AMO-Bench 2025

Divide each side of an equilateral triangle $ABC$ into four equal parts. Through each division point, draw lines parallel to the other two sides. The sides of triangle$ABC$ together with these parallel lines define $15$ lattice points. Find the smallest integer $n$ such that, no matter which $n$ points are chosen among these $15$ lattice points, there always exist three points that form an isosceles triangle.

Let $p$ be a prime number and let $a$ and $b$ be non-negative integers. Determine all ordered triples $(p, a, b)$ for which the expression $p^a + p^b$ is a perfect power.

Find all positive integers $\mathbf{n}$ such that for any: ${a}_{n} \geq {a}_{n - 1} \geq {a}_{n - 2} \geq \cdots \cdots {a}_{2} \geq {a}_{1} > 0,$ satisfying $\mathop{\sum }\limits_{{k = 1}}^{n}{a}_{k} = \mathop{\sum }\limits_{{k = 1}}^{n}\frac{1}{{a}_{k}}$, the inequality $\mathop{\prod }\limits_{{k = 1}}^{n}{a}_{k}^{k} \geq 1$ holds.

Let $x_1,x_2,\ldots,x_{2025}$ be real numbers in the interval $[0,1]$, and let $f(x)$ be a real-valued function defined on $[0,1]$. Find the minimum possible value of $$\max_{x_1,x_2,\ldots,x_{2025}\in[0,1]} \left| \sum_{k=1}^{2025} f(x_k) - \left(\sum_{k=1}^{2025} x_k\right)^2 \right|.$$

The sequence $\{a_n\}_{n=1}^{\infty}$ consists of positive terms, with $a_1=7$, $a_2=2$, and satisfies the recurrence relation $$8a_{n+2}^4=3+4a_{n+1}+a_n (n\in\mathbb{N}^*).$$ Find the general term formula for this sequence.

Let ${x}_{1},{x}_{2},\cdots ,{x}_{2024}$ be positive real numbers such that ${x}_{k} + {x}_{m} \geq {km}$ for any $1 \leq k < m \leq {2024}$. Find the minimum value of ${x}_{1} + {x}_{2} + \cdots + {x}_{2024}$.

$$\left\lfloor \frac{n!}{(n+1)(n+2)} \right\rfloor$$ represents the greatest integer less than or equal to $\frac{n!}{(n+1)(n+2)}$. Find all possible values of $$\left\lfloor \frac{n!}{(n+1)(n+2)} \right\rfloor - \left\lfloor \frac{1}{32} \cdot \frac{n!}{(n+1)(n+2)} \right\rfloor \times 32.$$

Given that $a$ is a positive integer and $b, c$ are positive real numbers, find the maximum value of $$\frac{16(\cos a)^5 - \cos(5a)}{2\cos a + \cos(3a)} \cdot \frac{a^2 b + b^2 c}{a^3 + b^3 + c^3}.$$

Given that $x, y, z$ are all positive real numbers, consider the equation in $w$: $$1121610\sqrt{3270} w^{9} - 407425607 w^{6} + 10360232 = 0$$ which has $k$ real roots $w_{1}, w_{2}, \ldots, w_{k}$, satisfying $$w_{1} < w_{2} < \cdots < w_{k}.$$ If $$m = (w_{1} + w_{k}) \cdot w_{\left\lfloor \tfrac{1 + k}{2} \right\rfloor},$$ (where $\left\lfloor \tfrac{1 + k}{2} \right\rfloor$ denotes the greatest integer not exceeding $\tfrac{1 + k}{2}$), find the minimum value of $$\frac{x^{3} + y^{3} + z^{3}}{xyz} + \frac{63m}{5(x+y+z)} \sqrt[3]{xyz}.$$

Let $x,y,z$ be positive real numbers such that ${xy} + {xz} + {yz} \neq 1$, $\frac{\left( {{x}^{2} - 1}\right) \left( {{y}^{2} - 1}\right) }{xy} + \frac{\left( {{x}^{2} - 1}\right) \left( {{z}^{2} - 1}\right) }{xz} + \frac{\left( {{z}^{2} - 1}\right) \left( {{y}^{2} - 1}\right) }{zy} = 4$. Find the minimum value of $9{x}^{2} + {13689}{y}^{2}{z}^{2} - {27378yz} + \frac{28561}{{\left( y + z\right) }^{2}}$.

Let $x_{1}, x_{2}, \ldots, x_{n}$ be positive numbers satisfying $$x_{1} + x_{2} + \cdots + x_{n} = n.$$ Find the largest positive integer $n$ such that $$\frac{1}{x_{1}^{2}} + \frac{1}{x_{2}^{2}} + \cdots + \frac{1}{x_{n}^{2}} \;\geq x_{1}^{2} + x_{2}^{2} + \cdots + x_{n}^{2}.$$

Space vectors $\overrightarrow{A_n}(x_{n}, y_{n}, z_{n}) (1 \leq n \leq 2052)$ satisfy that $\sqrt{1+x_{n}}, \sqrt{1+y_{n}}, \sqrt{1+z_{n}} (1 \leq n \leq 2052)$ are all real numbers, and $$\sum_{n=1}^{2052} x_{n}^{3} = \sum_{n=1}^{2052} z_{n}^{3} = \sum_{n=1}^{2052} y_{n} - 2052 = 0,$$ $$\sum_{n=1}^{2052} 4z_{n} = \sum_{n=1}^{2052} x_{n}^{2} = 2736, \sum_{n=1}^{2052} y_{n}^{2} = 4210703, \sum_{n=1}^{2052} y_{n}^{3} = 8640361530.$$

Find the maximum value of $$\frac{\lvert \overrightarrow{A}_{1234}\rvert}{\lvert \overrightarrow{A}_{432}\rvert}.$$

Let $\{a_k\}$ be a sequence of non-negative integers, and $\{b_k\}$ be a sequence of positive real numbers. Given that: 1.$$\left(\sum_{n=1}^{2023} a_n b_n\right)\left( \sum_{k=0}^{673} \frac{1}{(k+675)(1348-k)} \right)= \sum_{k=1}^{674}\frac{1}{k(2k-1)}$$ 2.$$\frac{\sum_{k=1}^{1364176}\frac{1}{k(2k-1)}}{\sum_{k=0}^{1364175} \frac{1}{(k+1364177)(2728352-k)}} = 1+\sum_{n=1}^{2023} a_n^2 b_n^2$$ 3.$$\left( \sum_{k=0}^{5519455420} \frac{1}{(k+5519455422)(11038910842-k)} \right)\left[\left(\sum^{2023}_{n=1}2a^3_nb^3_n \right)-1\right] =\sum_{k=1}^{5519455421} \frac{1}{k(2k-1)}$$ Find the minimum value of $$\sum_{n=1}^{2023} a_n^4 b_n^4$$

Let $\{a_n\}$ be a sequence of positive integers, and let $[x]$ denote the greatest integer less than or equal to $x$. Consider the system of equations in $x,y$: $$\big([y] \cdot x^2 + 2n + 2\big)\left(\sum_{k=0}^{n-1} \frac{1}{(k+n+1)(2n-k)}\right) = \sum_{k=1}^{n} \frac{1}{k(2k-1)},$$ $$4\big([x]y^2\big)^5 + 3n(4-n)\big([x]y^2\big)^3 - n^2(6+n)\big([x]y^2\big)^2 + 2n^2(4-3n)\big([x]y^2\big) = 8n^3.$$

Suppose there are $a_n$ distinct real solutions $(x,y)$ to this system. Determine whether there exist three positive prime numbers $p,k,q$ with $p > k > q$ such that $$a_p < a_k < a_q.$$ If no such primes exist, explain why; if they do exist, find the minimum value of $p+k+q$.

Let $a, b, c, d$ be real numbers such that $$4a^4 + \frac{c^4}{16} + b^4 + b^3d^2 + 27 + \frac{d^2}{4} + a^2c^2 = 3c^2 + 24a^2 + 8b^2 + 4d^2.$$ Find the minimum value of $$\frac{4a^4 + \frac{c^4}{16} + a^2c^2 + 12a^2 + \frac{3}{2}c^2 + b^4 + \frac{d^4}{4} + 10 + b^2d^2 + 2b^2 + d^2 }{\sqrt{2a^2b^2 + a^2d^2 + 2a^2 + \frac{b^2c^2}{4} + \frac{c^2d^2}{8} + \frac{c^2}{4} + 3b^2 + 3 + \frac{3}{2}d^2}}.$$

Let $n$ be a positive integer such that $2 \leq n \leq 5$. For each such $n$, let $A_nB_nC_nD_n$ be a rectangle with vertex $A_n$ at $(-n, 1 + 6 \sqrt{2} - \sqrt{2}n)$, and vertex $B_n$ the reflection of $A_n$ through the origin. The line segments $A_nC_n$ and $B_nD_n$ are parallel to the $x$-axis, and the line segments $A_nD_n$ and $B_nC_n$ are parallel to the $y$-axis. Let $P_i (1 \leq i \leq 36)$ be 36 distinct points on the curve $(y - |x|)^2 + x^2 = 1$. For each point $P_i$, let the distances to the 16 sides of the four rectangles be denoted by $r_{i_1}, r_{i_2}, \ldots, r_{i_{16}}$. Let $d_i = \min \Big( r_{i_k} \Big) , (1 \leq k \leq 16)$, and suppose that these values satisfy the condition $3 \sum_{i=1}^{36}d_{i}^{3} -15\sum_{i=1}^{36} d_{i}^{2} = -1200$. Find the maximum value of $\sum_{i=1}^{36} d_i$.

Let $\{x_1, x_2, \ldots, x_{2025}\} = \{1, 2, \ldots, 2025\}$. Find the greatest positive integer less than $$\sum_{i=1}^{2024} \frac{x_i}{x_{i+1}}.$$

Let complex numbers $z_1, z_2, \ldots, z_{2025}$ satisfy $$\sum_{k=1}^{2025} z_k = 0, \sum_{k=1}^{2025} |z_k|^2 = 1.$$ Find the minimum possible value of $$\max_{1 \le i < j \le 2025} |z_i - z_j|.$$

Let $a_1 < a_2 < \cdots < a_{38}$ be positive integers with $$a_1 + a_2 + \cdots + a_{38} = 1997.$$ Define $$b_k = \min\!\left(\sum_{i=1}^{k} a_i, \sum_{i=k+1}^{38} a_i\right),$$ and let $$S = \max(b_1, b_2, \ldots, b_{38}),$$ where $b_{38} = 0$. Find the minimum possible value of $S$.

A school has newly formed $3$ classes. It is known that for any two students belonging to different classes, in the third class there are exactly $10$ students who know both of them and $10$ students who know neither of them. Find the total number of students across the three classes.

Given that $x, y, z$ are integers satisfying the system $$\begin{cases} x^2 - 7y^4 = z,\\[6pt] 7z^4+ \frac{\sum_{k=1}^{z} \frac{1}{k(2k-1)}}{\sum_{k=0}^{ z-1} \frac{1}{(k+z+1)(2z-k)}} = \frac{\sum_{i=0}^{174545562} \frac{1}{i(2i-1)}}{\sum_{i=0}^{174545561} \frac{1}{(i+174545563)(349091124-i)}} \end{cases}$$ find the maximum value of $x+y+z$.

Let variables $x, y, c$ satisfy $x+y+c > \sqrt{13}$ with $x, y \ge 0$. Find the minimum value of $$\frac{(2+x^2)\,(4+y^2)\,\big(49+6c^2+4c\sqrt{3+3c^2}\big)} {\big(7\sqrt{3} c+14\sqrt{1+c^2}\big)\,(x+y)}.$$

It is known that the equation in $u, v$, $u^2 - 3v^4 = 1$, has at most four positive integer solutions; and the equation in $\lambda, \iota$, $4\lambda^2 - 3\iota^2 = 1$, has at most four positive integer solutions. If a prime $p$ satisfies $p \equiv 5 \pmod{8}$, and integers $x, y$ satisfy $$\left(\frac{1+x^2}{2} + \frac{\sqrt{ 9x^4 + \frac{4x^6}{3} + 9 + 18x^2\,}}{6}\right)^{ 1/3} + \left(\frac{1+x^2}{2} - \frac{\sqrt{ 9x^4 + \frac{4x^6}{3} + 9 + 18x^2\,}}{6}\right)^{ 1/3} = 2py^2 - x^3,$$ find the maximum value of $x+y$.

Let $n$ be an integer with $n>2$. Real numbers $a_1, a_2, \ldots, a_n$ satisfy $$\sum_{k=1}^{n} a_k = 2n, \sum_{k=1}^{n} k\,|a_k| = 4n.$$ Find the minimum value of $a_1^2 + a_2^2 + \cdots + a_n^2$.

Given a positive integer $k$, find all positive integers $n$ that can be expressed in the form $$\frac{a}{b}+\frac{a+k}{b+k},$$ where $a, b \in \mathbb{N}^{+}$.

Find the number of positive integer solutions $(x, y, z)$ to the equation $$x^{999} + y^{1000} = z^{1001}.$$

Find the number of integer solutions to $$x^2 + y^2 + z^2 + 3(xy + yz + zx) + 5(x + y + z) + 1 = 0.$$

For each positive integer $k \ge 2$, define $f(k)$ to be the product of the distinct prime factors of $k$. All the coefficients of the polynomial function $9P(x)$ are rational numbers. Suppose $45 \le P(2025) \le 81$, and there are infinitely many positive integers $n$ for which $P(n) = f(n)$. How many polynomials $P(x)$ satisfy these conditions?

Let $a$ be a prime greater than $100$ with $a \equiv 3 \pmod{5}$. For complex numbers $z_n$ $(1 \le n \le a)$, both the real and imaginary parts are nonzero. For each $n$, let $d_n$ be the distance from the point $(n,\lvert z_n\rvert)$ to the line $$y=\frac{202}{5}x+2025.$$ Choose one complex number at random from $\{z_n\}_{n=1}^a$; let $p$ be the probability that its modulus is a positive integer. Given $$\bigg(2a-\sum_{i=1}^{a} i^{d_i}\bigg)^{ 2} +\bigg(3+\sum_{k=2}^{a}(2k+2)-\sum_{i=1}^{a} i^{2d_i}\bigg)^{ 2} +\bigg(\frac{7}{2}+\sum_{j=2}^{a}\!\big(3j^3+3j+\tfrac{1}{2}\big)-\sum_{i=1}^{2022} i^{3d_i}\bigg)^{ 2}=0,$$ find the maximum value of $p$.

Let $p, q$ be positive integers satisfying the system of congruences: $$\begin{cases} p \equiv 3 \pmod{911}, \\ q \equiv 2 \pmod{997}, \\ p \equiv 738 \pmod{q}; \end{cases}$$ Let $\{ a_n \}, \{ b_n \}, \{ c_n \}$ be periodic sequences of rational numbers satisfying the equations: $$\left[ 4b_n + 1 - 3\sin(\pi a_n) \right]^2 + \left[ 4c_n - 1 + 3\sin(\pi a_n) \right]^2 = 0 (\forall n \in \mathbb{N})$$ and $$\left| \sum_{i=1}^{p} b_i^3 \right|+\left| \sum_{i=1}^{q} c_i^3 \right|+\left| p - 3 \sum_{i=1}^{p} b_i \right|+\left| q - 3 \sum_{i=1}^{q} c_i^2 \right| = 0.$$ Find the minimum value of $p + q + 2a_p^2 + b_q^2 + c_{p+q}^2 + (a_p + a_{q-1})^2 + (b_p + c_q)^2$.

Let $(a_i, b_i)$, for $1 \leq i \leq 2025$, be 2025 distinct points on the curve $$3x^2 + 3y^2 = 8 + 4xy + 2(x + y)|x - y|$$ Let $d_i$ be the distance from the point $(a_i, b_i)$ to the line $x + y = 6$, and let $\left\lfloor d_i \right\rfloor$ denote the greatest integer not exceeding $d_i$.

It is given that: $$r_i = \left\lfloor \frac{21 - 15\sqrt{2}}{2} \left( d_i^3 - 3d_i^2 - 6d_i^2 \cdot \sqrt{2} + 18d_i + 6d_i \cdot \sqrt{2} + \frac{28}{3} + \frac{16\sqrt{2}}{3} \right) \right\rfloor$$ and that: $$\sum_{i=1}^{2025} r_i^2 = 2043, \sum_{i=1}^{2025} r_i = 2021$$ Find the maximum value of $$\sum_{i=1}^{2025} r_i^4$$

Let $x$ and $y$ be positive rational numbers that satisfy: $$1 + \sqrt{1 - 2y} = y + \frac{4x - 2 + 2\sqrt{1 - 2x}}{3 - 5x - \sqrt{1 - 2x}} + \sqrt[3]{\frac{1 + x^2}{2} + \frac{1}{6} \cdot \sqrt{9x^4 + \frac{4x^6}{3} + 9 + 18x^2}}+ \sqrt[3]{\frac{1 + x^2}{2} - \frac{1}{6} \cdot \sqrt{9x^4 + \frac{4x^6}{3} + 9 + 18x^2}}$$ Given that $kx$ and $ky$ are both positive integers, find the minimum value of the positive integer $k$.

An album contains 25 photos, each showing three people, subject to the following conditions: (1) In each photo, the person on the left is a full brother of the person in the middle, and the person on the right is the daughter of the person in the middle. (2) The person in the middle is different in each of the 25 photos. (3) No two people who appear in the album have a child in common. What is the minimum number of distinct people that must appear in this album?

Find $C_{\max}$ such that $\forall x, y, z \in \mathbb{R}$,$$f = 1 + |x+y+z| + |xy+yz+zx| + |xyz| \geq C(|x| + |y| + |z|)$$

In a game, a positive integer $n$ not exceeding 10000 is initially written on a blackboard. Two players, Player 1 and Player 2, take turns making moves, with Player 1 going first. A move consists of erasing a positive integer $k$ greater than 1 from the blackboard and replacing it with five non-negative integers that sum to $k$ and whose pairwise differences are at most 1. For example, the integer 11 could be replaced with 2, 2, 2, 2, 3. We also stipulate that 1 couldn't be replaced with five integers. The game ends when a player cannot make a valid move, at which point the other player wins. Assuming both players use an optimal strategy and Player 1 wins, how many possible values of $n$ are there?

On a 19*19 board, a black or a white stone is placed on each of the 361 squares. An operation consists of choosing one row and one column and reversing the color of all 37 stones in that row and column. Find the largest integer n such that, regardless of the initial arrangement of stones, it is always possible to perform a sequence of such operations so that the number of black stones on the board is at least n.

In $\triangle ABC$ with $AB < AC < BC$, let $O$ and $I$ be its circumcenter and incenter, respectively. Let $M_b$ and $M_c$ be the midpoints of the arc $AC$ (not containing $B$) and the arc $AB$ (not containing $C$), respectively, on the circumcircle $\odot O$. Let $S_b$ be a point on $\odot O$ such that $IS_b \perp BS_b$, and let $S_c$ be a point on $\odot O$ such that $IS_c \perp CS_c$. Let $H_a$ be the intersection of lines $M_bS_c$ and $M_cS_b$. Points $H_b$ and $H_c$ are defined analogously.It is given that the lines $AH_a, BH_b, CH_c$ are pairwise non-parallel. Let $G$ be the centroid of the triangle formed by these lines (if the three lines are concurrent, $G$ is defined as their point of concurrency). Let the side lengths of $\triangle ABC$ be $BC = a$, $CA = b$, and $AB = c$. Find the product of the distances from point $G$ to the lines $IH_a, IH_b, IH_c$. (Expressed in terms of $a, b, c$ and simplified completely.)

In a $d$-dimensional Euclidean space, there are two definitions:

1. Encapsulation: For a point $p$, if there exists a ball centered at $p$ with sufficiently small radius such that every point within this ball (except $p$ itself) belongs to at least one convex set $K_i$ (where $i=1, \dots, n$), then we say point $p$ is encapsulated by these convex sets;
2. Contact: For a set $K$ and a point $p$, if $p$ does not lie inside $K$ but any neighborhood of $p$ (no matter how small) contains points from $K$, we say point $p$ contacts set $K$ (This is equivalent to saying $p$ is a boundary point of $K$). Let $K_1, K_2, K_3$ be three convex sets in $d$-dimensional space. Let $S$ be the set of points encapsulated by these three convex sets. The maximum possible value of $|S|$ is a function theat depends only on the dimension $d$. Based on the above definitions and known conditions, the maximum value can be derived. Let $f(d, 3) = \max |S|$. Please provide the expression for $f(d, 3)$: $f(d, 3) =$

Given positive integers $n$ and $t$ with $1 \leq t < n$. There are $n$ cups, all initially upright. An operation consists of choosing $t$ cups and turning them over. If it is possible to turn all $n$ cups upside down in $m$ operations, find the minimum value of $m$. (expressed in terms of $n$ and $t$)

Let $m$ be an odd integer with $m \geq 3$. Construct a positive integer $x$ such that for some positive integers $a$ and $b$, the sequence of consecutive integers $a+1, a+2, \ldots, a+b$ contains exactly 20232024 terms $y$ satisfying $y \equiv -1 \pmod{m}$ and $y^2 \equiv 1 \pmod{x}$, and exactly 2023 terms $z$ satisfying $z \equiv 1 \pmod{m}$ and $z^2 \equiv 1 \pmod{x}$.

A point M on an infinite number line moves one unit to the left or to the right at each step. The probability of moving to the right is $p (p > \frac{1}{2})$, and the probability of moving to the left is $1-p$. The point M starts at the origin $x = 0$. Let $x$ be the probability that M is at position $x = 3$ after $n$ moves, where $n \geq 2$, and let $y$ be the probability that M is at position $x = 4$ after $m$ moves, where $m \geq 2$. Find the conditions on $m$ and $n$ such that $x > (1-p)y$.

Let $f(x),g(x),h(x)$ be functions defined on the set of positive integers. Let $\left\lfloor \tfrac{x}{2}\right\rfloor$ denote the greatest integer less than or equal to $\tfrac{x}{2}$. Define $f(x)$ to be the number of integers in the list $1,2,3,\dots,\left\lfloor \tfrac{x}{2}\right\rfloor$ that are coprime to $x$. If $p_{1},p_{2},\dots,p_{k}$ are distinct primes and $y=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots p_{k}^{\alpha_{k}}$, then $$g(y)=\max\{ p_{1}^{\alpha_{1}}, p_{2}^{\alpha_{2}},\,\dots, p_{k}^{\alpha_{k}}\,\}.$$ For a positive integer $z$, let $h(z)$ be the least positive integer $m$ such that $z\mid m!$. Find all integers $b>1$ such that $$f(b)=h\!\big(g(b^{2})\big).$$

Let $a=\log_3\bigg(\frac{1}{2}\sum\limits_{n=0}^{676}(-1)^n{2028 \choose 3n}\bigg)$. Calculate the sum of distinct prime factors of $a^2+(a-1)^4$.

Find the largest positive integer $n$ that satisfies the following property: there exists a sequence of $n$ positive integers ${a}_{1},\cdots ,{a}_{n}$ such that ${a}_{i} \leq {2}^{2025}\left( {\forall 1 \leq i \leq n}\right)$, and for any $1 \leq i < j \leq n$, there is no non-empty proper subset $A \subset \{ i,i + 1,\cdots ,j\}$ such that $$\mathop{\sum }\limits_{{k \in A}}{a}_{k} = \mathop{\sum }\limits_{{k \in \{ i,i + 1,\cdots ,j\} - A}}{a}_{k}. \tag{1}$$

There are $n\times n\times n$ mud blocks on the ground. A miner starts digging through one of the blocks. Each time, he can only dig a face-adjacent block until all blocks are dug out. If he digs the blocks consecutively and for $n$ times these blocks are all in a line, then the miner is described as to have dug a straight line'' once. Find the maximum possible number the miner can dig a straight line''.

Assume that graph $G$ has $50$ vertices, and for any two non-adjacent vertices, they have exactly two common neighbors. Vertices $v_1$ and $v_2$ are adjacent and have no common neighbors, the degree of $v_1$ is $d(v_1)=11$. The parameter $M$ is defined as: if such a graph $G$ exists, then $M=max(d(v_2)-d(v_1))$; otherwise, $M=0$. Let $a$ be the smallest positive real root of the equation $x^{6}-4x^{5}+7x^{4}-25x^{3}+38x^{2}-6x+34=M(2x^3-4x^2+3x-19-M)$, find the remainder of the closest integer to $a^{25}$ divided by $1000$.

A city is going to be built as a $10\times 10$ square grid of blocks. Now, the city is planning to set up security booths in some blocks. To ensure the safety of the entire city, city safety regulations require that for any block (regardless of whether it has a security booth itself), at least one of its four adjacent blocks (up, down, left, right) must have a security booth. How many security booths are needed at a minimum to satisfy the safety regulations?

In the Cartesian plane, there is a lamp at every integer lattice point, each with two states: on'' and off''. The lamp board features a special switching system: pressing any lamp toggles the states of all lamps in its entire row and entire column (on to off, and off to on). Define $f(x)=x^2-18x+23$ and $g(x)=37x+18$. The equation $f(f(x))=g(x)$ has real roots $a_1,a_2,a_3,a_4$ with $a_1 < a_2 < a_3 < a_4$. You can operate the switches at coordinates $(x,y)$ where $1\leq x\leq 17$ and $1\leq y\leq a_2+a_3$. Regardless of the initial state of the lamp board, what is the minimum number of lamps within this operable rectangular region that you can guarantee will be lit after performing operations using this system?

Let the polynomial $f(x)=x^6 - 11x^5 + 28x^4 + 2x^3 + 57x^2 + 13x + 30$, and let $\alpha$ be the smallest positive real root of the equation $f(x)=0$. Set $N$ as the remainder of the closest integer to $\alpha^{35}$ divided by 1498. A logistics company warehouses pallets with serial numbers $m=1,2,\dots, M$. To simplify the sampling-judgment'' process, three quality inspectors use their respective cyclic-power residue'' rules: Inspector A uses a cycle of $7$ pallets'' rule to record the remainder of serial number $m$ divided by $7$. If this remainder is a non-zero quadratic residue of modulo $7$ (there exists an $r$ coprime to $7$ such that $r^2\equiv m\pmod7$), it is marked as passed''; Inspector B uses a cycle of $11$ pallets'' rule to record the remainder of serial number $m$ divided by $11$. If this remainder is a non-zero cubic residue of modulo $11$ (there exists an $r$ coprime to $11$ such that $r^3\equiv m\pmod{11}$), it is marked as passed''; Inspector C uses a cycle of $13$ pallets'' rule to record the remainder of serial number $m$ divided by $13$. If this remainder is a non-zero quartic residue of modulo $13$ (there exists an $r$ coprime to $13$ such that $r^4\equiv m\pmod{13}$), it is marked as passed''. A serial number $m$ is marked as qualified'' if and only if all three inspectors mark it as passed''. Let the total number of qualified serial numbers be counted within the range $1 \le m \le M$, where the integer $M$ satisfies $M-724=N$. How many ``qualified'' serial numbers are there in total?