Given a set $I=\{(x_1,x_2,x_3,x_4)|x_i\in\{1,2,\cdots,11\}\}$. $A\subseteq I$, satisfying that for any $(x_1,x_2,x_3,x_4),(y_1,y_2,y_3,y_4)\in A$, there exists $i,j(1\leq i<j\leq4)$, $(x_i-x_j)(y_i-y_j)<0$. Find the maximum value of $|A|$.

Prove that every prime of the form $4k+1$ is the hypotenuse of a rectangular triangle with integer sides.

The letters of the word ‘GAUSS’ and the digits in the number ‘1998’ are each cycled separately and then numbered as shown. 1. AUSSG 9981 2. USSGA 9819 3. SSGAU 8199 etc. If the pattern continues in this way, what number will appear in front of GAUSS 1998? $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 9 \qquad \textbf{(D)}\ 16 \qquad \textbf{(E)}\ 20$

Lucy starts by writing $s$ integer-valued $2022$-tuples on a blackboard. After doing that, she can take any two (not necessarily distinct) tuples $\mathbf{v}=(v_1,\ldots,v_{2022})$ and $\mathbf{w}=(w_1,\ldots,w_{2022})$ that she has already written, and apply one of the following operations to obtain a new tuple: \begin{align*} \mathbf{v}+\mathbf{w}&=(v_1+w_1,\ldots,v_{2022}+w_{2022}) \\ \mathbf{v} \lor \mathbf{w}&=(\max(v_1,w_1),\ldots,\max(v_{2022},w_{2022})) \end{align*} and then write this tuple on the blackboard. It turns out that, in this way, Lucy can write any integer-valued $2022$-tuple on the blackboard after finitely many steps. What is the smallest possible number $s$ of tuples that she initially wrote?

Compute the number of positive integers $n\le1000$ such that $\text{lcm}(n,9)$ is a perfect square. (Recall that $\text{lcm}$ denotes the least common multiple.)

Let $q$ be a fixed odd prime. A prime $p$ is said to be [i]orange [/i] if for every integer $a$ there exists an integer $r$ such that $r^q \equiv a$ (mod $p$). Prove that there are infinitely many [i]orange [/i] primes.

Let $S$ be the Sierpiński triangle. What can we say about the Hausdorff dimension of the elevation sets $f^{-1}(y)$ for typical continuous real functions defined on $S$? (A property is satisfied for typical continuous real functions on $S$ if the set of functions not having this property is of the first Baire category in the metric space of continuous $S\rightarrow\mathbb{R}$ functions with the supremum norm.) (translated by Miklós Maróti)

Evaluate $ \int_{\minus{}\frac{1}{2}}^{\frac{1}{2}} \frac{x}{\{(2x\plus{}1)\sqrt{x^2\minus{}x\plus{}1}\plus{}(2x\minus{}1)\sqrt{x^2\plus{}x\plus{}1}\}\sqrt{x^4\plus{}x^2\plus{}1}}\ dx$.

Let $S$ be the set of all real numbers $x$ such that $0 \le x \le 2016 \pi$ and $\sin x < 3 \sin(x/3)$. The set $S$ is the union of a finite number of disjoint intervals. Compute the total length of all these intervals.

Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. What is the least positive integer $n$ such that performing the sequence of transformations transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself? $\textbf{(A) } 359 \qquad \textbf{(B) } 360\qquad \textbf{(C) } 719 \qquad \textbf{(D) } 720 \qquad \textbf{(E) } 721$

On side $AB$ of a scalene triangle $ABC$ there are points $M$, $N$ such that $AN=AC$ and $BM=BC$. The line parallel to $BC$ through $M$ and the line parallel to $AC$ through $N$ intersect at $S$. Prove that $\measuredangle{CSM} = \measuredangle{CSN}$.

We call polynomial $S(x)\in\mathbb{R}[x]$ sadeh whenever it's divisible by $x$ but not divisible by $x^2$. For the polynomial $P(x)\in\mathbb{R}[x]$ we know that there exists a sadeh polynomial $Q(x)$ such that $P(Q(x))-Q(2x)$ is divisible by $x^2$. Prove that there exists sadeh polynomial $R(x)$ such that $P(R(x))-R(2x)$ is divisible by $x^{1401}$.

a) Prove that when you divide any prime number by $30$, the remainder is either $1$ or is a prime number! b) Does this also apply when dividing a prime number by $60$? Justify your answer!

On the plane there is a finite set of points $Z$ with the property that no two distances of the points of the set $Z$ are equal. We connect the points $ A, B $ belonging to $ Z $ if and only if $ A $ is the point closest to $ B $ or $ B $ is the point closest to $ A $. Prove that no point in the set $Z$ will be connected to more than five others.

$n$ is a natural number larger than $3$ and denote all positive coprime numbers with $n$ as $1= b_1 < b_2 < \cdots b_k$. For a positive integer $m$ which is larger than $3$ and is coprime with $n$, let $A$ be the set of tuples $(a_1,a_2, \cdots a_k)$ satisfying the condition. $$\textbf{Condition}: \text{For all integers } i, 0 \le a_i < m \text{ and } a_1b_1 + a_2b_2 + \cdots a_kb_k \text{ is a mutiple of } n$$ For elements of $A$, show that the difference of number of elements such that $a_1 = 1$ and the number of elements such that $a_2 = 2$ maximum $1$

Let ABCD be a square inscribed in a circle k and P be an arbitrary point of that circle. Prove that at least one of the numbers PA, PB, PC and PD is not rational.

Suppose the real number $\lambda \in \left( 0,1\right),$ and let $n$ be a positive integer. Prove that the modulus of all the roots of the polynomial $$f\left ( x \right )=\sum_{k=0}^{n}\binom{n}{k}\lambda^{k\left ( n-k \right )}x^{k}$$ are $1.$

A finite sequence of decimal digits from $\{0,1,\cdots, 9\}$ is said to be [i]common[/i] if for each sufficiently large positive integer $n$, there exists a positive integer $m$ such that the expansion of $n$ in base $m$ ends with this sequence of digits. For example, $0$ is common because for any large $n$, the expansion of $n$ in base $n$ is $10$, whereas $00$ is not common because for any squarefree $n$, the expansion of $n$ in any base cannot end with $00$. Determine all common sequences. [i]Proposed by Wong Jer Ren[/i]

Consider a sequence denoted by $F_n$ of non-square numbers . $F_1=2$,$F_2=3$,$F_3=5$ and so on . Now , if $m^2\leq F_n<(m+1)^2$ . Then prove that $m$ is the integer closest to $\sqrt{n}$.

In a parallelogram $ABCD$, let $M$ be the point on the $BC$ side such that $MC = 2BM$ and let $N$ be the point of side $CD$ such that $NC = 2DN$. If the distance from point $B$ to the line $AM$ is $3$, calculate the distance from point $N$ to the line $AM$.

The excircle of right-angled triangle $ABC$ ($\angle B =90^o$) touches side $BC$ at point $A_1$ and touches line $AC$ in point $A_2$. Line $A_1A_2$ meets the incircle of $ABC$ for the first time at point $A'$, point $C'$ is defined similarly. Prove that $AC||A'C'$.

Let $a,b,c$ be the sidelengths of a triangle. Prove that $$2<\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}<\sqrt{6}.$$

Prove that there are no integers $x, y, z$ satisfying the equation $$x^2+y^2-z^2=xyz-2.$$ [i]Proposed by Navid Safaei[/i]

Find all pairs of distinct primes $(p,q)$ such that $p$ and $q$ are both prime factors of $p^3+q^2+1$, and are the only such prime factors. [i]Proposed by Takeda Shigenori[/i]

Let $ f(x)$ be a monic polynomial of degree $ 1991$ with integer coefficients. Define $ g(x) \equal{} f^2(x) \minus{} 9.$ Show that the number of distinct integer solutions of $ g(x) \equal{} 0$ cannot exceed $ 1995.$