problem stringlengths 10 7.47k | answer stringlengths 1 1.22k |
|---|---|
Points $D, E$, and $F$ are chosen on the sides $A C, A B$, and $B C$ of isosceles triangle $A B C (A B=B C)$ such that $D E = D F$ and $\angle B A C = \angle F D E$.
Prove that $A E + F C = A C$. | AE+FC=AC |
14. The perimeter of a rectangle is 20 decimeters. If it is cut along the line connecting the midpoints of the longer sides, it is divided into two identical smaller rectangles, and the sum of their perimeters is 6 decimeters more than the original perimeter. The area of the original rectangle is $\qquad$ square decime... | 21 |
9. Arrange $1,2, \cdots, 7$ in a row such that the sum of the first term, the sum of the first two terms, the sum of the first three terms, ... up to the sum of all seven terms are not divisible by 3. Then there are $\qquad$ different arrangements.
(Thailand) | 360 |
Group Event 6
The following shows the graph of $y=p x^{2}+5 x+p . A=(0,-2)$, $B=\left(\frac{1}{2}, 0\right), C=(2,0), O=(0,0)$.
G6.1 Find the value of $p$. | -2 |
The situation of the roots of the quadratic equation in $x$, $x^{2}-(m+1)x+3m-6=0$, is ( ).
A: Two equal real roots
B: Two distinct real roots
C: Two real roots
D: No real roots | \text{C} |
The sum of the two perfect cubes that are closest to $500$ is $343+512=855$. Find the sum of the two perfect cubes that are closest to $2008$. | 3925 |
Suppose $x_1, x_2, x_3$ are the roots of polynomial $P(x) = x^3 - 6x^2 + 5x + 12$
The sum $|x_1| + |x_2| + |x_3|$ is
(A): $4$ (B): $6$ (C): $8$ (D): $14$ (E): None of the above. | 8 |
4. In 2016, the world record for completing a 5000m three-legged race was 19 minutes and 6 seconds. It was set by Damian Thacker and Luke Symonds in Sheffield.
What was their approximate average speed in $\mathrm{km} / \mathrm{h}$ ?
A 10
B 12
C 15
D 18
E 25 | 15 |
Given that the radius of a sector is 3cm, and the central angle is 2 radians, then the area of the sector is \_\_\_\_\_\_ cm2. | 9 |
Cube [ Properties of parts obtained by cutting ]
The surface of a 3 x 3 x 3 Rubik's cube consists of 54 cells. What is the maximum number of cells that can be marked so that the marked cells do not share any vertices? | 14 |
# Problem 6.
Find the smallest natural number $n$, for which the number $n+2018$ is divisible by 2020, and the number $n+2020$ is divisible by 2018. | 2034142 |
International Mathematical Olympiad National Selection Test
Malaysia 2020 Round 1 Primary
Time: 2.5 hours [hide=Rules]
$\bullet$ For each problem you have to submit the answer only. The answer to each problem is a non-negative integer.
$\bullet$ No mark is deducted for a wrong answer.
$\bullet$ The maximum number of p... | 9 |
## Problem Statement
Find the angle between the planes:
$$
\begin{aligned}
& 3 x-y+2 z+15=0 \\
& 5 x+9 y-3 z-1=0
\end{aligned}
$$ | \frac{\pi}{2} |
Nonnegative real numbers $p_{1},\ldots,p_{n}$ and $q_{1},\ldots,q_{n}$ are such that $p_{1}+\cdots+p_{n}=q_{1}+\cdots+q_{n}$
Among all the matrices with nonnegative entries having $p_i$ as sum of the $i$-th row's entries and $q_j$ as sum of the $j$-th column's entries, find the maximum sum of the entries on the main di... | \sum_{i=1}^n \min(p_i, q_i) |
For a positive integer $n$, let $d_n$ be the units digit of $1 + 2 + \dots + n$. Find the remainder when
\[\sum_{n=1}^{2017} d_n\]is divided by $1000$. | 69 |
Given a triangle $\triangle ABC$ with angles $A$, $B$, $C$ and their respective opposite sides $a$, $b$, $c$, such that $b^2 + c^2 - a^2 = \sqrt{3}bc$.
(1) If $\tan B = \frac{\sqrt{6}}{12}$, find $\frac{b}{a}$;
(2) If $B = \frac{2\pi}{3}$ and $b = 2\sqrt{3}$, find the length of the median on side $BC$. | \sqrt{7} |
The arithmetic mean of a list of numbers is the sum of them divided by the number of elements in the list. For example, the arithmetic mean of the list 3, 3, 4, 5, and 10 is
$$
\frac{3+3+4+5+10}{5}=5
$$
The arithmetic mean of 5 distinct positive integers is equal to 11. What is the maximum possible value of a number ... | 45 |
The coordinates of the center of the circle described by \\(\rho=5\cos \theta-5 \sqrt {3}\sin \theta\\) are ( ).
A: \\(\left(5, \dfrac {4\pi}{3}\right)\\)
B: \\(\left(5, \dfrac {\pi}{3}\right)\\)
C: \\(\left(5, \dfrac {\pi}{3}\right)\\)
D: \\(\left(5, \dfrac {5\pi}{3}\right)\\) | D |
In an increasing geometric sequence $\{a_n\}$, it is known that $a_1=1$, and $$\frac {a_{3}+a_{4}}{a_{1}+a_{2}}=4,$$ then the value of $S_{5}$ is \_\_\_\_\_. | 31 |
Let $P=\{y|y=-x^2+1, x\in \mathbb{R}\}$, and $Q=\{y|y=2^x, x\in \mathbb{R}\}$. Then ( )
A: $P\subseteq Q$
B: $Q\subseteq P$
C: $\complement_{\mathbb{R}}P\subseteq Q$
D: $Q\subseteq \complement_{\mathbb{R}}P$ | C |
4. Let $n$ be a natural number. Determine (in terms of $n$) the smallest natural number $m$ for which it is possible to color exactly $n$ cells of an infinite square grid such that the following condition holds:
- Every rectangle with sides along the grid lines, whose two opposite corners are colored, contains at most... | =1forn=1=[\frac{n}{5}]+2forn\geqslant2 |
1. In an online store, two types of New Year's gifts are sold. The first type of gift contains a toy, 3 chocolate, and 15 caramel candies and costs 350 rubles. The second type of gift contains 20 chocolate and 5 caramel candies and costs 500 rubles. Eugene wants to buy an equal number of caramel and chocolate candies, ... | 3750 |
3. Two spheres with radii 2 and 1 are placed inside a cone. The larger sphere touches the base and the lateral surface. Calculate the volume of the cone. | \frac{64\pi}{3} |
Let $n \geqslant 2$ be an integer, and let $A_{n}$ be the set
$$ A_{n}=\left\{2^{n}-2^{k} \mid k \in \mathbb{Z}, 0 \leqslant k<n\right\} . $$
Determine the largest positive integer that cannot be written as the sum of one or more (not necessarily distinct) elements of $A_{n}$. (Serbia) Answer. $(n-2) 2^{n}+1$. | (n-2) 2^{n}+1 |
$30 \cdot 24$ Let $a_{n}=4^{2 n+1}+3^{n+2}(n=1,2,3, \cdots), p$ is the smallest prime that can divide infinitely many terms of $a_{1}, a_{2}$, $a_{3}, \cdots$, $q$ is the smallest prime that can divide every term of $a_{1}, a_{2}, a_{3}, \cdots$, then $p \cdot q$ is
(A) $5 \cdot 7$.
(B) $7 \cdot 13$.
(C) $13 \cdot 13$.... | D |
The point $P\left( \frac{3}{2},y_{0}\right)$ on the parabola $y^{2}=ax(a > 0)$ is at a distance of $2$ from the focus $F$. Find the value of $a$. | 2 |
A travel enthusiast plans to choose two countries to visit from 3 Asian countries \\(A_1, A_2, A_3\\) and 3 European countries \\(B_1, B_2, B_3\\).
\\((1)\\) If two countries are randomly selected from these 6 countries, what is the probability that both countries are from Asia?
\\((2)\\) If one country is randomly s... | \dfrac{2}{9} |
For all $x_{1}$, $x_{2}\in \left[1,e\right]$, when $x_{1} \lt x_{2}$, $ln\frac{x_{1}}{x_{2}}＜a(x_{1}-x_{2})$, then the maximum value of the real number $a$ is ( )
A: $\frac{1}{e^{2}}$
B: $\frac{1}{e}$
C: $\frac{\sqrt{e}}{e}$
D: $1$ | B |
16. $[\mathbf{9}]$ Solve for $x$ :
$$
x\lfloor x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor\rfloor=122 \text {. }
$$ | \frac{122}{41} |
7.1. Given a rectangular parallelepiped $2 \times 3 \times 2 \sqrt{3}$. What is the smallest value that the sum of the distances from an arbitrary point in space to all eight of its vertices can take?
+1=x f(x)$, then $f(x)=$ | \frac{x-3}{x^{2}-x+4} |
45. Given an arithmetic progression consisting of four integers. It is known that the largest of these numbers is equal to the sum of the squares of the other three numbers. Find these numbers. | 2,1,0,-1 |
Example 2.29. Prove that the integral
$$
I=\int_{0}^{\infty} \frac{d x}{\left(1+x^{2}\right)\left(1+x^{a}\right)}
$$
does not depend on $a$. | \frac{\pi}{4} |
Given a point \\(P(2, -\frac{5\pi}{3})\\) in the polar coordinate plane, the polar coordinates and Cartesian coordinates of the point symmetric to \\(P\\) with respect to the pole are ( )
A: \\(\left(2, \frac{\pi}{3}\right),\left(1, \sqrt{3}\right) \\)
B: \\(\left(2, -\frac{\pi}{3}\right),\left(1, -\sqrt{3}\right)... | \text{D} |
II. (40 points) Given the function $f: \mathbf{R}_{+} \rightarrow \mathbf{R}_{+}$ satisfies $f(f(n))=2016 n-215 f(n)$.
Find $f(n)$. | f(n)=9n |
The complex number $z=\frac{2-i}{2+i}$ (where $i$ is the imaginary unit) corresponds to a point in the complex plane located in the quadrant:
A: First quadrant
B: Second quadrant
C: Third quadrant
D: Fourth quadrant | \text{D} |
A3. A square, a circle, and an equilateral triangle have the same perimeter of $36 \mathrm{~cm}$. Which of these shapes has the largest area?
(A) triangle
(B) $\mathrm{circle}$
(C) square
(D) All shapes have the same area.
(E) It is not possible to determine. | 103.2\mathrm{~}^{2} |
Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate
\[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by... | \binom{2n-1}{n-1} n! |
Given a sector with a radius of $10cm$ and a perimeter of $45cm$, find the central angle of the sector in radians. | 2.5 |
1. The function $f(x)(x \neq 1)$ defined on $\mathbf{R}$ satisfies $f(x)+2 f\left(\frac{x+2002}{x-1}\right)=4015-x$. Then $f(2004)=(\quad)$. | 2005 |
There are 232 people working at location A and 146 people working at location B. If $x$ people are transferred from location B to location A, then the number of people working at location A is three times the number of people working at location B. Among the following equations, the correct one is ( )
A: $3(323+x)=146... | \text{D} |
The Hangzhou Asian Games will be held on September 23, 2023. A sports brand sponsor has developed a new type of sports equipment and plans to produce and install 360 units in 15 days to be delivered to the designated venue for athletes to use. Due to the inability to allocate enough skilled workers to complete the inst... | 4 |
If the equation $x^3 - x + 1 = 0$ has a root in the interval $(a, b)$ (where $a, b$ are integers, and $b - a = 1$), then $a + b = \_\_\_\_\_\_\_\_$. | -3 |
## Task A-1.4.
In container $A$, there are four kilograms of chickpeas, of which $45 \%$ are chickpeas. In container $B$, there are five kilograms of chickpeas, of which $48 \%$ are chickpeas. In container $C$, there is one kilogram of chickpeas. A certain portion is transferred from this container to container $A$, a... | 80 |
Given the functions $f(x) = \frac{1}{2}mx^2 + 1$, $g(x) = 2\ln x - (2m+1)x - 1$ ($m \in \mathbb{R}$), and $h(x) = f(x) + g(x)$,
(1) If the tangents of the function $h(x)$ at $(1, f(1))$ and $(3, f(3))$ are parallel, find the value of the real number $m$;
(2) Determine the intervals of monotonicity for $h(x)$. | \frac{2}{3} |
2. (5 points) At the World Meteorologists Conference, each participant in turn announced the average monthly temperature in their hometown. At this moment, all the others recorded the product of the temperatures in their and the speaker's cities. In total, 36 positive and 36 negative numbers were recorded. What is the ... | 3 |
## Task 14/89
Given a parallelogram $A B C D$ with $E$ as the midpoint of side $C D$. In what ratio does the line segment $B E$ divide the diagonal $A C$? | AS:SC=2:1 |
Given $z\_1=\frac{16}{a+5}-(10-a^{2})i$, $z\_2=\frac{2}{1-a}+(2a-5)i$, where $a \in R$ and $i$ is the imaginary unit. If $z\_1+z\_2$ is a real number, find the value of the real number $a$. | a=3 |
Given a sequence of positive terms ${a_n}$ with the sum of the first $n$ terms denoted as $S_n$. It is known that $S_n$ is the arithmetic mean of $a_n^2$ and $a_n$.
(I) Find the general term formula for the sequence ${a_n}$.
(II) If ${a_{k_n}}$ is a member of ${a_1, a_2, …a_n, …}$ and forms a geometric sequence ${a_{k_... | T_n = 2^{n+1} - 2 |
3 [ Common Tangent to Two Circles ]
Circles $S 1$ and $S 2$ with centers $O 1$ and $O 2$ respectively touch each other externally; a line is tangent to the circles at different points $A$ and $B$ respectively. It is known that the intersection point of the diagonals of quadrilateral $O 1 A B O 2$ lies on one of the ci... | 1:2 |
The function $f(x)$ is a decreasing function on the interval $[-1, 1]$. $\alpha$ and $\beta$ are two acute angles of a triangle, and $\alpha \neq \beta$. Among the following inequalities, the correct one is ( )
A: $f(\sin \alpha) > f(\cos \beta)$
B: $f(\cos \alpha) f(\sin \beta)$
D: $f(\sin \alpha) < f(\sin \beta)... | \text{C} |
[ $\underline{\text { equations in integers }}]$ [GCD and LCM. Mutual simplicity ]
The last digit in the notation of a natural number is 2016 times less than the number itself. Find all such numbers.
# | 4032,8064,12096,16128 |
8. (6th Irish Mathematical Olympiad 1993) Real numbers $\alpha, \beta$ satisfy the system of equations
$$
\left\{\begin{array}{l}
\alpha^{3}-3 \alpha^{2}+5 \alpha-17=0, \\
\beta^{3}-3 \beta^{2}+5 \beta+11=0 .
\end{array}\right.
$$
Find $\alpha+\beta$. | \alpha+\beta=2 |
Given point $P(4,2a+10)$, if point $P$ lies on the $x$-axis, then $a=$______. | -5 |
13. Given the sequence $\left\{a_{n}\right\}$ satisfies $a_{1} 、 a_{2}+1 、 a_{3}$ forming an arithmetic sequence, and for any positive integer $n$, we have
$$
S_{n}=\frac{1}{2} a_{n+1}-2^{n}+\frac{3}{2},
$$
where $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Find
(1) the values of $... | a_{n}=3^{n-1}-2^{n} |
A certain radio station only starts broadcasting news at the beginning of each hour and half-hour, each lasting for 5 minutes. If a person turns on the radio without knowing the time, the probability of hearing the news is ( ).
A: $ \boxed{\frac {1}{14}}$
B: $ \boxed{\frac {1}{12}}$
C: $ \boxed{\frac {1}{7}}$
D: $ ... | D |
How many solutions does the system have:
$ \{\begin{matrix}&(3x+2y) *(\frac{3}{x}+\frac{1}{y})=2\\ & x^2+y^2\leq 2012\\ \end{matrix} $
where $ x,y $ are non-zero integers | 102 |
If point $P$ is in the second quadrant, and its distance to the $x$-axis is $3$, and its distance to the $y$-axis is $1$, then the coordinates of point $P$ are ____. | (-1,3) |
Let $S$ be a randomly selected four-element subset of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. Let $m$ and $n$ be relatively prime positive integers so that the expected value of the maximum element in $S$ is $\dfrac{m}{n}$. Find $m + n$. | 41 |
Example 1. Calculate approximately the value of the function $f(x)=$ $=\sqrt{x^{2}+x+3}$ at $x=1.97$ | 2.975 |
Given that $P(x, y)$ is a point on the circle defined by $(x+1)^2 + y^2 = 1$, find the maximum value of $2x + 3y$. | \sqrt{13} - 2 |
Solve the following equation:
$$
\sqrt{x-\sqrt{1-x}}+\sqrt{x}=1
$$ | \frac{16}{25} |
Let $E$ and $F$ be the respective midpoints of $BC,CD$ of a convex quadrilateral $ABCD$. Segments $AE,AF,EF$ cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$. | 6 |
Example 1. Find the rotor of the vector $\mathbf{v}=(x+z) \mathbf{i}+(y+z) \mathbf{j}+\left(x^{2}+z\right) \mathbf{k}$. | rot{}=-{i}-(2x-1){j} |
Given $z-\left(2-3i\right)=\frac{1}{i}$ (where $i$ is the imaginary unit), which quadrant does the point corresponding to the complex number $\overline{z}$ lie in the complex plane?
A: Quadrant IV
B: Quadrant III
C: Quadrant II
D: Quadrant I | D |
$(1)$ Given a complex number $z=(m^{2}-8m+15)+(m^{2}-4m+3)i$, where $m\in R$. If $z$ is a purely imaginary number, find the value of the real number $m$;$(2)$ Given a complex number $z=\frac{{2+4mi}}{{1+i}}$ ($m\in R$, $i$ is the imaginary unit). Let $\overline{z}$ be the conjugate of $z$, and the point corresponding t... | (-\frac{1}{2}, \frac{1}{2}) |
If the graph of the function $f(x) = \sin \omega x \cos \omega x + \sqrt{3} \sin^2 \omega x - \frac{\sqrt{3}}{2}$ ($\omega > 0$) is tangent to the line $y = m$ ($m$ is a constant), and the abscissas of the tangent points form an arithmetic sequence with a common difference of $\pi$.
(Ⅰ) Find the values of $\omega$ and... | \frac{11\pi}{3} |
Given the complex number $z$ that satisfies the equation $(2+i)z=-i$ ($i$ is the imaginary unit), the point corresponding to $z$ in the complex plane is located in ( )
A: the first quadrant
B: the second quadrant
C: the third quadrant
D: the fourth quadrant | \text{C} |
Given the function $f(x) = a(x - 5)^2 + 6\ln x$, where $a \in \mathbb{R}$, the tangent line to the curve $y = f(x)$ at the point $(1, f(1))$ intersects the $y$-axis at the point $(0, 6)$.
(1) Find the equation of the tangent line to the function $f(x)$ at the point $(1, f(1))$.
(2) Find the extreme values (maximum an... | 2 + 6\ln 3 |
Given that the terminal side of angle \\(\alpha\\) passes through \\(P(4,-3)\\).
\\((1)\\) Find the value of \\(2\sin \alpha - \cos \alpha\\);
\\((2)\\) Find the coordinates of point \\(P\\), the intersection of the terminal side of angle \\(\alpha\\) with the unit circle. | \left(\frac{4}{5}, -\frac{3}{5}\right) |
2. Rationalize and simplify the fraction: $\frac{9+2 \sqrt{2}}{4+\sqrt{162}}$ | \frac{\sqrt{2}}{2} |
There are 1000 doors $D_1, D_2, . . . , D_{1000}$ and 1000 persons $P_1, P_2, . . . , P_{1000}$.
Initially all the doors were closed. Person $P_1$ goes and opens all the doors.
Then person $P_2$ closes door $D_2, D_4, . . . , D_{1000}$ and leaves the odd numbered doors open. Next $P_3$ changes the state of every third... | 31 |
1. $n$ houses with house numbers 1 to $n$ are located on SMO Street, where $n$ is a natural number. The houses with odd numbers are on the left-hand side of the street, the houses with even numbers are on the right-hand side. The letter carrier Quirin wants to deliver a newspaper to each house. How many ways does he ha... | 2\cdot(\frac{n}{2}!)^{2} |
A certain product increased in price by 5% in 2002 compared to 2001, increased again by 10% in 2003 compared to 2002, and decreased by 12% in 2004 compared to 2003. Thus, the price in 2004 compared to 2001 was ( )
A: Increased by 3%
B: Increased by 1.64%
C: Increased by 1.2%
D: Decreased by 1.2% | \text{B} |
2. Given a function $f(x)$ defined on $\mathbf{R}$ that satisfies
$$
\begin{array}{l}
f(x+1)=f(-x), \\
f(x)=\left\{\begin{array}{ll}
1, & -1<x \leqslant 0 \\
-1, & 0<x \leqslant 1 .
\end{array}\right.
\end{array}
$$
Then $f(f(3.5))=$ $\qquad$ | -1 |
In the expansion of $({x-\frac{2}{x}})^6$, the constant term is ( )
A: $-20$
B: $20$
C: $-160$
D: $160$ | C |
2. Solve the equation $\log _{3}\left(\log _{2} x+12\right)+2=4$ and write the solution as a fraction. | \frac{1}{8} |
How many alphabetic sequences (that is, sequences containing only letters from $a\cdots z$) of length $2013$ have letters in alphabetic order? | \binom{2038}{25} |
Convert the binary number $1011011_{(2)}$ to decimal. | 91_{(10)} |
If the directrix of a parabola is $x = -7$, then the standard equation of the parabola is ( )
A: $x^2 = -28y$
B: $y^2 = 28x$
C: $y^2 = -28x$
D: $x^2 = 28y$ | \text{B} |
It is known that point P has coordinates (2, 3), and the symmetric point of P with respect to the y-axis is ____. | (-2, 3) |
9. The real solution to the equation $x^{[x]}=\frac{9}{2}$ is $\qquad$ $([x]$ denotes the greatest integer not exceeding the real number $x$). | \frac{3 \sqrt{2}}{2} |
Given the sequence $\{a_n\}$ satisfies $a_1=1$, $a_{n+1}-a_n=(-1)^{n+1}\frac{1}{n(n+2)}$, then the sum of the first $40$ terms of the sequence $\{(-1)^n a_n\}$ is
A: $\frac{19}{20}$
B: $\frac{325}{462}$
C: $\frac{41}{84}$
D: $\frac{20}{41}$ | D |
2. Points $X, Y$, and $Z$ lie on a circle with center $O$ such that $X Y=12$. Points $A$ and $B$ lie on segment $X Y$ such that $O A=A Z=Z B=B O=5$. Compute $A B$. | 2\sqrt{13} |
3. In the Cartesian coordinate system, the set of points $(x, y)$ that satisfy $x<y,|x|<3,|y|<3$, and make the equation $\left(x^{3}-y^{3}\right) t^{4}+$ $(3 x+y) t^{2}+\frac{1}{x-y}=0$ have no real roots is denoted as $N$. Then the area of the region formed by the point set $N$ is $\qquad$ | \frac{81}{5} |
In the diagram, points $A, B, C$, and $D$ are on a circle. Philippe uses a ruler to connect each pair of these points with a line segment. How many line segments does he draw?
 | 6 |
3. In triangle $\triangle A B C$, the base $A B$ of which lies on the x-axis, altitudes $A M, B N$ and $C K$ are drawn. Find the length of the base $A B$, if the coordinates of points $M(2,2)$ and $N(4,4)$ are known. | 4\sqrt{5} |
Problem 4. In triangle $A B C$ with the ratio of sides $A B: A C=5: 4$, the bisector of angle $B A C$ intersects side $B C$ at point $L$. Find the length of segment $A L$, if the length of the vector $4 \cdot \overrightarrow{A B}+5 \cdot \overrightarrow{A C}$ is 2016. | 224 |
11. (This question is worth 20 points) Let $F_{1}, F_{2}$ be the left and right foci of the ellipse $\frac{x^{2}}{2}+y^{2}=1$, respectively. Suppose a line $l$ that does not pass through the focus $F_{1}$ intersects the ellipse at two distinct points $A, B$. The distance from the focus $F_{2}$ to the line $l$ is $d$. I... | (\sqrt{3},2) |
Given $(3-x)^4 = a + a_1x + a_2x^2 + a_4x^4$, then $a - a_1 + a_2 - a_3 + a_4$ equals to ( )
A: 256
B: 120
C: 136
D: 16 | \text{A} |
65. There is a four-digit number, its product with 6 is a perfect cube, and its quotient with 6 is a perfect square; then this four-digit number is $\qquad$. | 7776 |
[ Proof by contradiction ] [ Examples and counterexamples. Constructions ]
There are several stones, the mass of each of which does not exceed 2 kg, and the total mass is 100 kg. From them, several stones are selected, the total mass of which differs from 10 kg by the smallest possible number $d$ for this set. What is... | \frac{10}{11} |
(2) Given two lines $l_{1}: y=2, l_{2}: y=4$, let the graph of the function $y=3^{x}$ intersect $l_{1} 、 l_{2}$ at points $A 、 B$, and the graph of the function $y=5^{x}$ intersect $l_{1} 、 l_{2}$ at points $C$ 、 $D$, then the coordinates of the intersection point of lines $A B$ and $C D$ are $\qquad$ . | (0,0) |
Which of the following sets of equations is a system of two linear equations?
A: $\left\{\begin{array}{l}x+y=4\\ x-y=1\end{array}\right.$
B: $\left\{\begin{array}{l}4x+3y=6\\ 2y+z=4\end{array}\right.$
C: $\left\{\begin{array}{l}x+y=5\\{x}^{2}+{y}^{2}=13\end{array}\right.$
D: $\left\{\begin{array}{l}\frac{1}{x}+y=4\... | A |
45. Calculate the determinant
$$
D=\left|\begin{array}{rrrr}
3 & 0 & 2 & 0 \\
2 & 3 & -1 & 4 \\
0 & 4 & -2 & 3 \\
5 & 2 & 0 & 1
\end{array}\right|
$$ | -54 |
## Problem 6.
Find all real solutions of the equation
$$
3^{x^{2}-x-y}+3^{y^{2}-y-z}+3^{z^{2}-z-x}=1
$$ | (x,y,z)=(1,1,1) |
5. Calculate $\frac{3}{1!+2!+3!}+\frac{4}{2!+3!+4!}+\cdots+\frac{2001}{1999!+2000!+2001!}$
The value is | \frac{1}{2}-\frac{1}{2001!} |
From a bag containing $a$ red balls and $b$ blue balls $(a$ and $b$ are both not less than $2)$, one ball is randomly drawn without replacement each time. Let "drawing a red ball for the first time" be $A_{1}$, and "drawing a blue ball for the first time" be $A_{2}$; "drawing a red ball for the second time" be $B_{1}$,... | D |
Evaluate the integral $\int_{0}^{\pi} (x-\sin x)dx$.
A: $\frac{\pi^2}{2}-2$
B: $\frac{\pi^2}{2}$
C: $\frac{\pi}{2}-2$
D: $\pi^2$ | A |
Four students and two teachers are taking a commemorative photo together. How many different seating arrangements are there for each of the following cases? (Please provide numerical answers)
(1) The teachers must sit in the middle;
(2) The teachers cannot sit at the ends, but must sit together;
(3) The teachers cannot... | 144 |
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