sungyub/dapo-math-17k-verl huggingface dataset:
user: The Imomi archipelago consists of $n \geqslant 2$ islands. Between each pair of distinct islands is a unique ferry line that runs in both directions, and each ferry line is operated by one of $k$ companies. It is known that if any one of the $k$ companies closes all its ferry lines, then it becomes impossible for a traveller, no matter where the traveller starts at, to visit all the islands exactly once (in particular, not returning to the island the traveller started at). Determine the maximal possible value of $k$ in terms of $n$. Please provide the value of $k$ when $n = 260$. The answer should be in the format of an integer.

Answer: 8
user: Determine all pairs $(a, p)$ of positive integers with $p$ prime such that $p^{a}+a^{4}$ is a perfect square. Please provide the sum of all $a + p$ for the solutions.

Answer: 30
user: Let $\mathbb{Z}_{>0}$ be the set of positive integers. Determine all functions $f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z}_{>0}$ such that
$$
f^{b f(a)}(a+1)=(a+1) f(b)
$$
holds for all $a, b \in \mathbb{Z}_{>0}$, where $f^{k}(n)=f(f(\cdots f(n) \cdots))$ denotes the composition of $f$ with itself $k$ times.Find the value of $f(5)$.

Answer: 4
user: At a tennis tournament there were $2n$ boys and $n$ girls participating. Every player played every other player. The boys won $\frac{7}{5}$ times as many matches as the girls. It is known that there were no draws. Find the smallest positive integer value of $n$ that satisfies the given condition.

Answer: 3
user: Let $ABC$ be an isosceles triangle with $AC=BC$, let $M$ be the midpoint of its side $AC$, and let $Z$ be the line through $C$ perpendicular to $AB$. The circle through the points $B$, $C$, and $M$ intersects the line $Z$ at the points $C$ and $Q$. Find the radius of the circumcircle of the triangle $ABC$ in terms of $m = CQ$. The original answer is in the form 4\frac{a}{b}m￥, please give the value of $a + b$ .

Answer: 5
user: Find all prime numbers $p, q, r$ such that $\frac{p}{q} - \frac{4}{r+1} = 1$. Please provide the sum of all $p+q+r$

Answer: 37
user: Find all primes $p$ such that there exist positive integers $x,y$ that satisfy $x(y^2-p)+y(x^2-p)=5p$. Please provide the sum of all such primes $p$.

Answer: 12
user: Let $n>3$ be a positive integer. Equilateral triangle $ABC$ is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$. Please provide the value of $m-d$ when $n = 5$. The answer should be an integer.

Answer: 21
user: Let the circles $k_1$ and $k_2$ intersect at two points $A$ and $B$, and let $t$ be a common tangent of $k_1$ and $k_2$ that touches $k_1$ and $k_2$ at $M$ and $N$ respectively. If $t\perp AM$ and $MN=2AM$, evaluate the angle $NMB$. The original answer is in the format of an angle in degrees. Please provide the integer part of the degrees measure.

Answer: 45
user: Find all ordered pairs $(a,b)$ of positive integers for which the numbers $\dfrac{a^3b-1}{a+1}$ and $\dfrac{b^3a+1}{b-1}$ are both positive integers. Please provide the sum of all $a + b$ values from the solutions.

Answer: 14
user: The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. When \(a = 10\), find the volume of the pyramid. Round it to the nearest whole number.

Answer: 300
user: Find four consecutive terms $a, b, c, d$ of an arithmetic progression and four consecutive terms $a_1,b_1,c_1,d_1$ of a geometric progression such that $a+a_1=27,b+b_1=27,c+c_1=39$, and $d+d_1=87.$ Please provide the sum of all terms in both progressions.

Answer: 180
user: The height of a truncated cone is equal to the radius of its base. The perimeter of a regular hexagon circumscribing its top is equal to the perimeter of an equilateral triangle inscribed in its base. Find the angle $\varphi$ between the cone's generating line and its base. Please provide the value of the parameter used in the arctan function, given that the original answer is in the format $\varphi=\arctan x$.Write down the value of $x$.

Answer: 4
user: Solve the system $$\left\{\begin{array}{l}x^2+y^2-2z^2=2a^2,\\x+y+2z=4(a^2+1),\\z^2-xy=a^2.\end{array}\right.$$ Please provide the sum of the elements for all solutions when $a=1$.

Answer: 12
user: Solve the system \(\left\{\begin{array}{l}x^3-y^3=2b,\\x^2y-xy^2=b.\end{array}\right.\) Determine the value of \(b\) for which there exists a solution \(x, y\).

Answer: 0
user: All rectangles that can be inscribed in an isosceles triangle with two of their vertices on the triangle’s base have the same perimeter. Construct the triangle such that the condition can be expressed as a simple integer calculation, e.g., height = base. Provide the sum of base and height if both are integers.

Answer: 10
user: What is the largest number of parts into which $n$ planes can divide space? Please provide the value of the expression when $n = 3$. The answer should be in the format \frac{m}{n}, directly give the value of m + n.

Answer: 9
user: Factor the expression $(b-c)^3+(c-a)^3+(a-b)^3$. The original answer is a product of three differences. Please provide the sum of the coefficients in the expression.

Answer: 9
user: Given an infinite cone, the measure of its unfolding's angle is equal to $\alpha$. A curve on the cone is represented on any unfolding by the union of line segments. Find the greatest integer $n$ such that $n\cdot \alpha < 180^{\circ }$. Please provide the integer value directly.

Answer: 179
user: Solve the equation: $$\begin{aligned}|x+1|-|x|+3|x-1|-2|x-2|=x+2.\end{aligned}$$ Please provide the sum of the minimum and maximum solutions.

Answer: 0
user: Find nonzero and nonequal integers $a, b, c$ so that $x( x- a) ( x- b) ( x- c) + 1$ factors into the product of two polynomials with integer coefficients. Provide the sum of $a, b, c$ if they are three consecutive integers.

Answer: 6
user: Solve in integers the equation $$x+y=x^2-xy+y^2.$$ Please provide the sum of all x and y values from the solutions.

Answer: 12
user: Divide $a^{2^k} − b^{2^k}$ by $(a + b)(a ^2 + b ^2 )(a ^4 + b^ 4 )\cdots (a^ {2 ^{k-1}} + b^ {2 ^{k-1}} )$. If $a=1$ and $b=-1$, what is the result of the division?

Answer: 1
user: Find three-digit numbers such that any of its positive integer powers ends with the same three digits and in the same order. Please provide the sum of all such numbers.

Answer: 503
user: Calculate (without calculators, tables, etc.) with accuracy to $0.00001$ the product \(\begin{pmatrix}1-\frac{1}{10}\end{pmatrix}\begin{pmatrix}1-\frac{1}{10^2}\end{pmatrix}\cdots\begin{pmatrix}1-\frac{1}{10^{99}}\end{pmatrix}.\) The original answer is a decimal number with five decimal places, please provide the value as an integer by removing the decimal point.

Answer: 89001
user: The sum of the reciprocals of three positive integers is equal to 1. What are all the possible such triples? Please provide the sum of all elements in these triples.

Answer: 30
user: Find all positive integer solutions of the equation $x^y=y^x\quad(x\neq y).$ Please provide the sum of all elements in the solutions.

Answer: 12
user: Find the first three figures after the decimal point in the decimal expression of the number \(\frac{0.123456789101112\ldots495051}{0.515049\ldots121110987654321}\). Please provide the value of the first three digits after the decimal point as a single integer.

Answer: 239
user: 99 straight lines divide a plane into n parts. Find all possible values of n less than 199. Please provide the sum of all possible values.

Answer: 298
user: Delete 100 digits from the number $1234567891011 . . . 9899100$ so that the remaining number were as big as possible. Provide the sum of the remaining digits as an integer.

Answer: 409
user: Find all rectangles that can be cut into 13 equal squares. Provide the sum of the dimensions of the rectangle.

Answer: 14
user: What greatest number of triples of points can be selected from $1955$ given points, so that each two triples have one common point? Please provide the value of \binom{1954}{2}.

Answer: 1908081
user: Find all two-digit numbers x the sum of whose digits is the same as that of $2x, 3x,$ etc., $9x$. Please provide the sum of all such two-digit numbers.

Answer: 486
user: In the decimal expression of a positive number, a, all decimals beginning with the third after the decimal point, are deleted (i.e., we take an approximation of a with accuracy to 0.01 with deficiency). The number obtained is divided by a and the quotient is similarly approximated with the same accuracy by a number b. What numbers b can be thus obtained? Please provide the sum of all possible values of b.

Answer: 51
user: Find the number of integer values of n for which $N = 20^n + 16^n − 3 ^n − 1$ is divisible by 323.

Answer: 323
user: Solve the system: $$\begin{cases}\frac{2x_1^2}{1+x_1^2}=x_2,\\[2ex]\frac{2x_2^2}{1+x_2^2}=x_3,\\[2ex]\frac{2x_3^2}{1+x_3^2}=x_1.\end{cases}$$ Provide the sum of all possible values of x_1 + x_2 + x_3.

Answer: 3
user: Represent 1957 as the sum of 12 positive integer summands $a_1, a_2, . . . , a_{12}$ for which the number $a_1! · a_2! · a_3! · · · · · a_{12}!$ is minimal. Please provide the sum of all summands.

Answer: 327
user: On a table lies a regular 100-gon whose vertices are numbered consecutively: $1, 2, \cdots , 100$. These numbers were then rewritten, in increasing order, according to the distance of the corresponding vertex from the front edge of the table. If vertices are at the same distance from the edge, the left number is written first, and then the right one. All possible sets of numbers corresponding to different positions of the 100-gon are written out. Calculate the sum of the numbers in the 13-th position from the left in all these sets. Please provide the value of the sum of these numbers multiplied by 2.

Answer: 20200
user: A circuit of several resistors connects clamps A and B. Each resistor has an input and an output clamp. What is the least number of resistors needed for a circuit design that will not be short or open if any 9 resistors between A and B break? Please provide the sum of the resistors needed for both principal circuit designs.

Answer: 200
user: Let x and y be unknown digits. The 200-digit number $89 252 525 . . . 2 525$ is multiplied by the number $\overline{444 x18 y27}$. It turns out that the 53-rd digit from the right of the product is 1, and the 54-th digit is 0. Please provide the value of x + y.

Answer: 10
user: The decimal expression of a 100-digit number consists of 1’s and 2’s. The Rule allows one to select arbitrarily 10 consecutive digits of which the first five may change places with the second five. Two numbers will be called similar if one can be obtained from the other one in several such operations. What greatest number of such 100-digit numbers can be selected so no two of them are similar? Please provide the integer value of the original answer.

Answer: 4084101
user: We have 1985 sets. Each of the sets has 45 elements, the union of any two sets has exactly 89 elements. How many elements has the union of all these 1985 sets? Please provide the final integer value of the total number of elements in the union.

Answer: 87341
user: 对任意的正整数 $n$, 记 $D_{n}$ 为 $n$ 的正约数全体， $f_{i}(n)(i=1,2)$ 为集合
$$
F_{i}(n)=\left\{a \in D_{n} \mid a \equiv i(\bmod 4)\right\}
$$
的元素个数. 求最小的正整数 $m$, 使得
$2 f_{1}(m)-f_{2}(m)=2017$. 请给出答案表达式中所有数字的和的计算结果。

Answer: 2023
user: There are given segments $AB, CD$ and a point O such that no three of the points $A, B, C, D, O$ are on one straight line. The endpoint of a segment is marked if the straight line passing through it and O does not intersect another segment. How many marked endpoints are there? Please provide the sum of all possible numbers of marked endpoints.

Answer: 6
user: Find all positive integers n such that $(n − 1)!$ is not divisible by $n^2$. When p is equal to 1,please provide the sum of all such integers.

Answer: 20
user: Denote by $d(N)$ the number of divisors of N (1 and N are also considered as divisors of N). Find all N such that $N/d(N) = p$ is a prime. Please provide the sum of all possible values of N.

Answer: 59
user: Number 4 has the following property: when divided by $q^2$, for any q, the remainder is less than $\frac {q^2}{2}$. Find all numbers with the same property and provide the sum of these numbers.

Answer: 5
user: How may 50 cities be connected by the least possible number of airlines so that from any city one could get to any other by changing airplanes not more than once (i.e., using two planes)? Please provide the number of airlines needed.

Answer: 49
user: Is there a quadrilateral $ABCD$ of area 1 such that for any point O inside it the area of at least one of the triangles $\triangle OAB, \triangle OBC, \triangle OCD,$ or $\triangle OAD$ is an irrational number? The original answer involves a trapezoid with bases m and $\sqrt[n]{p}$. Please provide the sum of m，n and p involved in the original example.

Answer: 6
user: Is there a real number h such that $[h\cdot1969^n]$ is not divisible by $[h\cdot1969^{n-1}]$ for any positive integer n? The original answer is in \frac{m^3}{n} format, please give the value of m + n.

Answer: 3937
user: Each pan of a balance has k weights, numbered from 1 to k. The left pan is heavier. It turns out that if we interchange the places of any two weights with the same number, then either the right pan becomes heavier or the two pans reach an equilibrium. For what k this is possible? Please provide the sum of all possible values of k.

Answer: 3
user: A closed non-selfintersecting broken line is drawn on an infinite chessboard along the sides of its squares. There are K black squares inside the broken line. What is the greatest area of the figure bounded by the broken line when $K=4$?

Answer: 17
user: In a soccer tournament, n teams take part. Every team plays with the other one only once. Find the greatest difference between the final scores of the team with neighboring final positions, and express this difference as a function of n. Please provide the value of this difference when n = 10.

Answer: 10
user: Solve in positive integers $3 \cdot 2^x + 1 = y^2$. Please provide the sum of all integers from the solutions in the form (x, y).

Answer: 21
user: Given a collection of stones. The mass of each stone is $≤ 2 kg$ and their total mass is equal to $100 kg$. We selected a set of stones whose total mass differs from $10 kg$ by the least possible for this set number d. What is the greatest value of d for every admissible collection of stones? The original answer is in $\frac{m}{n}$ format, please give the value of $m + n$.

Answer: 21
user: On an $m × n$ rectangular piece of graph paper there are several squares whose sides are on the vertical and horizontal lines of the paper. It is known that no two squares coincide and no square is situated inside another one. What is the maximal number of such squares? Please provide the value when $m=3,n=4$.

Answer: 12
user: Consider a triangle. The radius of the inscribed circle is equal to $\frac 43$ ; the lengths of the triangle’s heights are integers whose sum is equal to 13. Find the lengths of the triangle’s sides.
If the sum of the squares of all sides is in the form of $\frac{m}{n}$, find the value of $m + n$.

Answer: 1871
user: $$\frac{2}{\sqrt{4-3\sqrt[4]{5}+2\sqrt{5}-\sqrt[4]{125}}}=a+\sqrt[b]{c}$$
Find the value of $a+b+c$.

Answer: 10
user: Find all positive integers n for which both $\frac{1}{n}$ and $\frac{1}{n+1}$ are finite decimal fractions. Provide the sum of all such n values.

Answer: 5
user: Find the total number of pairs of integers $(x, y)$ that satisfy the equation $x ^2 = y ^2 + 2y + 13$.

Answer: 4
user: The numbers $1, 2, . . . , 1983$ stand at the vertices of a regular 1983-gon. Any of the axes of symmetry of the 1983-gon divides the numbers which do not stand at the vertices through which the axis passes (if any) into two sets: on either side of the axis. Let us call an arrangement of numbers good with respect to a given axis of symmetry if every number of one set is greater than the number symmetrical to it. How many such good arrangements exist with respect to any axis of symmetry? Please provide the number of such arrangements.

Answer: 1
user: Three circles of radii $3, 4, 5$ are externally tangent to one another. The common tangent to the first two circles is drawn through the point at which they are tangent to each other. Find the length of this tangent contained inside the circle of radius 5. The original answer is in $\frac{m\sqrt{3}}{n}$ format, please give the value of $m + n$.

Answer: 47
user: Given the equation $\begin{aligned}\frac{x-49}{50}+\frac{x-50}{49}=\frac{49}{x-50}+\frac{50}{x-49}\end{aligned}$, if the sum of all possible values of $x$ is $\frac{b}{a}$, find the value of $a+b$.

Answer: 14801
user: Suppose the "complexity" of a number is the minimum possible length of a sequence (if any) that begins with 0 and ends with that number, with each subsequent term either equal to half of the previous term or equal to the sum of the previous term and the previous term, which is equal to 1. (The length of the empty sequence is assumed to be 0.) Find the number with the largest "complexity" among all numbers of the form $\frac m {2^{50}}$, where $m = 1, 3, 5, . . . , 2^{50} − 1.$. If the number $\frac{a}{b}+\frac{c}{b}\cdot\left(\frac{1}{2}\right)^{50}$ has the largest "complexity", equal to 101, find the value of $a+b+c$.

Answer: 6
user: Solve the equation:
$$(1 + x + x^ 2 )(1 + x + . . . + x^{10}) = (1 + x + . . . + x^6 ) ^2.$$
Please write the sum of all solutions for $x$.

Answer: -1
user: Given that in a convex quadrilateral $ABCD$, $\angle BAC=30^\circ, \angle ACD=40^\circ,\angle ADB=50^\circ,\angle CBD=60^\circ$ and $\angle ABC+\angle ADC>180^\circ.$, if $\angle A=\angle C=a^\circ, \angle B=b^\circ, \angle D=c^\circ.$, find the value of $a+b+c$.

Answer: 290
user: In the representation of numbers A and B as decimal fractions, the lengths of their minimal periods are equal to 6 and 12, respectively. What might the length of the minimal period in the similar representation of $A + B$ be? Find all answers and provide the sum of these possible lengths.

Answer: 16
user: Given n points on a plane, no three of them lie on a straight line. There is a straight line through each pair of points. When $n=5$, what is the minimum number of pairs of non-parallel lines among these lines?

Answer: 5
user: A fly is flying inside a regular tetrahedron with side a. The shortest distance for the fly to fly over every face and return to the initial position is $\frac{a}{\sqrt{m}}$. Find the value of $m$.

Answer: 10
user: In a round goblet whose section is the graph of the function $y = x^4$, a cherry — a ball of radius r — is dropped. What is the largest r for which the ball can touch the lowest point of the bottom? (In plain math words: what is the maximal radius of the disc lying in the domain $y ≥ x^4$ and containing the origin?) The original answer is in the format $\frac{m}{n}\sqrt[3]{2}$, please give the value of $m + n$.

Answer: 7
user: Let us refer to a rectangular of size$1 × k$ for any natural k a strip.For $n <= a$ and for $n > 2 · b + 1$can cut a $1995 × n$ rectangle into pairwise different strips, Find the value of $a+b$.

Answer: 2993
user: 已知 $\triangle A B C$ 及其边 $B C$ 上的一点 $D$ 满足： $A B=2 B D$ ， $A C=3 C D$ ，且以 $A 、 D$ 为焦点可以作一个椭圆 $\Gamma$ 同时经过 $B 、 C$ 两点，求 $\Gamma$ 的离心率的$\sqrt{21}$倍。

Answer: 3
user: 已知抛物线 $y^{2}=2 p x$ 及定点 $A(a, b), B(-a, 0),\left(a b \neq 0, b^{2} \neq 2 p a\right). M$ 是抛物线上的点，设直线 $A M, B M$ 与抛物线的另一交点分别为 $M_{1}, M_{2}$ . 
求证：当 $M$ 点在抛物线上变动时（只要 $M_{1}, M_{2}$ 存在且 $M_{1} \neq M_{2}$ ）直线 $M_{1} M_{2}$ 恒过一个定点.那么当a=3时，求出这个定点的横坐标.

Answer: 3
user: 给定正整数 $n=3$. 求最大的实数 $\lambda$, 使得不等式 $a_{n}^{2} \geqslant \lambda\left(a_{1}+a_{2}+\cdots+a_{n-1}\right)+2 a_{n}$ 对任何满足 $a_{1}<a_{2}<\cdots<a_{n}$ 的正整数 $a_{1}, a_{2}, \cdots, a_{n}$ 均成立.

Answer: 1
user: 已知复数数列 $\left\{z_{n}\right\}$ 满足 $z_{1}=1, z_{n+1}=\overline{z_{n}}+1+n \mathrm{i}(n=1,2,3 \cdots)$ ，其中 $i$ 为虚数单位， $\overline{z_{n}}$ 表示 $z_{n}$ 的共轨复数，则 $z_{2015}$ 的实部为 $\qquad$.

Answer: 2015
user: 设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的最小值的$\sqrt{3}$是 $\qquad$.

Answer: -2
user: 函数 $y=x+\sqrt{x^{2}-3 x+2}$ 的最小值

Answer: 1
user: 设函数 $f(x)$ 的定义域为 $\mathbf{R}$, 且当 $x \geq 0$ 时, $f(x)=|x-2|+a$ (其中 $a$ 为实数). 若 $f(x)$ 为奇函数, 则不等式 $f(x) \geq 1$ 的最小值

Answer: -3
user: 设集合 $S$ 是 $\{0,1, \cdots, 98\}$ 的 $m(m \geqslant 3)$元子集，满足对任意的 $x, y \in S$ ，均存在 $z \in$ $S$, 使得 $x+y \equiv 2 z(\bmod 99)$. 求 $m$ 的所有可能值。请给出所有可能值的和。

Answer: 155
user: 已知 $f(x)$ 是定义在 $(0,+
fty)$ 上的减函数。若 $f\left(2 a^{2}+a+1\right)<f\left(3 a^{2}-4 a+1\right)$成立，请给出 $a$ 的取值范围端点之和的整数部分。

Answer: 6
user: 若对一切 $\theta \in \mathbf{R}$ ，复数 $z=(a+\cos \theta)+(2 a-\sin \theta) i$ 的模不超过 2 ，则实数 $a$ 的取值范围的端点值之和。

Answer: 0
user: 在椭圆 $\Gamma$ 中, $A$ 为长轴的一个端点, $B$ 为短轴的一个端点, $F_{1}, F_{2}$ 为两个焦点. 若 $\overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}}+\overrightarrow{B F_{1}} \cdot \overrightarrow{B F_{2}}=0$, 求 $5\tan \angle A B F_{1} \cdot \tan \angle A B F_{2}$ 的值。

Answer: -1
user: 不等式 $|x|^{3}-2 x^{2}-4|x|+3<0$ 的解集是 $x\in\left(a,b\right)\bigcup\left(c,d\right)$请问$a+d=\qquad$.

Answer: 0
user: 已知复数 $Z_{1} 、 Z_{2}$ 满足 $\left|Z_{1}\right|=2,\left|Z_{2}\right|=3$, 若它们所对应的向量的夹角为 $60^{\circ}$, 则 $7\sqrt{133}\left|\frac{Z_{1}+Z_{2}}{Z_{1}-Z_{2}}\right|=$ $\qquad$

Answer: 133
user: 设 $f(x)$ 是定义在 $\mathbf{R}$ 上的奇函数, 且当 $x \geqslant 0$ 时, $f(x)=x^{2}$ 。若对任意的 $x \in[a, a+2]$ ，不等式 $f(x+a) \geqslant 2 f(x)$ 恒成立，则实数 $a$ 的最小值的$\sqrt{2}$倍是
$\qquad$.

Answer: 2
user: 已知一个正三棱柱的各条棱长均为 $3$, 则其外接球的体积的$\frac{2 \sqrt{21}}{\pi}$为 $\qquad$.

Answer: 147
user: 记 [ $x$ ] 表示不超过实数 $x$ 的最大整数. 设 $x_{1}, x_{2}, \cdots, x_{n} \in \mathbf{R}$, 且 $\left[x_{1}\right],\left[x_{2}\right], \cdots,\left[x_{n}\right]$ 为 $1, 2, \cdots, n$ 的一个排列, 其中, $n \geqslant 2$ 为给定整数. 当$n=6$时， 求 $\sum\limits_{i=1}^{n-1}\left[x_{i+1}-x_{i}\right]$ 的最大值

Answer: 5
user: 对有限非空实数集 $X$, 记 $X$ 的元素个数为 $|X|$ ，均有 
$f(X)=\frac{1}{|X|} \sum\limits_{a \in X} a$. 
集合对 $(A, B)$ 满足 
$A \cup B=\{1,2, \cdots, 100\}, A \cap B=\varnothing$, 
且 $1 \leqslant|A| \leqslant 98$. 
任取 $p \in B$ ，令 
$A_{p}=A \cup\{p\}, B_{p}=\{x \mid x \in B, x \neq p\}$. 
对所有满足上述条件的集合对 $(A, B)$ 与 $p \in$ $B$ ，求 $\left(f\left(A_{p}\right)-f(A)\right)\left(f\left(B_{p}\right)-f(B)\right)$ 的最大值时的$\frac{a}{b}$中的a+b.

Answer: 821
user: 已知 
$f(x)=\lg (x+1)-\frac{1}{2} \log _{3} x$. 
解方程: $f(x)=0$;

Answer: 9
user: 已知 $a_{1}, a_{2}, \cdots, a_{10}$ 和 $b_{1}, b_{2}, \cdots, b_{10}$ 均为 $1,2, \cdots, 10$ 的排列，即集合
$$
\begin{aligned}
& \left\{a_{1}, a_{2}, \cdots, a_{10}\right\} \\
& =\left\{b_{1}, b_{2}, \cdots, b_{10}\right\} \\
& =\{1,2, \cdots, 10\}.
\end{aligned}
$$
记 $c_{k}=a_{k}^{2}+b_{k}^{2}(k=1,2, \cdots, 10)$. 设 $c_{1}$, $c_{2}, \cdots, c_{10}$ 中的最大数为 $M$, 最小数为 $m$. 求
 $M$ 的最小可能值;

Answer: 101
user: 记 $[x]$ 表示不超过实数 $x$ 的最大整数。已知正整数数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=a$ ，且对任意正整数 $n$ ，均有 $$ a_{n+1}=a_{n}+2\left[\sqrt{a_{n}}\right] $$ (1) 若 $a=8$, 求最小的正整数 $n$, 使得 $a_{n}$为完全平方数； (2) 若 $a=2017$, 求最小的正整数 $n$, 使得 $a_{n}$ 为完全平方数。 请提供两个部分答案的和。

Answer: 87
user: $\sum\limits_{k=0}^{2022} C_{2022}^{k} \cos \frac{(1011-k) \pi}{2}$ 的值为$2^a$ ，$a=\qquad$.

Answer: 1011
user: 已知集合 $S=\{(-1,0),(1,0),(0,1),(0,-1)\}$, 甲虫第一天在原点 $O(0,0)$, 第 $n+1$ 天从第 $n$天的位置出发沿向量 $\frac{1}{4^{n}} \boldsymbol{v}$ 移动, 其中 $\boldsymbol{v} \in S$, 用 $S_{n}$ 表示第 $n$ 天甲虫可能在多少个不同的位置上，则 $S_{2023}=4^a$ ，请问$a=\qquad$

Answer: 2022
user: Determine the maximal length $L$ of a sequence $a_{1}, \\ldots, a_{L}$ of positive integers satisfying both the following properties: - every term in the sequence is less than or equal to $2^{2023}$, and - there does not exist a consecutive subsequence $a_{i}, a_{i+1}, \\ldots, a_{j}$ (where $1 \\leqslant i \\leqslant j \\leqslant L$ ) with a choice of signs $s_{i}, s_{i+1}, \\ldots, s_{j} \\in\\{1,-1\\}$ for which $$s_{i} a_{i}+s_{i+1} a_{i+1}+\\cdots+s_{j} a_{j}=0$$ Please provide the sum of the base and exponent used in the expression for the maximal length $L$.

Answer: 2026