quadrant:
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | \text{D} |
The solution set of the inequality $\lg(x-1) < 2$ is. | (1, 101) |
A ray of light starts from point $A(-3, 9)$ and reflects off the $x$-axis onto circle $C$: $(x-2)^2 + (y-3)^2 = 1$. Find the shortest path length. | 12 |
The equation $x^2+y^2+2ax-2ay=0$ (where $a\neq 0$) represents a circle that is symmetric with respect to ( )
A: the x-axis
B: the y-axis
C: the line $x-y=0$
D: the line $x+y=0$ | D |
In the arithmetic sequence $\{a_n\}$, if $a_2 + a_4 + a_9 + a_11 = 32$, then $a_6 + a_7 =$ ( )
(A) 9 (B) 12 (C) 15 (D) 16 | \text{D} |
3.355. $\sin ^{4} \frac{\pi}{8}+\cos ^{4} \frac{3 \pi}{8}+\sin ^{4} \frac{5 \pi}{8}+\cos ^{4} \frac{7 \pi}{8}$. | \frac{3}{2} |
A prism has at least \_\_\_\_\_\_ faces, the prism with the least number of faces has \_\_\_\_\_\_ edges, \_\_\_\_\_\_ lateral edges, and \_\_\_\_\_\_ vertices. | 5; 9; 3; 6 |
10. The solution set of the inequality $\sqrt{a x+b}>x \quad(a>0, b>0)$ is | [-\frac{b}{},\frac{+\sqrt{^{2}+4b}}{2}) |
16. Two bullets are placed in two consecutive chambers of a 6-chamber pistol. The cylinder is then spun. The pistol is fired but the first shot is a blank. Let $p$ denote the probability that the second shot is also a blank if the cylinder is spun after the first shot and let $\mathrm{q}$ denote the probability that th... | 89 |
(4) The number of solutions to the equation $\pi^{x-1} x^{2}+\pi^{x^{2}} x-\pi^{x^{2}}=x^{2}+x-1$ is
$\qquad$ , where $\pi$ is the ratio of a circle's circumference to its diameter. | 2 |
(8) Let $A=\left\{a \mid a=2^{x} 3^{y} 5^{z}, x, y, z \in \mathbf{N}\right\}, B=\{b \mid b \in A, 1 \leqslant b \leqslant 10\}$, find the number of subsets of $B$ that include the elements 1 and 10. | 128 |
Alice is counting up by fives, starting with the number $3$. Meanwhile, Bob is counting down by fours, starting with the number $2021$. How many numbers between $3$ and $2021$, inclusive, are counted by both Alice and Bob? | 101 |
Given \\(a > 0\\), if points \\(A(a,0)\\), \\(B(0,a)\\), \\(C(-4,0)\\), \\(D(6,0)\\), and \\(E(0,-6)\\) satisfy that the circumcircle of \\(\triangle ABC\\) is tangent to line \\(DE\\), then the value of \\(a\\) is \_\_\_\_\_\_. | 2 \sqrt {5} |
Given the sets $A=\{1,3\}$, $B=\{x|0<\lg(x+1)<\frac{1}{2}, x\in Z\}$, then $A\cup B=(\ )$
A: $\{1,3\}$
B: $\{1,2,3\}$
C: $\{1,3,4\}$
D: $\{1,2,3,4\}$ | B |
Example 7 In the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$, the degree of the dihedral angle $A-B D_{1}-A_{1}$ is | 60^{\circ} |
7. If real numbers $a, b, c$ satisfy
$$
a+2 b+3 c=6, a^{2}+4 b^{2}+9 c^{2}=12 \text {, }
$$
then $a b c=$ . $\qquad$ | \frac{4}{3} |
## Problem 1
Calculate the sum
$$
S=[\lg 1]+[\lg 2]+[\lg 3]+\cdots+\left[\lg 10^{2014}\right]
$$ | \frac{10}{9}(2013\cdot10^{2014}-2014\cdot10^{2013}+1)+2014 |
7. In a math competition, there are three problems: $A$, $B$, and $C$. Among the 39 participants, each person answered at least one question correctly. Among those who answered $A$ correctly, those who only answered $A$ are 5 more than those who answered other questions as well; among those who did not answer $A$ corre... | 23 |
Let's determine the parameters $p$ and $q$ such that the roots of equation (1) are real (not necessarily distinct) and can be arranged in such a way that (2) is satisfied.
$$
x^{4} + p x^{2} + q = 0
$$
$$
\frac{x_{2}}{x_{1}} = \frac{x_{3}}{x_{2}} = \frac{x_{4}}{x_{3}}
$$ | p<0p^{2}=4q |
10.3. In the distant times of stagnation in the Soviet Union, 15 and 20 kopeck coins were in circulation. Schoolboy Valera had a certain amount of money only in such coins. Moreover, the number of 20 kopeck coins was greater than the number of 15 kopeck coins. Valera spent one-fifth of all his money, paying two coins f... | 2 |
## Problem Statement
Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.
$$
\rho=6 \cos \varphi, 0 \leq \varphi \leq \frac{\pi}{3}
$$ | 2\pi |
10,11
The side of the base $ABC$ of the pyramid $TABCD$ is 4, the lateral edge $TA$ is perpendicular to the plane of the base. Find the area of the section of the pyramid by a plane passing through the midpoints of the edges $AC$ and $BT$ and parallel to the median $BD$ of the face $BCT$, given that the distance from ... | \frac{6}{\sqrt{5}} |
5・146 Find integer coefficient polynomials $P(x)$ and $Q(x)$, such that
$$\frac{P(\sqrt{2}+\sqrt{3}+\sqrt{5})}{Q(\sqrt{2}+\sqrt{3}+\sqrt{5})}=\sqrt{3}+\sqrt{2}$$ | P(x)=3 x^{4}-20 x^{2}+24, Q(x)=4 x^{3} |
Solve the inequality system: $\left\{\begin{array}{l}{2x-1≥x+2}\\{x+5＜4x-1}\end{array}\right.$. | x \geq 3 |
Find the equation of the circle with its center at the right focus of the ellipse $\frac{x^2}{169} + \frac{y^2}{144} = 1$ and tangent to the asymptotes of the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$. The options are:
A: $x^2 + y^2 - 10x + 9 = 0$
B: $x^2 + y^2 - 10x - 9 = 0$
C: $x^2 + y^2 + 10x + 9 = 0$
D: $x^2 ... | A |
A foreign investor plans to invest in 3 different projects across four candidate cities, with no more than 2 projects in the same city. The number of different investment plans available to the investor is ( )
A: 60
B: 42
C: 36
D: 16 | A |
Translate the function $y=\sin \omega x$ (where $\omega > 0$) to the right by $\frac{\pi}{4}$ units. The resulting graph passes through the point $\left(\frac{3\pi}{4},0\right)$. The minimum value of $\omega$ is ____.
A: $\frac{1}{3}$
B: $1$
C: $\frac{5}{3}$
D: $2$ | D |
Calculate the value of $\lg \frac{5}{2} + 2\lg2 - \left(\frac{1}{2}\right)^{-1}$. | -1 |
10.008. In an isosceles triangle with a lateral side equal to 4 cm, a median of the lateral side is drawn. Find the base of the triangle if the median is 3 cm. | \sqrt{10} |
The solution to the linear equation $8x=2x-6$ is ( ).
A: $x=1$
B: $x=0$
C: $x=-2$
D: $x=-1$ | D |
Simplify first, then evaluate: $\frac{3}{2}x^2y-[x^2y-3(2xy-x^2y)-xy]$, where $\left(x-1\right)^{2}+|y+2|=0$. | -9 |
2. How many ways are there to arrange 5 identical red balls and 5 identical blue balls in a line if there cannot be three or more consecutive blue balls in the arrangement? | 126 |
16. In $\triangle A B C, A B=33 \mathrm{~cm}, A C=21 \mathrm{~cm}$ and $B C=x \mathrm{~cm}$, where $x$ is an integer. $D$ is a point on the segment $A B$ and $E$ is a point on the segment $A C$ such that $A D=D E=E C=y \mathrm{~cm}$, where $y$ is again an integer. Find $x$.
(2 marks)
在 $\triangle A B C$ 中, $A B=33 \mat... | 30 |
In an information technology exam, the scores (in points) of 8 students in a hobby group were as follows: 7, 10, 9, 8, 7, 9, 9, 8. The mode of this set of data is
A: 7
B: 8
C: 9
D: 10 | \text{C} |
1.6.11 Find all $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any $x, y \in \mathbf{R}$, we have
$$
f\left(x^{2}+y+f(y)\right)=2 y+f(x)^{2} .
$$ | f(x)=x |
45th Putnam 1984 Problem A2 Evaluate 6/( (9 - 4)(3 - 2) ) + 36/( (27 - 8)(9 - 4) ) + ... + 6 n /( (3 n+1 - 2 n+1 )(3 n - 2 n ) ) + ... . Solution | 2 |
7. Given three numbers $a, b, c$. It is known that the arithmetic mean of the numbers $a$ and $b$ is 5 more than the arithmetic mean of all three numbers. And the arithmetic mean of the numbers $a$ and $c$ is 8 less than the arithmetic mean of all three numbers. By how much does the arithmetic mean of the numbers $b$ a... | 3 |
Compute
\[\frac{(10^4+324)(22^4+324)(34^4+324)(46^4+324)(58^4+324)}{(4^4+324)(16^4+324)(28^4+324)(40^4+324)(52^4+324)}.\] | 373 |
11.10 a) Find the reduced quadratic trinomial $P(x)$, for which the graph $\mathrm{y}=P(x)$ is symmetric with respect to the Oy axis and is tangent to the line $y=x$.
b) Let $P(x)$ be a quadratic trinomial with a positive leading coefficient. Prove that if the equation $P(x)=x$ has a unique solution, then the equation... | P(x)=x^{2}+1/4 |
8. For any real number $x$, $[x]$ represents the greatest integer not exceeding $x$. When $0 \leqslant x \leqslant 100$, the range of $f(x)=[3 x]+[4 x]+[5 x]+[6 x]$ contains $\qquad$ different integers. | 1201 |
3. Given the equation $x^{2}-1999 x+a=0$ has two prime roots. Then the constant $a=$ $\qquad$ . | 3994 |
The value of $\frac{3 \times 4}{6}$ is
(A) 1
(B) 2
(C) 3
(D) 4
(E) 6 | 2 |
If the complex number $z=1-mi$ (where $i$ is the imaginary unit, $m\in\mathbb{R}$), and $z^2=-2i$, then the imaginary part of the complex number $z$ is. | -1 |
Calculate:$(1)-4.7+0.9$;$(2)-\frac{1}{2}-(-\frac{1}{3})$;$(3)(-1\frac{1}{9})×(-0.6)$;$(4)0\times \left(-5\right)$. | 0 |
For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ with $a > 0$, $b > 0$, its asymptotic line equation is $y = x$. Determine the eccentricity $e$ of the hyperbola. | \sqrt{2} |
Given that one asymptote of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1 (a > 0, b > 0)$ passes through the point $(2, \sqrt{3})$, and one focus of the hyperbola is $F(- \sqrt{7}, 0)$, find the equation of the hyperbola. | \frac{x^{2}}{4} - \frac{y^{2}}{3} = 1 |
Find all the pairs positive integers $(x, y)$ such that
$$
\frac{1}{x}+\frac{1}{y}+\frac{1}{[x, y]}+\frac{1}{(x, y)}=\frac{1}{2}
$$
where $(x, y)$ is the greatest common divisor of $x, y$ and $[x, y]$ is the least common multiple of $x, y$. | (8,8),(9,24),(24,9),(5,20),(20,5),(12,15),(15,12),(8,12),(12,8),(6,12),(12,6) |
5. A race on an unpredictable distance is held as follows. On a circular running track, two points $A$ and $B$ are randomly selected (using a spinning arrow), after which the athletes run from $A$ to $B$ along the shorter arc. A spectator has bought a ticket to the stadium and wants the athletes to run past his seat (t... | \frac{1}{4} |
To obtain the graph of the function $y=\sin \left(2x-\frac{\pi}{3}\right)$, it is only necessary to shift all points on the graph of the function $y=\sin 2x$
A: Shift to the left by $\frac{\pi}{3}$ units
B: Shift to the right by $\frac{\pi}{3}$ units
C: Shift to the left by $\frac{\pi}{6}$ units
D: Shift to the... | \text{D} |
In a box, there are 10 ping-pong balls of the same size but different colors, including 5 red balls, 3 white balls, and 2 blue balls. Now, a ball is randomly drawn from the box each time. If a blue ball is drawn, the process ends. If a ball that is not blue is drawn, it is put back into the box, and another ball is dra... | \frac {153}{1000} |
## Task 3
Draw a line segment that is $4 \mathrm{~cm}$ long, and below it, draw a second one that is $3 \mathrm{~cm}$ longer! Draw a third line segment that is three times as long as the first one! | 7\mathrm{~},12\mathrm{~} |
8. Let $n$ be a positive integer. For positive real numbers $a_{1}, a_{2}, \cdots, a_{n}$, we have $a_{1}+a_{2}+\cdots+a_{n}=1$. Let $A$ be the minimum of the following $n$ numbers $\frac{a_{1}}{1+a_{1}}, \frac{a_{2}}{1+a_{1}+a_{2}}, \cdots, \frac{a_{n}}{1+a_{1}+a_{2}+\cdots+a_{n}}$. Find the maximum value of $A$ when ... | 1-\frac{1}{\sqrt[n]{2}} |
5.2. Given three points $A(2 ; 1), B(3 ;-1), C(-4 ; 0)$, which are vertices of an isosceles trapezoid $A B D C$. Find the coordinates of point $D$, if $\overline{A B}=k \overline{C D}$. | (-1.4;-5.2) |
Determine the range of the function $f(x) = \cos\left(x - \frac{\pi}{3}\right)$, where $x \in \left[0, \frac{\pi}{2}\right]$. | \left[\frac{1}{2}, 1\right] |
## Task B-2.3.
Solve the equation in the set of real numbers
$$
4^{2 x+\sqrt{-1+x^{2}}}-5 \cdot 2^{2 x-1+\sqrt{-1+x^{2}}}=6
$$ | 1 |
Exercise 2. The government of Bosnia and Herzegovina has decided to implement a new license plate system. Each license plate must contain 8 digits, each of which can be $0,1,2,3,4,5,6,7,8$ or 9. In addition, two distinct license plates must always have at least two different digits. For example, if it puts the plate 00... | 10^7 |
2.2. In a triangle, one of the angles is less than $40^{\circ}$, and another is less than $80^{\circ}$. Find the cosine of the third angle if its sine is $\frac{5}{8}$. | -\frac{\sqrt{39}}{8} |
4. A. Xiao Qian and Xiao Ling each have several RMB notes with integer yuan denominations. Xiao Qian says to Xiao Ling: "If you give me 2 yuan, my money will be $n$ times yours"; Xiao Ling says to Xiao Qian: "If you give me $n$ yuan, my money will be 2 times yours", where $n$ is a positive integer. Then the number of p... | D |
2. $\prod_{i=1}^{2018} \sin \frac{i \pi}{2019}=$ $\qquad$ | \frac{2019}{2^{2018}} |
Among the four students A, B, C, and D participating in competitions in mathematics, writing, and English, each subject must have at least one participant (and each participant can only choose one subject). If students A and B cannot participate in the same competition, the total number of different participation schem... | 30 |
In the diagram with seven cells, draw stars so that all the following conditions are met:
1. There are a total of 21 stars.
2. Each cell contains at least one star.
3. The cells labeled $A, B, C$ together contain 8 stars.
4. The cells labeled $A$ and $B$ together contain fewer stars than the cell labeled $C$.
5. The c... | A=1,B=2,C=5,D=6,E=3,F=3,G=1 |
The equation $x^2+kx-k=0$ has two distinct real roots $x_1$ and $x_2$, and it satisfies $1 < x_1 < 2 < x_2 < 3$. The range of the real number $k$ is ( )
A: $(- \frac {9}{2}, -4)$
B: $(4, \frac {9}{2})$
C: $(-6, -4)$
D: $(-4, \frac {4}{3})$ | \text{A} |
Among the numbers -5, -3, -1, 2, 4, 6, the largest product obtained by multiplying any three of them is __________. | 90 |
A certain school organized a practical activity for handcrafting. The slant height of a cone made by a student is $30$ cm, and the radius of the base circle is $10$ cm. The central angle of the unfolded diagram of the lateral surface of this cone is ( ).
A: $30^{\circ}$
B: $60^{\circ}$
C: $90^{\circ}$
D: $120^{\cir... | D |
2. (7 points) From Monday to Wednesday, the gnome eats semolina porridge for breakfast, from Thursday to Saturday - rice porridge, and on Sunday he makes himself an omelet. On even-numbered days of the month, the gnome tells the truth, and on odd-numbered days - lies. On which of the first ten days of August 2016 could... | OnTuesday,August2,onWednesday,August3,onFriday,August5,onMonday,August8 |
Given the function $f(x)= \frac{2x}{|x|+1}$ ($x\in R$), interval $M=[a,b]$ (where $a < b$), and set $N={y|y=f(x),x\in M}$, determine the number of real number pairs $(a,b)$ that make $M=N$ true. | 3 |
5. Let positive integers $a, b, c, d$ satisfy $a>b>c>d$, and $a+b+c+d=2004, a^{2}-b^{2}+c^{2}-d^{2}=2004$. Then the minimum value of $a$ is . $\qquad$ | 503 |
Problem 11.2. Vera has a set of weights of different masses, each weighing an integer number of grams. It is known that the lightest weight in the set weighs 71 times less than all the other weights combined. It is also known that the two lightest weights in the set together weigh 34 times less than all the other weigh... | 35 |
Which of the following calculations is correct?
A: $a^{3}+a^{2}=a^{5}$
B: $a^{3}\cdot a^{2}=a^{5}$
C: $(a^{2})^{3}=a^{5}$
D: $a^{10}\div a^{2}=a^{5}$ | B |
Given the circle \\(O: x^2 + y^2 = 9\\), and a line \\(l\\) intersects the circle \\(O\\) at points \\(M\\) and \\(N\\), with \\(|MN| = 4\\), then \\(\overrightarrow{MN} \cdot \overrightarrow{MO} = \\) ( )
A: \\(2\\)
B: \\(3\\)
C: \\(4\\)
D: \\(8\\) | D |
Given an ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ ($a>0$, $b>0$) with its left focus $F$ and right vertex $A$, upper vertex $B$. If the distance from point $F$ to line $AB$ is $\frac{5\sqrt{14}}{14}b$, find the eccentricity of the ellipse. | \frac{2}{3} |
If the value of the fraction $\frac{x^2-9}{x+3}$ is $0$, then the value of $x$ is:
A: $3$
B: $3$ or $-3$
C: $-3$
D: $0$ | \text{A} |
Example 1 Given that $a, b$ are positive real numbers, the equations
$$
x^{2}+a x+2 b=0 \text{ and } x^{2}+2 b x+a=0
$$
both have real roots. Find the minimum value of $a^{2}+b^{2}$. | 20 |
Given a complex number $z=4-3i$, find $\overline{z}=$____ and $|z|=$____. | 5 |
Given a sequence ${a_n}$ that satisfies the equation $$a_{1}+2a_{2}+2^{2}a_{3}+…+2^{n-1}a _{ n }^{}= \frac {n^{2}}{2}(n∈N^{*})$$, find the general term $a_n$. | a_n = (n - \frac{1}{2}) \cdot \frac{1}{2^{n-1}} |
In the expansion of $(3x-1)^{6}$, the coefficient of $x^{2}$ is ________ (use numbers to answer). | 135 |
13. 6:10 ** Given the parabola $y^{2}=4 a x(0<a<1)$ with focus $F$, a circle is drawn above the $x$-axis with center $A(a+4,0)$ and radius $|A F|$, intersecting the parabola at two different points $M$ and $N$. Let $P$ be the midpoint of $M N$.
(1) Find the value of $|M F|+|N F|$;
(2) Does there exist a value of $a$ su... | 8 |
8. Variant 1.
On side $A B$ of parallelogram $A B C D$, a point $F$ is chosen, and on the extension of side $B C$ beyond vertex $B$, a point $H$ is chosen such that $A B / B F = B C / B H = 5$. Point $G$ is chosen so that $B F G H$ is a parallelogram. $G D$ intersects $A C$ at point $X$. Find $A X$, if $A C = 100$. | 40 |
Real numbers $x, y$ satisfy the inequality $x^2 + y^2 \le 2$. Orove that $xy + 3 \ge 2x + 2y$ | xy + 3 \ge 2x + 2y |
Subject 2. Find the real numbers $\mathrm{x}$ and $\mathrm{y}$ for which we have
$$
\sqrt{4 x^{2}-12 x+25}+\sqrt{4 x^{2}+12 x y+9 y^{2}+64} \leq 12
$$ | \frac{3}{2},-1 |
A point $M$ is chosen inside the square $ABCD$ in such a way that $\angle MAC = \angle MCD = x$ . Find $\angle ABM$. | 45^\circ |
4. Two players, A and B, are playing a game that is a best-of-seven series, where the first to win four games is declared the winner, and the series ends. If in each game, both players have a $\frac{1}{2}$ probability of winning, then the expected value of the number of games by which the winner exceeds the loser is $\... | \frac{35}{16} |
Find all positive integers $x$ and $y$ for which
$$
\frac{1}{x}+\frac{1}{y}=\frac{1}{4}
$$
(A. Bohiniková)
Hint. Can both unknowns be greater than, for example, 14 at the same time? | (5,20),(6,12),(8,8),(12,6),(20,5) |
Standa and Jana received two three-digit numbers. Standa added a decimal point after the first digit in the first number, and after the second digit in the second number, then added the resulting decimal numbers and got the result 50.13. Jana added a decimal point after the second digit in the first number, and after t... | 765 |
10. (20 points) Point $A$ lies on the line $y=\frac{12}{5} x-9$, and point $B$ lies on the parabola $y=x^{2}$. What is the minimum length of segment $A B$? | \frac{189}{65} |
29. The product of some prime numbers is 30030. Is their sum a prime or a composite number? | 41 |
4. Dima recorded his digital password in code: P, AR, AK, RU, SS. The key to the code: different digits are represented by different letters, and the same digits by the same letters, two letters together represent a two-digit number, numbers are separated by commas, the difference between any two adjacent numbers is th... | 5,12,19,26,33 |
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