Navvye/Olympiad-Problems-CSV huggingface dataset:
1990 China National Olympiad
Let $x$ be a natural number. We call $\{x_0,x_1,\dots ,x_l\}$ a factor link of $x$ if the sequence $\{x_0,x_1,\dots ,x_l\}$ satisfies the following conditions:

(1) $x_0=1, x_l=x$;

(2) $x_{i-1}<x_i, x_{i-1}|x_i, i=1,2,\dots,l$ .

Meanwhile, we define $l$ as the length of the factor link  $\{x_0,x_1,\dots ,x_l\}$. Denote by $L(x)$ and $R(x)$ the length and the number of the longest factor link of $x$ respectively. For $x=5^k\times 31^m\times 1990^n$, where $k,m,n$ are natural numbers, find the value of $L(x)$ and $R(x)$.
1990 China National Olympiad
A function $f(x)$ defined for $x\ge 0$ satisfies the following conditions:

i. for $x,y\ge 0$, $f(x)f(y)\le x^2f(y/2)+y^2f(x/2)$;

ii. there exists a constant $M$($M>0$), such that $|f(x)|\le M$ when $0\le x\le 1$.

Prove that $f(x)\le x^2$.
1990 China National Olympiad
Given a positive integer number $a$ and two real numbers $A$ and $B$, find a necessary and sufficient condition on $A$ and $B$ for the following system of equations to have integer solution:

$$ \left\{\begin{array}{cc} x^2+y^2+z^2=(Ba)^2\\ x^2(Ax^2+By^2)+y^2(Ay^2+Bz^2)+z^2(Az^2+Bx^2)=\dfrac{1}{4}(2A+B)(Ba)^4\end{array}\right. $$
1990 China National Olympiad
Given a finite set $X$, let $f$ be a rule such that $f$ maps every even-element-subset $E$ of $X$ (i.e. $E \subseteq X$, $|E|$ is even) into a real number $f(E)$. Suppose that $f$ satisfies the following conditions:

(I) there exists an even-element-subset $D$ of $X$ such that $f(D)>1990$;

(II) for any two disjoint even-element-subsets $A,B$ of $X$, equation $f(A\cup B)=f(A)+f(B)-1990$ holds.

Prove that there exist two subsets $P,Q$ of $X$ satisfying:

(1) $P\cap Q=\emptyset$, $P\cup Q=X$;

(2) for any non-even-element-subset $S$ of $P$ (i.e. $S\subseteq P$, $|S|$ is odd), we have $f(S)>1990$;

(3) for any even-element-subset $T$ of $Q$, we have $f(T)\le 1990$.
1990 China National Olympiad
A convex $n$-gon and its $n-3$ diagonals which have no common point inside the polygon form a subdivision graph. Show that if and only if $3|n$, there exists a subdivision graph that can be drawn in one closed stroke. (i.e. start from a certain vertex, get through every edges and diagonals exactly one time, finally back to the starting vertex.)
1989 China National Olympiad
We are given two point sets $A$ and $B$ which are both composed of finite disjoint arcs on the unit circle. Moreover, the length of each arc in $B$ is equal to $\dfrac{\pi}{m}$ ($m \in \mathbb{N}$). We denote by $A^j$ the set obtained by a counterclockwise rotation of $A$ about the center of the unit circle for $\dfrac{j\pi}{m}$ ($j=1,2,3,\dots$). Show that there exists a natural number $k$ such that $l(A^k\cap B)\ge \dfrac{1}{2\pi}l(A)l(B)$.(Here $l(X)$ denotes the sum of lengths of all disjoint arcs in the point set $X$)
1989 China National Olympiad
Let $x_1, x_2, \dots ,x_n$ ($n\ge 2$) be positive real numbers satisfying $\sum^{n}_{i=1}x_i=1$. Prove that:$$\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}.$$
1989 China National Olympiad
Let $S$ be the unit circle in the complex plane (i.e. the set of all complex numbers with their moduli equal to $1$).

We define function $f:S\rightarrow S$ as follow: $\forall z\in S$,

$ f^{(1)}(z)=f(z), f^{(2)}(z)=f(f(z)), \dots,$

$f^{(k)}(z)=f(f^{(k-1)}(z)) (k>1,k\in \mathbb{N}), \dots$

We call $c$ an $n$-period-point of $f$ if $c$ ($c\in S$) and $n$ ($n\in\mathbb{N}$) satisfy:

$f^{(1)}(c) \not=c, f^{(2)}(c) \not=c, f^{(3)}(c) \not=c, \dots, f^{(n-1)}(c) \not=c, f^{(n)}(c)=c$.

Suppose that $f(z)=z^m$ ($z\in S; m>1, m\in \mathbb{N}$), find the number of $1989$-period-point of $f$.
1989 China National Olympiad
Given a triangle $ABC$, points $D,E,F$ lie on sides $BC,CA,AB$ respectively. Moreover, the radii of incircles of $\triangle AEF, \triangle BFD, \triangle CDE$ are equal to $r$. Denote by $r_0$ and $R$ the radii of incircles of $\triangle DEF$ and $\triangle ABC$ respectively. Prove that $r+r_0=R$.
1989 China National Olympiad
Given $1989$ points in the space, any three of them are not collinear. We divide these points into $30$ groups such that the numbers of points in these groups are different from each other. Consider those triangles whose vertices are points belong to three different groups among the $30$. Determine the numbers of points of each group such that the number of such triangles attains a maximum.
1989 China National Olympiad
Find all functions $f:(1,+\infty) \rightarrow (1,+\infty)$ that satisfy the following condition:

for arbitrary $x,y>1$ and $u,v>0$, inequality $f(x^uy^v)\le f(x)^{\dfrac{1}{4u}}f(y)^{\dfrac{1}{4v}}$ holds.
1988 China National Olympiad
Let $r_1,r_2,\dots ,r_n$ be real numbers. Given $n$ reals $a_1,a_2,\dots ,a_n$ that are not all equal to $0$, suppose that inequality

$$r_1(x_1-a_1)+ r_2(x_2-a_2)+\dots + r_n(x_n-a_n)\leq\sqrt{x_1^2+ x_2^2+\dots + x_n^2}-\sqrt{a_1^2+a_2^2+\dots +a_n^2}$$holds for arbitrary reals $x_1,x_2,\dots ,x_n$. Find the values of $r_1,r_2,\dots ,r_n$.
1988 China National Olympiad
Given two circles $C_1,C_2$ with common center, the radius of $C_2$ is twice the radius of $C_1$. Quadrilateral $A_1A_2A_3A_4$ is inscribed in $C_1$. The extension of $A_4A_1$ meets $C_2$ at $B_1$; the extension of $A_1A_2$ meets $C_2$ at $B_2$; the extension of $A_2A_3$ meets $C_2$ at $B_3$; the extension of $A_3A_4$ meets $C_2$ at $B_4$. Prove that $P(B_1B_2B_3B_4)\ge 2P(A_1A_2A_3A_4)$, and in what case the equality holds? ($P(X)$ denotes the perimeter of quadrilateral $X$)
1988 China National Olympiad
Given a finite sequence of real numbers $a_1,a_2,\dots ,a_n$ ($\ast$), we call a segment $a_k,\dots ,a_{k+l-1}$ of the sequence ($\ast$) a “long”(Chinese dragon) and $a_k$ “head” of the “long” if the arithmetic mean of $a_k,\dots ,a_{k+l-1}$ is greater than $1988$. (especially if a single item $a_m>1988$, we still regard $a_m$ as a “long”). Suppose that there is at least one “long” among the sequence ($\ast$), show that the arithmetic mean of all those items of sequence ($\ast$) that could be “head” of a certain “long” individually is greater than $1988$.
1988 China National Olympiad
(1) Let $a,b,c$ be positive real numbers satisfying $(a^2+b^2+c^2)^2>2(a^4+b^4+c^4)$. Prove that $a,b,c$ can be the lengths of three sides of a triangle respectively.

(2) Let $a_1,a_2,\dots ,a_n$ be $n$ ($n>3$) positive real numbers satisfying $(a_1^2+a_2^2+\dots +a_n^2)^2>(n-1)(a_1^4+ a_2^4+\dots +a_n^4)$. Prove that any three of $a_1,a_2,\dots ,a_n$ can be the lengths of three sides of a triangle respectively.
1988 China National Olympiad
Given three tetrahedrons $A_iB_i C_i D_i$ ($i=1,2,3$), planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) are drawn through $B_i ,C_i ,D_i$ respectively, and they are perpendicular to edges $A_i B_i, A_i C_i, A_i D_i$ ($i=1,2,3$) respectively. Suppose that all nine planes $\alpha _i,\beta _i,\gamma _i$ ($i=1,2,3$) meet at a point $E$, and points $A_1,A_2,A_3$ lie on line $l$. Determine the intersection (shape and position) of the circumscribed spheres of the three tetrahedrons.
1988 China National Olympiad
Let $n$ ($n\ge 3$) be a natural number. Denote by $f(n)$ the least natural number by which $n$ is not divisible (e.g. $f(12)=5$). If $f(n)\ge 3$, we may have $f(f(n))$ in the same way. Similarly, if $f(f(n))\ge 3$, we may have $f(f(f(n)))$, and so on. If $\underbrace{f(f(\dots f}_{k\text{ times}}(n)\dots ))=2$, we call $k$ the “length” of $n$ (also we denote by $l_n$ the “length” of $n$). For arbitrary natural number $n$ ($n\ge 3$), find $l_n$ with proof.
1987 China National Olympiad
Let $n$ be a natural number. Prove that a necessary and sufficient condition for the equation $z^{n+1}-z^n-1=0$ to have a complex root whose modulus is equal to $1$ is that $n+2$ is divisible by $6$.
1987 China National Olympiad
We are given an equilateral triangle ABC with the length of its side equal to $1$. There are $n-1$ points on each side of the triangle $ABC$ that equally divide the side into $n$ segments. We draw all possible lines that pass through any two of all those $3(n-1)$ points such that they are parallel to one of three sides of triangle $ABC$. All such lines divide triangle $ABC$ into some lesser triangles whose vertices are called nodes. We assign a real number for each node such that the following conditions are satisfied:

(I) real numbers $a,b,c$ are assigned to $A,B,C$ respectively;

(II) for any rhombus that is consisted of two lesser triangles that share a common side, the sum of the numbers of vertices on its one diagonal is equal to that of vertices on the other diagonal.

1) Find the minimum distance between the node with the maximal number to the node with the minimal number;

2) Denote by $S$ the sum of the numbers of all nodes, find $S$.
1987 China National Olympiad
Some players participate in a competition. Suppose that each player plays one game against every other player and there is no draw game in the competition. Player $A$ is regarded as an excellent player if the following condition is satisfied: for any other player $B$, either $A$ beats $B$ or there exists another player $C$ such that $C$ beats $B$ and $A$ beats $C$. It is known that there is only one excellent player in the end, prove that this player beats all other players.
1987 China National Olympiad
Five points are arbitrarily put inside a given equilateral triangle $ABC$ whose area is equal to $1$. Show that we can draw three equilateral triangles within triangle $ABC$ such that the following conditions are all satisfied:

i) the five points are covered by the three equilateral triangles;

ii) any side of the three equilateral triangles is parallel to a certain side of the triangle $ABC$;

iii) the sum of the areas of the three equilateral triangles is not larger than $0.64$.
1987 China National Olympiad
Let $A_1A_2A_3A_4$ be a tetrahedron. We construct four mutually tangent spheres $S_1,S_2,S_3,S_4$ with centers $A_1,A_2,A_3,A_4$ respectively. Suppose that there exists a point $Q$ such that we can construct two spheres centered at $Q$ satisfying the following conditions:

i) One sphere with radius $r$ is tangent to $S_1,S_2,S_3,S_4$;

ii) One sphere with radius $R$ is tangent to every edges of tetrahedron $A_1A_2A_3A_4$.

Prove that $A_1A_2A_3A_4$ is a regular tetrahedron.
1987 China National Olympiad
Sum of $m$ pairwise different positive even numbers and $n$ pairwise different positive odd numbers is equal to $1987$. Find, with proof, the maximum value of $3m+4n$.
1986 China National Olympiad
We are given $n$ reals $a_1,a_2,\cdots , a_n$ such that the sum of any two of them is non-negative. Prove that the following statement and its converse are both true: if $n$ non-negative reals $x_1,x_2,\cdots ,x_n$ satisfy $x_1+x_2+\cdots +x_n=1$, then the inequality $a_1x_1+a_2x_2+\cdots +a_nx_n\ge a_1x^2_1+ a_2x^2_2+\cdots + a_nx^2_n$ holds.
1986 China National Olympiad
In $\triangle ABC$, the length of altitude $AD$ is $12$, and the bisector $AE$ of $\angle A$ is $13$. Denote by $m$ the length of median $AF$. Find the range of $m$ when $\angle A$ is acute, orthogonal and obtuse respectively.
1986 China National Olympiad
Let $Z_1,Z_2,\cdots ,Z_n$ be complex numbers satisfying $|Z_1|+|Z_2|+\cdots +|Z_n|=1$. Show that there exist some among the $n$ complex numbers such that the modulus of the sum of these complex numbers is not less than $1/6$.
1986 China National Olympiad
Given a $\triangle ABC$ with its area equal to $1$, suppose that the vertices of quadrilateral $P_1P_2P_3P_4$ all lie on the sides of $\triangle ABC$. Show that among the four triangles $\triangle P_1P_2P_3, \triangle  P_1P_2P_4, \triangle P_1P_3P_4, \triangle P_2P_3P_4$ there is at least one whose area is not larger than $1/4$.
1986 China National Olympiad
Given a sequence $1,1,2,2,3,3,\ldots,1986,1986$, determine, with proof, if we can rearrange the sequence

so that for any integer $1\le k \le 1986$ there are exactly $k$ numbers between the two “$k$”s.
1986 China National Olympiad
Suppose that each point on the plane is colored either white or black. Show that there exists an equilateral triangle with the side length equal to $1$ or $\sqrt{3}$ whose three vertices are in the same color.
2020 China Northern MO
The function $f(x)=x^2+ \sin x$ and the sequence of positive numbers $\{ a_n \}$ satisfy $a_1=1$, $f(a_n)=a_{n-1}$, where $n \geq 2$. Prove that there exists a positive integer $n$ such that $a_1+a_2+ \dots + a_n > 2020$.
2020 China Northern MO
In $\triangle ABC$, $AB>AC$. Let $O$ and $I$ be the circumcenter and incenter respectively. Prove that if $\angle AIO = 30^{\circ}$, then $\angle ABC = 60^{\circ}$.
2020 China Northern MO
A set of $k$ integers is said to be a complete residue system modulo $k$ if no two of its elements are congruent modulo $k$. Find all positive integers $m$ so that there are infinitely many positive integers $n$ wherein $\{ 1^n,2^n, \dots , m^n \}$ is a complete residue system modulo $m$.
2020 China Northern MO
Two students $A$ and $B$ play a game on a $20 \text{ x } 20$ chessboard. It is known that two squares are said to be adjacent if the two squares have a common side. At the beginning, there is a chess piece in a certain square of the chessboard. Given that $A$ will be the first one to move the chess piece, $A$ and $B$ will alternately move this chess piece to an adjacent square. Also, the common side of any pair of adjacent squares can only be passed once. If the opponent cannot move anymore, then he will be declared the winner (to clarify since the wording wasn’t that good, you lose if you can’t move). Who among $A$ and $B$ has a winning strategy? Justify your claim.
2020 China Northern MO
Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.
2020 China Northern MO
For all positive real numbers $a,b,c$, prove that

$$\frac{a^3+b^3}{ \sqrt{a^2-ab+b^2} } + \frac{b^3+c^3}{ \sqrt{b^2-bc+c^2} } + \frac{c^3+a^3}{ \sqrt{c^2-ca+a^2} } \geq 2(a^2+b^2+c^2)$$
2020 China Northern MO
Given $a,b,c \in \mathbb{R}$ satisfying $a+b+c=a^2+b^2+c^2=1$, show that $\frac{-1}{4} \leq ab \leq \frac{4}{9}$.
2020 China Northern MO
Are there infinitely many positive integers $n$ such that $19|1+2^n+3^n+4^n$? Justify your claim.
2020 China Northern MO
In $\triangle ABC$, $\angle BAC = 60^{\circ}$, point $D$ lies on side $BC$, $O_1$ and $O_2$ are the centers of the circumcircles of $\triangle ABD$ and $\triangle ACD$, respectively. Lines $BO_1$ and $CO_2$ intersect at point $P$. If $I$ is the incenter of $\triangle ABC$ and $H$ is the orthocenter of $\triangle PBC$, then prove that the four points $B,C,I,H$ are on the same circle.
2020 China Northern MO
It is known that subsets $A_1,A_2, \cdots , A_n$ of set $I=\{1,2,\cdots ,101\}$ satisfy the following condition

$$\text{For any } i,j \text{ } (1 \leq i < j \leq n) \text{, there exists } a,b \in A_i \cap A_j \text{ so that } (a,b)=1$$Determine the maximum positive integer $n$.



*$(a,b)$ means $\gcd (a,b)$
2019 China Northern MO
Find all positive intengers $x,y$, satisfying:

$$3^x+x^4=y!+2019.$$
2019 China Northern MO
Two circles $O_1$ and $O_2$ intersect at $A,B$. Diameter $AC$ of $\odot O_1$ intersects $\odot O_2$ at $E$, Diameter $AD$ of $\odot O_2$ intersects $\odot O_1$ at $F$. $CF$ intersects $O_2$ at $H$, $DE$ intersects $O_1$ at $G,H$. $GH\cap O_1=P$. Prove that $PH=PK$.
2019 China Northern MO
$n(n\geq2)$ is a given intenger, and $a_1,a_2,...,a_n$ are real numbers. For any $i=1,2,\cdots ,n$,

$$a_i\neq -1,a_{i+2}=\frac{a_i^2+a_i}{a_{i+1}+1}.$$Prove: $a_1=a_2=\cdots=a_n$.

(Note: $a_{n+1}=a_1,a_{n+2}=a_2$.)
2019 China Northern MO
A manager of a company has 8 workers. One day, he holds a few meetings.

(1)Each meeting lasts 1 hour, no break between two meetings.

(2)Three workers attend each meeting.

(3)Any two workers have attended at least one common meeting.

(4)Any worker cannot leave until he finishes all his meetings.

Then, how long does the worker who works the longest work at least?
2019 China Northern MO
Two circles $O_1$ and $O_2$ intersect at $A,B$. Bisector of outer angle $\angle O_1AO_2$ intersects $O_1$ at $C$, $O_2$ at $D$. $P$ is a point on $\odot(BCD)$, $CP\cap O_1=E,DP\cap O_2=F$. Prove that $PE=PF$.
2019 China Northern MO
For nonnegative real numbers $a,b,c,x,y,z$, if$a+b+c=x+y+z=1$, find the maximum value of $(a-x^2)(b-y^2)(c-z^2)$.
2019 China Northern MO
There are $n$ cities in Qingqiu Country. The distance between any two cities are different. The king of the country plans to number the cities and set up two-way air lines in such ways:

The first time, set up a two-way air line between city 1 and the city nearest to it.

The second time, set up a two-way air line between city 2 and the city second nearest to it.

...

The $n-1$th time, set up a two-way air line between city $n-1$ and the city farthest to it.

Prove: The king can number the cities in a proper way so that he can go to any other city from any city by plane.
2019 China Northern MO
For positive intenger $n$, define $f(n)$: the smallest positive intenger that does not divide $n$.

Consider sequence $(a_n): a_1=a_2=1, a_n=a_{f(n)}+1(n\geq3)$.

For example, $a_3=a_2+1=2,a_4=a_3+1=3$.

(a) Prove that there exists a positive intenger $C$, for any positive intenger $n$, $a_n\leq C$.

(b) Are there positive intengers $M$ and $T$, satisfying that for any positive intenger $n\geq M$, $a_n=a_{n+T}$.
2018 China Northern MO
In triangle $ABC$, let the circumcenter, incenter, and orthocenter be $O$, $I$, and $H$ respectively. Segments $AO$, $AI$, and $AH$ intersect the circumcircle of triangle $ABC$ at $D$, $E$, and $F$. $CD$ intersects $AE$ at $M$ and $CE$ intersects $AF$ at $N$. Prove that $MN$ is parallel to $BC$.
2018 China Northern MO
Let $a$,$b$,$c$ be nonnegative reals such that

$$a^2+b^2+c^2+ab+\frac{2}{3}ac+\frac{4}{3}bc=1$$Find the maximum and minimum value of $a+b+c$.
2018 China Northern MO
Let $p$ be a prime such that $3|p+1$. Show that $p|a-b$ if and only if $p|a^3-b^3$
2018 China Northern MO
In each square of a $4$ by $4$ grid, you put either a $+1$ or a $-1$. If any 2 rows and 2 columns are deleted, the sum of the remaining 4 numbers is nonnegative. What is the minimum number of $+1$'s needed to be placed to be able to satisfy the conditions
2018 China Northern MO
A right triangle has the property that it's sides are pairwise relatively prime positive integers and that the ratio of it's area to it's perimeter is a perfect square. Find the minimum possible area of this triangle.
2018 China Northern MO
Let $H$ be the orthocenter of triangle $ABC$. Let $D$ and $E$ be points on $AB$ and $AC$ such that $DE$ is parallel to $CH$. If the circumcircle of triangle $BDH$ passes through $M$, the midpoint of $DE$, then prove that $\angle ABM=\angle ACM$
2018 China Northern MO
If $a$,$b$,$c$ are positive reals, prove that

$$\frac{a+bc}{a+a^2}+\frac{b+ca}{b+b^2}+\frac{c+ab}{c+c^2} \geq 3$$
2018 China Northern MO
2 players A and B play the following game with A going first: On each player's turn, he puts a number from 1 to 99 among 99 equally spaced points on a circle (numbers cannot be repeated), and the player who completes 3 consecutive numbers that forms an arithmetic sequence around the circle wins the game. Who has the winning strategy? Explain your reasoning.
2018 China Northern MO
Let $p$ be a prime. We say $p$ is good if and only if for any positive integer $a,b,$ such that $$a\equiv b (\textup{mod}p)\Leftrightarrow a^3\equiv b^3 (\textup{mod}p).$$Prove that

(1)There are infinite primes $p$ which are good;

(2)There are infinite primes $p$ which are not good.
2018 China Northern MO
$A,B,C,D,E$ lie on $\odot O$ in that order,and $$BD \cap CE=F,CE \cap AD=G,AD \cap BE=H,BE \cap AC=I,AC \cap BD=J.$$Prove that $\frac{FG}{CE}=\frac{GH}{DA}=\frac{HI}{BE}=\frac{IJ}{AC}=\frac{JF}{BD}$ when and only when $F,G,H,I,J$ are concyclic.
2018 China Northern MO
For $n(n\geq3)$ positive intengers $a_1,a_2,\cdots,a_n$. Put the numbers on a circle. In each operation, calculate difference between two adjacent numbers and take its absolute value. Put the $n$ numbers we get on another ciecle (do not change their order). Find all $n$, satisfying that no matter how $a_1,a_2,\cdots,a_n$ are given, all numbers on the circle are equal after limited operations.
2018 China Northern MO
For $a_1 = 3$, define the sequence $a_1, a_2, a_3, \ldots$ for $n \geq 1$ as $$na_{n+1}=2(n+1)a_n-n-2.$$Prove that for any odd prime $p$, there exist positive integer $m,$ such that $p|a_m$ and $p|a_{m+1}.$
2018 China Northern MO
Prove that there exist infinite positive integer $n,$ such that $2018 | \left( 1+2^n+3^n+4^n \right).$
2017 China Northern MO
A sequence \(\{a_n\}\) is defined as follows: \(a_1 = 1\), \(a_2 = \frac{1}{3}\), and for all \(n \geq 1,\) \(\frac{(1+a_n)(1+a_{n+2})}{(1+a_n+1)^2} = \frac{a_na_{n+2}}{a_{n+1}^2}\).



Prove that, for all \(n \geq 1\), \(a_1 + a_2 + ... + a_n < \frac{34}{21}\).
2017 China Northern MO
Prove that there exist infinitely many integers \(n\) which satisfy \(2017^2 | 1^n + 2^n + ... + 2017^n\).
2017 China Northern MO
Let \(D\) be the midpoint of side \(BC\) of triangle \(ABC\). Let \(E, F\) be points on sides \(AB, AC\) respectively such that \(DE = DF\). Prove that \(AE + AF = BE + CF \iff \angle EDF = \angle BAC\).
2017 China Northern MO
Let \(Q\) be a set of permutations of \(1,2,...,100\) such that for all \(1\leq a,b \leq 100\), \(a\) can be found to the left of \(b\) and adjacent to \(b\) in at most one permutation in \(Q\). Find the largest possible number of elements in \(Q\).
2017 China Northern MO
Triangle \(ABC\) has \(AB > AC\) and \(\angle A = 60^\circ \). Let \(M\) be the midpoint of \(BC\), \(N\) be the point on segment \(AB\) such that \(\angle BNM = 30^\circ\). Let \(D,E\) be points on \(AB, AC\) respectively. Let \(F, G, H\) be the midpoints of \(BE, CD, DE\) respectively. Let \(O\) be the circumcenter of triangle \(FGH\). Prove that \(O\) lies on line \(MN\).
2017 China Northern MO
Find all integers \(n\) such that there exists a concave pentagon which can be dissected into \(n\) congruent triangles.
2017 China Northern MO
Let \(S(n)\) denote the sum of the digits of the base-10 representation of an natural number \(n\). For example. \(S(2017) = 2+0+1+7 = 10\). Prove that for all primes \(p\), there exists infinitely many \(n\) which satisfy \(S(n) \equiv n \mod p\).
2017 China Northern MO
Let \(n>1\) be an integer, and let \(x_1, x_2, ..., x_n\) be real numbers satisfying \(x_1, x_2, ..., x_n \in [0,n]\) with \(x_1x_2...x_n = (n-x_1)(n-x_2)...(n-x_n)\). Find the maximum value of \(y = x_1 + x_2 + ... + x_n\).
2017 China Northern MO
Define sequence $(a_n):a_1=\text{e},a_2=\text{e}^3,\text{e}^{1-k}a_n^{k+2}=a_{n+1}a_{n-1}^{2k}$ for all $n\geq2$, where $k$ is a positive real number. Find $\prod_{i=1}^{2017}a_i$.
2017 China Northern MO
Positive intenger $n\geq3$. $a_1,a_2,\cdots,a_n$ are $n$ positive intengers that are pairwise coprime, satisfying that there exists $k_1,k_2,\cdots,k_n\in\{-1,1\}, \sum_{i=1}^{n}k_ia_i=0$. Are there positive intengers $b_1,b_2,\cdots,b_n$, for any $k\in\mathbb{Z}_+$, $b_1+ka_1,b_2+ka_2,\cdots,b_n+ka_n$ are pairwise coprime?
2017 China Northern MO
Length of sides of regular hexagon $ABCDEF$ is $a$. Two moving points $M,N$ moves on sides $BC,DE$, satisfy that $\angle MAN=\frac{\pi}{3}$. Prove that $AM\cdot AN-BM\cdot DN$ is a definite value.
2017 China Northern MO
Define $S_r(n)$: digit sum of $n$ in base $r$. For example, $38=(1102)_3,S_3(38)=1+1+0+2=4$. Prove:

(a) For any $r>2$, there exists prime $p$, for any positive intenger $n$, $S_{r}(n)\equiv n\mod p$.

(b) For any $r>1$ and prime $p$, there exists infinitely many $n$, $S_{r}(n)\equiv n\mod p$.
2017 China Northern MO
On Qingqing Grassland, there are 7 sheep numberd $1,2,3,4,5,6,7$ and 2017 wolves numberd $1,2,\cdots,2017$. We have such strange rules:

(1) Define $P(n)$: the number of prime numbers that are smaller than $n$. Only when $P(i)\equiv j\pmod7$,  wolf $i$ may eat sheep $j$ (he can also choose not to eat the sheep).

(2) If wolf $i$ eat sheep $j$, he will immediately turn into sheep $j$.

(3) If a wolf can make sure not to be eaten, he really wants to experience life as a sheep.

Assume that all wolves are very smart, then how many wolves will remain in the end?