problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Consider polynomial functions $ax^2 -bx +c$ with integer coefficients which have two distinct zeros in the open interval $(0,1).$ Exhibit with proof the least positive integer value of $a$ for which such a polynomial exists. | 5 |
The full score for this question is 10 points.
Given the function $f(x) = x^2 + 2x + a\ln x$.
(1) If $f(x)$ is a monotonic function on the interval $(0,1)$, find the range of values for $a$;
(2) If for all $t \geq 1$, $f(2t - 1) \geq 2f(t) - 3$, find the range of values for $a$. | a \leq 2 |
## Aufgabe 2 - 181222
Man ermittle alle diejenigen reellen Zahlen $x$, für die durch $k=\frac{x}{x^{2}-5 x+7}$ eine ganze Zahl $k$ definiert ist.
| x_{1}=0,x_{2,3}=3\\sqrt{2},x_{4}=\frac{7}{2},x_{5}=2,x_{6}=\frac{7}{3},x_{7}=3 |
Problem 7.5. A rectangular strip of length 16 was cut into two strips of lengths 9 and 7. These two strips were placed on the table as shown in the figure.
It is known that the area of the part of the table covered only by the left strip is 27, and the area of the part of the table covered only by the right strip is 1... | 13.5 |
I4.1 If the curve $y=2 x^{2}-8 x+a$ touches the $x$-axis, find $a$. | 8 |
In a basketball shooting game, the rules are as follows: each round consists of up to three shots, stopping when the first shot is made. Scoring is as follows: making the first shot scores 8 points; missing the first but making the second scores 6 points; missing the first two but making the third scores 4 points; miss... | 6 |
4.5.5 ** If for all positive integers $m, n$ the inequality
$$
\left(\frac{1}{m+n}\right)^{2} \leqslant k\left(\frac{1}{m^{2}}+\frac{1}{n^{2}}\right)
$$
holds, find the minimum value of the real number $k$. | \frac{1}{8} |
Problem 5. In the cells of an $8 \times 8$ chessboard, there are 8 white and 8 black chips such that no two chips are in the same cell. Additionally, no column or row contains chips of the same color. For each white chip, the distance to the black chip in the same column is calculated. What is the maximum value that th... | 32 |
10.2 Find the number conjugate to the number $z=\frac{2 \sqrt{3}-i}{\sqrt{3}+i}+3$. | 4.25+0.75\sqrt{3}i |
Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on th... | 8 |
1. A circle inscribed in a right angle of an isosceles right triangle divides the hypotenuse into 3 equal parts. Find the radius of the circle if the leg of the triangle is 1. | \frac{1}{3} |
Two numbers are added together to give a sum of 100. If one addend is decreased by 48 and the other addend remains unchanged, the new sum is ______. | 52 |
Given that $α$ is an angle in the third quadrant and $\cos α=-\frac{12}{13}$, find the value of $\tan α$. | \frac{5}{12} |
A group of children riding on bicycles and tricycles rode past Billy Bob's house. Billy Bob counted $7$ children and $19$ wheels. How many tricycles were there?
$\mathrm{(A)}\ 2 \qquad\mathrm{(B)}\ 4 \qquad\mathrm{(C)}\ 5 \qquad\mathrm{(D)}\ 6 \qquad\mathrm{(E)}\ 7$ | 5 |
If the function $f(x) = \log_{a}(x + \sqrt{x^2 + 2a^2})$ is an odd function, find the value of $a$. | \frac{\sqrt{2}}{2} |
A square has an area of 25 . A rectangle has the same width as the square. The length of the rectangle is double its width. What is the area of the rectangle?
(A) 25
(B) 12.5
(C) 100
(D) 50
(E) 30 | 50 |
336. Using Euler's theorem, find such values of $x$ for which $a x-b$ is divisible by $c$, if $(a, c)=1$. | ^{\varphi()-1}+ |
Example 1 In the tetrahedron $A B C D$, it is known that
$$
A B=A C=A D=D B=5, B C=3, C D=4 \text {. }
$$
Then the volume of the tetrahedron is $\qquad$ . | 5 \sqrt{3} |
The equation of the hyperbola that passes through the focus of the parabola $y^2=4x$ and has asymptotes given by the equations $x \pm y = 0$ is ( )
A: $x^2-y^2=4$
B: $x^2-y^2=2$
C: $x^2-y^2=1$
D: $x^2-y^2=-1$ | \text{C} |
7. Triangle $A B C$ is the right angled triangle with the vertex $C$ at the right angle. Let $P$ be the point of reflection of $C$ about $A B$. It is known that $P$ and two midpoints of two sides of $A B C$ lie on a line. Find the angles of the triangle. | 30,60,90 |
10.039. The lines containing the non-parallel sides of an isosceles trapezoid intersect at a right angle (Fig. 10.39). Find the lengths of the sides of the trapezoid if its area is \(12 \mathrm{~cm}^{2}\) and the height is \(2 \mathrm{~cm}\).
=\ln \left( {{e}^{x}}+{{e}^{-x}} \right)+{{x}^{2}}$, then the range of $x$ for which $f\left( 2x \right) > f\left( x+3 \right)$ holds is $(\quad)$
A: $\left( -1,3 \right)$
B: $\left( -\infty ,-3 \right)\bigcup \left( 3,+\infty \right)$
C: $\left( -3,3 \right)$
D: $\left( -\infty... | D |
If a uniform dice (a cubic toy with faces numbered 1, 2, 3, 4, 5, 6 respectively) is thrown twice in succession, the probability that the sum of the numbers facing up is 4 is. | \frac{1}{12} |
The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\text{nine}}.$ What is the remainder when $N$ is divided by $5?$
$\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 2\qquad\textbf{(D) } 3\qquad\textbf{(E) }4$ | 3 |
26 Find all functions $f: \mathbf{R} \rightarrow \mathbf{R}$, such that for any real numbers $x, y$, we have
$$
f(f(x)+y)=2 x+f(f(y)-x) .
$$ | f(x)=x+ |
\section*{Problem 2 - 071022}
Given a rectangle \(A B C D\). The midpoint of \(A B\) is \(M\). Connect \(C\) and \(D\) with \(M\) and \(A\) with \(C\). The intersection point of \(A C\) and \(M D\) is \(S\).
Determine the ratio of the area of the rectangle \(A B C D\) to the area of the triangle \(\triangle S M C\)! | 6:1 |
3. Simplify the fraction $\frac{2 x^{6}+5 x^{4}-3 x^{3}+2 x^{2}-12 x-14}{4 x^{6}-4 x^{4}-6 x^{3}-3 x^{2}+25 x-28}$. After simplification, the degrees of the polynomials in the numerator and the denominator should be reduced.
# | \frac{x^{3}+2x+2}{2x^{3}-3x+4} |
3. Solve the system of equations:
$$
\left\{\begin{array}{l}
3 x y-5 y z-x z=3 y \\
x y+y z=-y \\
-5 x y+4 y z+x z=-4 y
\end{array}\right.
$$ | (2,-\frac{1}{3},-3),(x,0,0),(0,0,z) |
The radius of circle $C: x^{2}+y^{2}-4x-2y+1=0$ is ( )
A: $4$
B: $2$
C: $\sqrt{2}$
D: $1$ | B |
Given the equation $\frac{a+2i}{i}=b+i$, where $i$ is the imaginary unit and $a,b \in \mathbb{R}$, find the value of $a+b$. | 1 |
5. The solution to the equation $\frac{x^{3}}{\sqrt{4-x^{2}}}+x^{2}-4=0$ is $\qquad$ | \sqrt{2} |
In $\triangle ABC$, the reciprocals of the three sides $a$, $b$, and $c$ form a geometric sequence. Prove that $B < \dfrac{\pi}{2}$. | B < \dfrac{\pi}{2} |
Which of the following statements about the function $f\left(x\right)=\sin |x|+|\sin x|$ is correct?
A: $f\left(x\right)$ is an even function.
B: $f\left(x\right)$ is monotonically increasing on the interval $(\frac{π}{2},π)$.
C: $f\left(x\right)$ has $4$ zeros in the interval $\left[-\pi ,\pi \right]$.
D: The maxi... | AD |
The circumference of a sector is 6cm, and its area is 2cm2. The central angle of the sector in radians is ( )
A: 1 or 4
B: 1 or 2
C: 2 or 4
D: 1 or 5 | \text{A} |
Authors: Kovaldji A.K., Senderov V.A.
Find all such natural $n$ that for some coprime $x$ and $y$ and natural $k>1$, the equality $3^{n}=x^{k}+y^{k}$ holds. | 2 |
If $1 < a < 3$ and $-4 < b < 2$, then the range of values for $a - |b|$ is \_\_\_\_\_\_\_. | -3, 3 |
Find a general formula, $a_n$, for the sequence 2, 22, 222, 2222, ...
A: $$a_{n}=10^{n}-8$$
B: $$a_{n}= \frac {10^{n}-1}{9}$$
C: $$a_{n}=2^{n}-1$$
D: $$a_{n}= \frac {2(10^{n}-1)}{9}$$ | D |
In a quadrangular pyramid, there can be at most ______ right-angled triangles among the four triangular lateral faces. | 4 |
## Task Condition
Calculate the area of the parallelogram constructed on vectors $a$ and $b$.
$a=6 p-q$
$b=5 q+p$
$|p|=\frac{1}{2}$
$|q|=4$
$(\widehat{p, q})=\frac{5 \pi}{6}$ | 31 |
Example 1 Let positive real numbers $x, y$ satisfy $x y=1$, find the range of the function $f(x, y)=\frac{x+y}{[x][y]+[x]+[y]+1}$ (where $[x]$ denotes the greatest integer not exceeding $x$) | {\frac{1}{2}}\cup[\frac{5}{6},\frac{5}{4}) |
If the quadratic function $y=x^{2}+2mx+\left(m+2\right)$ has two distinct zeros, then the range of $m$ is ()
A: $(-2,1)$
B: $(-1,2)$
C: $(-\infty ,-1)\cup \left(2,+\infty \right)$
D: $(-\infty ,-2)\cup \left(1,+\infty \right)$ | C |
1. Find the solutions and the number of solutions to the following congruence equations by direct computation:
(i) $x^{5}-3 x^{2}+2 \equiv 0(\bmod 7)$;
(ii) $3 x^{4}-x^{3}+2 x^{2}-26 x+1 \equiv 0(\bmod 11)$;
(iii) $3 x^{2}-12 x-19 \equiv 0(\bmod 28)$;
(iv) $3 x^{2}+18 x-25 \equiv 0(\bmod 28)$;
(v) $x^{2}+8 x-13 \equiv ... | x \equiv 1,5(\bmod 7) |
A6. On a wheel lock, we can set a 3-digit code. For each of the three positions, we can choose one of the digits from 0 to 9. What is the maximum number of different codes we can set?
(A) 30
(B) 100
(C) 300
(D) $9^{3}$
(E) 1000 | 1000 |
2. Find the value of the expression $\cos ^{4} \frac{7 \pi}{24}+\sin ^{4} \frac{11 \pi}{24}+\sin ^{4} \frac{17 \pi}{24}+\cos ^{4} \frac{13 \pi}{24}$. | \frac{3}{2} |
II. (25 points) Given a two-digit number, whose tens and units digits are $p$ and $q$ respectively, the quadratic function $y=x^{2}+q x+p$ intersects the $x$-axis at two distinct points $A$ and $B$, with the vertex at $C$, and $S_{\triangle A B C} \leqslant 1$.
(1) Find the range of $q^{2}-4 p$;
(2) Find all such two-d... | 23,65,34,86 |
Given a point $(x, y)$ is above the line $Ax + By + C = 0$ ($A \neq 0, B \neq 0$), then the value of $Ax + By + C$ is ( )
A: The same sign as $A$
B: The opposite sign of $A$
C: The same sign as $B$
D: The opposite sign of $B$ | \text{C} |
Given set A = {x ||x - 2| < a} and set B = {x | (2x - 1)/(x + 2) ≤ 1}, and A is a subset of B, find the range of real number a. | a \in (-\infty, 1] |
Juan and Mary play a two-person game in which the winner gains 2 points and the loser loses 1 point. If Juan won exactly 3 games and Mary had a final score of 5 points, how many games did they play?
(A) 7
(B) 8
(C) 4
(D) 5
(E) 11 | 7 |
Task 2. (10 points) A numerical sequence is given:
$$
x_{0}=\frac{1}{n} ; x_{k}=\frac{1}{n-k}\left(x_{0}+x_{1}+\ldots+x_{k-1}\right) ; k=1,2, \ldots, n-1
$$
Find $S_{n}=x_{0}+x_{1}+\ldots+x_{n-1}$, if $n=2022$. | 1 |
Given \\(m, n \in \mathbb{R}_{+}\\) and \\(m > n\\):
\\((1)\\) If \\(n > 1\\), compare the size of \\(m^{2}+n\\) and \\(mn+m\\), and explain the reason;
\\((2)\\) If \\(m+2n=1\\), find the minimum value of \\( \dfrac {2}{m} + \dfrac {1}{n}\\). | 8 |
Given $x+y+z=m$, prove that $x^2+y^2+z^2 \geq \frac{m^2}{3}$. | x^2+y^2+z^2 \geq \frac{m^2}{3} |
Given: $a \in \mathbb{R}$, $b \in \mathbb{R}$. If the set $\left\{a, \frac{b}{a}, 1\right\} = \left\{a^2, a+b, 0\right\}$, then the value of $a^{2015} + b^{2015}$ is ( )
A: -2
B: -1
C: 1
D: 2 | \text{B} |
6. On the sides $A B$ and $A C$ of an acute-angled triangle $A B C$, two equal rectangles $A M N B$ and $A P Q C$ are constructed outwardly. Find the distance between the vertices $N$ and $Q$ of the rectangles, if the lengths of the sides $A B$ and $A C$ are $2 \sqrt{2}$ and 1, respectively, and the angle at vertex $A$... | 3\sqrt{2+\sqrt{2}} |
Given that the terminal side of angle $\alpha$ passes through the point $P\left( \frac {1}{2}, \frac { \sqrt {3}}{2}\right)$, then $\sin\alpha=$ （ ）
A: $\frac {1}{2}$
B: $\sqrt {3}$
C: $\frac { \sqrt {3}}{2}$
D: $- \frac { \sqrt {3}}{2}$ | C |
Given the parametric equations of line $l$ as $\begin{cases}x=t \\ y=mt\end{cases} (t$ is the parameter$)$, and the parametric equations of circle $C$ as $\begin{cases}x=\cos \alpha \\ y=1+\sin \alpha\end{cases} (\alpha$ is the parameter$)$.
(Ⅰ) If the chord length of the intersection between line $l$ and circle $C$ i... | (x-1)^{2}+(y- \dfrac{1}{2})^{2}= \dfrac{1}{4} |
12.309. The base of the prism is a rectangle. The lateral edge forms equal angles with the sides of the base and is inclined to the plane of the base at an angle $\alpha$. Find the angle between the lateral edge and the side of the base. | \arccos\frac{\sqrt{2}\cos\alpha}{2} |
Let $U=\{0,1,2,3,4\}$, $A=\{0,1,2,3\}$, $B=\{2,3,4\}$. Then, $(C_UA) \cup (C_UB) =$ ( )
A: $\{0,1,2,3,4\}$
B: $\{0,1,4\}$
C: $\{0,1\}$
D: $\{0\}$ | \text{B} |
Given $a \lt b$, which of the following expressions may not hold true?
A: $a-1 \lt b-1$
B: $-2a \gt -2b$
C: $2a+1 \lt 2b+1$
D: $ma \gt mb$ | D |
121. Solve the Cauchy problem for the equation $y^{\prime \prime}=1+x+x^{2}+$ $+x^{3}$, if $y=1$ and $y^{\prime}=1$ when $x=0$. | \frac{x^{2}}{2}+\frac{x^{3}}{6}+\frac{x^{4}}{12}+\frac{x^{5}}{20}+x+1 |
2. Given that when $0 \leqslant \theta \leqslant \frac{\pi}{2}$, it always holds that $\cos ^{2} \theta+2 m \sin \theta-2 m-2<0$. Then the range of real number $m$ is $\qquad$. | m>-\frac{1}{2} |
Solve the following system of equations:
$$
\begin{aligned}
& 37 x+92 y=5043, \\
& 92 x+37 y=2568 .
\end{aligned}
$$ | x=7,\quady=52 |
7. Design a computer operation program: (1) Initial value $x=3, y=0$. (2) $x=x+2$. (3) $y=y+x$. (4) If $y \geqslant 10000$, then proceed to (5); otherwise continue from (2).
(5) Print $x$. (6) Stop running. What is the result printed by this program?
$\qquad$ | 201 |
Given that the function $f(x)=\frac{x+a}{{x}^{2}+1}$ defined on the interval $(-1,1)$ is an odd function.$(1)$ Find the analytical expression of the function $f(x)$;$(2)$ Determine the monotonicity of the function $f(x)$ on the interval $(-1,1)$ and prove it using the definition method.$(3)$ Solve the inequality relate... | (0, \frac{1}{3}) |
4.1 For Eeyore's Birthday, Winnie-the-Pooh, Owl, and Piglet decided to give balloons. Winnie-the-Pooh prepared twice as many balloons as Piglet, and Owl prepared three times as many balloons as Piglet. When Piglet was carrying his balloons, he was in a hurry, stumbled, and some of the balloons burst. Eeyore received a ... | 1 |
Given an arithmetic-geometric sequence $\{a\_n\}$, where all terms are positive numbers, and $a\_1=1$, $a\_1+a\_2+a\_3=7$, find the general term formula $a\_n=$ \_\_\_\_\_\_ and the sum of the first $n$ terms $S\_n=$ \_\_\_\_\_\_. | 2^n-1 |
6. (10 points) There are 2015 beauties, each of whom is either an angel or a demon; angels always tell the truth, demons sometimes tell the truth and sometimes lie.
The 1st says: There is exactly 1 angel among us.
The 3rd says: There are exactly 3 angels among us.
The 2013th says: There are exactly 2013 angels among u... | 3 |
If the set $M=\{s||s| \lt 4\}$, $N=\{x\left|\right.3x\geqslant -1\}$, then $M\cap N=\left(\ \ \right)$
A: $\{x\left|\right.0\leqslant x \lt 4\}$
B: $\left\{{x|{-\frac{1}{3}≤x＜4}}\right\}$
C: $\left\{{x|{-4＜x＜-\frac{1}{3}}}\right\}$
D: $\varnothing $ | B |
Given that the terminal side of angle \\(\theta\\) passes through the point \\((4,-3)\\), then \\(\cos \theta =\\) ?
A: \\( \dfrac {4}{5} \\)
B: \\(- \dfrac {4}{5} \\)
C: \\( \dfrac {3}{5} \\)
D: \\(- \dfrac {3}{5} \\) | A |
If "Labor Day" is on Monday, then "National Day" is on what day of the week? | \text{Sunday} |
In the Cartesian coordinate system $xy$, the parametric equation of curve $C_1$ is
\[
\begin{cases}
x=a\cos t \\
y=1+a\sin t
\end{cases}
\]
where $t$ is the parameter, $a > 0$. In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the equation of curve $C_2$ is $\rho=4\... | 1 |
13. (16 points) Does there exist a smallest positive integer $t$, such that the inequality
$$
(n+t)^{n+t}>(1+n)^{3} n^{n} t^{t}
$$
holds for any positive integer $n$? Prove your conclusion. | 4 |
2. On a basketball court, there are 5 people practicing. The tactic is for Player A to start the game (the 1st pass), and after 6 passes (with each person having an equal chance of passing and receiving the ball), it returns to Player A, who then attempts a 3-point shot. The number of different passing methods is ( ).
... | D |
Given an arithmetic sequence $\{a\_n\}$ with a common difference greater than $0$, the sum of the first three terms is $-3$, and the product of the first three terms is $8$.
(I) Find the general term formula for the arithmetic sequence $\{a\_n\}$.
(II) Find the sum of the first $10$ terms of the sequence $\{|a\_n|\}$... | 105 |
On a circular table are sitting $ 2n$ people, equally spaced in between. $ m$ cookies are given to these people, and they give cookies to their neighbors according to the following rule.
(i) One may give cookies only to people adjacent to himself.
(ii) In order to give a cookie to one's neighbor, one must eat a coo... | 2^n |
Let $f(x)$ be a non-decreasing function on $[0,1]$, that is, for $0 \leqslant x\_1 \leqslant x\_2 \leqslant 1$, we have $f(x\_1) \leqslant f(x\_2)$, and it satisfies the following conditions: $(1) f(0) = 0$; $(2) f(\frac{x}{3}) = \frac{1}{2}f(x)$; $(3) f(1 - x) = 1 - f(x)$. Find $f(\frac{1}{2016})$.
A: $\frac{1}{3}$
B... | \text{C} |
In a two-dimensional space, the one-dimensional measure of a circle (circumference $l=2\pi r$) and its two-dimensional measure (area $S=\pi r^{2}$); in a three-dimensional space, the two-dimensional measure of a sphere (surface area $S=4\pi r^{2}$) and its three-dimensional measure (volume $V= \dfrac {4}{3}\pi r^{3}$);... | 2\pi r^{4} |
Let $x\in \mathbb{R}$. Determine the correct relationship between "x=1" and "the complex number $z=(x^2-1)+(x+1)i$ is a pure imaginary number":
A) Sufficient but not necessary condition
B) Necessary but not sufficient condition
C) Necessary and sufficient condition
D) Neither sufficient nor necessary condition | C |
A moving point P draws two tangents PA and PB to the circle $x^2+y^2=1$, with the points of tangency being A and B, respectively, and $\angle APB=90^\circ$. Find the equation of the trajectory of point P. | x^2+y^2=2 |
Point P is outside a regular hexagon ABCDEF, and O is the center of ABCDEF. Then $\vec{PA} + \vec{PB} + \vec{PC} + \vec{PD} + \vec{PE} + \vec{PF}$ equals to ( )
A: $\vec{PO}$
B: $3\vec{PO}$
C: $6\vec{PO}$
D: $\vec{0}$ | \text{C} |
## Problem Statement
Calculate the definite integral:
$$
\int_{1}^{64} \frac{6-\sqrt{x}+\sqrt[4]{x}}{\sqrt{x^{3}}-7 x-6 \sqrt[4]{x^{3}}} d x
$$ | 4\ln(\frac{2}{2\sqrt{2}+1}) |
The maximum value of the function $f(x) = x^3 - 3x$, where $x \in [-1, 3]$, is ___. | 18 |
You are traveling in a foreign country whose currency consists of five different-looking kinds of coins. You have several of each coin in your pocket. You remember that the coins are worth 1, 2, 5, 10, and 20 florins, but you have no idea which coin is which and you don't speak the local language. You find a vending ma... | 4 |
【Example 3.41】Solve the equation
$$
\mathrm{C}_{2 n+1}^{1} x^{n}-\mathrm{C}_{2 n+1}^{3} x^{n-1}+\mathrm{C}_{2 n+1}^{5} x^{n-2}-\cdots+(-1)^{n}=0
$$ | \cot^{2}\frac{\pi}{2n+1},\cot^{2}\frac{2\pi}{2n+1},\cdots,\cot^{2}\frac{n\pi}{2n+1} |
Given the hyperbola $C$ with the equation $9y^{2} - 16x^{2} = 144$.
(1) Find the lengths of the transverse and conjugate axes, the coordinates of the foci, and the eccentricity of the hyperbola $C$.
(2) Find the equation of the hyperbola that has the same asymptotes as hyperbola $C$ and passes through point $P(6, 4)$... | \frac{x^{2}}{27} - \frac{y^{2}}{48} = 1 |
A chord of length $6$ passes through the left focus $F_1$ of the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$. Find the perimeter of $\Delta ABF_2$ (where $F_2$ is the right focus).
A: $28$
B: $22$
C: $14$
D: $12$ | A |
If four different numbers are chosen from 5, 6, 7, 8, 9 to replace the $\square$ 's below, what is the smallest possible sum of the two 2-digit numbers?
(A) 123
(B) 125
(C) 126
(D) 134
(E) 161 | 125 |
If the internal angles A, C, B of $\triangle ABC$ form an arithmetic sequence, and the area of $\triangle ABC$ is $2\sqrt{3}$, then the minimum value of side AB is ______. | 2\sqrt{2} |
1. What is the value of $123^{2}-23^{2}$ ?
A 10000
B 10409
C 12323
D 14600
E 15658 | 14600 |
If $-4a^{2m}b^{3}$ and $5a^{6}b^{n+1}$ are like terms, then $m+n=$____. | 5 |
8. On the line $x=\sqrt{3}$, find the point $M$ through which two tangents to the graph of the function $y=x^{2} / 4$ pass, such that the angle between them is $60^{\circ}$. | M_{1}(\sqrt{3};0),M_{2}(\sqrt{3};-10/3) |
6. In tetrahedron $ABCD$, $\triangle ABD$ is an equilateral triangle, $\angle BCD=90^{\circ}, BC=CD=1, AC=\sqrt{3}, E$ and $F$ are the midpoints of $BD$ and $AC$ respectively. Then the cosine of the angle formed by line $AE$ and $BF$ is $\qquad$ | \frac{\sqrt{2}}{3} |
Example 1 Let $a, b, c$ be non-zero real numbers, and
$$
a b=2(a+b), b c=3(b+c), c a=4(a+c) \text {. }
$$
Then $a+b+c=$ $\qquad$ | \frac{1128}{35} |
8,9
In trapezoid $ABCD$, points $K$ and $M$ are the midpoints of the bases $AB=5$ and $CD=3$. Find the area of the trapezoid if triangle $AMB$ is a right triangle and $DK$ is the height of the trapezoid. | 8 |
13. Let the function $f(x)=\sin x+\sqrt{3} \cos x+1$,
(I) Find the maximum and minimum values of the function $f(x)$ on $\left[0, \frac{\pi}{2}\right]$;
(II) If real numbers $a, b, c$ satisfy $a f(x)+b f(x-c)=1$ for any $x \in \mathbb{R}$, find the value of $\frac{b \cos c}{a}$. | -1 |
5. On a coordinate plane with the origin at point $O$, a parabola $y=x^{2}$ is drawn. Points $A, B$ are marked on the parabola such that $\angle A O B$ is a right angle. Find the smallest possible value of the area of triangle $A O B$. | 1 |
Example 6 Let $S=\{1,2,3, \cdots, 280\}$. Find the smallest natural number $n$ such that every subset of $S$ with $n$ elements contains 5 pairwise coprime numbers. | 217 |
Problem 1. A cup of Coca-Cola, three sandwiches, and seven croissants cost 127 denars. A cup of Coca-Cola, four sandwiches, and ten croissants cost 166 denars. How much together does one cup of Coca-Cola, one sandwich, and one croissant cost? | 49 |
In the trapezoid $K L M N$, the base $K L$ has a length of $40 \mathrm{~cm}$ and the base $M N$ has a length of $16 \mathrm{~cm}$. Point $P$ lies on the segment $K L$ such that the segment $N P$ divides the trapezoid into two parts with equal areas.
Determine the length of the segment $K P$.
(L. Hozová)
Hint. Modify... | 28 |
7. In how many different ways can all the letters of the word ANAGRAM be arranged so that no two $A$s are ever next to each other?
The use of a pocket calculator or any reference materials is not allowed. | 240 |
6. At a round table, there are 2023 people. Each person is either a knave or a knight. Knights always tell the truth, and knaves always lie. The first person says: "There is at least one knave at this table." The person to their left says: "There is at least one knight at this table." The third person says: "There are ... | 506 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.