problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Solve the following equations using appropriate methods:
1. $x^{2}+4x-1=0$;
2. $\left(x-1\right)^{2}=3\left(x-1\right)$. | 4 |
11. As shown in the figure, there are 10 cards, each card has a number from $0 \sim 9$, and there are no duplicates. Xiaoyu and Xiaomu each take 5 cards and start playing a card game. The rules of the game are as follows:
(1) Both players calculate the sum of the numbers on all 5 cards they have as their initial score.... | 20 |
Problem 3. Given is a rectangle with side lengths $2023 \mathrm{~cm}$ and $1309 \mathrm{~cm}$. The rectangle is divided into equal squares with the largest possible side length expressed in centimeters as a natural number. Determine the sum of the perimeters of the divided squares. | 89012\mathrm{~} |
B3. We must choose a seven-character password that contains at least one letter and at least one digit. We can choose from the characters $A, B, C, G, J, M, R, Z$ and 3.
a) How many possible choices are there for the password if the characters cannot be repeated?
b) How many possible choices are there for the passwor... | 2685816 |
A store sells a certain model of glasses. First, the price is increased by 9 times the purchase price, and then a "50% discount plus a 20 yuan taxi fare" promotion is advertised. Selling a pair of these glasses still yields a profit of 300 yuan. What is the purchase price of this model of glasses? | 80 |
If the binary quadratic equation with respect to $x$ and $y$: $x^2+2xy+8y^2+14y+m=0$ (where $m$ is a constant) represents two lines, then the value of the constant $m$ is ______. | m=7 |
8. If $(x+1)(y+1)=2$, then $\arctan x+\arctan y=$ | \frac{\pi}{4} |
## Problem Statement
Calculate the limit of the function:
$\lim _{x \rightarrow 1} \frac{x^{2}-1}{2 x^{2}-x-1}$ | \frac{2}{3} |
Given that F1 and F2 are the left and right foci of the hyperbola $$\frac {x^{2}}{a^{2}}$$ - $$\frac {y^{2}}{b^{2}}$$ = 1, respectively. If there exists a point A on the hyperbola such that ∠F1AF2 = 90° and |AF1| = 2|AF2|, find the eccentricity of the hyperbola. | \sqrt {5} |
Person A and Person B start from locations A and B, respectively, and walk towards each other. They meet at a point 10 kilometers away from location A. After meeting, they continue to walk, reach locations B and A respectively, and then immediately turn back. They meet again at a point 3 kilometers away from location B... | 27 \text{ kilometers} |
On the same stretch of road, the speed going uphill is $V_1$, and the speed going downhill is $V_2$. The average speed for the round trip is ____. | \frac{2V_1 V_2}{V_1 + V_2} |
43rd Putnam 1982 Problem B1 ABC is an arbitary triangle, and M is the midpoint of BC. How many pieces are needed to dissect AMB into triangles which can be reassembled to give AMC? Solution | 2 |
The domain of the function $y=x^{2}-2x$ is $\{0,1,2,3\}$. What is its range?
A: $\{y|-1\leqslant y\leqslant 3\}$
B: $\{y|0\leqslant y\leqslant 3\}$
C: $\{0,1,2,3\}$
D: $\{-1,0,3\}$ | D |
15$\cdot$18 Mapping $f:\{a, b, c, d\} \rightarrow\{1,2,3\}$. If $10<f(a) \cdot f(b)$. $f(c) \cdot f(d)<20$, the number of such mappings is
(A) 23.
(B) 24.
(C) 25.
(D) 26.
(2nd "Hope Cup" National Mathematics Invitational Competition, 1991) | 25 |
1. The range of the function $f(x)=\frac{\left(x-x^{3}\right)\left(1-6 x^{2}+x^{4}\right)}{\left(1+x^{2}\right)^{4}}$ is . $\qquad$ | [-\frac{1}{8},\frac{1}{8}] |
25.35. Find the first digit of the number $2^{400}$. | 2 |
7. Points $A(2,-4)$ and $B(3,1.5)$ are given. Point $C$ is centrally symmetric to point $B$ with respect to point $O(0,0)$, while point $D$ is axially symmetric to point $A$ with respect to the $y$-axis. Determine the area of quadrilateral $A B C D$.
No use of a pocket calculator or any reference materials is allowed.... | 20 |
Berpow S.l.
At the alumni meeting, 45 people attended. It turned out that any two of them who had the same number of acquaintances among those present were not acquainted with each other. What is the maximum number of pairs of acquaintances that could have been among those who attended the meeting? | 870 |
## Task 4 - 090614
A working group received a sum of exactly $240 \mathrm{M}$ as an award for very good performance.
If this money had been distributed evenly among all members of the working group, each member would have received an integer amount (in Marks). However, the members decided to spend the $240 \mathrm{M}... | 15 |
2. Let $a_{1}=1, a_{n+1}=2 a_{n}+n^{2}$, then the general term formula $a_{n}$ $=$ $\qquad$ | 7 \times 2^{n-1}-n^{2}-2 n-3 |
Use the Euclidean algorithm to find the greatest common divisor (GCD) of the two numbers 102 and 238. The options are:
A: 38
B: 34
C: 28
D: 24 | B |
Determine all positive integers $n$ for which the equation
\[ x^n + (2+x)^n + (2-x)^n = 0 \]
has an integer as a solution. | n=1 |
In an isosceles trapezoid, the bases are 40 and 24, and its diagonals are perpendicular to each other. Find the area of the trapezoid.
# | 1024 |
Express $0.0000907$ in scientific notation as ( ).
A: $9.07\times 10^{-4}$
B: $9.07\times 10^{-6}$
C: $9.07\times 10^{5}$
D: $9.07\times 10^{-5}$ | D |
1.220 students take a percentage-based test (scores are counted in integers), and no more than three students have the same score. Then the minimum number of scores that exactly three students have the same score is ( ).
(A) 17
(B) 18
(C) 19
(D) 20 | B |
The expression $a+1+a-2+a+3+a-4$ is equal to
(A) $10 a$
(B) 0
(C) $4 a-2$
(D) $4 a+2$
(E) $-2 a$ | 4a-2 |
Five workers have to choose one day off among three consecutive days, ensuring that there is at least one person resting each day. Calculate the number of different arrangements possible. | 150 |
Given: $\sin \left(\alpha+ \frac {\pi}{4}\right)+2\sin \left(\alpha- \frac {\pi}{4}\right)=0$.
$(1)$ Find the value of $\tan \alpha$;
$(2)$ If $\tan \left( \frac {\pi}{4}-\beta\right)= \frac {1}{3}$, find the value of $\tan (\alpha+\beta)$. | 1 |
1. A mathematician and a physicist started running on a running track towards the finish line at the same time. After finishing, the mathematician said: "If I had run twice as fast, I would have beaten the physicist by 12 seconds." And after finishing, the physicist said: "If I had run twice as fast, I would have beate... | 16 |
If the function $y=f(x)$ has an inverse function $y=f^{-1}(x)$, and the graph of the function $y=2x-f(x)$ passes through the point $(2,1)$, then the graph of the function $y=f^{-1}(x)-2x$ must pass through the point. | (3,-4) |
10. Given that the product of a root of the quadratic trinomial $a x^{2}+b x+b$ and a root of the quadratic trinomial $a x^{2}+a x+b$ is equal to 1. Then the sum of the squares of these two roots is $\qquad$ | 3 |
$22 \cdot 11$ Point $P$ is outside the plane of $\triangle A B C$, $P A \perp$ plane $A B C$, and $S_{1}, S_{2}$ represent the areas of $\triangle A B C$ and $\triangle P B C$ respectively, then the value of $S_{1}: S_{2}$
(A) is greater than 0, and no more than 1.
(B) is greater than 0 and less than 1.
(C) is greater... | B |
Given that the circumradius R of △ABC is within the interval [1, +∞), and AB = AC = 1, determine the maximum value of the area of △ABC. | \frac{\sqrt{3}}{4} |
5. Find all values of the parameter $b$, for which for any value of the parameter $a \in[-1 ; 2]$ the inequality $\operatorname{tg}^{2} x+4(a+b) \operatorname{tg} x+a^{2}+b^{2}-18<0$ is satisfied for each $x \in[-\pi / 4 ; \pi / 4]$. | (-2;1) |
A function is defined as $f(x)=\sin (ωx+φ)$ where $ω > 0$ and $|φ| < \dfrac {π}{2}$. It is known that the function is monotonically decreasing in the interval $[\dfrac {5π}{12}, \dfrac {11π}{12}]$.
1. Find the analytic expression for $f(x)$.
2. Let $g(x)$ be the function obtained by first shifting the graph of $y=f(x)... | 0 < m < \dfrac {1}{2} |
Which of the following statements are correct?
A: If $a \gt b$, then $ac^{2} \gt bc^{2}$
B: If $a \gt b$, then $a-1 \gt b-2$
C: If $\frac{a}{{c}^{2}} \gt \frac{b}{{c}^{2}}$, then $a \gt b$
D: If $a \gt b$, then $a^{2} \gt b^{2}$ | B \text{ and } C |
4. The equation $6 x+\sqrt{3 x-2}+2 k-3=0$ has real roots, the range of values for $k$ is $\qquad$ - | k \leqslant -\frac{1}{2} |
In the geometric sequence $\{a_n\}$, if $a_3=3$ and $a_7=6$, then $a_{11}=$? | 12 |
Given that $a$ and $b \in [-1,1]$, find the probability that the inequality $x^{2}-2ax+b \geqslant 0$ holds true for all $x \in R$. | \frac{1}{3} |
Given that the sum of the first $n$ terms of the sequence ${a_n}$ is $S_n = An^2 + Bn + q (A \neq 0)$, then $q = 0$ is the ( ) condition for ${a_n}$ to be an arithmetic sequence.
A: Necessary but not sufficient
B: Sufficient but not necessary
C: Necessary and sufficient
D: Neither necessary nor sufficient | C |
8. Given three positive real numbers $a, b, c$ with $a c=9$. It is known that for all real numbers $x, y$ with $x y \neq 0$ the inequality $\frac{a}{x^{2}}+\frac{b}{x y}+\frac{c}{y^{2}} \geq 0$ holds. What is the maximum possible value for $b$?
(A) 1
(B) 3
(C) 6
(D) 9
(E) There is no such $b$. | 6 |
In the plane of triangle $A B C$, we take a point $P$ and connect it with the vertices of the triangle. The lines $P A, P B$, and $P C$ intersect the sides of the triangle at points $A_{1}, B_{1}$, and $C_{1}$. Show that
$$
\frac{P A}{A A_{1}}+\frac{P B}{B B_{1}}+\frac{P C}{C C_{1}}=2
$$ | 2 |
9. (15 points) A store purchased a batch of pens. If it sells 20 pens at a retail price of 7 yuan each and 15 pens at a retail price of 8 yuan each, the profit earned is the same. What is the purchase price of each pen in yuan?
---
Please note that the translation retains the original format and structure, including ... | 4 |
5. In a school there are 300 boys and 300 girls, divided into 5 classes, each with the same number of students. It is known that there are at least 33 boys and 33 girls in each class. A boy and a girl from the same class may form a group to enter a contest, and each student may only belong to one group. What is the max... | 192 |
Given that $a$, $b$, $m$, and $n$ are all positive numbers, and it's known that $a + b = 1$ and $mn = 2$, find the minimum value of the expression $(am + bn)(bm + an)$. | 2 |
## Task 3 - 340723
In the triangles $A B C$, let the measures of the interior angles at $A, B$, and $C$ be denoted by $\alpha, \beta$, and $\gamma$ respectively. Furthermore, assume that the perpendicular bisector of side $A B$ and the angle bisector passing through $A$ intersect at a point $D$ that lies on side $B C$... | \beta=30,45 |
1.1. Arina wrote down a number and the number of the month of her birthday, multiplied them and got 248. In which month was Arina born? Write the number of the month in your answer.
 | 8 |
In triangle $\triangle ABC$, if $\tan(A-B-\pi)=\frac{1}{2}$ and $\tan(3\pi-B)=\frac{1}{7}$, then $2A-B=$ （ ）
A: $\frac{\pi}{4}$
B: $\frac{5\pi}{4}$
C: $-\frac{3\pi}{4}$
D: $\frac{\pi}{4}$ or $\frac{5\pi}{4}$ | C |
Given the function $f(x) = x^2 - 2x$, where $a-1 \leq x \leq a+1$, $a \in \mathbb{R}$. Let the set $M = \{(m, f(n)) | m, n \in [a-1, a+1]\}$, if the area of the plane region enclosed by all points in $M$ is $S$, then the minimum value of $S$ is. | 2 |
5. || $\log _{2} x|+1| \geqslant \log _{2} 4 x$ The solution is
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | 0<x\leqslant\frac{\sqrt{2}}{2} |
Equilateral triangle $ABC$ has side length $6$. Circles with centers at $A$, $B$, and $C$ are drawn such that their respective radii $r_A$, $r_B$, and $r_C$ form an arithmetic sequence with $r_A<r_B<r_C$. If the shortest distance between circles $A$ and $B$ is $3.5$, and the shortest distance between circles $A$ and $C... | \frac{29\pi}{24} |
Auto: GToooanov A.C.
The sum of the digits in the decimal representation of a natural number $n$ is 100, and the sum of the digits of the number $44 n$ is 800. What is the sum of the digits of the number $3 n$? | 300 |
Given vectors $\overrightarrow {a}=\{2,-3,1\}$, $\overrightarrow {b}=\{2,0,-2\}$, $\overrightarrow {c}=\{-1,-2,0\}$ and $\overrightarrow {r}=2 \overrightarrow {a}-3 \overrightarrow {b}+ \overrightarrow {c}$, find the coordinates of the vector $\overrightarrow {r}$. | (-3, -8, 8) |
1. Determine the value of the expression
$$
S=\frac{1}{1+x+x y}+\frac{1}{1+y+y z}+\frac{1}{1+z+z x}
$$
if for the real numbers $x, y, z$ it holds that $x y z=1$. | 1 |
The parabola $y=x^{2}-\left(2m-1\right)x-6m$ intersects the $x$-axis at points $(x_{1}$,$0)$ and $(x_{2}$,$0)$, where $x_{1}x_{2}=x_{1}+x_{2}+49$. To make this parabola pass through the origin, it should be shifted to the right by ____ units. | 4 |
Given that $$C_{ n }^{ 2 }= C_{ n-1 }^{ 2 }+ C_{ n-1 }^{ 3 }(n≥2，n∈N^{*})$$, find the value of $n$. | 5 |
24. Let $a$ and $b$ be two integers. Suppose that $\sqrt{7-4 \sqrt{3}}$ is a root of the equation $x^{2}+a x+b=0$. Find the value of $b-a$. | 5 |
79. - Tell me, famous Pythagoras, how many students attend your school and listen to your discourses?
- In all, 一 replied the philosopher: - half study mathematics, a quarter study music, a seventh part remains in silence, and, in addition, there are three women. | 28 |
There are three table tennis associations, A, B, and C, with 27, 9, and 18 athletes, respectively. Six athletes are to be selected for a competition using stratified sampling.
(I) Determine the number of athletes to be selected from each association;
(II) The selected 6 athletes are numbered A$_{1}$, A$_{2}$, A$_{3}$... | \frac{3}{5} |
7. In the Cartesian coordinate system, the parabola
$$
y=x^{2}+m x-\frac{3}{4} m^{2}(m>0)
$$
intersects the $x$-axis at points $A$ and $B$. If the distances from points $A$ and $B$ to the origin are $O A$ and $O B$, respectively, and satisfy $\frac{1}{O B}-\frac{1}{O A}=\frac{2}{3}$, then the value of $m$ is $\qquad$. | 2 |
12 Given $\sin (x+\sin x)=\cos (x-\cos x), x \in[0, \pi]$, then $x=$
$\qquad$ . | \frac{\pi}{4} |
(Ⅰ) Given $\frac {4\sin\alpha-2\cos\alpha}{5\cos\alpha +3\sin\alpha } = \frac {5}{7}$, find the value of $\sin\alpha \cdot \cos\alpha$;
(Ⅱ) Calculate the value of $\frac { \sqrt {1-2\sin 10^\circ \cos 10^\circ }}{\cos 10^\circ - \sqrt {1-\cos^2 170^\circ }}$. | 1 |
Let $n$ be a positive integer. Mariano divides a rectangle into $n^2$ smaller rectangles by drawing $n-1$ vertical lines and $n-1$ horizontal lines, parallel to the sides of the larger rectangle. On every step, Emilio picks one of the smaller rectangles and Mariano tells him its area. Find the least positive integer $... | 2n - 1 |
How many integers exist from 0 to 999999, in the decimal representation of which there are no two adjacent identical digits?
# | 597871 |
8. Three consecutive positive integers less than 20 are, in ascending order, prime, even and triangular. What is the product of these three integers?
A 6
B 60
С 990
D 1786
E 2730 | 2730 |
A city decided to build a fast road to construct a three-dimensional road network in the city. In order to complete the project half a year ahead of schedule, the work efficiency needs to be increased by $40\%$. How many months were originally planned to complete this project? | 21 |
## problem statement
Write the canonical equations of the line.
$$
\begin{aligned}
& x-2 y+z-4=0 \\
& 2 x+2 y-z-8=0
\end{aligned}
$$ | \frac{x-4}{0}=\frac{y}{3}=\frac{z}{6} |
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 |
Given the function $f(x)=|x-a|$ ($a$ is a constant), if $f(x)$ is an increasing function in the interval $[1,+\infty)$, then the range of values for $a$ is _______. | (-\infty,1] |
Given that the point $P(\cos α, \sin α)$ is on the line $y = 2x$, find the value of $\sin 2α$ which is equal to $(\ \ \ \ )$.
A: $- \dfrac {4}{5}$
B: $\dfrac {4}{5}$
C: $- \dfrac {3}{5}$
D: $\dfrac {3}{5}$ | B |
Let $n>2,n \in \mathbb{N}$ and $a>0,a \in \mathbb{R}$ such that $2^a + \log_2 a = n^2$. Prove that: \[ 2 \cdot \log_2 n>a>2 \cdot \log_2 n -\frac{1}{n} . \]
[i]Radu Gologan[/i] | 2 \log_2 n - \frac{1}{n} < a < 2 \log_2 n |
8. (10 points) In a plane, there are 5 points, among which no 3 points are collinear. Connecting line segments with these points as endpoints, apart from these 5 points, these line segments have at least $\qquad$ intersection points. | 1 |
66. Given the password: $3 \times \overline{A B C P Q R}=4 \times \overline{P Q R A B C}$, where each letter represents a decimal digit, translate this password into an equation is $\qquad$ . | 3\times571428=4\times428571 |
Given that $a$ is a positive integer, point $P(4,2-a)$ lies in the first quadrant, then $a=\_\_\_\_\_\_$. | a = 1 |
The positional relationship between the line containing a lateral edge of the prism and the plane containing a lateral face that does not include this lateral edge is ( )
A: Parallel
B: Intersect
C: Parallel or Intersect
D: Do not intersect | \text{A} |
For non-negative real numbers $a,$ $b$ let $A(a, b)$ be their arithmetic mean and $G(a, b)$ their geometric mean. We consider the sequence $\langle a_n \rangle$ with $a_0 = 0,$ $a_1 = 1$ and $a_{n+1} = A(A(a_{n-1}, a_n), G(a_{n-1}, a_n))$ for $n > 0.$
(a) Show that each $a_n = b^2_n$ is the square of a rational number ... | \left| b_n - \frac{2}{3} \right| < \frac{1}{2^n} |
For two non-intersecting space lines $a$ and $b$, there must exist a plane $\alpha$ such that ( )
A: $a \subset \alpha$, $b \subset \alpha$
B: $a \perp \alpha$, $b \perp \alpha$
C: $a \subset \alpha$, $b \perp \alpha$
D: $a \subset \alpha$, $b \parallel \alpha$ | \text{D} |
The distance from the focus to the asymptote of the hyperbola $\frac {x^{2}}{4}- \frac {y^{2}}{8}=1$ is $\_\_\_\_\_\_$. | 2 \sqrt {2} |
Given a geometric sequence $\{a_n\}$ where $a_1 = 3$ and $a_4 = 81$, find the general term $a_n$. | 3^n |
In the function $y=\dfrac{\sqrt{x+4}}{x}$, the range of the independent variable $x$ is:
A: $x \gt 0$
B: $ x\geqslant -4$
C: $ x\geqslant -4$ and $x\neq 0$
D: $ x \gt 0$ and $x\neq -1$ | C |
Among the following four functions, the one that has $\pi$ as its smallest positive period and is monotonically decreasing in the interval $\left(\frac{\pi}{2}, \pi\right)$ is ( )
A: $y=\sin 2x$
B: $y=2|\cos x|$
C: $y=\cos \frac{x}{2}$
D: $y=\tan (-x)$ | D |
Given the function $f(x)=(a-1)(a^x-a^{-x})$ where $0<a<1$:
1. Determine the parity of $f(x)$ and provide proof.
2. Prove that $f(x)$ is an increasing function on $\mathbb{R}$ using its definition.
3. If $f(2at^2-a^2-a)+f(6at-1) \leq 0$ for any $t \in \left[0, \frac{1}{2}\right]$, find the range of values for $a$. | \left(0, \frac{1}{2}\right] |
When the vertex of a parabolic arch bridge is 2 meters above the water surface, the width of the water is measured to be 8 meters. When the water level rises by $\frac{1}{2}$ meter, the width of the water is \_\_\_\_\_\_ meters. | 4\sqrt{3} |
Task 3. (15 points) The function $f(x)$ satisfies the condition: for any real numbers $a$ and $b$, the equality $f\left(\frac{a+2 b}{3}\right)=\frac{f(a)+2 f(b)}{3}$ holds. Find the value of the function $f(2022)$, if $f(1)=1, f(4)=7$.
# | 4043 |
19. In a jar, there are red, green, and blue balls. It is known that there are 6 balls other than the red ones; 8 balls other than the green ones; and 4 balls other than the blue ones. Then the total number of balls in the jar is ( ).
(A) 6
(B) 8
(C) 9
(D) 10
(E) 12 | C |
If the complex number $z$ is a root of the equation $x^{2}+2x+2=0$, then $|{z}^{2}+{(\overline{z})}^{2}|$ is equal to:
A: $0$
B: $1$
C: $\sqrt{2}$
D: $\sqrt{3}$ | A |
Given two non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, the number of true statements among the following is ( ).
(1) The direction of $2\overrightarrow{a}$ is the same as the direction of $\overrightarrow{a}$, and the magnitude of $2\overrightarrow{a}$ is twice the magnitude of $\overrightarrow{a}$;... | C |
The contrapositive of the proposition "If $x=2$, then $x^2-3x+2=0$" is ( )
A: If $x \neq 2$, then $x^2-3x+2 \neq 0$
B: If $x^2-3x+2=0$, then $x=2$
C: If $x^2-3x+2 \neq 0$, then $x \neq 2$
D: If $x \neq 2$, then $x^2-3x+2=0$ | \text{C} |
Given that the lines $x - my + 2m = 0$ and $x + 2y - m = 0$ are perpendicular, find the real number $m$. | \frac{1}{2} |
The function $y=\sin (2x+ \frac {π}{3})$ is monotonically decreasing in the interval $(\quad \quad)$ on $[0,π]$.
A: $[0, \frac {5}{12}π]$
B: $[ \frac {π}{12}, \frac {7}{12}π]$
C: $[ \frac {5}{12}π, \frac {11}{12}π]$
D: $[ \frac {π}{6}, \frac {π}{2}]$ | \text{B} |
Given a set of data of two variables with a linear correlation relationship $(x\_1,y\_1)$, $(x\_2,y\_2)$, $...(x\_n,y\_n)$, the regression line equation is $\hat{y}=bx+0.2$. If the average of $x\_1$, $x\_2$, $...$, $x\_n$ is $\overline{x}=4$, and the average of $y\_1$, $y\_2$, $...$, $y\_n$ is $\overline{y}=5$, then th... | 2.6 |
2. Petya bought two books. The first one was $50 \%$ more expensive than the second. By what percentage is the second book cheaper than the first? | 33\frac{1}{3} |
Example 4 Find the smallest positive integer $n$, such that for any sequence of $n$ positive integers $a_{1}, a_{2}, \cdots, a_{n}$ satisfying $\sum_{i=1}^{n} a_{i}=2007$, there must be a sum of some consecutive terms equal to 30.
(Fourth China Southeast Mathematical Olympiad) | 1018 |
Given proposition $p$: for all $x$ in $\mathbb{R}$, $2^x > 0$, then the negation of $p$ is
A: $\neg p$: There exists an $x \notin \mathbb{R}$ such that $2^x \leqslant 0$
B: $\neg p$: There exists an $x \in \mathbb{R}$ such that $2^x \leqslant 0$
C: $\neg p$: There exists an $x \in \mathbb{R}$ such that $2^x 0$ | \text{B} |
4. A doctor told Mikael to take a pill every 75 minutes. He took his first pill at 11:05. At what time did he take his fourth pill?
A $12: 20$
B 13:35
C 14:50
D 16:05
E 17:20 | 14:50 |
5. The circle inscribed in the right triangle ABC touches the legs CA and CB at points P and Q, respectively. The line PQ intersects the line passing through the center of the inscribed circle and parallel to the hypotenuse at point N. M is the midpoint of the hypotenuse. Find the measure of angle MCN. | 90 |
In all figures of this question, we see a triangle $ABC$ divided into four parts; in these triangles, $D$ is the midpoint of $AB$, $E$ is the midpoint of $AC$, and $FG$ measures $\frac{1}{2} BC$.
=$ . | 0 |
9. (40 points) What is the maximum number of numbers that can be chosen among the natural numbers from 1 to 3000 such that the difference between any two of them is different from 1, 4, and 5? | 1000 |
8. If positive numbers $a, b, c$ satisfy $\frac{a}{b+c}=\frac{b}{a+c}-\frac{c}{a+b}$, prove that $\frac{b}{a+c} \geqslant \frac{\sqrt{17}-1}{4}$. | \frac{\sqrt{17}-1}{4} |
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