GAIC Math 2024

Let $S=\left\{ 1,2,\cdots 2024 \right\}$, if the set of any $n$ pairwise prime numbers in $S$ has at least one prime number, the minimum value of $n$ is \underline{\hspace{2cm}}.

Let $A_l = (4l+1)(4l+2) \cdots \left(4(5^5+1)l\right)$. Given a positive integer $l$ such that $5^{25l} \mid A_l$ and $5^{25l+1} \nmid A_l$, the minimum value of $l$ satisfying these conditions is \underline{\hspace{2cm}}.

Sasha collects coins and stickers, with fewer coins than stickers, but at least 1 coin. Sasha chooses a positive number $t > 1$ (not necessarily an integer). If he increases the number of coins by a factor of $t$, then he will have a total of 100 items in his collection. If he increases the number of stickers by a factor of $t$, then he will have a total of 101 items in his collection. If Sasha originally had more than 50 stickers, then he originally had \underline{\hspace{2cm}} stickers.

Let $n$ be a positive integer. An integer $k$ is called a "fan" of $n$ if and only if $0 \leqslant k \leqslant n-1$ and there exist integers $x$, $y$, $z \in \mathbf{Z}$ such that $x^2+y^2+z^2 \equiv 0 (\mathrm{mod} n)$ and $xyz \equiv k (\mathrm{mod} n)$. Let $f(n)$ denote the number of fans of $n$. Then f(2020) = \underline{\hspace{2cm}}.

Four positive integers satisfy $a^3=b^2$, $c^5=d^4$, and $c-a=77$. Then, $d-b=$ \underline{\hspace{2cm}}.

The smallest $n$ such that both $3n+1$ and $5n+1$ are perfect squares is \underline{\hspace{2cm}}.

Find the largest positive integer $n$ such that the product of the numbers $n, n+1, n+2,\cdots, n+100$ is divisible by the square of one of these numbers.

Given a positive integer $x$ with $m$ digits in its decimal representation, and let $x^3$ have $n$ digits. Which of the following options cannot be the value of $m + n$?

\begin{align*} \text{A)}\ & 2022 & \text{B)}\ & 2023\ \text{C)}\ & 2024 & \text{D)}\ & 2025\ \end{align*}

Positive integers $a$, $b$, and $c$ satisfy $a > b > c > 1$, and also satisfy $abc \mid (ab - 1)(bc - 1)(ca - 1)$. There are \underline{\hspace{2cm}} possible sets of $(a, b, c)$.

There are \underline{\hspace{2cm}} sets of positive integers $a \leq b \leq c$ such that $ab - c$, $bc - a$, and $ca - b$ are all powers of 2.

Define the function $f(x) =[x[x]]$, where $[x]$ represents the largest integer not exceeding $x$. For example, $[-2.5] = -3$.For a positive integer $n$, let $a_n$ be the number of elements in the range set of $f(x)$ when $x \in [0,n)$. Then the minimum value of $\frac{a_n + 90}{n}$ is \underline{\hspace{2cm}}.

If a positive integer's sum of all its positive divisors is twice the number itself, then it is called a perfect number. If a positive integer $n$ satisfies both $n-1$ and $\frac{n(n+1)}{2}$ being perfect numbers, then $n=$ \underline{\hspace{2cm}}.

Try to find all prime numbers with the shape $p^p+1$($p$ is a natural number) that have no more than 19 digits and what the sum of these prime numbers is.

For any positive integer $q_0$, consider a sequence $q_1,q_2,\cdots,q_n$ defined by $q_i=\left(q_{i-1}-1\right)^3+3\left(i=1,2,\cdots,n\right)$. If every $q_i\left(i=1,2,\cdots,n\right)$ is a power of prime, then the maximum possible value of $n$ is \underline{\hspace{2cm}}.

The first digit before the decimal point in the decimal representation of $(\sqrt{2} + \sqrt{5})^{2000}$ is \underline{\hspace{2cm}} and after the decimal point is \underline{\hspace{2cm}}.

$N$ is a 5-digit number composed of 5 different non-zero digits, and $N$ is equal to the sum of all three digits formed by 3 different digits in these 5 digits, then the sum of all such 5-digit $N$ is \underline{\hspace{2cm}}.

If the last three digits of a positive integer $n$ cubed are 888, then the minimum value of $n$ is \underline{\hspace{2cm}}.

$a$, $b$ are both two-digit positive integers, $100a+b$ and $201a+b$ are both four-digit perfect squares, then $a+b=$ \underline{\hspace{2cm}}.

For a positive integer $n$, which can be uniquely expressed as the sum of the squares of 5 or fewer positive integers (where two expressions with different summation orders are considered the same, such as $3^2+4^2$ and $4^2+3^2$ are considered the same expression of 25), then the sum of all the $n$ that satisfy the conditions is \underline{\hspace{2cm}}.

Given that the product of the digits of a natural number $x$ is equal to $44x-86868$, and the sum of its digits is a perfect cube. Then the sum of all such natural numbers $x$ is \underline{\hspace{2cm}}.

Given $a$ is a prime number and $b$ is a positive integer such that $9(2a+b)^2=509(4a+511b)$, we need to find the values of $a$ and $b$.

Let $\varphi(n)$ denote the number of natural numbers coprime to and less than $n$. Then, when $\varphi(pq)=3p+q$, what is the sum of $p$ and $q$?

Consider the sequence $\{S_n\}$ constructed as follows: $S_1=\{1,1\}$, $S_2=\{1,2,1\}$, $S_3=\{1,3,2,3,1\}$, and in general, if $S_k=\{a_1,a_2,\cdots,a_n\}$, then $S_{k+1}=\{a_1,a_1+a_2,a_2,a_2+a_3,\cdots,a_{n-1}+a_n,a_n\}$. What is the number of terms equal to $1988$ in $S_{1988}$?

For a natural number $n$, let $S(n)$ denote the sum of its digits. For example, $S(611)=6+1+1=8$. Let $a$, $b$, and $c$ be three-digit numbers such that $a+b+c=2005$, and let $M$ be the maximum value of $S(a)+S(b)+S(c)$. How many sets $(a,b,c)$ satisfy $S(a)+S(b)+S(c)=M$?

For a natural number $n$, let $K(n,0)=\varnothing$. For any non-negative integers $m$ and $n$, define $K(n,m+1)$ as the set of elements $k$ such that $1\leq k\leq n$ and $K(k,m)\cap K(n-k,m)=\varnothing$, then the set $K(2004,2004)$ contains \underline{\hspace{2cm}} elements.

Let $T=\{0,1,2,3,4,5,6\}$ and $M=\left\{\left.\frac{a_1}{7}+\frac{a_2}{7^2}+\frac{a_3}{7^3}+\frac{a_4}{7^4}\right|a_i\in T,i=1,2,3,4\right\}$. If the elements of $M$ are arranged in descending order, then the 2005th number is \underline{\hspace{2cm}}. \begin{align*} \text{A)}\ & \frac{5}{7}+\frac{5}{7^2}+\frac{6}{7^3}+\frac{3}{7^4}& \text{B)}\ & \frac{5}{7}+\frac{5}{7^2}+\frac{6}{7^3}+\frac{2}{7^4}\ \text{C)}\ & \frac{1}{7}+\frac{1}{7^2}+\frac{0}{7^3}+\frac{4}{7^4}& \text{D)}\ & \frac{1}{7}+\frac{1}{7^2}+\frac{0}{7^3}+\frac{3}{7^4} \ \end{align*}

Mutually prime positive integers $p_n, q_n$ satisfy $\frac{P_n}{q_n}=1+\frac12+\frac13+\cdots+\frac1n$. The sum of all positive integers $n$ such that $3|p_n$ is \underline{\hspace{2cm}}.

Given $x$ and $y$ are prime numbers. The sum of the values of $y$ in the solutions of the indeterminate equation $x^2-y^2=xy^2-19$ is \underline{\hspace{2cm}}.

The number of non-zero integer pairs $(a,b)$ for which $\left(a^3+b\right)\left(a+b^3\right)=(a+b)^4$ holds is \underline{\hspace{2cm}}.

If the sum of the digits of a natural number $\alpha$ equals 7, then $a$ is called an "auspicious number". Arrange all "auspicious numbers" in ascending order $a_1,a_2,a_3,\cdots$, if $a_n=2005$, then $a_{5n}$=\underline{\hspace{2cm}}.

The number of integers $n$ in the interval $1\leq n\leq10^6$ such that the equation $n=x^y$ has non-negative integer solutions $x,y$, and $x\neq n$ is \underline{\hspace{2cm}}.

Let $p$ be a real number. If all three roots of the cubic equation $5x^3 - 5(p+1)x^2 + (71p - 1)x + 1 = 66p$ are natural numbers, then the sum of all possible values of $p$ is \underline{\hspace{2cm}}.

The number of triples of positive integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 = 2005$ and $a \leq b \leq c$ is \underline{\hspace{2cm}}.

The maximum positive integer $k$ that satisfies $1991^k \mid 1990^{19911992} + 1992^{19911990}$ is \underline{\hspace{2cm}}.

Let $a$ and $b$ be positive integers such that $79 \mid (a + 77b)$ and $77 \mid (a + 79b)$. Then the smallest possible value of the sum $a + b$ is \underline{\hspace{2cm}}.

Let $a_i, b_i \ (i=1,2,\ldots,n)$ be rational numbers such that for any real number $x$, we have $x^2 + x + 4 = \sum_{i=1}^{n} (a_{i}x + b_{i})^{2}$. Then the minimum possible value of $n$ is \underline{\hspace{2cm}}.

Let $\alpha$ and $\beta$ be the two roots of the equation $x^2 - x - 1 = 0$. Define $\alpha_n = \frac{\alpha^n - \beta^n}{\alpha - \beta}$ for $n = 1, 2, \ldots$. For some positive integers $a$ and $b$, with $a < b$, if for any positive integer $n$, $b$ divides $a_n - 2na^n$, then the sum of all such positive integers $b$ is \underline{\hspace{2cm}}.

Let $n$ be a natural number greater than $3$ such that $1+C_n^1+C_n^2+C_n^3$ divides $2^{2000}$. Then, the sum of all such $n$ satisfying this condition is \underline{\hspace{2cm}}.

In the decimal representation, the product of the digits of $k$ equals $\frac{25}{8}k-211$. Then the sum of all positive integers $k$ satisfying this condition is \underline{\hspace{2cm}}.

Let $n$ be an integer, and let $p(n)$ denote the product of its digits (in decimal representation). Then the sum of all $n$ such that $10\:p(n)=n^{2}+4n-2005$ is \underline{\hspace{2cm}}.

There are some positive integers with more than two digits, such that each pair of adjacent digits forms a perfect square. Then the sum of all positive integers satisfying the above conditions is \underline{\hspace{2cm}}.

Let $\alpha$ be an integer, and $|\alpha|\leq2005$. The number of values of $\alpha$ that make the system of equations $\begin{cases}x^{2}=y+\alpha,\\y^{2}=x+\alpha\end{cases}$ have integer solutions is \underline{\hspace{2cm}}.

Divide the set $S=\{1,2,\cdots,2006\}$ into two disjoint subsets $A$ and $B$ such that:

(1) $B \in A$;

(2) If $a\in A$ and $b\in B$ with $a+b\in S$, then $a+b\in B$;

(3) If $a\in A$, $b\in B$, and $a b\in S$, then $a b\in A$.

The number of elements in set $A$ is \underline{\hspace{2cm}}.

Let $S$ be a finite set of integers. Suppose that for any two distinct elements $p, q \in S$, there exist three elements $a$, $b$, $c \in S$ (not necessarily distinct, and $a \neq 0$) such that the polynomial $F(x) = \alpha x^2 + bx + c$ satisfies $F(p) = F(q) = 0$. The maximum number of elements in $S$ is \underline{\hspace{2cm}}

A natural number whose last four digits are 2022 and is divisible by 2003 has a minimum value of _____.

The number of positive integer solution in $\left(x^{2}+2\right)\left(y^{2}+3\right)\left(z^{2}+4\right)=60 x y z$ is _____.

The number of integers satisfying the condition that $x^2+5n+1$ is a perfect square is known to be _____.

If p, q, r are prime numbers such that $p+q+r=1000$, then the remainder when $p^{2} q^{2} r^{2}$ is divided by 48 is _____.

Given $x, y, z$ are integers, and $10x^{3}+20y^{3}+2006 xyz=2007z^{3}$, then the maximum of $x+y+z$ is _____.

For a positive integer $n$, if the first two digits of $5^{n}$ and $2^{n}$ are the same, denoted as $a$ and $b$ respectively, then the value of the two-digit number $\overline{ab}$ is _____.

The remainder when $\frac{2020 \times 2019 \times \cdots \times 1977}{44!}$ is divided by 2021 is _____.

Given the sequence $\{a_n\}: a_1=1, a_{n+1}=\frac{\sqrt{3}a_n+1}{\sqrt{3}-a_n}$, then $\sum\limits_{n=1}^{2022}a_n=\_\_\_\_\_$.

Given $2bx^2 + ax + 1 - b \geq 0$ holds for $x \in [-1, 1]$, find the maximum value of $a + b$.

Given $x,y \in [0,+\infty)$, and satisfying $x^3+y^3+6xy=8$. Then the minimum value of $2x^2+y^2=\_\_\_\_\_$.

Given $f(x)$ and $g(x)$ are two quadratic functions with the coefficient of the quadratic term being 1 for both. If $g(6)=35,\frac{f(-1)}{g(-1)}=\frac{f(1)}{g(1)}=\frac{21}{20}$, then $f(6)=\_\_\_\_\_$.

Given $(n + 1)^{a+1} - n^{a+1} < n a^(a + 1) < n a^{n+1} - (n - 1)^{a+1} (-1 < a < 0) \textcircled{1}$. Let$x=\sum\limits_{k=4}^{106}\frac{1}{\sqrt[3]{k}}$, then the integer part of x is _____.

Let $a_1=\frac{\pi}{6}, a_n \in (0,\frac{\pi}{2})$, and $\tan a_{n+1}\cdot \cos a_n=1 (n\geq 1)$. If $\prod\limits_{k=1}^m \sin a_k=\frac{1}{100}$, then $m=$ _____.

Let $y=f(x)$ be a strictly monotonically increasing function, and let its inverse function be $y=g(x)$. Let $x_1, x_2$ be the solutions to the equations $f(x)+x=2$ and $g(x)+x=2$ respectively. Then $x_1+x_2=\_\_\_\_\_$.

Let $x_0>0, x_0 \neq \sqrt{3}, Q(x_0,0), P(0,4)$, and the line PQ intersects the hyperbola $x^2-\frac{y^2}{3}=1$ at points A and B. If $\overrightarrow{PQ}=t\overrightarrow{QA}=(2-t)\overrightarrow{QB}$, then $x_0=\_\_\_\_\_$.

Assuming sequence ${F_n}$ satisfying: $F_1=F_2=1, F_{n+1}=F_n+F_{n-1} (n\geq 2)$. Then the number of sets of positive integers $(x,y)$ that satisfy $5F_x-3F_y=1$ is

For some positive integers $n$, there exists a positive integer $k\geq 2$ such that for positive integers $x_1,x_2,...,x_k$ satisfying the given condition, $\sum\limits_{i=1}^{k-1}x_ix_{i+1}=n$, $\sum\limits_{i=1}^{k}x_i=2019$ the number of such positive integers is _____.

Considering all non-increasing functions $f:\{1,2,\cdots,10\} \rightarrow \{1,2,\cdots,10\}$, some of these functions have fixed points, while others do not. The difference in the number of these two types of functions is _____.

Given an integer coefficient polynomial $P(x)$ satisfying: $P(-1)=-4, P(-3)=-40, P(-5)=-156$. The maximum number of solutions x for $P(P(x))=x^2$ is _____.

Given hyperbola $\Gamma: \frac{x^2}{a^2}-\frac{y^2}{b^2}=1$ passes the point $M(3,\sqrt{2})$, line $l$ passes its right focus $F(2,0)$ and cross the right branch of $\Gamma$ at points $A$ and $B$, and cross the y-axis at point P. If $\overrightarrow{PA}=m\overrightarrow{AF}, \overrightarrow{PB}=n\overrightarrow{BF}$, then $m+n=$ _____.

Let positive real numbers $x_1, x_2, x_3, x_4$ satisfying $x_1x_2+x_2x_3+x_3x_4+x_4x_1=x_1x_3+x_2x_4$. Then the minimum of $f=\frac{x_1}{x_2}+\frac{x_2}{x_3}+\frac{x_3}{x_4}+\frac{x_4}{x_1}$ is _____.

Given sequence $\{a_n\}$ satisfying $a_1=a, a_{n+1}=2(a_n+\frac{1}{a_n})-3$. If $a_{n+1} > a_n (n \in Z_{+})$. The range of real number $a$ is _____.

Given positive number $\alpha, \beta, \gamma, \delta$ satisfying $\alpha+\beta+\gamma+\delta=2\pi$, and $k=\frac{3\tan\alpha}{1+\sec \alpha}=\frac{4\tan \beta}{1+\sec \beta}=\frac{5\tan \gamma}{1+\sec \gamma}=\frac{6\tan\delta}{1+\sec\delta}$, then $k$= _____.

Let $A=\{1,2,\cdots,6\}$, function $f:A \rightarrow A$. Mark $p(f)=f(1)\cdots f(6)$. Then the number of functions that make $p(f)|36$ is _____.

If unit complex number $a, b$ satisfy $a\bar{b}+\bar{a}b = \sqrt{3}$, then $|a-b|=$ _____.

The right focus $F_1$ of the ellipse $\Gamma_1: \frac{x^2}{24}+\frac{y^2}{b^2}=1 (0<b<2\sqrt{6})$ coincides with the focus of the parabola $\Gamma_2: y^2=4px(p \in Z_{+})$. The line $l$ passing through the point $F_1$ with a positive integer slope intersects the ellipse $\Gamma_1$ at points A and B, and intersects the parabola $\Gamma_2$ at points C and D. If $13|AB|=\sqrt{6}|CD|$, then $b^2+p=$ _____.

Given that $p(x)$ is a quintic polynomial. If $x=0$ is a triple root of $p(x)+1=0$ and $x=1$ is a triple root of $p(x)-1=0$, then the coefficient of the $x^3$ term in the expression of $p(x)$ is _____.

Set $X=\{1,2,\cdots,10\}$, mapping $f:X\rightarrow X$ satisfy:\ (1) $f\circ f = I_x$, where, $f\circ f$ is a composite mapping, $I_x$ is an identity mapping on X.\ (2) $|f(i)-i|=2$, for any $i\in X$.\ Then the number of mapping $f$ is _____.

Given a integer coefficient polynomial of degree 2022 with leading coefficient is 1, how many roots can it possibly have in the interval (0,1) as maximum?

The system of equations $\left\{\begin{array}{l}x^{2} y+y^{2} z+z^{2}=0, \\ z^{3}+z^{2} y+z y^{3}+x^{2} y=\frac{1}{4}\left(x^{4}+y^{4}\right)\end{array}\right.$ has how many sets of real solution $(x,y,z)$ ?.

Set $x, y, z$ are real numbers, satisfying $x^{2}+y^{2}+z^{2}=1$. Then the maximum and minimum of $(x-y)(y-z)(x-z)$ are

If there are a total of 95 numbers, each of which can take any value in +1 or -1, and it is known that the sum of the pairwise products of these 95 numbers is positive, then what is the minimum possible value of this positive sum?

A monotonically increasing sequence of positive integers, starting from the third term, with each subsequent term being the sum of the preceding two terms. If its seventh term is 120, then its eighth term is _____.

Sequence $\left\{a_{n}\right\}$ satisfy $a_{0}=0, a_{1}=1$, and for any positive integer $\mathrm{n}$, we have $a_{2 n}=a_{n^{\prime}} a_{2 n+1}=a_{n}+1$, then $a_{2024}=$_____.

Non-negative real numbers $\mathrm{x}, \mathrm{y}, \mathrm{z}$ satisfy $4 x^{2}+4 y^{2}+z^{2}+2 z=3$, then the minimum of $5 \mathrm{x}+4 \mathrm{y}+3 \mathrm{z}$ is _____.

Real numbers $x, y$ satisfy $2 x-5 y \leq-6$ and $3 x+6 y \leq 25$, then the maximum value of $9 x+y$ is _____.

The sum of the maximum element and minimum element in set $\left\{\left.\frac{3}{a}+b \right\rvert 1 \leq a \leq b \leq 2\right\}$ is _____.

A monotonically increasing function $\mathrm{f}(\mathrm{x})$ defined on $R^{+}$ satisfies $\mathrm{f}\left(\mathrm{f}(\mathrm{x})+\frac{2}{\mathrm{x}}\right)=-1$ consistently holds within its domain, then $\mathrm{f}(1)=$_____.

Given set: $A=\left\{x+y \left\lvert \frac{x^{2}}{9}+y^{2}=1\right., x+y \in \mathbf{Z}_{+}\right\}, B=\{2 x+y \mid x, y \in A, x<y\}$, $C=\{2 x+y \mid x, y \in A, x>y\}$. Then the sum of all elements in $B \cap C$ is

Given a hyperbola $\Gamma: \frac{x^{2}}{7}-\frac{y^{2}}{5}=1$, a line $l: a x+b y+1=0$ intersects $\Gamma$ at point $A$. A tangent to $\Gamma$ drawn through point $A$ is perpendicular to the line $l$. Then $\frac{7}{a^{2}}-\frac{5}{b^{2}}=$_____.

In the Cartesian coordinate plane $x O y$, point $A(a, 0), B(0, b), C(0,4)$, moving point $D$ satisfies $|C D|=1$. If the maximum value of $|\overrightarrow{O A}+\overrightarrow{O B}+\overrightarrow{O D}|$ is 6, then the minimum value of $a^{2}+b^{2}$ is _____.

Given $n$ is positive integer, for $i=1,2, \cdots, n$, positive integer $a_{i}$ and positive even number $b_{i}$ satisfy $0<\frac{a_{i}}{b_{i}}<1$, and for any positive integers $i_{1}, i_{2}\left(1 \leq i_{1}<i_{2} \leq n\right), a_{i_{1}} \neq a_{i_{2}}$ and $b_{i_{1}} \neq b_{i_{2}}$ at least one holds true. If for any positive integer $n$ and all positive integers $a_{i}$ and positive even numbers $b_{i}$ satisfy the above conditions, we all have $\frac{\sum_{i=1}^{n} b_{i}}{n^{\frac{3}{2}}} \geq c$, then the maximum value of real number $c$ is _____.

In a cube $ABCD-A_1B_1C_1D_1$, $AA_1=1$, E, F are the midpoints of edges $CC_1, DD_1$, then the area of the cross-section obtained by the plane AEF intersecting the circumscribed sphere of the cube is _____.

In tetrahedron $ABCD$, triangle $ABC$ is an equilateral triangle, $\angle BCD=90^{\circ}$, $BC=CD=1$,$AC=\sqrt{3}$,$E$ and $F$ are the midpoints of edges $BD$ and $AC$ respectively. Then the cosine of the angle formed by lines $AE$ and $BF$ is _____.

Let P be a point inside triangle ABC, and 2$\overrightarrow{PA} = \overrightarrow{PB} + \overrightarrow{PC}=0$. If $\angle BAC = \frac{\pi}{3}, BC=2$, then the maximum value of $\overrightarrow{PB}\cdot\overrightarrow{PC}$ is _____.

A rectangular solid whose length, width, and height are all natural numbers, and the sum of all its edge lengths equals its volume, is called a "perfect rectangular solid. The maximum value of the volume of a perfect rectangular solid is _____.

In the convex quadrilateral $A B C D$ inscribed in a circle, if $\overrightarrow{A B}+3 \overrightarrow{B C}+2 \overrightarrow{C D}+4 \overrightarrow{D A}=0$, and $|\overrightarrow{A C}|=4$, then the maximum of $|\overrightarrow{A B}|+|\overrightarrow{B C}|$ is _____.

Given that the edge length of the cube $A B C D-A_{1} B_{1} C_{1} D_{1}$ is $1$, where, $E$ is the middle point of $A B$, $F$ is the middle point of $C C_{1}$. Then the distance from point $D$ to the plane passing through the three points $D_{1}, E, F$ is _____.

Given that the vertices of triangle $\triangle OAB$ are $O(0,0)$, $A(4,4\sqrt{3})$, and $B(8,0)$, with the incenter denoted as $I$, let $\Gamma$ be a circle passing through points $A$ and $B$, intersecting circle $\odot I$ at points $P$ and $Q$. If the tangents drawn through points $P$ and $Q$ are perpendicular, then the radius of circle $\Gamma$ is _____.

A person has some $2 \times 5 \times 8$ bricks and some $2 \times 3 \times 7$ bricks, as well as a $10 \times 11 \times 14$ box. All bricks and the box are rectangular prisms. He wants to pack all the bricks into the box so that the bricks can fill the entire box. The number of bricks he can fit into the box is _____ pieces.

Let $a$ be an acute angle not exceeding $45^\circ$. If $\cot 2a - \sqrt{3} = \sec a$, then $a =$ _____ degrees.

Known that there is a regular 200-gon $A_{1}A_{2} \ldots A_{200}$, connecting the diagonals $A_{i}A_{i+9}(\mathrm{i}=1,2, \ldots, 200)$, where $A_{i+200}=A_{i}(i=1,2, \ldots, 9)$. Then there are a total of _____ distinct intersection points inside the regular 200-gon.

The distance between the highest point of the ellipse obtained by counterclockwise rotating the ellipse $\frac{x^{2}}{2}+y^{2}=1$ about the origin by 45 degrees and the origin is:_____.

A convex $\mathrm{n}$-gon with interior angles of $\mathrm{n}$ degrees each, all integers, and all different. The degree measure of the largest interior angle is three times the degree measure of the smallest interior angle. The maximum value that $n$ can take is _____.

In triangle $\mathrm{ABC}$ with its incenter $\mathrm{I}$, if $3\overrightarrow{IA} + 4\overrightarrow{IB} + 5\overrightarrow{IC} = \overrightarrow{0}$, then the measure of angle $\mathrm{C}$ is _____.

Given the circle $x^2 + y^2 = 4$ and the point $\mathrm{P}(2,1)$, two mutually perpendicular lines are drawn through point $\mathrm{P}$, intersecting the circle at points $\mathrm{A}, \mathrm{B}$ and $\mathrm{C}, \mathrm{D}$ respectively. Point $\mathrm{A}$ lies inside the line segment $PB$, and point $\mathrm{D}$ lies inside the line segment $PC$. The maximum area of quadrilateral $ABCD$ is _____.

Given that the area of triangle $\mathrm{ABC}$ is 1, and $\mathrm{BC}=1$, when the product of the three altitudes of this triangle is maximized, $\sin A =$

If each face of a tetrahedron is not an isosceles triangle, then it has at least_____ distinct edge lengths.

In tetrahedron $ABCD$, where $AC=15$, $BD=18$, $E$ is the trisect point of $AD$ closer to $A$, $F$ is the trisect point of $BC$ closer to $C$, and $EF=14$. Then, the cosine value of the angle between edges $AC$ and $BD$ is_____.

Given that each face of the tetrahedron has edges of lengths $\sqrt{2}$, $\sqrt{3}$, and $2$, the volume of this tetrahedron is _____.

Given that the line $l$ intersects two parabolas $\Gamma_{1}: y^{2}=2px (p>0)$ and $\Gamma_{2}: y^{2}=4px$ at four distinct points $A\left(x_{1}, y_{1}\right)$, $B\left(x_{2}, y_{2}\right)$, $D\left(x_{3}, y_{3}\right)$, and $E\left(x_{4}, y_{4}\right)$, where $y_{4}<y_{2}<y_{1}<y_{3}$. Let $l$ intersect the $x$-axis at point $M$. Given that $AD = 6 BE$, then the value of $\frac{AM}{ME}$ is _____.

Given that $\triangle ABC$ is an acute-angled triangle, with $A$, $B$, and $C$ as its internal angles, the minimum value of $2 \cot A+3 \cot B+4 \cot C$ is _____.

There are 10 points in the plane, with no three points lying on the same line. Using these 10 points as vertices of triangles, such that any two triangles have at most one common vertex, we can form at most _____ triangles.

Given that quadrilateral $ABCD$ is a parallelogram, with the lengths of $AB$, $AC$, $AD$, and $BD$ being distinct integers, then the minimum perimeter of quadrilateral $ABCD$ is _____.

When the two ends of a strip of paper are glued together, forming a loop, it is called a circular ring. Cutting along the bisector of the paper strip will result in two circular rings. When a strip of paper is twisted 180 degrees and then the ends are glued together again, forming a loop, it is called a Mobius strip. Cutting along the bisector of the Mobius strip will result in a longer loop-like structure. If cut along the bisector of this longer loop-like structure, it will yield _____ loop-like structure again.

In the Cartesian coordinate system xOy, let $\Gamma_{1}$ be the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ and $\Gamma_{2}$ be the parabola $y^{2}=\frac{1}{2}ax$. They intersect at points A and B, and P is the rightmost point of $\Gamma_{1}$. If points O, A, P, and B are concyclic, then the eccentricity of $\Gamma_{1}$ is_____.

Given the circle $\Gamma: x^{2}+y^{2}=1$, points A and B are two points symmetric about the x-axis on the circle. M is any point on the circle $\Gamma$ distinct from A and B. If MA and MB intersect the x-axis at points P and Q respectively, then the product of the abscissas of P and Q is _____.

If three points are randomly chosen from the vertices of a regular 17-sided polygon, what is the probability that the chosen points form an acute-angled triangle?

A rook piece moves through each square of a $2023\times 2023$ grid paper once, each time moving only one square (i.e., from the current square to an adjacent square). If the squares are numbered from 1 to $n^2$ according to the order in which the rook piece reaches them, let M denote the maximum difference in numbers between adjacent squares. Then the minimum possible value of $M$ is?

Using the 24-hour clock, the probability of the sum of four digits at a certain moment being smaller than the sum of the four digits at 20:21 is _____.

A six-digit number $N=\overline{a_1a_2...a_{6}}$ composed of non-repeating digits from 1 to 6 satisfies the condition $|a_{k+1}-a_{k}| \neq 1, (k\in \{1,2, \cdots, 5\})$. Then the number of such six-digit numbers is _____.

Given $M=\{1,2, \ldots, 8\}, A, B$ are two distincet subsets of set $M$, satisfying

(1) The number of elements in set $A$ is fewer than the number of elements in set $B$.

(2) The smallest element in set $A$ is larger than the largest element in set $B$.

Then the total number of ordered pairs $(A, B)$ satisfying these conditions is _____.

Using six different colors to color each edge of the regular tetrahedron $ABCD$, each edge can only be colored with one color and edges sharing a vertex cannot have the same color. The probability that all edges have different colors is _____.

The king summons two wizards into the palace. He demands Wizard A to write down 100 positive real numbers on cards (allowing duplicates) without revealing them to Wizard B. Then, Wizard B must accurately write down all of these 100 positive real numbers. Otherwise, both wizards will be beheaded. Wizard A is allowed to provide a sequence of numbers to Wizard B, where each number is either one of the 100 positive real numbers or a sum of some of them. However, he cannot tell Wizard B which are the numbers on the cards and which are the sums of numbers on the cards. Ultimately, the king decides to pull off the same number of beards from both wizards based on the count of these numbers. Without the ability to communicate beforehand, the question is: How many beard pulls does each wizard need to endure at least to ensure their own survival?

Using $1 \times 1$, $2 \times 2$, and $3 \times 3$ tiles to cover a $23 \times 23$ floor (without overlapping or leaving gaps), what is the minimum number of $1 \times 1$ tiles needed? (Assuming each tile cannot be divided into smaller tiles).

In a number-guessing game, the host has prearranged a permutation of the numbers 1 to 100, and participants also need to provide a permutation of these 100 numbers. Interestingly, as long as the permutation provided by a participant has at least one number whose position matches that of the host's permutation, it is considered a successful guess. How many participants are needed to ensure that at least one person guesses correctly?

There is a stack of 52 face-down playing cards on the table. Mim takes 7 cards from the top of this stack, flips them over, and puts them back at the bottom, calling it one operation. The question is: how many operations are needed at least to make all the playing cards face-down again?

There are two segments of length $3n (0 \leq n \leq 1011)$. How many different shapes of triangles can be formed from these 2024 segments? (Congruent triangles are considered the same.)

Let $A$ and $B$ be two subsets of the set $\{1,2, \ldots, 20\}$, where $A \cap B = \varnothing$, and if $n \in A$, then $2n + 2 \in B$. Let $M(A)$ denote the sum of the elements of $A$. The maximum value of $M(A)$ is:

Alice and Bob are playing a game. They write down four expressions on four cards: $x+y$, $x-y$, $x^2+xy+y^2$, and $x^2-xy+y^2$. They place these four cards face down on the table, then randomly choose one card to reveal its expression. Alice can pick two of the four cards and hand the other two to Bob, then all four cards are revealed. Alice can assign a value (real number) to one of the variables $x$ or $y$ and inform Bob of which variable she has assigned and what value. Afterwards, Bob assigns a value (real number) to the other variable. Finally, they each calculate the product of the values on their two cards, and the person with the larger product wins. Who has a winning strategy? A. Alice B. Bob C. Neither of them has a winning strategy.

Find the smallest integer $k > 2$ such that any partition of $\{2,3,\ldots,k\}$ into two sets must contain at least one set containing $a$, $b$, and $c$ (which are allowed to be the same), satisfying $ab=c$.

On a plane, there are 2019 points. Drawing circles passing through these points, a drawing method is called "$k$-good" if and only if drawing $k$ circles divides the plane into several closed shapes, and there is no closed shape containing two points. Then, find the minimum value of $k$ , it ensures that no matter how these 2019 points are arranged, there always exists a $k$-good drawing method.

Zheng flips an unfair coin 5 times. If the probability of getting exactly 1 head is equal to the probability of getting exactly 2 heads and is nonzero, then the probability of getting exactly 3 heads is ____.

Let $a_{1}, a_{2}, \ldots, a_{6}$ be any permutation of $\{1,2, \ldots, 6\}$. If the sum of any three consecutive numbers cannot be divided by 3, then the number of such permutations is _____.

Given an integer $n > 2$. Now, there are $n$ people playing a game of "Passing Numbers". It is known that some people are friends (friendship is mutual), and each person has at least one friend. The rules of the game are as follows: each person first writes down a positive real number, and the $n$ positive real numbers written by everyone are all different; then, for each person, if he has $k$ friends, he divides the number he wrote by $k$, and tells all his friends the result obtained; finally, each person writes down the sum of all the numbers he hears. The question is: what is the minimum number of times that someone writes down different numbers?

A restaurant can offer 9 types of appetizers, 9 types of main courses, 9 types of desserts, and 9 types of wines. A company is having a dinner party at this restaurant, and each guest can choose one appetizer, one main course, one dessert, and one type of wine. It is known that any two people's choices of the four dishes are not completely identical, and it is also impossible to find four people on the spot who have three identical choices but differ pairwise in the fourth choice (for example, there are no 9 people who have the same appetizer, main course, and dessert, but differ pairwise in the wine). Then, at most how many guests can there be?

Alice and Bob are playing hide-and-seek. Initially, Bob selects a point $B$ inside a unit square (without informing Alice). Then, Alice sequentially selects points $P_0, P_1, \ldots, P_n$ on the plane. After each selection of a point $P_k (1 \leq k \leq n$, and at this point, Alice has not yet chosen the next point), Bob informs Alice which of the points $P_k$ and $P_{k-1}$ is closer to point $B$. After Alice selects $P_n$ and receives Bob's response, she chooses a final point $A$. If the distance between $A$ and $B$ does not exceed $\frac{1}{2020}$, Alice wins. Otherwise, Bob wins. When $n=18$, which of the following options is correct? A. Alice cannot guarantee victory. B. Alice can guarantee victory.

Anna, Carl take turns selecting numbers from the set $\{1,2, \cdots, p-1\}$ (where $p$ is a prime greater than 3). Anna goes first, and each number can only be selected once. Each number chosen by Anna is multiplied by the number Carl selects next. Carl wins if, after any round, the sum of all products computed so far is divisible by $p$. Anna wins if, after all numbers are chosen, Carl has not won. Which of the following options is correct?

A. Anna has a winning strategy. B. Carl has a winning strategy. C. Both players have no winning strategy.

Xiao Ming is playing a coin game with three doors. Each time he opens a door, it costs him 2 coins. After opening the first door, he can see the second door. Upon opening the second door, two equally likely options appear: either return to the outside of the first door or proceed to the third door. Upon opening the third door, three equally likely options appear: either return to the outside of the first door, stay in place and need to reopen the third door, or pass the game. If Xiao Ming wants to pass the game, on average, he needs to spend how many coins?

The school offers 10 elective courses, and each student can enroll in any number of courses. The director selects $k$ students, where although each student's combination of courses is different, any two students have at least one course in common. At this point, it is found that any student outside these $k$ students cannot be classmates with these $k$ students regardless of how they enroll (having one course in common is enough to be classmates). Then $k=$ _____.

Let $n$ be a positive integer. Now, a frog starts jumping from the origin of the number line and makes $2^{n}-1$ jumps. The process satisfies the following conditions:

(1) The frog will jump to each point in the set $\left\{1,2,3, \cdots, 2^{n}-1\right\}$ exactly once, without missing any.

(2) Each time the frog jumps, it can choose a step length from the set $\left\{2^{0}, 2^{1}, 2^{2}, \cdots\right\}$, and it can jump either left or right.

Let $T$ be the reciprocal sum of the step lengths of the frog. When $n=2024$, the minimum value of $T$ is _____.

Given that there are 66 dwarves with a total of 111 hats, each hat belonging to a specific dwarf, and each hat is dyed in one of 66 colors. During the holiday, each dwarf wears his own hat. It is known that for any holiday, the colors of the hats worn by all dwarves are different. For any two holidays, there is at least one dwarf who wears hats of different colors on the two occasions. The question is: how many holidays can the dwarves celebrate at most?

There are $n$ cards, each labeled with the numbers $1 , 2, \cdots, n$. These $n$ cards are distributed among 17 people, with each person receiving at least 1 card. Then, there is always one person who receives cards with numbers $x$ and $y$, where $x>y$, and $118x \leqslant 119y$. The smallest positive integer $n$ that satisfies this condition is _____.