Navvye/Olympiad-Problems-CSV huggingface dataset:
1978 Canada National Olympiad
The sides $AD$ and $BC$ of a convex quadrilateral $ABCD$ are extended to meet at $E$.  Let $H$ and $G$ be the midpoints of $BD$ and $AC$, respectively.  Find the ratio of the area of the triangle $EHG$ to that of the quadrilateral $ABCD$.
1978 Canada National Olympiad
Eve and Odette play a game on a $3\times 3$ checkerboard, with black checkers and white checkers. The rules are as follows:



$\text{I.}$ They play alternately.



$\text{II.}$ A turn consists of placing one checker on an unoccupied square of the board.



$\text{III.}$ In her turn, a player may select either a white checker or a black checker and need not always use the same colour.



$\text{IV.}$ When the board is full, Eve obtains one point for every row, column or diagonal that has an even number of black checkers, and Odette obtains one point for very row, column or diagonal that has an odd number of black checkers.



$\text{V.}$ The player obtaining at least five of the eight points WINS.



$\text{(a)}$ Is a $4-4$ tie possible? Explain.



$\text{(b)}$ Describe a winning strategy for the girl who is first to play.
1978 Canada National Olympiad
Sketch the graph of $x^3 + xy + y^3 = 3$.
1977 Canada National Olympiad
If $f(x) = x^2 + x$, prove that the equation $4f(a) = f(b)$ has no solutions in positive integers $a$ and $b$.
1977 Canada National Olympiad
Let $O$ be the centre of a circle and $A$ a fixed interior point of the circle different from $O$.  Determine all points $P$ on the circumference of the circle such that the angle $OPA$ is a maximum.



[asy]

import graph;



unitsize(2 cm);



pair A, O, P;



A = (0.5,0.2);

O = (0,0);

P = dir(80);



draw(Circle(O,1));

draw(O--A--P--cycle);



label("$A$", A, E);

label("$O$", O, S);

label("$P$", P, N);

[/asy]
1977 Canada National Olympiad
$N$ is an integer whose representation in base $b$ is $777$.  Find the smallest integer $b$ for which $N$ is the fourth power of an integer.
1977 Canada National Olympiad
Let

$$p(x) = a_n x^n + a_{n - 1} x^{n - 1} + \dots + a_1 x + a_0$$

and

$$q(x) = b_m x^m + a_{m - 1} x^{m - 1} + \dots + b_1 x + b_0$$

be two polynomials with integer coefficients.  Suppose that all the coefficients of the product $p(x) \cdot q(x)$ are even but not all of them are divisible by 4.  Show that one of $p(x)$ and $q(x)$ has all even coefficients and the other has at least one odd coefficient.
1977 Canada National Olympiad
A right circular cone has base radius 1 cm and slant height 3 cm is given.  $P$ is a point on the circumference of the base and the shortest path from $P$ around the cone and back to $P$ is drawn (see diagram).  What is the minimum distance from the vertex $V$ to this path?



[asy]

import graph;



unitsize(1 cm);



filldraw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(Circle((0,0),1)),gray(0.9),nullpen);

draw(yscale(0.3)*(arc((0,0),1.5,0,180)),dashed);

draw(yscale(0.3)*(arc((0,0),1.5,180,360)));

draw((1.5,0)--(0,4)--(-1.5,0));

draw((0,0)--(1.5,0),Arrows);

draw(((1.5,0) + (0.3,0.1))--((0,4) + (0.3,0.1)),Arrows);

draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,0,180)),dashed);

draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,180,360)));



label("$V$", (0,4), N);

label("1 cm", (0.75,-0.5), N);

label("$P$", (-1.5,0), SW);

label("3 cm", (1.7,2));

[/asy]
1977 Canada National Olympiad
Let $0 < u < 1$ and define

$$u_1 = 1 + u, \quad u_2 = \frac{1}{u_1} + u, \quad \dots, \quad u_{n + 1} = \frac{1}{u_n} + u, \quad n \ge 1.$$

Show that $u_n > 1$ for all values of $n = 1$, 2, 3, $\dots$.
1977 Canada National Olympiad
A rectangular city is exactly $m$ blocks long and $n$ blocks wide (see diagram).  A woman lives in the southwest corner of the city and works in the northeast corner.  She walks to work each day but, on any given trip, she makes sure that her path does not include any intersection twice.  Show that the number $f(m,n)$ of different paths she can take to work satisfies $f(m,n) \le 2^{mn}$.



[asy]

unitsize(0.4 cm);



for(int i = 0; i <= 11; ++i) {

  draw((i,0)--(i,7));

}



for(int j = 0; j <= 7; ++j) {

  draw((0,j)--(11,j));

}



label("$\underbrace{\hspace{4.4 cm}}$", (11/2,-0.5));

label("$\left. \begin{array}{c} \vspace{2.4 cm} \end{array} \right\}$", (11,7/2));

label("$m$ blocks", (11/2,-1.5));

label("$n$ blocks", (14,7/2));

[/asy]
1976 Canada National Olympiad
Given four weights in geometric progression and an equal arm balance, show how to find the heaviest weight using the balance only twice.
1976 Canada National Olympiad
Suppose

$$ n(n+1)a_{n+1}=n(n-1)a_n-(n-2)a_{n-1}

$$

for every positive integer $ n\ge1$. Given that $ a_0=1,a_1=2$, find

$$ \frac{a_0}{a_1}+\frac{a_1}{a_2}+\frac{a_2}{a_3}+\dots+\frac{a_{50}}{a_{51}}.

$$
1976 Canada National Olympiad
Two grade seven students were allowed to enter a chess tournament otherwise composed of grade eight students. Each contestant played once with each other contestant and received one point for a win, one half point for a tie and zero for a loss. The two grade seven students together gained a total of eight points and each grade eight student scored the same number of points as his classmates. How many students for grade eight participated in the chess tournament? Is the solution unique?
1976 Canada National Olympiad
Let $ AB$ be a diameter of a circle, $ C$ be any fixed point between $ A$ and $ B$ on this diameter, and $ Q$ be a variable point on the circumference of the circle. Let $ P$ be the point on the line determined by $ Q$ and $ C$ for which $ \frac{AC}{CB}=\frac{QC}{CP}$. Describe, with proof, the locus of the point $ P$.
1976 Canada National Olympiad
Prove that a positive integer is a sum of at least two consecutive positive integers if and only if it is not a power of two.
1976 Canada National Olympiad
If $ A,B,C,D$ are four points in space, such that

$$ \angle ABC=\angle BCD=\angle CDA=\angle DAB=\pi/2,

$$

prove that $ A,B,C,D$ lie in a plane.
1976 Canada National Olympiad
Let $ P(x,y)$ be a polynomial in two variables $ x,y$ such that $ P(x,y)=P(y,x)$ for every $ x,y$ (for example, the polynomial $ x^2-2xy+y^2$ satisfies this condition). Given that $ (x-y)$ is a factor of $ P(x,y)$, show that $ (x-y)^2$ is a factor of $ P(x,y)$.
1976 Canada National Olympiad
Each of the 36 line segments joining 9 distinct points on a circle is coloured either red or blue. Suppose that each triangle determined by 3 of the 9 points contains at least one red side. Prove that there are four points such that the 6 segments connecting them are all red.
1975 Canada National Olympiad
Simplify



$ \left(\frac {1 \cdot 2 \cdot 4 + 2 \cdot 4 \cdot 8 + \cdots + n \cdot 2n \cdot 4n}{1 \cdot 3 \cdot 9 + 2 \cdot 6 \cdot 18 + \cdots + n \cdot 3n \cdot 9n}\right)^{\frac {1}{3}}$
1975 Canada National Olympiad
A sequence of numbers $ a_1, a_2, a_3, ...$ satisfies



(i) $ a_1 = \frac{1}{2}$

(ii) $ a_1+a_2 + \cdots + a_n = n^2 a_n \ (n \geq 1)$



Determine the value of $ a_n \ (n \geq 1)$.
1975 Canada National Olympiad
For each real number $ r$, $ [r]$ denotes the largest integer less than or equal to $ r$, e.g. $ [6] = 6, [\pi] = 3, [-1.5] = -2$. Indicate on the $ (x,y)$-plane the set of all points $ (x,y)$ for which $ [x]^2 + [y]^2 = 4$.
1975 Canada National Olympiad
For a positive number such as 3.27, 3 is referred to as the integral part of the number and .27 as the decimal part. Find a positive number such that its decimal part, its integral part, and the number itself form a geometric progression.
1975 Canada National Olympiad
$ A,B,C,D$ are four "consecutive" points on the circumference of a circle and $ P, Q, R, S$ are points on the circumference which are respectively the midpoints of the arcs $ AB,BC,CD,DA$. Prove that $ PR$ is perpendicular to $ QS$.
1975 Canada National Olympiad
(i) 15 chairs are equally placed around a circular table on which are name cards for 15 guests. The guests fail to notice these cards until after they have sat down, and it turns out that no one is sitting in the correct seat. Prove that the table can be rotated so that at least two of the guests are simultaneously correctly seated.



(ii) Give an example of an arrangement in which just one of the 15 guests is correctly seated and for which no rotation correctly places more than one person.
1975 Canada National Olympiad
A function $ f(x)$ is periodic if there is a positive number $ p$ such that $ f(x+p) = f(x)$ for all $ x$. For example, $ \sin x$ is periodic with period $ 2 \pi$. Is the function $ \sin(x^2)$ periodic? Prove your assertion.
1975 Canada National Olympiad
Let $ k$ be a positive integer. Find all polynomials



$$ P(x) = a_0 + a_1 x + \cdots + a_n x^n,$$



where the $ a_i$ are real, which satisfy the equation



$$ P(P(x)) = \{ P(x) \}^k$$
1974 Canada National Olympiad
i) If $x = \left(1+\frac{1}{n}\right)^{n}$ and $y=\left(1+\frac{1}{n}\right)^{n+1}$, show that $y^{x}= x^{y}$.



ii) Show that, for all positive integers $n$, $$1^{2}-2^{2}+3^{2}-4^{2}+\cdots+(-1)^{n}(n-1)^{2}+(-1)^{n+1}n^{2}= (-1)^{n+1}(1+2+\cdots+n).$$
1974 Canada National Olympiad
Let $ABCD$ be a rectangle with $BC=3AB$. Show that if $P,Q$ are the points on side $BC$ with $BP = PQ = QC$, then $$\angle DBC+\angle DPC = \angle DQC.$$
1974 Canada National Olympiad
Let $$f(x) = a_{0}+a_{1}x+a_{2}x^{2}+\cdots+a_{n}x^{n}$$ be a polynomial with coefficients satisfying the conditions: $$0\le a_{i}\le a_{0},\quad i=1,2,\ldots,n.$$ Let $b_{0},b_{1},\ldots,b_{2n}$ be the coefficients of the polynomial



\begin{align*}\left(f(x)\right)^{2}&= \left(a_{0}+a_{1}x+a_{2}x^{2}+\cdots a_{n}x^{n}\right)\\ &= b_{0}+b_{1}x+b_{2}x^{2}+\cdots+b_{2n}x^{2n}. \end{align*}



Prove that $b_{n+1}\le \frac{1}{2}\left(f(1)\right)^{2}$.
1974 Canada National Olympiad
Let $n$ be a fixed positive integer. To any choice of real numbers satisfying $$0\le x_{i}\le 1,\quad i=1,2,\ldots, n,$$ there corresponds the sum $$\sum_{1\le i<j\le n}|x_{i}-x_{j}|.$$ Let $S(n)$ denote the largest possible value of this sum. Find $S(n)$.
1974 Canada National Olympiad
Given a circle with diameter $AB$ and a point $X$ on the circle different from $A$ and $B$, let $t_{a}$, $t_{b}$ and $t_{x}$ be the tangents to the circle at $A$, $B$ and $X$ respectively. Let $Z$ be the point where line $AX$ meets $t_{b}$ and $Y$ the point where line $BX$ meets $t_{a}$. Show that the three lines $YZ$, $t_{x}$ and $AB$ are either concurrent (i.e., all pass through the same point) or parallel.



Invalid URL
1974 Canada National Olympiad
An unlimited supply of 8-cent and 15-cent stamps is available. Some amounts of postage cannot be made up exactly, e.g., 7 cents, 29 cents. What is the largest unattainable amount, i.e., the amount, say $n$, of postage which is unattainable while all amounts larger than $n$ are attainable? (Justify your answer.)
1974 Canada National Olympiad
A bus route consists of a circular road of circumference 10 miles and a straight road of length 1 mile which runs from a terminus to the point $Q$ on the circular road (see diagram).



Invalid URL



It is served by two buses, each of which requires 20 minutes for the round trip. Bus No. 1, upon leaving the terminus, travels along the straight road, once around the circle clockwise and returns along the straight road to the terminus. Bus No. 2, reaching the terminus 10 minutes after Bus No. 1, has a similar route except that it proceeds counterclockwise around the circle. Both buses run continuously and do not wait at any point on the route except for a negligible amount of time to pick up and discharge passengers. A man plans to wait at a point $P$ which is $x$ miles ($0\le x < 12$) from the terminus along the route of Bus No. 1 and travel to the terminus on one of the buses.



Assuming that he chooses to board that bus which will bring him to his destination at the earliest moment, there is a maximum time $w(x)$ that his journey (waiting plus travel time) could take.



Find $w(2)$; find $w(4)$.



For what value of $x$ will the time $w(x)$ be the longest?



Sketch a graph of $y = w(x)$ for $0\le x < 12$.
1973 Canada National Olympiad
(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$; i.e. find a single inequality equivalent to the two simultaneous inequalities.



(ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$.



(iii) Give a rational number between $11/24$ and $6/13$.



(iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10.



(v) Without the use of logarithm tables evaluate $$\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.$$
1973 Canada National Olympiad
Find all real numbers that satisfy the equation $|x+3|-|x-1|=x+1$. (Note: $|a| = a$ if $a\ge 0$; $|a|=-a$ if $a<0$.)
1973 Canada National Olympiad
Prove that if $p$ and $p+2$ are prime integers greater than 3, then 6 is a factor of $p+1$.
1973 Canada National Olympiad
The figure shows a (convex) polygon with nine vertices. The six diagonals which have been drawn dissect the polygon into the seven triangles: $P_{0}P_{1}P_{3}$, $P_{0}P_{3}P_{6}$, $P_{0}P_{6}P_{7}$, $P_{0}P_{7}P_{8}$, $P_{1}P_{2}P_{3}$, $P_{3}P_{4}P_{6}$, $P_{4}P_{5}P_{6}$. In how many ways can these triangles be labeled with the names $\triangle_{1}$, $\triangle_{2}$, $\triangle_{3}$, $\triangle_{4}$, $\triangle_{5}$, $\triangle_{6}$, $\triangle_{7}$ so that $P_{i}$ is a vertex of triangle $\triangle_{i}$ for $i = 1, 2, 3, 4, 5, 6, 7$? Justify your answer.



Invalid URL
1973 Canada National Olympiad
For every positive integer $n$, let $$h(n) = 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}.$$ For example, $h(1) = 1$, $h(2) = 1+\frac{1}{2}$, $h(3) = 1+\frac{1}{2}+\frac{1}{3}$. Prove that for $n=2,3,4,\ldots$ $$n+h(1)+h(2)+h(3)+\cdots+h(n-1) = nh(n).$$
1973 Canada National Olympiad
If $A$ and $B$ are fixed points on a given circle not collinear with centre $O$ of the circle, and if $XY$ is a variable diameter, find the locus of $P$ (the intersection of the line through $A$ and $X$ and the line through $B$ and $Y$).
1973 Canada National Olympiad
Observe that

$$\frac{1}{1}= \frac{1}{2}+\frac{1}{2};\quad \frac{1}{2}=\frac{1}{3}+\frac{1}{6};\quad \frac{1}{3}=\frac{1}{4}+\frac{1}{12};\quad \frac{1}{4}= \frac{1}{5}+\frac{1}{20}. $$

State a general law suggested by these examples, and prove it.



Prove that for any integer $n$ greater than 1 there exist positive integers $i$ and $j$ such that

$$\frac{1}{n}= \frac{1}{i(i+1)}+\frac{1}{(i+1)(i+2)}+\frac{1}{(i+2)(i+3)}+\cdots+\frac{1}{j(j+1)}. $$

Remark.It seems that this is a two-part problem.
1972 Canada National Olympiad
Given three distinct unit circles, each of which is tangent to the other two, find the radii of the circles which are tangent to all three circles.
1972 Canada National Olympiad
Let $a_1,a_2,\ldots,a_n$ be non-negative real numbers. Define $M$ to be the sum of all products of pairs $a_ia_j$ $(i<j)$, $\textit{i.e.}$, $$ M = a_1(a_2+a_3+\cdots+a_n)+a_2(a_3+a_4+\cdots+a_n)+\cdots+a_{n-1}a_n.  $$ Prove that the square of at least one of the numbers $a_1,a_2,\ldots,a_n$ does not exceed $2M/n(n-1)$.
1972 Canada National Olympiad
a) Prove that $10201$ is composite in all bases greater than 2.



b) Prove that $10101$ is composite in all bases.
1972 Canada National Olympiad
Describe a construction of quadrilateral $ABCD$ given:



(i) the lengths of all four sides;



(ii) that $AB$ and $CD$ are parallel;



(iii) that $BC$ and $DA$ do not intersect.
1972 Canada National Olympiad
Prove that the equation $x^3+11^3=y^3$ has no solution in positive integers $x$ and $y$.
1972 Canada National Olympiad
Let $a$ and $b$ be distinct real numbers. Prove that there exist integers $m$ and $n$ such that $am+bn<0$, $bm+an>0$.
1972 Canada National Olympiad
a) Prove that the values of $x$ for which $x=(x^2+1)/198$ lie between $1/198$ and $197.99494949\cdots$.



b) Use the result of problem a) to prove that $\sqrt{2}<1.41\overline{421356}$.



c) Is it true that $\sqrt{2}<1.41421356$?
1972 Canada National Olympiad
During a certain election campaign, $p$ different kinds of promises are made by the different political parties ($p>0$). While several political parties may make the same promise, any two parties have at least one promise in common; no two parties have exactly the same set of promises. Prove that there are no more than $2^{p-1}$ parties.
1972 Canada National Olympiad
Four distinct lines $L_1,L_2,L_3,L_4$ are given in the plane: $L_1$ and $L_2$ are respectively parallel to $L_3$ and $L_4$. Find the locus of a point moving so that the sum of its perpendicular distances from the four lines is constant.
1972 Canada National Olympiad
What is the maximum number of terms in a geometric progression with common ratio greater than 1 whose entries all come from the set of integers between 100 and 1000 inclusive?
2021 China Girls Math Olympiad
Let $n \in \mathbb{N}^+,$ $x_1,x_2,...,x_{n+1},p,q\in  \mathbb{R}^+ $ , $p<q$ and $x^p_{n+1}>\sum_{i=1}^{n}x^p_{i}.$ Prove that

$(1)x^q_{n+1}>\sum_{i=1}^{n}x^q_{i};$

$(2)\left(x^p_{n+1}-\sum_{i=1}^{n}x^p_{i}\right)^{\frac{1}{p}}<\left(x^q_{n+1}-\sum_{i=1}^{n}x^q_{i}\right)^{\frac{1}{q}}.$
2021 China Girls Math Olympiad
In acute triangle $ABC$ ($AB \neq AC$), $I$ is its incenter and $J$ is the $A$-excenter. $X, Y$ are on minor arcs $\widehat{AB}$ and $\widehat{AC}$ respectively such that $\angle{AXI}=\angle{AYJ}=90^{\circ}$. $K$ is on line $BC$ such that $KI=KJ$.

Proof that line $AK$ bisects $\overline{XY}$.
2021 China Girls Math Olympiad
Find the smallest positive integer $n$, such that one can color every cell of a $n \times n$ grid in red, yellow or blue with all the following conditions satisfied:

(1) the number of cells colored in each color is the same;

(2) if a row contains a red cell, that row must contain a blue cell and cannot contain a yellow cell;

(3) if a column contains a blue cell, it must contain a red cell but cannot contain a yellow cell.
2021 China Girls Math Olympiad
Call a sequence of positive integers $(a_n)_{n \ge 1}$ a "CGMO sequence" if $(a_n)_{n \ge 1}$ strictly increases, and for all integers $n \ge 2022$, $a_n$ is the smallest integer such that there exists a non-empty subset of $\{a_{1}, a_{2}, \cdots, a_{n-1} \}$ $A_n$ where $a_n \cdot \prod\limits_{a \in A_n} a$ is a perfect square.



Proof: there exists $c_1, c_2 \in \mathbb{R}^{+}$ s.t. for any "CGMO sequence" $(a_n)_{n \ge 1}$ , there is a positive integer $N$ that satisfies any $n \ge N$, $c_1 \cdot n^2 \le a_n \le c_2 \cdot n^2$.
2021 China Girls Math Olympiad
Proof that if $4$ numbers (not necessarily distinct) are picked from $\{1, 2, \cdots, 20\}$, one can pick $3$ numbers among them and can label these $3$ as $a, b, c$ such that $ax \equiv b \;(\bmod\; c)$ has integral solutions.
2021 China Girls Math Olympiad
Given a finite set $S$, $P(S)$ denotes the set of all the subsets of $S$. For any $f:P(S)\rightarrow \mathbb{R}$ ,prove the following inequality:$$\sum_{A\in P(S)}\sum_{B\in P(S)}f(A)f(B)2^{\left| A\cap B \right|}\geq 0.$$
2021 China Girls Math Olympiad
In an acute triangle $ABC$, $AB \neq AC$, $O$ is its circumcenter. $K$ is the reflection of $B$ over $AC$ and $L$ is the reflection of $C$ over $AB$. $X$ is a point within $ABC$ such that $AX \perp BC, XK=XL$. Points $Y, Z$ are on $\overline{BK}, \overline{CL}$ respectively, satisfying $XY \perp CK, XZ \perp BL$.



Proof that $B, C, Y, O, Z$ lie on a circle.
2021 China Girls Math Olympiad
Let $m, n$ be positive integers, define:

$f(x)=(x-1)(x^2-1)\cdots(x^m-1)$, $g(x)=(x^{n+1}-1)(x^{n+2}-1)\cdots(x^{n+m}-1)$.



Show that there exists a polynomial $h(x)$ of degree $mn$ such that $f(x)h(x)=g(x)$,  and its $mn+1$ coefficients are all positive integers.
2020 China Girls Math Olympiad
In the quadrilateral $ABCD$, $AB=AD$, $CB=CD$, $\angle ABC =90^\circ$. $E$, $F$ are on $AB$, $AD$ and $P$, $Q$ are on $EF$($P$ is between $E, Q$), satisfy $\frac{AE}{EP}=\frac{AF}{FQ}$. $X, Y$ are on $CP, CQ$ that satisfy $BX \perp CP, DY \perp CQ$. Prove that $X, P, Q, Y$ are concyclic.
2020 China Girls Math Olympiad
Let $n$ be an integer and $n \geq 2$, $x_1, x_2, \cdots , x_n$ are arbitrary real number, find the maximum value of $$2\sum_{1\leq i<j \leq n}\left \lfloor x_ix_j \right \rfloor-\left (  n-1 \right )\sum_{i=1}^{n}\left \lfloor x_i^2 \right \rfloor  $$
2020 China Girls Math Olympiad
There are $3$ classes with $n$ students in each class, and the heights of all $3n$ students are pairwise distinct. Partition the students into groups of $3$ such that in each group, there is one student from each class. In each group, call the tallest student the tall guy. Suppose that for any partition of the students, there are at least 10 tall guys in each class, prove that the minimum value of $n$ is $40$.
2020 China Girls Math Olympiad
Let $p,q$ be primes, where $p>q$. Define $t=\gcd(p!-1,q!-1)$. Prove that $t\le p^{\frac{p}{3}}$.
2020 China Girls Math Olympiad
Find all the real number sequences $\{b_n\}_{n \geq 1}$ and $\{c_n\}_{n \geq 1}$ that satisfy the following conditions:

(i) For any positive integer $n$, $b_n \leq c_n$;

(ii) For any positive integer $n$, $b_{n+1}$ and $c_{n+1}$ is the two roots of the equation $x^2+b_nx+c_n=0$.
2020 China Girls Math Olympiad
Let $p, q$ be integers and $p, q > 1$ , $gcd(p, \,6q)=1$. Prove that:$$\sum_{k=1}^{q-1}\left \lfloor \frac{pk}{q}\right\rfloor^2 \equiv 2p \sum_{k=1}^{q-1}k\left\lfloor \frac{pk}{q} \right\rfloor (mod \, q-1)$$
2020 China Girls Math Olympiad
Let $O$ be the circumcenter of triangle $\triangle ABC$, where $\angle BAC=120^{\circ}$. The tangent at $A$ to $(ABC)$ meets the tangents at $B,C$ at $(ABC)$ at points $P,Q$ respectively. Let $H,I$ be the orthocenter and incenter of $\triangle OPQ$ respectively. Define $M,N$ as the midpoints of arc $\overarc{BAC}$ and $OI$ respectively, and let $MN$ meet $(ABC)$ again at $D$. Prove that $AD$ is perpendicular to $HI$.
2020 China Girls Math Olympiad
Let $n$ be a given positive integer. Let $\mathbb{N}_+$ denote the set of all positive integers.



Determine the number of all finite lists $(a_1,a_2,\cdots,a_m)$ such that:

(1) $m\in \mathbb{N}_+$ and $a_1,a_2,\cdots,a_m\in \mathbb{N}_+$ and $a_1+a_2+\cdots+a_m=n$.

(2) The number of all pairs of integers $(i,j)$ satisfying $1\leq i<j\leq m$  and $a_i>a_j$ is even.



For example, when $n=4$, the number of all such lists $(a_1,a_2,\cdots,a_m)$ is $6$, and these lists are $(4),$ $(1,3),$ $(2,2),$ $(1,1,2),$ $(2,1,1),$ $(1,1,1,1)$.
2019 China Girls Math Olympiad
Let $ABCD$ be a cyclic quadrilateral with circumcircle $\odot O.$ The lines tangent to $\odot O$ at $A,B$ intersect at $L.$ $M$ is the midpoint of the segment $AB.$ The line passing through $D$ and parallel to $CM$ intersects $ \odot (CDL) $ at $F.$ Line $CF$ intersects $DM$ at $K,$ and intersects $\odot O$ at $E$ (different from point $C$).

Prove that $EK=DK.$
2019 China Girls Math Olympiad
Find integers $a_1,a_2,\cdots,a_{18}$, s.t. $a_1=1,a_2=2,a_{18}=2019$, and for all $3\le k\le 18$, there exists $1\le i<j<k$ with $a_k=a_i+a_j$.
2019 China Girls Math Olympiad
For a sequence, one can perform the following operation: select three adjacent terms $a,b,c,$ and change it into $b,c,a.$ Determine all the possible positive integers $n\geq 3,$ such that after finite number of operation, the sequence $1,2,\cdots, n$ can be changed into $n,n-1,\cdots,1$ finally.
2019 China Girls Math Olympiad
Given parallelogram $OABC$ in the coodinate with $O$ the origin and $A,B,C$ be lattice points. Prove that for all lattice point $P$ in the internal or boundary of $\triangle ABC$, there exists lattice points $Q,R$(can be the same) in the internal or boundary of $\triangle OAC$ with $\overrightarrow{OP}=\overrightarrow{OQ}+\overrightarrow{OR}$.
2019 China Girls Math Olympiad
Let $p$ be a prime number such that $p\mid (2^{2019}-1) .$ The sequence $a_1,a_2,...,a_n$ satisfies the following conditions: $a_0=2, a_1=1 ,a_{n+1}=a_n+\frac{p^2-1}{4}a_{n-1}$ $(n\geq 1).$ Prove that $p\nmid (a_n+1),$ for any $n\geq 0.$
2019 China Girls Math Olympiad
Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+

\sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$