problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
1. 147 Simplify
$$
(2+1)\left(2^{2}+1\right)\left(2^{4}+1\right)\left(2^{8}+1\right) \cdots \cdots\left(2^{256}+1\right) .
$$ | 2^{512}-1 |
[ Pairing and grouping; bijections $]$ [ Decimal number system ]
Find the last digit of the number $1^{2}+2^{2}+\ldots+99^{2}$. | 0 |
2. Given three vertices of a parallelogram are $(4,2),(5,7),(-3,4)$. Then the fourth vertex cannot be
A. $(12,5)$
B. $(-2,9)$
C. $(-4,-1)$
D. $(3,7)$ | D |
13. $[8]$ Find the smallest positive integer $n$ such that $\frac{5^{n+1}+2^{n+1}}{5^{n}+2^{n}}>4.99$. | 7 |
Let's determine the trinomial whose third power is:
$$
x^{6}-6 x^{5}+15 x^{4}-20 x^{3}+15 x^{2}-6 x+1
$$ | x^{2}-2x+1 |
Prove by mathematical induction that $n(n+1)(2n+1)$ is divisible by 6. When deducing the case for $n=k+1$ from the induction hypothesis, the original expression for $n=k+1$ should be represented as ( )
A: $k(k+1)(2k+1)+6(k+1)$
B: $6k(k+1)(2k+1)$
C: $k(k+1)(2k+1)+6(k+1)^2$
D: None of the above | C |
Given two points P(4, -9) and Q(-2, 3), find the ratio in which the line PQ divides the directed line segment $\overrightarrow {PQ}$ at the point where it intersects the y-axis. | 2:1 |
Let $P \in \mathbb{K}[X]$ and $(a, b) \in \mathbb{K}^{2}$ such that $a \neq b$. Knowing that the remainder of the Euclidean division of $P$ by $X-a$ is 1 and that the remainder of the Euclidean division of $P$ by $X-b$ is -1, determine the remainder of the Euclidean division of $P$ by $(X-a)(X-b)$. | R(X)=\frac{2}{-b}X+\frac{b+}{b-} |
13. The real numbers $a$ and $b$, where $a>b$, are solutions to the equation $3^{2 x}-10 \times 3^{x+1}+81=$ 0 . What is the value of $20 a^{2}+18 b^{2}$ ? | 198 |
Given a positive sequence $\{a_n\}$ with the sum of its first $n$ terms being $S_n$, if both $\{a_n\}$ and $\{\sqrt{S_n}\}$ are arithmetic sequences with equal common differences, then $S_0=$
A: $50$
B: $100$
C: $1500$
D: $2500$ | D |
14. Teacher $D$ placed cards with the numbers $1, 2, \cdots, 13$ in a clockwise order on a circle. Initially, all cards are face up. Each time, one card can be flipped to face down (once a card is face down, it cannot be flipped again). Teacher $D$'s rule for flipping cards is: if a face-up card has the number $A$, and... | 26624 |
Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions
(a) $0\le A<B<C\le99$,
(b) there exist integers $a$, $b$, and $c$, and prime $p$ where $0\le b<a<c<p$,
(c) $p$ divides $A-a$, $B-b$, and $C-c$, and
(d) each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithm... | 272 |
## Problem II - 1
In the basement of the castle, 7 gnomes guard their treasure. The treasure is behind 12 doors, each with 12 locks. All the locks are different. Each gnome has keys to some of the locks. Any three gnomes together have keys to all the locks. Prove that among all the gnomes, they have at least 336 keys. | 336 |
Calculate:
$(1) \sqrt[3]{-27} + \sqrt{(-6)^{2}} + (\sqrt{5})^{2}$;
$(2) (-\sqrt{3})^{2} - \sqrt{16} - |1-\sqrt{2}|$. | -\sqrt{2} |
## Task 2
At the ABC Mathematics Olympiad, 45 students from Erich-Weinert-Oberschule received a certificate. At the Juri-Gagarin-Oberschule, 17 fewer students received a certificate.
How many students from the Juri-Gagarin-Oberschule received a certificate? | 28 |
2. As shown in Figure $1, D$ and $E$ are on the extensions of sides $B A$ and $B C$ of equilateral $\triangle A B C$, respectively, with $A D=A C+$ $C E, \angle B D C=10^{\circ}$. Then the degree measure of $\angle C D E$ is ( ).
(A) $30^{\circ}$
(B) $40^{\circ}$
(C) $45^{\circ}$
(D) $35^{\circ}$ | B |
[ Dirichlet's Principle (continued). ] [Partitions into pairs and groups; bijections]
Before a clairvoyant lies a deck of 36 cards face down (4 suits, 9 cards of each suit). He names the suit of the top card, after which the card is revealed to him. Then he names the suit of the next card, and so on. The task of the c... | 23 |
Given that the sequence $\{a_n\}$ is an arithmetic sequence, $a_1=t^2-t$, $a_2=4$, $a_3=t^2+t$.
$(1)$ Find the general formula for the sequence $\{a_n\}$;
$(2)$ If the sequence $\{a_n\}$ is increasing and the sequence $\{b_n\}$ satisfies $\log_2 b_n=a_n$, find the sum of the first $n$ terms $S_n$ of the sequence $\{... | \frac{(6n-5)\cdot 4^{n+1}+20}{9} |
Given real numbers $a$, $b$, $x$, $y$ satisfy $ax + by = 3$ and $ay - bx = 5$, find the value of $(a^2 + b^2)(x^2 + y^2)$. | 34 |
$2 \cdot 35$ subset $X$ of the set $\{00,01, \cdots, 98,99\}$ satisfies: in any infinite sequence of digits, there are two adjacent digits that form an element of $X$. What is the minimum number of elements that $X$ should contain? | 55 |
Which of the following derivative operations is correct? ( )
A: (3x2+2)′=6x+2
B: (3x2+2)′=6x+2
C: $$(- \frac {1}{x})^{′}= \frac {1}{x^{2}}$$
D: \[（2e）x\]′=（2e）x | \text{C} |
10.3. In a row from left to right, all natural numbers from 1 to 37 are written in such an order that each number, starting from the second to the 37th, divides the sum of all numbers to its left: the second divides the first, the third divides the sum of the first and second, and so on, the last divides the sum of the... | 2 |
## PROBLEM 46. INSCRIBED DECAGON
The problem of inscribing a regular decagon in a circle has aroused keen interest among mathematicians since ancient Greece. Euclid wrote about it and, in particular, gave a very beautiful method for inscribing a decagon. First, we will show that the side $x$ of a regular decagon inscr... | \frac{r}{2}(\sqrt{5}-1) |
II. (50 points) The player numbers of a sports team are uniquely selected from the positive integers 1 to 100. If the number of any player is neither the sum of the numbers of any other two players nor twice the number of another player, what is the maximum number of players this sports team can have? | 50 |
1. The number of pairs of skew lines among all the edges of a regular pentagonal prism is $\qquad$ pairs. | 50 |
## Problem 3
Solve the equation in the set $\mathbb{R}$: $\sqrt{x-2}+\sqrt[3]{4-x}=2$. | x_1=3,x_2=14-6\sqrt{3},x_3=14+6\sqrt{3} |
1. $[2]$ Find the minimum of $x^{2}-2 x$ over all real numbers $x$. | -1 |
10.5. The tsar has eight sons, and they are all fools. Every night the tsar sends three of them to guard the golden apples from the firebird. The tsareviches cannot catch the firebird, blame each other for it, and therefore no two of them agree to go on guard duty a second time. For how many nights can this continue at... | 8 |
7.4. Find the volume of a regular triangular pyramid, the midpoint of the height of which is at a distance of 2 and $\sqrt{7}$ from the lateral face and the lateral edge, respectively. Round your answer to two decimal places if necessary. | 112\sqrt{7} |
For $ a\in\mathbb{R}$, find the minimum value of $ \int_0^{\frac {\pi}{2}} \left|\frac {\sin 2x}{1 \plus{} \sin ^ 2 x} \minus{} a\cos x\right| dx$. | 3\ln(2) - 2\ln(5) |
Given a right-angled triangle with two side lengths of $3$ and $4$, the square of the third side length is $(\ \ \ \ )$.
A: $7$
B: $14$
C: $25$
D: $7$ or $25$ | D |
In △ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. Given that a=2, c= $\sqrt {2}$, cosA=- $\frac { \sqrt {2}}{4}$, find the value of b. | 1 |
7.1. $\quad$ A cyclist planned to travel from point $A$ to point $F$ in 5 hours, moving at a constant speed. With the planned speed, he traveled halfway, then increased his speed by $25 \%$. He continued to point $B$ with the new speed. How long did the entire journey take? | 4.5 |
17. At 10:28:06, the time displayed on a digital watch is as shown in the figure. During the time from 10:00 to 10:30, the number of times when all six digits on the digital watch are different is $\qquad$.
保留源文本的换行和格式，直接输出翻译结果如下：
17. At 10:28:06, the time displayed on a digital watch is as shown in the figure. Durin... | 90 |
"Given that quadrilateral $ABCD$ is a rectangle, it follows that the diagonals of quadrilateral $ABCD$ are equal." The major premise required to complete the above reasoning is ( )
A: The diagonals of a square are equal
B: The diagonals of a rectangle are equal
C: The diagonals of an isosceles trapezoid are equal
D:... | \text{B} |
Find all pairs $(p,n)$ so that $p$ is a prime number, $n$ is a positive integer and \[p^3-2p^2+p+1=3^n \] holds. | (2, 1), (5, 4) |
The coordinates of the vertex of the parabola $y=\left(x-2\right)^{2}+5$ are ______. | (2,5) |
Problem 11.3. Let $n$ be a fixed positive integer. The positive integers $a, b$, $c$ and $d$ are less than or equal to $n, d$ is the largest one and they satisfy the equality
$$
(a b+c d)(b c+a d)(a c+b d)=(d-a)^{2}(d-b)^{2}(d-c)^{2}
$$
a) Prove that $d=a+b+c$.
b) Find the number of the quadruples $(a, b, c, d)$ whi... | \frac{n(n-1)(n-2)}{6} |
Observe the following equations:
1=1
3+5=8
7+9+11=27
13+15+17+19=64
According to this pattern, the 6th equation should be. | 31+33+35+37+39+41=216 |
Let $a=6^{0.7}$, $b=0.7^{6}$, $c=\log_{0.7}6$, then the relationship among $a$, $b$, and $c$ is ( )
A: $c<b<a$
B: $c<a<b$
C: $b<a<c$
D: $a<c<b$ | \text{A} |
11th Balkan 1994 Problem 3 What is the maximum value f(n) of |s 1 - s 2 | + |s 2 - s 3 | + ... + |s n-1 - s n | over all permutations s 1 , s 2 , ... , s n of 1, 2, ... , n? Solution | f(2m)=2m^2-1,\,f(2m+1)=2m^2+2m-1 |
Given the natural $n$. We shall call [i]word [/i] sequence from $n$ letters of the alphabet, and [i]distance [/i] $\rho(A, B)$ between [i]words [/i] $A=a_1a_2\dots a_n$ and $B=b_1b_2\dots b_n$ , the number of digits in which they differ (that is, the number of such $i$, for which $a_i\ne b_i$). We will say that the ... | n + 1 |
Let $i$ be the imaginary unit, set $M={z|iz=1}$, $N={z|z+i=1}$, then the product of the elements in set $M$ and $N$ is $($ $)$
A: $(-1+i)$
B: $(-1-i)$
C: $i$
D: $(-i)$ | B |
## Task A-2.5.
Let $n$ be a natural number. If a regular $n$-gon is divided into $n-2$ triangles by drawing $n-3$ diagonals that do not have common interior points, we say that we have obtained a triangulation. A triangulation of an $n$-gon, some of whose vertices are red, is good if each of those $n-2$ triangles has ... | \lceil\frac{n}{2}\rceil |
The coordinates of the vertex of the parabola $y=-(x-1)^{2}+3$ are ( ).
A: $(1,3)$
B: $(-1,3)$
C: $(-1,-3)$
D: $(1,-3)$ | A |
Problem 6. An excursion group of 6 tourists is visiting attractions. At each attraction, three people take photos, while the others photograph them. After what minimum number of attractions will each tourist have photos of all other participants in the excursion?
Answer: after 4 attractions. | 4 |
The missing number in the geometric sequence $2,6,18,54$, , 486 is
(A) 72
(B) 90
(C) 108
(D) 162
(E) 216 | 162 |
Let $p_{1}, p_{2}, \ldots, p_{30}$ be a permutation of the numbers $1,2, \ldots, 30$. For how many permutations does the equality $\sum_{k=1}^{30}\left|p_{k}-k\right|=450$ hold?
Answer: $(15 !)^{2}$. | (15!)^{2} |
Problem 7.8. In a chess tournament, 30 chess players participated, each playing against each other exactly once. A win was awarded 1 point, a draw - $1 / 2$, and a loss - 0. What is the maximum number of chess players who could have ended up with exactly 5 points by the end of the tournament? | 11 |
Given a triangle with side lengths of $2$, $a$, and $3$, where $a$ is an odd number, find the value of $a$. | 3 |
Given $\dfrac{1}{1+i} = \dfrac{1}{2} - ni$ where $n$ is a real number and $i$ is the imaginary unit, then $n=$ ______. | \dfrac{1}{2} |
A certain shooter fires $5$ shots in a row. The set of data formed by the number of rings hit (the number of rings is an integer) has a median of $8$, a unique mode of $9$, and a range of $3$. The average of this set of data is ( ).
A: $7.6$
B: $7.8$
C: $8$
D: $8.2$ | B |
6. Given that the function $f(x)$ is defined and increasing on $(0,+\infty)$, and satisfies $f(x) \cdot f\left(f(x)+\frac{1}{x}\right)=1$. Then $f(1)=$ ).
(A) 1
(B) 0
(C) $\frac{1+\sqrt{5}}{2}$
(D) $\frac{1-\sqrt{5}}{2}$ | D |
1. Let set $M=\{-2,0,1\}, N=\{1,2,3,4,5\}$, mapping $f: M \rightarrow N$ such that for any $x \in M$, $x+f(x)+x f(x)$ is an odd number, then the number of such mappings $f$ is ( ).
A. 45
B. 27
C. 15
D. 11 | 45 |
8.6. Ostap Bender put new tires on the car "Gnu Antelope". It is known that the front tires of the car wear out after 25,000 km, while the rear tires wear out after 15,000 km (the tires are the same both in the front and in the rear, but the rear ones wear out more). After how many kilometers should Ostap Bender swap t... | 9375 |
7. Find the smallest positive integer which cannot be expressed in the form $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$ where $a, b, c, d$ are non-negative integers.
(1 mark)
求最小的正整數, 該數不能表達成 $\frac{2^{a}-2^{b}}{2^{c}-2^{d}}$ （其中 $a 、 b 、 c 、 d$ 為非負整數）的形式。
(1 分) | 11 |
18. Determine the area of triangle $ABC$, if $A(1 ; 2)$, $B(-2 ; 5)$ and $C(4 ;-2)$. | 1.5 |
How many of the first ten numbers of the sequence $121, 11211, 1112111, \ldots$ are prime numbers?
$\textbf{(A) } 0 \qquad \textbf{(B) }1 \qquad \textbf{(C) }2 \qquad \textbf{(D) }3 \qquad \textbf{(E) }4$ | \textbf{(A)}0 |
1. If the function $f(x)=\lg \left(a x^{2}-4 x+a-3\right)$ has a range of $\mathbf{R}$, then the range of real number $a$ is
$\qquad$ | 0\leqslant\leqslant4 |
The solution set of the inequality $|2x^2-1| \leq 1$ is ( )
A: $\{x|-1\leq x\leq 1\}$
B: $\{x|-2\leq x\leq 2\}$
C: $\{x|0\leq x\leq 2\}$
D: $\{x|-2\leq x\leq 0\}$ | A |
11.A. In the Cartesian coordinate system $x O y$, the graph of the linear function $y = kx + b (k \neq 0)$ intersects the positive half-axes of the $x$-axis and $y$-axis at points $A$ and $B$, respectively, and makes the area of $\triangle O A B$ equal to $|O A| + |O B| + 3$.
(1) Express $k$ in terms of $b$;
(2) Find t... | 7+2 \sqrt{10} |
In the Cartesian coordinate system $xOy$, the curve $C_{1}$ is defined by $\begin{cases}x=t\cos \alpha \\ y=t\sin \alpha\end{cases} (t$ is a parameter, $t\neq 0)$, where $0\leqslant \alpha < \pi$. In the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis, the curve $C_{2}$ is defin... | 8 |
6. The graph of the inverse proportion function $y=\frac{k}{x}$ is an axially symmetric figure, one of its axes of symmetry is the graph of which of the following linear functions? ( ).
(A) $y=-k x$
(B) $y=|k| x$
(C) $y=\frac{-k}{|k|} x$
(D) $y=k x$ | C |
A baker uses $6\tfrac{2}{3}$ cups of flour when she prepares $\tfrac{5}{3}$ recipes of rolls. She will use $9\tfrac{3}{4}$ cups of flour when she prepares $\tfrac{m}{n}$ recipes of rolls where m and n are relatively prime positive integers. Find $m + n.$ | 55 |
Let points $A = (0, 0)$, $B = (1, 2)$, $C=(3, 3)$, and $D = (4, 0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left(\frac{p}{q}, \frac{r}{s}\right)$, where these fractions are in lowest terms. What is $p+q+r+s$?
$\textbf{(A)}\ 5... | 58 |
4. Points $P, Q$ are located on the sides $A B$ and $A C$ of triangle $A B C$ such that $A P: P B=1: 3, A Q: Q C=1: 1$. Point $M$ is chosen on side $B C$ completely at random. Find the probability that the area of triangle $A B C$ does not exceed the area of triangle $P Q M$ by more than six times. Find the expected va... | P(A)=\frac{5}{6};M_{X}=\frac{1}{4} |
8.2. Find the value of the expression $a^{3}+12 a b+b^{3}$, given that $a+b=4$. | 64 |
Let $$f(x)= \begin{cases} \sin(\pi x), & x\geq 0 \\ \cos\left( \frac {\pi x}{2}+ \frac {\pi}{3}\right), & x<0 \end{cases}$$ then $$f(f( \frac {15}{2}))$$ equals to （ ）
A: $$\frac {1}{2}$$
B: $$\frac { \sqrt {3}}{2}$$
C: $$- \frac {1}{2}$$
D: $$- \frac { \sqrt {3}}{2}$$ | B |
1. Let the arithmetic sequence $\left\{a_{n}\right\}$ satisfy $3 a_{8}=5 a_{13}$, and $a_{1}>0, S_{n}$ be the sum of the first $n$ terms, then the $n$ that makes $S_{n}$ the largest is $($ )
A. 18
B. 19
C. 20
D. 21 | 20 |
Given $|a-2|+(b+\frac{1}{2})^2=0$, find the value of the polynomial $(2ab^{2}+a^{2}b)-(3ab^{2}+a^{2}b-1)$. | \frac{1}{2} |
6. Define the sequence $\left\{a_{n}\right\}: a_{1}=1, a_{2}=2, a_{n+2}=a_{n}+a_{n+1}, n \in \mathbf{N}_{+}$, then $\left[\frac{a_{2}}{a_{1}}\right] \cdot\left\{\frac{a_{3}}{a_{2}}\right\} \cdot\left\{\frac{a_{4}}{a_{3}}\right\} \cdot \cdots$
- $\left\{\frac{a_{99}}{a_{98}}\right\} \cdot\left[\frac{a_{98}}{a_{2}}\right... | 1 |
1. Let $a-b=2+\sqrt{3}, b-c=2-\sqrt{3}$, then the value of $a^{2}+b^{2}+c^{2}-a b-b c-c a$ is | 15 |
Given that $\alpha \in \left( \pi ,\frac{3}{2}\pi \right)$ and $\cos \alpha =-\frac{4}{5}$, then $\tan \left(\frac{\pi }{4}-\alpha \right)$ equals to ( ).
A: $7$
B: $\frac{1}{7}$
C: $-\frac{1}{7}$
D: $-7$ | B |
We need to move a pawn from the bottom left corner of a chessboard to the opposite corner, moving one step to the right or up at a time. How many such paths pass through any of the four center squares? | 2450 |
Given a point on the line $y=x+2$ that draws a tangent to the circle $(x-4)^{2}+(y+2)^{2}=1$, the minimum length of the tangent is ( ).
A: $4 \sqrt {2}$
B: $\sqrt {31}$
C: $\sqrt {33}$
D: $4 \sqrt {2}-1$ | B |
7. Randomly throw three dice. The probability that the sum of the numbers on two of the dice is 7 is $\qquad$ . | \frac{5}{12} |
Given the function $f(x)=\sin \frac {\pi x}{3}$, then $f(1)+f(2)+\ldots+f(2010)=$ ()
A: $- \frac { \sqrt {3}}{2}$
B: $0$
C: $ \frac { \sqrt {3}}{2}$
D: $ \sqrt {3}$ | B |
4. Solve the equation $\frac{4}{\sqrt{\log _{3}(81 x)}+\sqrt{\log _{3} x}}+\sqrt{\log _{3} x}=3$. | 243 |
Given $i$ as the imaginary unit, which of the following statements is incorrect?
A: The vector corresponding to complex number $z_{1}$ is $\overrightarrow{O{Z_1}}$, and the vector corresponding to complex number $z_{2}$ is $\overrightarrow{O{Z_2}}$. If $|z_{1}+z_{2}|=|z_{1}-z_{2}|$, then $\overrightarrow{O{Z_1}}⊥\over... | \text{D} |
Let $C_1$ be a circle with centre $O$, and let $AB$ be a chord of the circle that is not a diameter. $M$ is the midpoint of $AB$. Consider a point $T$ on the circle $C_2$ with diameter $OM$. The tangent to $C_2$ at the point $T$ intersects $C_1$ at two points. Let $P$ be one of these points. Show that $PA^2+PB^2=4PT^2$... | PA^2 + PB^2 = 4PT^2 |
6. An increasing sequence of positive integers $\left\{a_{n}\right\}$ has the following property: for any $n \in \mathbf{N}_{+}$, it holds that $a_{n+2}=a_{n+1}+a_{n}$, and $a_{7}=120, a_{8}$ is ( ).
A. Uncertain
B. Equal to 194
C. Equal to 200
D. Equal to 216 | 194 |
The magic school stipulates that students must wear school uniforms, with shirts being either short-sleeved or long-sleeved, and pants being either shorts or long pants. One day, among the 150 students who arrived at school, 90 wore long pants, 41 wore long sleeves, and 32 wore both short sleeves and shorts. How many s... | 13 |
There are 6 measuring cups labeled A, B, C, D, E, and F, with capacities of 16 mL, 18 mL, 22 mL, 23 mL, 24 mL, and 34 mL, respectively. Some cups are filled with alcohol, some are filled with distilled water, and one cup is left empty. The volume of alcohol is twice the volume of distilled water. Then, the cups filled ... | \text{D} |
## Task B-1.1.
Calculate
$$
\frac{20182019^{2}-20182018^{2}}{20182018 \cdot 20182020-20182017 \cdot 20182019}
$$ | 1 |
In the Shanghai High School Entrance Exam (Gaokao) reform scheme, each high school student is required to choose 3 out of 6 subjects including Physics, Chemistry, Biology, Politics, History, and Geography (3 science subjects and 3 humanities subjects) for level tests. Student Ding has good grades in science and decided... | 10 |
A finite sequence of numbers $(a_1,\cdots,a_n)$ is said to be alternating if $$a_1>a_2~,~a_2<a_3~,~a_3>a_4~,~a_4<a_5~,~\cdots$$ $$\text{or ~}~~a_1<a_2~,~a_2>a_3~,~a_3<a_4~,~a_4>a_5~,~\cdots$$ How many alternating sequences of length $5$ , with distinct numbers $a_1,\cdots,a_5$ can be formed such that $a_i\in\{1,2,\cdo... | 32 \times \binom{20}{5} |
Problem 3. Marco was born between 1300 and 1400, and Darko between 1400 and 1500. Both were born on April 6th, in years that are perfect squares of natural numbers. Both lived for 110 years. In which year, while they were alive, were their ages on April 7th perfect squares of natural numbers. | 1469 |
Given the parabola $\Gamma$: $y^{2}=2px(p > 0)$ with focus $F$. If a line passing through point $F$ with a slope of $1$ intersects the parabola $\Gamma$ at points $M$ and $N$, and the area of $\triangle MON$ is $S_{\triangle MON}= \frac { \sqrt {2}}{2}$.
$(1)$ Find the equation of the parabola $\Gamma$;
$(2)$ If po... | 8 |
SI. 1 Let $[x]$ be the largest integer not greater than $x$. If $a=\left[(\sqrt{3}-\sqrt{2})^{2009}\right]+16$, find the value of $a$. | 16 |
As shown in the figure, points $C$, $E$, and $B$ are collinear, $CB \perp AB$, $AE \parallel DC$, $AB=8$, $CE=5$. Then the area of $\triangle AED$ is | 20 |
Consider 2022 lines in the plane, no two of which are parallel and no three of which are concurrent. Let $E$ be the set of their intersection points. We want to assign a color to each point in $E$ such that any two points on the same line, whose segment connecting them contains no other point of $E$, are of different c... | 3 |
12. (5 points) In the multiplication problem shown below, $A, B, C, D, E, F, G, H, I$ each represent different single-digit numbers. Then the five-digit number “FIGAA” is $\qquad$
$$
\begin{array}{r}
A B C \\
\times \quad D C \\
\hline B E A \\
\hline F I G A A
\end{array}
$$ | 15744 |
The sums of the first $n$, first $2n$, and first $3n$ terms of a geometric sequence are $A$, $B$, and $C$ respectively. Then ( )
A: $B^2 = AC$
B: $A + C = 2B$
C: $B(B-A) = A(C-A)$
D: $B(B-A) = C(C-A)$ | C |
Given a linear function $y=(m-3)x+6+2m$, if $y$ decreases as the independent variable $x$ increases, the range of $m$ is ____. | m < 3 |
In the polar coordinate system, given point A (2, $\frac{\pi}{2}$), and point B lying on the line $l: \rho \cos \theta + \rho \sin \theta = 0$ (where $0 \leq \theta \leq 2\pi$), find the polar coordinates of point B when the length of segment AB is the shortest. | (\sqrt{2}, \frac{3\pi}{4}) |
Given $(1-2x)^{9}=a\_{0}+a\_{1}x+a\_{2}x^{2}+…+a\_{9}x^{9}$, find the value of $a\_{0}+a\_{1}+a\_{2}+…+a\_{9}$. | -1 |
18. In a track and field meet, 216 athletes participate in the 100-meter dash. The venue has 6 lanes, with 1 athlete in each lane. Each race has 6 participants, and only the first-place finisher in each group advances to the next round. It will take ( ) races to determine the champion.
(A) 36
(B) 42
(C) 43
(D) 60
(E) 7... | 43 |
Given that the center of circle $C$ is the intersection point of the lines $x+y+1=0$ and $x-y-1=0$, and that the line $3x+4y-11=0$ intersects with circle $C$ at points $A$ and $B$ with $|AB|=6$, find the equation of circle $C$. | x^{2}+(y+1)^{2}=18 |
Given an ellipse C with its center at the origin that passes through point A (2, 3) and has a right focus at point F (2, 0).
(1) Find the equation and eccentricity e of ellipse C;
(2) If a line l parallel to OA intersects the ellipse at some points, find the range of the y-intercept of line l on the y-axis. | [-4\sqrt{3}, 4\sqrt{3}] |
$2 \cdot 19$ If $m, n$ are the roots of the quadratic equation $x^{2}+1994 x+7=0$, then the value of $\left(m^{2}+1993 m+6\right)\left(n^{2}+1995 n+8\right)$ is
(A) 2000.
(B) 1994.
(C) 1986.
(D) 7.
(Chinese "Zu Chongzhi Cup" Mathematics Invitational Competition, 1994) | 1986 |
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