task_type stringclasses 1
value | problem stringlengths 23 3.94k | answer stringlengths 1 231 | problem_tokens int64 8 1.39k | answer_tokens int64 1 200 |
|---|---|---|---|---|
math | The incircle of $ABC$ touches the sides $BC,CA,AB$ at $A' ,B' ,C'$ respectively. The line $A' C'$ meets the angle bisector of $\angle A$ at $D$. Find $\angle ADC$. | 90^\circ | 56 | 4 |
math | (solved by Lucas Boczkowski). Find all $n \in\{1,2, \ldots, 999\}$ such that $n^{2}$ is equal to the cube of the sum of the digits of $n$. | 1,27 | 53 | 4 |
math | The base of a triangle is $a$, and its height is $m$. Draw a square on the base such that two of its vertices fall on the sides of the triangle. In the smaller triangle that is formed, inscribe a square in the same manner, and continue this process to infinity. What is the sum of the areas of the squares? | \frac{^{2}}{+2} | 71 | 10 |
math | [ Tasks with constraints $]$
A New Year's garland hanging along the school corridor consists of red and blue bulbs. Next to each red bulb, there is definitely a blue one. What is the maximum number of red bulbs that can be in this garland if there are 50 bulbs in total?
# | 33 | 64 | 2 |
math | ## Task 3
$1 \mathrm{~m}=\ldots \mathrm{cm} ; \quad 40 \mathrm{~mm}=\ldots \mathrm{cm} ; \quad 10 \mathrm{~cm}=\ldots \mathrm{dm}$ | 1 | 61 | 1 |
math | 12. If to a certain three-digit number, first the digit 7 is appended on the left, and then on the right, the first of the resulting four-digit numbers will be 3555 more than the second. Find this three-digit number. | 382 | 54 | 3 |
math | If $x$ is a positive integer, then the
$$
10 x^{2}+5 x, \quad 8 x^{2}+4 x+1, \quad 6 x^{2}+7 x+1
$$
are the lengths of the sides of a triangle, all of whose angles are less than $90^{\circ}$, and whose area is an integer. | 2x(x+1)(2x+1)(6x+1) | 86 | 16 |
math | ## Task 2 - 050812
For which real numbers $a$ and $b$ is the equation
$$
\frac{1}{a}+\frac{1}{b}=\frac{(a+b) \cdot(a-b)}{a b} \quad \text{ satisfied? }
$$ | +b=0or=b+1 | 68 | 7 |
math | João and Maria each have a large jug with one liter of water. On the first day, João puts $1 \mathrm{ml}$ of water from his jug into Maria's jug. On the second day, Maria puts $2 \mathrm{ml}$ of water from her jug into João's jug. On the third day, João puts $3 \mathrm{ml}$ of water from his jug into Maria's jug, and s... | 900 | 111 | 3 |
math | 11. Find the value of
$$
\frac{2011^{2}+2111^{2}-2 \times 2011 \times 2111}{25}
$$ | 400 | 48 | 3 |
math | Let $ABCD$ be a rectangle of sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $A$ cuts $BD$ at point $H$. We call $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the measure of the segment $MN$. | 2.16 | 77 | 4 |
math | 13.421 If a two-digit number is divided by a certain integer, the quotient is 3 and the remainder is 8. If the digits in the dividend are swapped while the divisor remains the same, the quotient becomes 2 and the remainder is 5. Find the original value of the dividend. | 53 | 65 | 2 |
math | 4. In the equality $a+b=c+d=e+f$ the letters represent different prime numbers less than 20. Determine at least one solution. | 5+19=7+17=11+13 | 31 | 15 |
math | ## Task A-4.5.
Determine the sum of all natural numbers $n$ less than 1000 for which $2^{n}+1$ is divisible by 11. | 50000 | 43 | 5 |
math | Example 2 Let the set $A=\{1,2, \cdots, 366\}$. If a binary subset $B=\{a, b\}$ of $A$ satisfies $17 \mid(a+b)$, then $B$ is said to have property $P$.
(1) Find the number of binary subsets of $A$ that have property $P$;
(2) The number of a set of binary subsets of $A$, which are pairwise disjoint and have property $P$... | 179 | 114 | 3 |
math | G5.1 If the roots of $x^{2}-2 x-P=0$ differ by 12 , find the value of $P$. | 35 | 32 | 2 |
math | Solve the following equation in the set of positive integers:
$$
\left(1+\frac{1}{a}\right)\left(1+\frac{1}{b}\right)\left(1+\frac{1}{c}\right)=2 \text {. }
$$ | (2,4,15),(2,5,9),(2,6,7),(3,3,8),(3,4,5) | 57 | 32 |
math | 2. A natural number, not ending in zero, had one of its digits erased. As a result, the number decreased by 6 times. Find all numbers for which this is possible. | 108or12awhen=1,2,3,4 | 39 | 16 |
math | 3 *. Find all pairs of natural numbers $x, y$ such that $x^{3}+y$ and $y^{3}+x$ are divisible by $x^{2}+y^{2}$. | 1 | 46 | 1 |
math | 2. It is known that the numbers $x, y, z$ form an arithmetic progression in the given order with a common difference $\alpha=\arcsin \frac{\sqrt{7}}{4}$, and the numbers $\frac{1}{\sin x}, \frac{4}{\sin y}, \frac{1}{\sin z}$ also form an arithmetic progression in the given order. Find $\sin ^{2} y$. | \frac{7}{13} | 93 | 8 |
math | The polynomial
$$
Q(x_1,x_2,\ldots,x_4)=4(x_1^2+x_2^2+x_3^2+x_4^2)-(x_1+x_2+x_3+x_4)^2
$$
is represented as the sum of squares of four polynomials of four variables with integer coefficients.
[b]a)[/b] Find at least one such representation
[b]b)[/b] Prove that for any such representation at least one of the four polyn... | Q(x_1, x_2, x_3, x_4) = (x_1 + x_2 - x_3 - x_4)^2 + (x_1 - x_2 + x_3 - x_4)^2 + (x_1 - x_2 - x_3 + x_4)^2 | 127 | 75 |
math | One, (20 points) Given the parabola $y=x^{2}$ and the moving line $y=(2 t-1) x-c$ have common points $\left(x_{1}, y_{1}\right)$ and $\left(x_{2}, y_{2}\right)$, and $x_{1}^{2}+x_{2}^{2}=t^{2}+2 t-3$.
(1) Find the range of values for $t$;
(2) Find the minimum value of $c$, and determine the value of $t$ when $c$ is at ... | \frac{11-6 \sqrt{2}}{4} | 131 | 15 |
math | 60 Given $z_{1}=x+\sqrt{5}+y i, z_{2}=x-\sqrt{5}+y i$, and $x, y \in \mathbf{R},\left|z_{1}\right|+\left|z_{2}\right|=6$.
$f(x, y)=|2 x-3 y-12|$ The product of the maximum and minimum values of is $\qquad$ | 72 | 95 | 2 |
math | Problem 5. Determine all natural numbers $n$ such that the numbers $3 n-4, 4 n-5$, and $5 n-3$ are prime numbers. | 2 | 38 | 1 |
math | 7,8
What is the maximum number of rooks that can be placed on an 8x8 chessboard so that they do not attack each other
# | 8 | 34 | 1 |
math | Let $p,q$ be positive integers. For any $a,b\in\mathbb{R}$ define the sets $$P(a)=\bigg\{a_n=a \ + \ n \ \cdot \ \frac{1}{p} : n\in\mathbb{N}\bigg\}\text{ and }Q(b)=\bigg\{b_n=b \ + \ n \ \cdot \ \frac{1}{q} : n\in\mathbb{N}\bigg\}.$$
The [i]distance[/i] between $P(a)$ and $Q(b)$ is the minimum value of $|x-y|$ as $x\i... | \frac{1}{2 \cdot \text{lcm}(p, q)} | 185 | 17 |
math | 7. Denote by $\langle x\rangle$ the fractional part of the real number $x$ (for instance, $\langle 3.2\rangle=0.2$ ). A positive integer $N$ is selected randomly from the set $\{1,2,3, \ldots, M\}$, with each integer having the same probability of being picked, and $\left\langle\frac{87}{303} N\right\rangle$ is calcula... | \frac{50}{101} | 138 | 10 |
math | 5. (7-8 grade) Maria Ivanovna is a strict algebra teacher. She only puts twos, threes, and fours in the grade book, and she never gives the same student two twos in a row. It is known that she gave Vovochka 6 grades for the quarter. In how many different ways could she have done this? Answer: 448 ways. | 448 | 84 | 3 |
math | 8.392. Find $\sin \alpha$, if $\cos \alpha=\operatorname{tg} \beta, \cos \beta=\operatorname{tg} \gamma, \cos \gamma=\operatorname{tg} \alpha\left(0<\alpha<\frac{\pi}{2}\right.$, $\left.0<\beta<\frac{\pi}{2}, 0<\gamma<\frac{\pi}{2}\right)$. | \sin\alpha=\frac{\sqrt{5}-1}{2} | 100 | 15 |
math | 2. The amount of 4800 denars should be equally divided among several friends. If three of the friends refuse their share, then the remaining friends will receive 80 denars more each.
How many friends participated in the distribution of this amount? | 15 | 54 | 2 |
math | How many ways are there to choose 6 numbers from the set $\{1,2, \ldots, 49\}$ such that there are at least two consecutive numbers among the 6? | 6924764 | 42 | 7 |
math | 11.5. Let $M$ be some set of pairs of natural numbers $(i, j), 1 \leq i<j \leq n$ for a fixed $n \geq 2$. If a pair $(i, j)$ belongs to $M$, then no pair $(j, k)$ belongs to it. What is the largest set of pairs that can be in the set $M$? | \frac{n^{2}}{4}forevenn,\frac{n^{2}-1}{4}foroddn | 87 | 25 |
math | 19. Let $x$ be a real number such that $x^{2}-15 x+1=0$. Find the value of $x^{4}+\frac{1}{x^{4}}$. | 49727 | 45 | 5 |
math | 14. $[\mathbf{9}]$ Given that $x$ is a positive real, find the maximum possible value of
$$
\sin \left(\tan ^{-1}\left(\frac{x}{9}\right)-\tan ^{-1}\left(\frac{x}{16}\right)\right) .
$$ | \frac{7}{25} | 69 | 8 |
math | 14. There is a large square containing two smaller squares. These two smaller squares can move freely within the large square (no part of the smaller squares can move outside the large square, and the sides of the smaller squares must be parallel to the sides of the large square). If the minimum overlapping area of the... | 189 | 118 | 3 |
math | 33. Let $f: \mathbb{N} \rightarrow \mathbb{Q}$ be a function, where $\mathbb{N}$ denotes the set of natural numbers, and $\mathbb{Q}$ denotes the set of rational numbers. Suppose that $f(1)=\frac{3}{2}$, and
$$
f(x+y)=\left(1+\frac{y}{x+1}\right) f(x)+\left(1+\frac{x}{y+1}\right) f(y)+x^{2} y+x y+x y^{2}
$$
for all nat... | 4305 | 145 | 4 |
math | Example 2 If the equation $\left(x^{2}-1\right)\left(x^{2}-4\right)=k$ has four non-zero real roots, and the four points corresponding to them on the number line are equally spaced, then $k=$ $\qquad$. | \frac{7}{4} | 57 | 7 |
math | Let's set $S_{n}=1^{3}+\cdots+n^{3}$ and $u_{n}=1+3+5+\cdots+(2 n-1)$ the sum of the first $n$ odd numbers.
1. By evaluating for small $n$, predict and then prove by induction a formula for $u_{n}$.
2. Using the formula found above for the sum of linear terms, find $u_{n}$ again.
3. Can we find a third way to derive a... | S_{n}=(\frac{n(n+1)}{2})^{2} | 160 | 18 |
math | 7. Given positive numbers $a, b$ satisfy $2 a+b=1$, then the maximum value of $4 a^{2}+b^{2}+4 \sqrt{a b}$ is $\qquad$ | \frac{1}{2}+\sqrt{2} | 47 | 12 |
math | Let $f, g$ be bijections on $\{1, 2, 3, \dots, 2016\}$. Determine the value of $$\sum_{i=1}^{2016}\sum_{j=1}^{2016}[f(i)-g(j)]^{2559}.$$
| 0 | 76 | 1 |
math | 11. The number of real solutions to the equation $\left(x^{2006}+1\right)\left(1+x^{2}+x^{4}+\cdots+\right.$ $\left.x^{2004}\right)=2006 x^{2005}$ is $\qquad$ | 1 | 70 | 1 |
math | Example 2 Let $S=\{1,2, \cdots, n\}, A$ be an arithmetic sequence with at least two terms and a positive common difference, all of whose terms are in $S$, and adding any other element of $S$ to $A$ does not form an arithmetic sequence with the same common difference as $A$. Find the number of such $A$ (here, a sequence... | [\frac{n^{2}}{4}] | 108 | 9 |
math | 3.254. $4 \cos \left(\alpha-\frac{\pi}{2}\right) \sin ^{3}\left(\frac{\pi}{2}+\alpha\right)-4 \sin \left(\frac{5}{2} \pi-\alpha\right) \cos ^{3}\left(\frac{3}{2} \pi+\alpha\right)$. | \sin4\alpha | 83 | 5 |
math | (Hungary 2017)(D) Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for all $x, y \in \mathbb{R}$,
$$
f(x y-1)+f(x) f(y)=2 x y-1
$$ | f(x)=xorf(x)=-x^2 | 70 | 11 |
math | 8. (1 mark) Given that $0.3010<\log 2<0.3011$ and $0.4771<\log 3<0.4772$. Find the leftmost digit of $12^{37}$.
(1 分) 設 $0.3010<\log 2<0.3011$ 及 $0.4771<\log 3<0.4772$, 求 $12^{37}$ 最左的一位數字。 | 8 | 129 | 1 |
math | 2. Six natural numbers (possibly repeating) are written on the faces of a cube, such that the numbers on adjacent faces differ by more than 1. What is the smallest possible value of the sum of these six numbers? | 18 | 46 | 2 |
math | 4. Given the sequence $\left\{a_{n}\right\}$, where $a_{1}=99^{\frac{1}{99}}, a_{n}=$ $\left(a_{n-1}\right)^{a_{1}}$. When $a_{n}$ is an integer, the smallest positive integer $n$ is $\qquad$ | 100 | 77 | 3 |
math | ## Task Condition
Derive the equation of the tangent line to the given curve at the point with abscissa $x_{0}$.
$y=3(\sqrt[3]{x}-2 \sqrt{x}), x_{0}=1$ | -2x-1 | 51 | 5 |
math | ## Task B-4.3.
For the function $f$, if $f(x)+3 f(x+1)-f(x) f(x+1)=5, f(1)=2017$, calculate what $f(2017)$ is. | 2017 | 56 | 4 |
math | 3. How many natural numbers have a product of digits 12, a sum of digits 12, and are divisible by 12? Determine the largest such number. | 32111112 | 37 | 8 |
math | 22. (12 points) Given the function
$$
f(x)=\ln (a x+1)+\frac{1-x}{1+x}(x \geqslant 0, a>0) \text {. }
$$
(1) If $f(x)$ has an extremum at $x=1$, find the value of $a$;
(2) If $f(x) \geqslant \ln 2$ always holds, find the range of values for $a$. | \geqslant1 | 109 | 6 |
math | 4. Given that the circumcenter, incenter, and orthocenter of a non-isosceles acute $\triangle ABC$ are $O, I, H$ respectively, and $\angle A=60^{\circ}$. If the altitudes of $\triangle ABC$ are $AD, BE, CF$, then the ratio of the circumradius of $\triangle OIH$ to the circumradius of $\triangle DEF$ is $\qquad$ . | 2 | 94 | 1 |
math | A teacher suggests four possible books for students to read. Each of six students selects one of the four books. How many ways can these selections be made if each of the books is read by at least one student? | 1560 | 43 | 4 |
math | Three consecutive odd numbers, the sum of the squares of which is a four-digit number, all of whose digits are equal. Which are these numbers? | 41,43,45\quador\quad-45,-43,-41 | 30 | 22 |
math | 27 Let $x_{1}, x_{2}, x_{3}, x_{4}$ denote the four roots of the equation
$$
x^{4}+k x^{2}+90 x-2009=0 \text {. }
$$
If $x_{1} x_{2}=49$, find the value of $k$. | 7 | 77 | 1 |
math | 2. The measures of the external angles of a right triangle (which are not right) are in the ratio $7: 11$. Determine the measures of the acute angles of that triangle. | 75,15 | 40 | 5 |
math | 3. On the blackboard, it is written
$$
1!\times 2!\times \cdots \times 2011!\times 2012!\text {. }
$$
If one of the factorials is erased so that the remaining product equals the square of some positive integer, then the erased term is . $\qquad$ | 1006! | 74 | 5 |
math | $3 \cdot 2$ Find $1 \cdot 1!+2 \cdot 2!+3 \cdot 3!+\cdots+(n-1)(n-1)!+n \cdot n$ !, where $n!=n(n-1)(n-2) \cdots 2 \cdot 1$. | (n+1)!-1 | 72 | 6 |
math | Example 7 Let $x=(15+\sqrt{220})^{19}+(15+\sqrt{200})^{22}$. Find the unit digit of the number $x$.
| 9 | 46 | 1 |
math | 9・53 Let $x, y, z$ be positive numbers and $x^{2}+y^{2}+z^{2}=1$, find
$$\frac{x}{1-x^{2}}+\frac{y}{1-y^{2}}+\frac{z}{1-z^{2}}$$
the minimum value. | \frac{3 \sqrt{3}}{2} | 71 | 12 |
math | Three positive reals $x , y , z $ satisfy \\
$x^2 + y^2 = 3^2 \\
y^2 + yz + z^2 = 4^2 \\
x^2 + \sqrt{3}xz + z^2 = 5^2 .$ \\
Find the value of $2xy + xz + \sqrt{3}yz$ | 24 | 81 | 2 |
math | 8. Arrange all positive integers whose sum of digits is 10 in ascending order to form the sequence $\left\{a_{n}\right\}$. If $a_{n}=2017$, then $n=$ $\qquad$ . | 120 | 53 | 3 |
math | 3rd ASU 1963 Problem 2 8 players compete in a tournament. Everyone plays everyone else just once. The winner of a game gets 1, the loser 0, or each gets 1/2 if the game is drawn. The final result is that everyone gets a different score and the player placing second gets the same as the total of the four bottom players.... | The\\3\player\won\against\the\\7\player | 99 | 15 |
math | ## Task B-1.4.
Fran decided to paint the fence with the help of his friends Tin and Luke. They estimated that it would take Tin 3 hours more than Fran to paint the fence alone, and Luke 2 hours less than Fran to paint the fence alone. Working together, each at their own pace, they would paint the fence in 4 hours. How... | 12, | 91 | 3 |
math | Find all natural numbers $n$ such that $n$ divides $2^{n}-1$. | 1 | 20 | 1 |
math | 1. The average score of the participants in a junior high school mathematics competition at a certain school is 75 points. Among them, the number of male participants is $80\%$ more than that of female participants, and the average score of female participants is $20\%$ higher than that of male participants. Therefore,... | 84 | 80 | 2 |
math | Problem 3. One notebook, 3 notepads, and 2 pens cost 98 rubles, while 3 notebooks and a notepad are 36 rubles cheaper than 5 pens. How much does each item cost, if the notebook costs an even number of rubles? (Each of these items costs a whole number of rubles.) | 4,22,14 | 75 | 7 |
math | Four, (20 points) A four-digit number, the sum of this four-digit number and the sum of its digits is 1999. Find this four-digit number and explain your reasoning. | 1976 | 42 | 4 |
math | An infinite rectangular stripe of width $3$ cm is folded along a line. What is the minimum possible area of the region of overlapping? | 4.5 \text{ cm}^2 | 28 | 10 |
math | 1. Given that $x$, $y$, $z$ are positive real numbers, and $x y z(x+y+z)=1$. Then the minimum value of $(x+y)(y+z)$ is $\qquad$ | 2 | 46 | 1 |
math | 5. The clock shows 00:00, at which the hour and minute hands of the clock coincide. Counting this coincidence as number 0, determine how much time (in minutes) will pass before they coincide for the 21st time. Round your answer to the nearest hundredth. | 1374.55 | 63 | 7 |
math | 8.2. In a cinema, five friends took seats numbered 1 to 5 (the leftmost seat is number 1). During the movie, Anya left to get popcorn. When she returned, she found that Varya had moved three seats to the right, Galia had moved one seat to the left, and Diana and Elia had swapped places, leaving the edge seat for Anya. ... | 3 | 94 | 1 |
math | # Problem 1. (2 points)
Petya came up with a quadratic equation $x^{2}+p x+q$, the roots of which are numbers $x_{1}$ and $x_{2}$. He told Vasya three out of the four numbers $p, q, x_{1}, x_{2}$, without specifying which was which. These turned out to be the numbers $1, 2, -6$. What was the fourth number? | -3 | 100 | 2 |
math | 7th Iberoamerican 1992 Problem B1 Let a n , b n be two sequences of integers such that: (1) a 0 = 0, b 0 = 8; (2) a n+2 = 2 a n+1 - a n + 2, b n+2 = 2 b n+1 - b n , (3) a n 2 + b n 2 is a square for n > 0. Find at least two possible values for (a 1992 , b 1992 ). | (1992\cdot1996,4\cdot1992+8),(1992\cdot1988,-4\cdot1992+8) | 123 | 43 |
math | $$
\begin{array}{l}
\text { 5. Let } \frac{\sin ^{4} \theta}{a}+\frac{\cos ^{4} \theta}{b}=\frac{1}{a+b}(a, b>0) \text {. } \\
\text { Then } \frac{\sin ^{8} \theta}{a^{3}}+\frac{\cos ^{8} \theta}{b^{3}}=
\end{array}
$$ | \frac{1}{(a+b)^{3}} | 104 | 12 |
math | 6. (15 points) From a homogeneous straight rod, a piece of length $s=60 \mathrm{~cm}$ was cut. By how much did the center of gravity of the rod move as a result? | 30\, | 47 | 4 |
math | 4. On a circle, 60 red points and one blue point are marked. All possible polygons with vertices at the marked points are considered. Which type of polygons is more numerous, and by how many: those with a blue vertex, or those without it? | 1770 | 54 | 4 |
math | 4. Determine the smallest natural number $n$ for which there exists a set
$$
M \subset\{1,2, \ldots, 100\}
$$
of $n$ elements that satisfies the conditions:
a) 1 and 100 belong to the set $M$,
b) for every $a \in M \backslash\{1\}$, there exist $x, y \in M$ such that $a=x+y$. | 9 | 102 | 1 |
math | Given $c \in\left(\frac{1}{2}, 1\right)$. Find the smallest constant $M$, such that for any integer $n \geqslant 2$ and real numbers $0<a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{n}$, if $\frac{1}{n} \sum_{k=1}^{n} k a_{k}=c \sum_{k=1}^{n} a_{k}$, then $\sum_{k=1}^{n} a_{k} \leqslant M \sum_{k=1}^{m} a_{k}$, w... | \frac{1}{1-c} | 170 | 8 |
math | II. (40 points) Given positive real numbers $x, y, z$ satisfying $x+y+z=1$. Try to find the minimum value of the real number $k$ such that the inequality
$$
\frac{x^{2} y^{2}}{1-z}+\frac{y^{2} z^{2}}{1-x}+\frac{z^{2} x^{2}}{1-y} \leqslant k-3 x y z
$$
always holds. | \frac{1}{6} | 107 | 7 |
math | 398. Given the distribution function $F(x)$ of a random variable $X$. Find the distribution function $G(y)$ of the random variable $Y=-(2 / 3) X+2$. | G(y)=1-F[\frac{3(2-y)}{2}] | 44 | 16 |
math | \section*{Problem 3A - 101043A}
Determine all positive real numbers \(c\) for which \(\left[\log _{12} c\right] \leq\left[\log _{4} c\right]\) holds.
Here, \([x]\) denotes the greatest integer not greater than \(x\). | \frac{1}{16}\leq\frac{1}{12}or\frac{1}{4}\leq | 79 | 28 |
math | Problems 14, 15 and 16 involve Mrs. Reed's English assignment.
A Novel Assignment
The students in Mrs. Reed's English class are reading the same 760-page novel. Three friends, Alice, Bob and Chandra, are in the class. Alice reads a page in 20 seconds, Bob reads a page in 45 seconds and Chandra reads a page in 3... | 456 | 269 | 3 |
math | 4. For any $x, y \in \mathbf{R}$, the function $f(x, y)$ satisfies
(1) $f(0, y)=y+1$;
(2) $f(x+1,0)=f(x, 1)$;
(3) $f(x+1, y+1)=f(x, f(x+1, y))$.
Then $f(3,2016)=$ . $\qquad$ | 2^{2019}-3 | 102 | 8 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(-3 ;-5 ; 6)$
$M_{2}(2 ; 1 ;-4)$
$M_{3}(0 ;-3 ;-1)$
$M_{0}(3 ; 6 ; 68)$ | \sqrt{573} | 88 | 7 |
math | 1. A new model car travels $4 \frac{1}{6}$ kilometers more on one liter of gasoline compared to an old model car. At the same time, its fuel consumption per 100 km is 2 liters less. How many liters of gasoline does the new car consume per 100 km? | 6 | 67 | 1 |
math | 5. There was a whole number of cheese heads on the kitchen. At night, rats came and ate 10 heads, and everyone ate equally. Several rats got stomachaches from overeating. The remaining seven rats the next night finished off the remaining cheese, but each rat could eat only half as much cheese as the night before. How m... | 11 | 78 | 2 |
math | ## Problem Statement
Find the distance from point $M_{0}$ to the plane passing through three points $M_{1}, M_{2}, M_{3}$.
$M_{1}(2 ; 3 ; 1)$
$M_{2}(4 ; 1 ;-2)$
$M_{3}(6 ; 3 ; 7)$
$M_{0}(-5 ;-4 ; 8)$ | 11 | 89 | 2 |
math | 1. [5] Evaluate $2+5+8+\cdots+101$. | 1751 | 20 | 4 |
math | Example 1 Factorize the polynomial $f(x)=x^{8}+x^{7}+1$ over the integers. | f(x)=(x^{2}+x+1)\cdot(x^{6}-x^{4}+x^{3}-x+1) | 27 | 30 |
math | 10. (10 points) Mom decides to take Xiao Hua on a car trip to 10 cities during the holiday. After checking the map, Xiao Hua is surprised to find: among any three of these 10 cities, either all have highways connecting them, or only two cities are not connected by a highway. Therefore, there are at least $\qquad$ highw... | 40 | 101 | 2 |
math | Example 2 A pedestrian and a cyclist are traveling south simultaneously on a road parallel to a railway. The pedestrian's speed is $3.6 \mathrm{~km} / \mathrm{h}$, and the cyclist's speed is $10.8 \mathrm{~km} / \mathrm{h}$. If a train comes from behind and takes $22 \mathrm{~s}$ to pass the pedestrian and $26 \mathrm{... | 286 | 135 | 3 |
math | 4. Let $A B C D$ be a trapezoid with $A B \| C D, A B=5, B C=9, C D=10$, and $D A=7$. Lines $B C$ and $D A$ intersect at point $E$. Let $M$ be the midpoint of $C D$, and let $N$ be the intersection of the circumcircles of $\triangle B M C$ and $\triangle D M A$ (other than $M$ ). If $E N^{2}=\frac{a}{b}$ for relatively... | 90011 | 142 | 5 |
math | 160. Which is greater: $5^{15}$ or $3^{23}$? | 3^{23}>5^{15} | 22 | 10 |
math | Example 8 Let the sequence $\left\{a_{n}\right\}$ satisfy $a_{1}=0$, and $a_{n}-2 a_{n-1}=n^{2}-3 \ (n=2,3, \cdots)$, find $a_{n}$. | 2^{n+2}-n^{2}-4n-3 | 63 | 14 |
math | Problem 4. Calculate: $\frac{2^{3} \cdot 4^{5} \cdot 6^{7}}{8^{9} \cdot 10^{11}}: 0.015^{7}$. | 1000 | 52 | 4 |
math | 9.48 The total weight of a pile of stones is 100 kilograms, where the weight of each stone does not exceed 2 kilograms. By taking out some of the stones in various ways and calculating the difference between the sum of the weights of these stones and 10 kilograms. Among all these differences, the minimum value of their... | \frac{10}{11} | 99 | 9 |
math | 5. In the sum $1+3+5+\ldots+k$ of consecutive odd natural numbers, determine the largest addend $k$ such that $1+3+5+\ldots+k=40000$.
## Tasks worth 10 points: | 399 | 58 | 3 |
math | Example 1 Suppose the lengths of the three sides of a triangle are integers $l$, $m$, and $n$, and $l > m > n$. It is known that
$$
\left\{\frac{3^{l}}{10^{4}}\right\}=\left\{\frac{3^{m}}{10^{4}}\right\}=\left\{\frac{3^{n}}{10^{4}}\right\},
$$
where $\{x\}=x-[x]$, and $[x]$ represents the greatest integer not exceedin... | 3003 | 151 | 4 |
math | $8 \cdot 4$ Let the sequence of positive integers $\left\{a_{n}\right\}$ satisfy
$$
a_{n+3}=a_{n+2}\left(a_{n+1}+2 a_{n}\right), n=1,2, \cdots
$$
and $a_{6}=2288$. Find $a_{1}, a_{2}, a_{3}$. | a_{1}=5,a_{2}=1,a_{3}=2 | 93 | 15 |
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