task_type stringclasses 4
values | problem stringlengths 21 5.23k | answer stringlengths 1 8.29k | problem_tokens int64 11 1.16k | answer_tokens int64 1 2.04k |
|---|---|---|---|---|
math | 13. (15 points) In the sequence $\left\{a_{n}\right\}$, $a_{n}=2^{n} a+b n-80\left(a, b \in \mathbf{Z}_{+}\right)$.
It is known that the minimum value of the sum of the first $n$ terms $S_{n}$ is obtained if and only if $n=6$, and $7 \mid a_{36}$. Find the value of $\sum_{i=1}^{12}\left|a_{i}\right|$. | 8010 | 124 | 4 |
math | 6. Find all values of the parameter $a$ for which the system $\left\{\begin{array}{l}4|x|+3|y|=12, \\ x^{2}+y^{2}-2 x+1-a^{2}=0\end{array} \quad\right.$ a) has exactly 3 solutions; b) has exactly 2 solutions. | )||=2;b)||\in{\frac{8}{5}}\bigcup(2;\frac{16}{5})\bigcup{\sqrt{17}} | 81 | 36 |
math | 5. In an isosceles trapezoid $A B C D$ with lateral sides $A B$ and $C D$, the lengths of which are 10, perpendiculars $B H$ and $D K$ are drawn from vertices $B$ and $D$ to the diagonal $A C$. It is known that the bases of the perpendiculars lie on segment $A C$ and $A H: A K: A C=5: 14: 15$. Find the area of trapezoi... | 180 | 120 | 3 |
math | 13.062. An apprentice turner is machining pawns for a certain number of chess sets. He wants to learn to produce 2 more pawns per day than he does now; then he would complete the same task 10 days faster. If he could learn to produce 4 more pawns per day than he does now, the time required to complete the same task wou... | 15 | 113 | 2 |
math | Let $ABCD$ be a unit square. $E$ and $F$ trisect $AB$ such that $AE<AF. G$ and $H$ trisect $BC$ such that $BG<BH. I$ and $J$ bisect $CD$ and $DA,$ respectively. Let $HJ$ and $EI$ meet at $K,$ and let $GJ$ and $FI$ meet at $L.$ Compute the length $KL.$ | \frac{6\sqrt{2}}{35} | 101 | 13 |
math | Task 12. (16 points)
The budget of the Petrovs consists of the following income items:
- parents' salary after income tax deduction - 56000 rubles;
- grandmother's pension - 14300 rubles;
- son's scholarship - 2500 rubles
Average monthly expenses of the family include:
- utility payments - 9800 rubles;
- food - 210... | 16740 | 185 | 5 |
math | Find all integers $n$ such that
$$
n^{4}+6 n^{3}+11 n^{2}+3 n+31
$$
is a perfect square.
(Xu Wandang) | n=10 | 47 | 4 |
math | 1. For natural numbers $m$ and $n$, $n>m>1$. The last three digits of $1978^{m}$ and $1978^{n}$ in decimal notation are the same in order. Find $m$ and $n$ such that $m+n$ is the smallest possible. (Kuba, 6 points) | =3,n=103 | 76 | 7 |
math | 4. (ROM) Solve the equation
$$ \cos ^{2} x+\cos ^{2} 2 x+\cos ^{2} 3 x=1 . $$ | x \in\{\pi / 2+m \pi, \pi / 4+m \pi / 2, \pi / 6+m \pi / 3 \mid m \in \mathbb{Z}\} | 39 | 48 |
math | 7. Given the sets
$$
\begin{array}{l}
A=\left\{(x, y) \mid x=m, y=-3 m+2, m \in \mathbf{Z}_{+}\right\}, \\
B=\left\{(x, y) \mid x=n, y=a\left(a^{2}-n+1\right), n \in \mathbf{Z}_{+}\right\} .
\end{array}
$$
Then the number of integers $a$ such that $A \cap B \neq \varnothing$ is $\qquad$. | 10 | 128 | 2 |
math | 2. A line $l$ is drawn through the right focus of the hyperbola $x^{2}-\frac{y^{2}}{2}=1$ intersecting the hyperbola at points $A$ and $B$. If a real number $\lambda$ makes $|A B|=\lambda$ such that there are exactly 3 lines $l$, then $\lambda=$ $\qquad$ . | 4 | 86 | 1 |
math | 4. Given the sequence $\left\{a_{n}\right\}$ satisfies:
$$
a_{1}=-2 \text {, }
$$
and $S_{n}=\frac{3}{2} a_{n}+n$ ( $S_{n}$ is the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$).
$f(x)$ is an odd function defined on $\mathbf{R}$, and satisfies
$$
f(2-x)=f(x) \text {. }
$$
Then $f\left(a_{2021}\rig... | 0 | 138 | 1 |
math | Example 1. If $p, q$ are both natural numbers, and the two roots of the equation $p x^{2}-$ $q x+1985=0$ are both prime numbers, then what is the value of $12 p^{2}+q$? (85 Beijing Mathematics Competition Question) | 414 | 70 | 3 |
math | Problem 7. In the queue for the school cafeteria, 16 schoolchildren are standing in such a way that boys and girls alternate. (The first is a boy, followed by a girl, then a boy again, and so on.) Any boy who is followed by a girl in the queue can swap places with her. After some time, it turned out that all the girls ... | 36 | 100 | 2 |
math | 8-5. Diligent Asey multiplied two three-digit numbers, while lazy Petya simply wrote them down one after the other. Petya's result turned out to be 7 times larger than Asey's. What numbers did she multiply? | 143143 | 53 | 6 |
math | 13. Let \( f(x, y) = \frac{a x^{2} + x y + y^{2}}{x^{2} + y^{2}} \), satisfying
\[
\max _{x^{2}+y^{2}+0} f(x, y) - \min _{x^{2}+y^{2}+0} f(x, y) = 2 \text{. }
\]
Find \( a \). | a=1 \pm \sqrt{3} | 101 | 10 |
math | 9. Fill the $3 \times 3$ grid with the numbers $1,2,3,4,5,6,7,8,9$ randomly, with each small square containing exactly one number, and all numbers being distinct. The probability that the sum of the numbers in each row and each column is odd is $\qquad$. | \frac{1}{14} | 72 | 8 |
math | Find a positive integer $n$ with five non-zero different digits, which satisfies to be equal to the sum of all the three-digit numbers that can be formed using the digits of $n$. | 35964 | 39 | 5 |
math | Circle $\omega_1$ of radius $1$ and circle $\omega_2$ of radius $2$ are concentric. Godzilla inscribes square $CASH$ in $\omega_1$ and regular pentagon $MONEY$ in $\omega_2$. It then writes down all 20 (not necessarily distinct) distances between a vertex of $CASH$ and a vertex of $MONEY$ and multiplies them all togeth... | 2^{20} + 1 | 102 | 8 |
math | 34.13. Find all continuous solutions of the functional equation
$$
f(x+y) f(x-y)=(f(x))^{2}
$$
(Lobachevsky).
## 34.4. Functional equations for differentiable functions | f(x)=^x | 52 | 5 |
math | The five-digit number $12110$ is divisible by the sum of its digits $1 + 2 + 1 + 1 + 0 = 5.$ Find the greatest five-digit number which is divisible by the sum of its digits | 99972 | 56 | 5 |
math | Three consecutive odd numbers, the sum of the squares of which is a four-digit number whose digits are equal. Which are these numbers? | 41,43,45 | 27 | 8 |
math | 1. The base of the prism is a square with side $a$. The length of the side $a$ is twice the length of the height of the prism. The numerical values of the surface area and volume of the prism are equal. Determine the lengths of the edges of this prism. | =8,=4 | 59 | 5 |
math | 3. Arrange $1,2,3,4,5,6$ randomly in a row, denoted as $a, b, c, d, e, f$, then the probability that $a b c+d e f$ is an even number is . $\qquad$ | \frac{9}{10} | 59 | 8 |
math | 11. (25 points) Let positive numbers $a, b$ satisfy $a+b=1$. Find
$$
M=\sqrt{1+2 a^{2}}+2 \sqrt{\left(\frac{5}{12}\right)^{2}+b^{2}}
$$
the minimum value. | \frac{5\sqrt{34}}{12} | 68 | 14 |
math | Example 4 Solve the equation $\sqrt{12-\frac{12}{x^{2}}}+\sqrt{x^{2}-\frac{12}{x^{2}}}=x^{2}$. | x = \pm 2 | 43 | 6 |
math | 2. Find all positive integers $n$ such that for any positive integers $a, b, c$ satisfying $a+b+c \mid a^{2}+b^{2}+c^{2}$, we have $a+b+c \mid a^{n}+b^{n}+c^{n}$.
(Li Changyong) | n=3k-1n=3k-2(k\in{N}^{*}) | 73 | 21 |
math | Among four numbers, the first three form an arithmetic sequence, the second three form a geometric sequence. The sum of the first and fourth number is 37, the sum of the second and third is 36. Which are these four numbers? | 12,16,20,25\quad | 51 | 13 |
math | Let's write digits in place of the letters $I, K, S$ so that the following subtraction is correct:
$K I S-$
SIK
$\overline{S K I}$ | 954-459=495 | 41 | 11 |
math | 24. (HUN 5) ${ }^{\mathrm{IMO6}}$ Father has left to his children several identical gold coins. According to his will, the oldest child receives one coin and one-seventh of the remaining coins, the next child receives two coins and one-seventh of the remaining coins, the third child receives three coins and one-seventh... | n=6,=36 | 109 | 7 |
math | ## Subiectul III.(20 puncte)
Să se rezolve ecuatiiile:
a) $\log _{9}\left(1+x^{2}+x^{3}\right)=2 \log _{4} x$.
b) $(5+\sqrt{24})^{x}+(5-\sqrt{24})^{x}=98$
| 4,\2 | 80 | 3 |
math | ## Task 4 - 050614
In a company, 1600 packages, each 1.6 dm long, 7 cm wide, and 45 mm high (external dimensions), are to be shipped.
Workers want to stack them in boxes with internal dimensions of 64 cm in length, 0.28 m in width, and 1.8 dm in height.
What is the smallest number of boxes needed to ship all these p... | 25 | 106 | 2 |
math | ## Task 2 - 140522
Anita and Peter were supposed to get 7 bottles of sparkling water for their group from the grocery store. They had an amount of money that would have been exactly enough for this. However, they could only get soda, which cost 15 pennies more per bottle than sparkling water. For their entire amount o... | 1.40 | 118 | 4 |
math | 7. (5 points) Two mischievous children walk against the direction of the moving escalator, from one end to the other. The boy took 100 seconds, and the girl took 300 seconds. It is known that when the escalator is stationary, the boy walks 3 meters per second, and the girl walks 2 meters per second. Then the length of ... | 150 | 91 | 3 |
math | 2. If the three sides of $\triangle A B C$ are $\sqrt{2}, \sqrt{3}, \sqrt{5}$, then its inradius $r=$ | \frac{\sqrt{2}+\sqrt{3}-\sqrt{5}}{2} | 37 | 20 |
math | 11. $x^{3}+k x-128=0$ has a root of multiplicity 2 . Find $k$. | -48 | 31 | 3 |
math | 3. Let $a, b \in \mathbf{Z}, m$ be a positive integer. If dividing $a, b$ by $m$ yields the same remainder, then $a$ and $b$ are said to be congruent modulo $m$, denoted as $a \equiv b(\bmod m)$. Among the following statements:
(1) If $a \equiv b(\bmod m), d$ is a positive divisor of $m$, then $a \equiv b(\bmod d)$;
(2... | (1)(4) | 250 | 5 |
math | 16. 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 less than or equal to $x$). | {\frac{1}{2}}\cup[\frac{5}{6},\frac{5}{4}) | 71 | 23 |
math | Let $a_1, a_2, \ldots, a_{2023}$ be nonnegative real numbers such that $a_1 + a_2 + \ldots + a_{2023} = 100$. Let $A = \left \{ (i,j) \mid 1 \leqslant i \leqslant j \leqslant 2023, \, a_ia_j \geqslant 1 \right\}$. Prove that $|A| \leqslant 5050$ and determine when the equality holds.
[i]Proposed by Yunhao Fu[/i] | |A| \leq 5050 | 148 | 12 |
math | ## 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 | 57 | 49 |
math | Let's determine the smallest positive number divisible by 999 that does not contain the digit 9. | 111888 | 22 | 6 |
math | 1. Several numbers were written on the board, their arithmetic mean was equal to $M$. They added the number 15, after which the arithmetic mean increased to $M+2$. After that, they added the number 1, and the arithmetic mean decreased to $M+1$. How many numbers were on the board initially? (Find all options and prove t... | 4 | 81 | 1 |
math | 15. (12 points) The sequence $\left\{a_{n}\right\}$ satisfies
$$
\begin{array}{l}
a_{1}=\frac{1}{2}, a_{n+1}=a_{n}^{2}+a_{n}(n \in \mathbf{N}), \\
b_{n}=\frac{1}{1+a_{n}}, S_{n}=b_{1}+b_{2}+\cdots+b_{n}, \\
P_{n}=b_{1} b_{2} \cdots b_{n} .
\end{array}
$$
Try to find the value of $2 P_{n}+S_{n}$. | 2 | 151 | 1 |
math | 12. A polyhedron, except for one vertex, the sum of the angles at the vertices of each face of the other vertices is $5160^{\circ}$, then the sum of the angles at the vertices of all faces of the polyhedron is $\qquad$ . | 5400 | 62 | 4 |
math | 2. Find the value of the expression $\frac{a}{b}+\frac{b}{a}$, where $a$ and $b$ are the largest and smallest roots of the equation $x^{3}-9 x^{2}+9 x=1$, respectively. | 62 | 58 | 2 |
math | 29th VMO 1991 Problem A1 Find all real-valued functions f(x) on the reals such that f(xy)/2 + f(xz)/2 - f(x) f(yz) ≥ 1/4 for all x, y, z. | f(x)=\frac{1}{2}forallx | 59 | 12 |
math | 12.033. A rectangular trapezoid with an acute angle $\alpha$ is circumscribed around a circle. Find the height of the trapezoid if its perimeter is $P$. | \frac{P\sin\alpha}{4\cdot\cos^{2}(\frac{\pi}{4}-\frac{\alpha}{2})} | 44 | 32 |
math | Five marbles are distributed at a random among seven urns. What is the expected number of urns with exactly one marble? | \frac{6480}{2401} | 26 | 13 |
math | What is the exact value of $2^{\log _{6} 18} \cdot 3^{\log _{6} 3}$? | 6 | 34 | 1 |
math | 1. Given $x$ and $y$ are real numbers, and $x^{2}+x y+y^{2}=3$. Let the maximum and minimum values of $x^{2}-x y+y^{2}$ be $m$ and $n$, respectively. Then the value of $m+n$ is $\qquad$ | 10 | 70 | 2 |
math | Example 8. Find the derivative of the function $y=\frac{\cos ^{2} x}{\sin x}$. | -\cosx(\operatorname{cosec}^{2}x+1) | 27 | 18 |
math | 9. Given that a line passing through the focus $F$ of the parabola $y^{2}=4 x$ intersects the parabola at points $M$ and $N$, and $E(m, 0)$ is a point on the $x$-axis. The extensions of $M E$ and $N E$ intersect the parabola at points $P$ and $Q$ respectively. If the slopes $k_{1}$ and $k_{2}$ of $M N$ and $P Q$ satisf... | 3 | 134 | 1 |
math | 7. In $\triangle A B C$, if $\tan \frac{A}{2}+\tan \frac{B}{2}=1$, then the minimum value of $\tan \frac{C}{2}$ is $\qquad$ . | \frac{3}{4} | 50 | 7 |
math | 1. Choose 5 numbers (repetition allowed) from $\{1,2, \cdots, 100\}$. Then the expected number of composite numbers taken is | \frac{37}{10} | 38 | 9 |
math | [ Prime numbers and their properties ] [ Examples and counterexamples. Constructions]
Find the smallest natural number $n$, for which the following condition is satisfied: if the number $p-$ is prime and $n$ is divisible by $p-1$, then $n$ is divisible by $p$. | 1806 | 62 | 4 |
math | 2. (16 points) A truck left the village of Mirny at a speed of 40 km/h. At the same time, a car left the city of Tikhaya in the same direction as the truck. In the first hour of the journey, the car traveled 50 km, and in each subsequent hour, it traveled 5 km more than in the previous hour. How many hours will it take... | 7 | 112 | 1 |
math | Example 1 Find all positive integer triples $(x, y, z)$ such that $y$ is a prime, $y$ and 3 are not divisible by $z$, and $x^{3}-y^{3}=$ $z^{2}$. | (8,7,13) | 54 | 8 |
math | (solved by Ambroise Marigot). For which integers $k$ does there exist a function $f: \mathbb{N} \rightarrow \mathbb{Z}$ satisfying $f(2006)=2007$ and:
$$
f(x y)=f(x)+f(y)+k f(\operatorname{GCD}(x, y))
$$
for all integers $x$ and $y$? | k=0ork=-1 | 92 | 6 |
math | 9.026. $\log _{0.3}(3 x-8)>\log _{0.3}\left(x^{2}+4\right)$. | x\in(\frac{8}{3};\infty) | 38 | 14 |
math | Example 8 Find the last 3 digits of $1 \times 3 \times 5 \times 7 \times \cdots \times 2005$. | 375 | 37 | 3 |
math | 10. How many ordered quadruples $(a, b, c, d)$ of positive odd integers are there that satisfy the equation $a+b+c+2 d=15 ?$ | 34 | 39 | 2 |
math | Example 3.35. Find the derivative of the function $u=x^{3} y^{3} z^{3}$ at the point $M(1,1,1)$ in the direction that forms angles of $60^{\circ}, 45^{\circ}, 60^{\circ}$ with the coordinate axes. | \frac{3}{2}(2+\sqrt{2}) | 72 | 13 |
math | There are $n$ lamps in a row. Some of which are on. Every minute all the lamps already on go off. Those which were off and were adjacent to exactly one lamp which was on will go on. For which $n$ one can find an initial configuration of lamps which were on, such that at least one lamp will be on at any time? | n \neq 1, 3 | 74 | 10 |
math |
G7 Let $A B C$ be an isosceles triangle with $A C=B C$. The point $D$ lies on the side $A B$ such that the semicircle with diameter $[B D]$ and center $O$ is tangent to the side $A C$ in the point $P$ and intersects the side $B C$ at the point $Q$. The radius $O P$ intersects the chord $D Q$ at the point $E$ such that... | \frac{6}{5} | 130 | 7 |
math | 20. List the natural numbers from $1,2,3, \cdots, 99,100$ that are both odd and composite in a row, such that no two adjacent numbers are coprime (if one line is not enough, continue on the second line, and if the second line is still not enough, continue on the third line). | 25,35,55,65,85,95,15,9,21,27,33,39,45,51,57,63,69,75,81,87,93,99,77,91,49 | 77 | 73 |
math | ## Task 6 - 250936
a) Is the term
$$
z=\sqrt{192+96 \cdot \sqrt{3}}+\sqrt{192-96 \cdot \sqrt{3}}
$$
defining a number?
b) If this is the case, is $z$ rational? | 24 | 75 | 2 |
math | How long does a ball dropped from a height of $h$ bounce? The collision number for the collision between the ground and the ball is $k$. For which balls is it useful to measure $k$ with the bouncing time? (The collision number is the ratio of the mechanical energy after and before the collision.) | \frac{1+\sqrt{k}}{1-\sqrt{k}}\cdot\sqrt{\frac{2h_{0}}{}} | 64 | 28 |
math | 1. Solve the inequality $\frac{64+\left(\log _{\frac{1}{5}}\left(x^{2}\right)\right)^{3}}{\log _{\frac{1}{5}}\left(x^{6}\right) \cdot \log _{5}\left(x^{2}\right)+5 \log _{5}\left(x^{6}\right)+14 \log _{\frac{1}{5}}\left(x^{2}\right)+2} \leq 0$. | x\in[-25;-\sqrt{5})\cup(-\frac{1}{\sqrt[3]{5}};0)\cup(0;\frac{1}{\sqrt[3]{5}})\cup(\sqrt{5};25] | 110 | 55 |
math | 2.014. $\frac{x-y}{x^{3 / 4}+x^{1 / 2} y^{1 / 4}} \cdot \frac{x^{1 / 2} y^{1 / 4}+x^{1 / 4} y^{1 / 2}}{x^{1 / 2}+y^{1 / 2}} \cdot \frac{x^{1 / 4} y^{-1 / 4}}{x^{1 / 2}-2 x^{1 / 4} y^{1 / 4}+y^{1 / 2}}$. | \frac{\sqrt[4]{x}+\sqrt[4]{y}}{\sqrt[4]{x}-\sqrt[4]{y}} | 132 | 30 |
math | 149 The unit digit of a positive integer $m$ is denoted by $f(m)$. $a_{n}=f\left(2^{n+1}-1\right)(n=1,2, \cdots)$, then $a_{1994}$ $=$ ـ. $\qquad$ | 7 | 70 | 1 |
math | 3.10. In the square $A B C D$, an isosceles triangle $A E F$ is inscribed; point $E$ lies on side $B C$, point $F$ lies on side $C D$, and $A E=A F$. The tangent of angle $A E F$ is 3. Find the cosine of angle $F A D$. | \frac{2\sqrt{5}}{5} | 81 | 12 |
math | 287. How many solutions does the system of equations generally have
$$
\begin{gathered}
a x^{2}+b x y+c y^{2}=d \\
a_{1} x^{2}+b_{1} x y+c_{1} y^{2}=d_{1} ?
\end{gathered}
$$
In particular, how many solutions does the system of question 280 have? | 4 | 92 | 1 |
math | 11. (20 points) Let $x, y, z \in(0,1)$, and $x^{2}+y^{2}+z^{2}=1$.
Determine the maximum value of $f=x+y+z-x y z$.
Translate the above text into English, please retain the original text's line breaks and format, and output the translation result directly. | \frac{8 \sqrt{3}}{9} | 83 | 12 |
math | The function $f$ defined by $\displaystyle f(x)= \frac{ax+b}{cx+d}$. where $a,b,c$ and $d$ are nonzero real numbers, has the properties $f(19)=19, f(97)=97$ and $f(f(x))=x$ for all values except $\displaystyle \frac{-d}{c}$. Find the unique number that is not in the range of $f$. | 58 | 96 | 2 |
math | 690. Find the divisibility rule for 2 in a number system with any odd base. | Aisdivisible2ifonlyifthesumofitsdigitsisdivisible2 | 21 | 17 |
math | let $p$and $q=p+2$ be twin primes. consider the diophantine equation $(+)$ given by
$n!+pq^2=(mp)^2$ $m\geq1$, $n\geq1$
i. if $m=p$,find the value of $p$.
ii. how many solution quadruple $(p,q,m,n)$ does $(+)$ have ? | 1 | 89 | 1 |
math | Find the number of ordered pairs of positive integers $(m,n)$ such that ${m^2n = 20 ^{20}}$. | 231 | 30 | 3 |
math | 13. [9] How many functions $f:\{0,1\}^{3} \rightarrow\{0,1\}$ satisfy the property that, for all ordered triples $\left(a_{1}, a_{2}, a_{3}\right)$ and $\left(b_{1}, b_{2}, b_{3}\right)$ such that $a_{i} \geq b_{i}$ for all $i, f\left(a_{1}, a_{2}, a_{3}\right) \geq f\left(b_{1}, b_{2}, b_{3}\right)$ ? | 20 | 129 | 2 |
math | Example 7 The three edge lengths of an isosceles tetrahedron are 3, $\sqrt{10}$, and $\sqrt{13}$. Then the radius of the circumscribed sphere of this tetrahedron is | 2 | 52 | 1 |
math | ## 263. Math Puzzle $4 / 87$
In a parking lot, there are cars, mopeds, and motorcycles with sidecars. There are a total of 16 vehicles with 50 wheels (excluding spare wheels). There are as many cars as motorcycles with sidecars.
How many vehicles of each type are parked? | 6 | 72 | 1 |
math | 11. Given the complex number $z$ satisfies $z \cdot \bar{z}-z-\bar{z}=3$, and $\arg (z-1)=\frac{\pi}{3}$, then $z=$ | 2+\sqrt{3}\mathrm{i} | 48 | 9 |
math | 9. Let $f, g: \mathbf{N} \rightarrow \mathbf{N}, f(n)=2 n+1, g(1)=3$ and
$$
g(n) \geqslant f[g(n-1)], \forall n \geqslant 2 .
$$
Find $g(n)$. | (n)=3\times2^{n-1}+2^{n-1}-1 | 73 | 19 |
math | 3. The sum of positive numbers $a, b, c$ and $d$ does not exceed 4. Find the maximum value of the expression
$$
\sqrt[4]{2 a^{2}+a^{2} b}+\sqrt[4]{2 b^{2}+b^{2} c}+\sqrt[4]{2 c^{2}+c^{2} d}+\sqrt[4]{2 d^{2}+d^{2} a}
$$ | 4\sqrt[4]{3} | 103 | 8 |
math | 【Question 11】A natural number has 10 different divisors (i.e., factors, which are natural numbers that can divide it), but its prime factors (i.e., factors that are prime numbers) are only 2 and 3. Therefore, this natural number is $\qquad$ ـ. | 162or48 | 66 | 6 |
math | 20. Let $x, y$ be acute angles such that $\sin y=2005 \cos (x+y) \sin x$. Find the greatest possible value of $\tan y$.
(3 marks)
Let $x, y$ be acute angles such that $\sin y=2005 \cos (x+y) \sin x$. Find the greatest possible value of $\tan y$. | \frac{2005\sqrt{2006}}{4012} | 86 | 21 |
math | 4. We will call a number anti-triangular if it can be written in the form $\frac{2}{n(n+1)}$ for some natural number $n$. For how many numbers $k(1000 \leqslant k \leqslant 2000)$ can the number 1 be written as the sum of $k$ anti-triangular numbers (not necessarily distinct)? | all1001possiblevaluesofkwork | 89 | 10 |
math | 3B. Determine the first two and the last two digits of the decimal representation of the number $x_{1001}$, if $x_{1}=2$ and $x_{n+1}=\frac{1}{\sqrt[10]{2}} x_{n}+\frac{\sqrt[10]{2}-1}{\sqrt[10]{2}}, n \in \mathbb{N}$. | 1025 | 91 | 4 |
math | Using only a compass and a ruler, reconstruct triangle $ABC$ given the following three points: point $M$ the intersection of its medians, point $I$ is the center of its inscribed circle and the point $Q_a$ is touch point of the inscribed circle to side $BC$. | \triangle ABC | 62 | 4 |
math | For which non-negative integers $ a<2007$ the congruence $ x^2\plus{}a \equiv 0 \mod 2007$ has got exactly two different non-negative integer solutions?
That means, that there exist exactly two different non-negative integers $ u$ and $ v$ less than $ 2007$, such that $ u^2\plus{}a$ and $ v^2\plus{}a$ are both divisi... | a \equiv 0, 8, 5, 2 \pmod{9} | 109 | 21 |
math | 12 Let $E$ be the intersection of the diagonals $AC$ and $BD$ of the cyclic quadrilateral $ABCD$. Given: $AC=BC, AD=5, BE=12, DE=3$, find $\angle BCD$. | 90 | 55 | 2 |
math | B3. The area of the triangle enclosed by the line with the positive half-axes of the coordinate axes is equal to 1. The line passes through the point $A\left(1, \frac{1}{2}\right)$. Write its equation. | \frac{x}{2}+1 | 55 | 8 |
math | (6) Let $n \geqslant 3$ be a positive integer. If there are $n$ lattice points $P_{1}, P_{2}, \cdots, P_{n}$ in the plane satisfying: when $\left|P_{i} P_{j}\right|$ is a rational number, there exists $P_{k}$ such that $\left|P_{i} P_{k}\right|$ and $\left|P_{j} P_{k}\right|$ are both irrational; when $\left|P_{i} P_{j... | 2005 | 208 | 4 |
math | $\begin{aligned} \text { 8. } 1+\frac{1}{1+2}+\cdots+\frac{1}{1+2+\cdots+2010} \\ =\end{aligned}$ | \frac{4020}{2011} | 50 | 13 |
math | Let $k$ be a positive integer. In the coordinate plane, circle $\omega$ has positive integer radius and is tangent to both axes. Suppose that $\omega$ passes through $(1,1000+k)$. Compute the smallest possible value of $k$.
[i]Proposed by Luke Robitaille | 58 | 65 | 2 |
math | Let's divide the sides $AB$, $BC$, and $CA$ of an equilateral triangle in the same ratio, and connect the division points. We again obtain an equilateral triangle. In what ratio should we divide the sides of the given triangle so that the area of the triangle formed by the division points is minimal? | \frac{}{2} | 66 | 6 |
math | Anton ran down a moving escalator and counted 30 steps. Then he decided to run up the same escalator at the same speed relative to the escalator and counted 150 steps. How many steps did he count when descending with a police officer on the stationary escalator?
# | 50 | 61 | 2 |
math | (England 1996) Find the positive integers $x, y, z$ such that $2^{x}+3^{y}=$ $z^{2}$. | (x,y,z)=(4,2,5) | 38 | 10 |
math | Define a \emph{crossword puzzle} to be a $15 \times 15$ grid of squares, each of which is either black or white. In a crossword puzzle, define a \emph{word} to be a sequence of one or more consecutive white squares in a row or column such that the squares immediately before and after the sequence both are either black ... | 4900 | 170 | 4 |
math | Find the value of $ t$ such that ${ \frac{\int_0^{\frac{\pi}{2}}(\sin x+t\cos x)dx}{\sqrt{\int_0^{\frac{\pi}{2}} (\sin x+t\cos x)^2dx}}}$ is maximized. | 2 \sqrt{\frac{2}{2 + \pi}} | 64 | 15 |
math | 9. Given $f(x)=2^{x} m+x^{2}+n x$. If
$$
\{x \mid f(x)=0\}=\{x \mid f(f(x))=0\} \neq \varnothing,
$$
then the range of values for $m+n$ is $\qquad$ . | [0,4) | 73 | 5 |
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