Navvye/Olympiad-Problems-CSV huggingface dataset:
2007 South africa National Olympiad
Let $ ABC$ be a triangle and $ PQRS$ a square with $ P$ on $ AB$, $ Q$ on $ AC$, and $ R$ and $ S$ on $ BC$. Let $ H$ on $ BC$ such that $ AH$ is the altitude of the triangle from $ A$ to base $ BC$. Prove that:



(a) $ \frac{1}{AH} +\frac{1}{BC}=\frac{1}{PQ}$

(b) the area of $ ABC$ is twice the area of $ PQRS$ iff $ AH=BC$
2007 South africa National Olympiad
Let $ Z$ and $ R$ denote the sets of integers and real numbers, respectively.

Let $ f: Z \rightarrow R$ be a function satisfying:

(i) $ f(n) \ge 0$ for all $ n \in Z$

(ii) $ f(mn)=f(m)f(n)$ for all $ m,n \in Z$

(iii) $ f(m+n) \le max(f(m),f(n))$ for all $ m,n \in Z$



(a) Prove that $ f(n) \le 1$ for all $ n \in Z$

(b) Find a function $ f: Z \rightarrow R$ satisfying (i), (ii),(iii) and $ 0<f(2)<1$ and $ f(2007) = 1$
2007 South africa National Olympiad
Prove that it is not possible to write numbers $ 1,2,3,...,25$ on the squares of $ 5$x$ 5$ chessboard such that any neighboring numbers differ by at most $ 4$.
2006 South africa National Olympiad
Reduce the fraction

$$\frac{2121212121210}{1121212121211}$$

to its simplest form.
2006 South africa National Olympiad
Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\hat{A}$.
2006 South africa National Olympiad
Determine all positive integers whose squares end in $196$.
2006 South africa National Olympiad
In triangle $ABC$, $AB=AC$ and $B\hat{A}C=100^\circ$. Let $D$ be on $AC$ such that $A\hat{B}D=C\hat{B}D$. Prove that $AD+DB=BC$.
2006 South africa National Olympiad
Find the number of subsets $X$ of $\{1,2,\dots,10\}$ such that $X$ contains at least two elements and such that no two elements of $X$ differ by $1$.
2006 South africa National Olympiad
Consider the function $f$ defined by

$$f(n)=\frac{1}{n}\left (\left \lfloor\frac{n}{1}\right \rfloor+\left \lfloor\frac{n}{2}\right \rfloor+\cdots+\left \lfloor\frac{n}{n}\right \rfloor \right )$$

for all positive integers $n$. (Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.) Prove that



(a) $f(n+1)>f(n)$ for infinitely many $n$.



(b) $f(n+1)<f(n)$ for infinitely many $n$.
2005 South africa National Olympiad
Five numbers are chosen from the diagram below, such that no two numbers are chosen from the same row or from the same column. Prove that their sum is always the same.



$$\begin{array}{|c|c|c|c|c|}\hline

1&4&7&10&13\\ \hline

16&19&22&25&28\\ \hline

31&34&37&40&43\\ \hline

46&49&52&55&58\\ \hline

61&64&67&70&73\\ \hline

\end{array}$$
2005 South africa National Olympiad
Let $F$ be the set of all fractions $m/n$ where $m$ and $n$ are positive integers with $m+n\le 2005$. Find the largest number $a$ in $F$ such that $a < 16/23$.
2005 South africa National Olympiad
A warehouse contains $175$ boots of size $8$, $175$ boots of size $9$ and $200$ boots of size $10$. Of these $550$ boots, $250$ are for the left foot and $300$ for the right foot. Let $n$ denote the total number of usable pairs of boots in the warehouse. (A usable pair consists of a left and a right boot of the same size.)



(a) Is $n=50$ possible?



(b) Is $n=51$ possible?
2005 South africa National Olympiad
The inscribed circle of triangle $ABC$ touches the sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$ respectively. Let $Q$ denote the other point of intersection of $AD$ and the inscribed circle. Prove that $EQ$ extended passes through the midpoint of $AF$ if and only if $AC = BC$.
2005 South africa National Olympiad
Let $x_1,x_2,\dots,x_n$ be positive numbers with product equal to 1. Prove that there exists a $k\in\{1,2,\dots,n\}$ such that

$$\frac{x_k}{k+x_1+x_2+\cdots+x_k}\ge 1-\frac{1}{\sqrt[n]{2}}.$$
2005 South africa National Olympiad
Consider the increasing sequence $1,2,4,5,7,9,10,12,14,16,17,19,\dots$ of positive integers, obtained by concatenating alternating blocks $\{1\},\{2,4\},\{5,7,9\},\{10,12,14,16\},\dots$ of odd and even numbers. Each block contains one more element than the previous one and the first element in each block is one more than the last element of the previous one. Prove that the $n$-th element of the sequence is given by $$2n-\Big\lfloor\frac{1+\sqrt{8n-7}}{2}\Big\rfloor.$$

(Here $\lfloor x\rfloor$ denotes the greatest integer less than or equal to $x$.)
2004 South africa National Olympiad
Let $a=1111\dots1111$ and $b=1111\dots1111$ where $a$ has forty ones and $b$ has twelve ones. Determine the greatest common divisor of $a$ and $b$.
2004 South africa National Olympiad
Fifty points are chosen inside a convex polygon having eighty sides such that no three of the fifty points lie on the same straight line. The polygon is cut into triangles such that the vertices of the triangles are just the fifty points and the eighty vertices of the polygon. How many triangles are there?
2004 South africa National Olympiad
Find all real numbers $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor\rfloor=88$.  The notation $\lfloor x\rfloor$ means the greatest integer less than or equal to $x$.
2004 South africa National Olympiad
Let $A_1$ and $B_1$ be two points on the base $AB$ of isosceles triangle $ABC$ (with $\widehat{C}>60^\circ$) such that $\widehat{A_1CB_1}=\widehat{BAC}$. A circle externally tangent to the circumcircle of triangle $\triangle A_1B_1C$ is tangent also to rays $CA$ and $CB$ at points $A_2$ and $B_2$ respectively. Prove that $A_2B_2=2AB$.
2004 South africa National Olympiad
For $n\ge 2$, find the number of integers $x$ with $0\le x<n$, such that $x^2$ leaves a remainder of $1$ when divided by $n$.
2004 South africa National Olympiad
The numbers $a_1,a_2$ and $a_3$ are distinct positive integers, such that



(i) $a_1$ is a divisor of $a_2+a_3+a_2a_3$;



(ii) $a_2$ is a divisor of $a_3+a_1+a_3a_1$;



(iii) $a_3$ is a divisor of $a_1+a_2+a_1a_2$.



Prove that $a_1,a_2$ and $a_3$ cannot all be prime.
2003 South africa National Olympiad
You have five pieces of paper. You pick one or more of them and cut each of them into five smaller pieces. Now you take one or more of the pieces from this lot and cut each of these into five smaller pieces. And so on. Prove that you will never have 2003 pieces.
2003 South africa National Olympiad
Given a parallelogram $ABCD$, join $A$ to the midpoints $E$ and $F$ of the opposite sides $BC$ and $CD$. $AE$ and $AF$ intersect the diagonal $BD$ in $M$ and $N$. Prove that $M$ and $N$ divide $BD$ into three equal parts.
2003 South africa National Olympiad
The first four digits of a certain positive integer $n$ are $1137$. Prove that the digits of $n$ can be shuffled in such a way that the new number is divisible by 7.
2003 South africa National Olympiad
In a given pentagon $ABCDE$, triangles $ABC$, $BCD$, $CDE$, $DEA$ and $EAB$ all have the same area. The lines $AC$ and $AD$ intersect $BE$ at points $M$ and $N$. Prove that $BM = EN$.
2003 South africa National Olympiad
Prove that the sum of the squares of two consecutive positive integers cannot be equal to a sum of the fourth powers of two consecutive positive integers.
2003 South africa National Olympiad
In $\Delta ABC$, the sum of the sides is $2s$ and the radius of the incircle is $r$. Three semicircles with diameters $AB$, $BC$ and $CA$ are drawn on the outside of $ABC$. A circle with radius $t$ touches all three semicircles. Prove that $$ \frac{s}{2} < t \leq \frac{s}{2} + \left(1 - \frac{\sqrt{3}}{2}\right)r. $$
2002 South africa National Olympiad
Given a quadrilateral $ABCD$ such that $AB^2 + CD^2 = AD^2 + BC^2$, prove that $AC \perp BD$.
2002 South africa National Olympiad
Find all triples of natural numbers $(a,b,c)$ such that $a$, $b$ and $c$ are in geometric progression and $a + b + c = 111$.
2002 South africa National Olympiad
A small square $PQRS$ is contained in a big square. Produce $PQ$ to $A$, $QR$ to $B$, $RS$ to $C$ and $SP$ to $D$ so that $A$, $B$, $C$ and $D$ lie on the four sides of the large square in order, produced if necessary. Prove that $AC = BD$ and $AC \perp BD$.
2002 South africa National Olympiad
How many ways are there to express 1000000 as a product of exactly three integers greater than 1? (For the purpose of this problem, $abc$ is not considered different from $bac$, etc.)
2002 South africa National Olympiad
In acute-angled triangle $ABC$, a semicircle with radius $r_a$ is constructed with its base on $BC$ and tangent to the other two sides. $r_b$ and $r_c$ are defined similarly. $r$ is the radius of the incircle of $ABC$. Show that $$ \frac{2}{r} = \frac{1}{r_a} + \frac{1}{r_b} + \frac{1}{r_c}. $$
2002 South africa National Olympiad
Find all rational numbers $a$, $b$, $c$ and $d$ such that $$ 8a^2 - 3b^2 + 5c^2 + 16d^2 - 10ab + 42cd + 18a + 22b - 2c - 54d = 42, $$ $$ 15a^2 - 3b^2 + 21c^2 - 5d^2 + 4ab +32cd - 28a + 14b - 54c - 52d = -22. $$
2001 South africa National Olympiad
$ABCD$ is a convex quadrilateral with perimeter $p$. Prove that $$ \dfrac{1}{2}p < AC + BD < p.  $$ (A polygon is convex if all of its interior angles are less than $180^\circ$.)
2001 South africa National Olympiad
Find all triples $(x,y,z)$ of real numbers that satisfy $$ \begin{aligned} & x\left(1 - y^2\right)\left(1 - z^2\right) + y\left(1 - z^2\right)\left(1 - x^2\right) + z\left(1 - x^2\right)\left(1 - y^2\right) \\ & = 4xyz \\ & = 4(x + y + z). \end{aligned}  $$
2001 South africa National Olympiad
For a certain real number $x$, the differences between $x^{1919}$, $x^{1960}$ and $x^{2001}$ are all integers. Prove that $x$ is an integer.
2001 South africa National Olympiad
$n$ red and $n$ blue points on a plane are given so that no three of the $2n$ points are collinear. Prove that it is always possible to split up the points into $n$ pairs, with one red and one blue point in each pair, so that no two of the $n$ line segments which connect the two members of a pair intersect.
2001 South africa National Olympiad
Starting from a given cyclic quadrilateral $\mathcal{Q}_0$, a sequence of quadrilaterals is constructed so that $\mathcal{Q}_{k + 1}$ is the circumscribed quadrilateral of $\mathcal{Q}_k$ for $k = 0,1,\dots$. The sequence terminates when a quadrilateral is reached that is not cyclic. (The circumscribed quadrilateral of a cylic quadrilateral $ABCD$ has sides that are tangent to the circumcircle of $ABCD$ at $A$, $B$, $C$ and $D$.) Prove that the sequence always terminates, except when $\mathcal{Q}_0$ is a square.
2001 South africa National Olympiad
The unknown real numbers $x_1,x_2,\dots,x_n$ satisfy $x_1 < x_2 < \cdots < x_n,$ where $n \geq 3$. The numbers $s$, $t$ and $d_1,d_2,\dots,d_{n - 2}$ are given, such that $$ \begin{aligned} s & = \sum\limits_{i = 1}^nx_i, \\ t & = \sum\limits_{i = 1}^nx_i^2,\\ d_i & = x_{i + 2} - x_i,\ \ i = 1,2,\dots,n - 2. \end{aligned}  $$ For which $n$ is this information always sufficient to determine $x_1,x_2,\dots,x_n$ uniquely?
2000 South africa National Olympiad
A number $x_n$ of the form 10101...1 has $n$ ones. Find all $n$ such that $x_n$ is prime.
2000 South africa National Olympiad
Solve for $x$, given $36x^4 + 36x^3 - 7x^2 - 6x + 1 = 0$.
2000 South africa National Olympiad
Let $c \geq 1$ be an integer, and define the sequence $a_1,\ a_2,\ a_3,\ \dots$ by $$ \begin{aligned} a_1 & = 2, \\ a_{n + 1} & = ca_n + \sqrt{\left(c^2 - 1\right)\left(a_n^2 - 4\right)}\textrm{ for }n = 1,2,3,\dots\ . \end{aligned} $$ Prove that $a_n$ is an integer for all $n$.
2000 South africa National Olympiad
$ABCD$ is a square of side 1. $P$ and $Q$ are points on $AB$ and $BC$ such that $\widehat{PDQ} = 45^{\circ}$. Find the perimeter of $\Delta PBQ$.
2000 South africa National Olympiad
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ (where $\mathbb{Z}$ is the set of all integers) such that $$ 2000f(f(x)) - 3999f(x) + 1999x = 0\textrm{ for all }x \in \mathbb{Z}. $$
2000 South africa National Olympiad
Let $A_n$ be the number of ways to tile a $4 \times n$ rectangle using $2 \times 1$ tiles. Prove that $A_n$ is divisible by 2 if and only if $A_n$ is divisible by 3.
1999 South africa National Olympiad
How many non-congruent triangles with integer sides and perimeter 1999 can be constructed?
1999 South africa National Olympiad
$A,\ B,\ C$ and $D$ are points on a given straight line, in that order. Show how to construct a square $PQRS$, with all of $P,\ Q,\ R$ and $S$ on the same side of $AD$, such that $A,\ B,\ C$ and $D$ lie on $PQ,\ SR,\ QR$ and $PS$ produced respectively.
1999 South africa National Olympiad
The bisector of $\angle{BAD}$ in the parallellogram $ABCD$ intersects the lines $BC$ and $CD$ at the points $K$ and $L$ respectively. Prove that the centre of the circle passing through the points $C,\ K$ and $L$ lies on the circle passing through the points $B,\ C$ and $D$.
1999 South africa National Olympiad
The sequence $L_1,\ L_2,\ L_3,\ \dots$ is defined by $$ L_1 = 1,\ \ L_2 = 3,\ \ L_n = L_{n - 1} + L_{n - 2}\textrm{ for }n > 2. $$ Prove that $L_p - 1$ is divisible by $p$ if $p$ is prime.
1999 South africa National Olympiad
Let $S$ be the set of all rational numbers whose denominators are powers of 3. Let $a$, $b$ and $c$ be given non-zero real numbers. Determine all real-valued functions $f$ that are defined for $x \in S$, satisfy $$ f(x) = af(3x) + bf(3x - 1) + cf(3x - 2) \textrm{ if }0 \leq x \leq 1, $$ and are zero elsewhere.
1999 South africa National Olympiad
You are at a point $(a,b)$ and you need to reach another point $(c,d)$. Both points are below the line $x = y$ and have integer coordinates. You can move in steps of length 1, either upwards of to the right, but you may not move to a point on the line $x = y$. How many different paths are there?
1998 South africa National Olympiad
Is $\log_{10}{8}$ rational?
1998 South africa National Olympiad
Find the maximum value of $$ \sin{2\alpha} + \sin{2\beta} + \sin{2\gamma} $$ where $\alpha,\beta$ and $\gamma$ are positive and $\alpha + \beta + \gamma = 180^{\circ}$.
1998 South africa National Olympiad
$A,\ B,\ C,\ D,\ E$ and $F$ lie (in that order) on the circumference of a circle. The chords $AD,\ BE$ and $CF$ are concurrent. $P,\ Q$ and $R$ are the midpoints of $AD,\ BE$ and $CF$ respectively. Two further chords $AG \parallel BE$ and $AH \parallel CF$ are drawn. Show that $PQR$ is similar to $DGH$.
1998 South africa National Olympiad
In a group of people, every two people have exactly one friend in common. Prove that there is a person who is a friend of everyone else.
1998 South africa National Olympiad
Prove that $$ \gcd{\left({n \choose 1},{n \choose 2},\dots,{n \choose {n - 1}}\right)} $$ is a prime if $n$ is a power of a prime, and 1 otherwise.
1998 South africa National Olympiad
You are given $n$ squares, not necessarily all of the same size, which have total area 1. Is it always possible to place them without overlapping in a square of area 2?
1997 South africa National Olympiad
From an initial triangle $\Delta A_0B_0C_0$, a sequence of triangles $\Delta A_1B_1C_1$, $A_2B_2C_2$, ... is formed such that, at each stage, $A_{k + 1}$, $B_{k + 1}$ and $C_{k + 1}$ are the points where the incircle of $\Delta A_kB_kC_k$ touches the sides $B_kC_k$, $C_kA_k$ and $A_kB_k$ respectively.



(a) Express $\angle A_{k + 1}B_{k + 1}C_{k + 1}$ in terms of $\angle A_kB_kC_k$.



(b) Deduce that, as $k$ increases, $\angle A_kB_kC_k$ tends to $60^{\circ}$.
1997 South africa National Olympiad
Find all natural numbers with the property that, when the first digit is moved to the end, the resulting number is $\dfrac{7}{2}$ times the original one.
1997 South africa National Olympiad
Find all solutions $x,y \in \mathbb{Z}$, $x,y \geq 0$, to the equation $$ 1 + 3^x = 2^y. $$
1997 South africa National Olympiad
Find all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy $$ f(m + f(n)) = f(m) + n $$ for all $m,n \in \mathbb{Z}$.
1997 South africa National Olympiad
A circle and a point $P$ higher than the circle lie in the same vertical plane. A particle moves along a straight line under gravity from $P$ to a point $Q$ on the circle. Given that the distance travelled from $P$ in time $t$ is equal to $\dfrac{1}{2}gt^2 \sin{\alpha}$, where $\alpha$ is the angle of inclination of the line $PQ$ to the horizontal, give a geometrical characterization of the point $Q$ for which the time taken from $P$ to $Q$ is a minimum.
1997 South africa National Olympiad
Six points are connected in pairs by lines, each of which is either red or blue. Every pair of points is joined. Determine whether there must be a closed path having four sides all of the same colour. (A path is closed if it begins and ends at the same point.)
1996 South africa National Olympiad
Find the highest power of $2$ that divides exactly into $1996!=1\times2\times\cdots\times1996$.
1996 South africa National Olympiad
Find all real numbers for which $3^x+4^x=5^x$.
1996 South africa National Olympiad
The sides of triangle $ABC$ has integer lengths. Given that $AC=6$ and $\angle BAC=120^\circ$, determine the lengths of the other two sides.
1996 South africa National Olympiad
In the Rainbow Nation there are two airways: Red Rockets and Blue Boeings. For any two cities in the Rainbow Nation it is possible to travel from the one to the other using either or both of the airways. It is known, however, that it is impossible to travel from Beanville to Mieliestad using only Red Rockets -  not directly nor by travelling via other cities. Show that, using only Blue Boeings, one can travel from any city to any other city by stopping at at most one city along the way.
1996 South africa National Olympiad
$ABC$ is a triangle with sides $1$, $2$ and $\sqrt3$. Determine the smallest possible area of an equilateral triangle with a vertex on each side of triangle $ABC$.
1996 South africa National Olympiad
The function $f$ is increasing and convex (i.e. every straight line between two points on the graph of $f$ lies above the graph) and satisfies $f(f(x))=3^x$ for all $x\in\mathbb{R}$. If $f(0)=0.5$ determine $f(0.75)$ with an error of at most $0.025$. The following are corrent to the number of digits given:



$$3^{0.25}=1.31607,\quad 3^{0.50}=1.73205,\quad 3^{0.75}=2.27951.$$
1995 South africa National Olympiad
Prove that there are no integers $m$ and $n$ such that

$$19m^2+95mn+2000n^2=1995.$$
1995 South africa National Olympiad
$ABC$ is a triangle with $\hat{A}<\hat{C}$, and $D$ is the point on $BC$ such that $B\hat{A}D=A\hat{C}B$. The perpendicular bisectors of $AD$ and $AC$ intersect in the point $E$. Prove that $B\hat{A}E=90^\circ$.
1995 South africa National Olympiad
Suppose that $a_1,a_2,\dots,a_n$ are the numbers $1,2,3,\dots,n$ but written in any order. Prove that

$$(a_1-1)^2+(a_2-2)^2+\cdots+(a_n-n)^2$$

is always even.
1995 South africa National Olympiad
Three circles, with radii $p$, $q$ and $r$ and centres $A$, $B$ and $C$ respectively, touch one another externally at points $D$, $E$ and $F$. Prove that the ratio of the areas of $\triangle DEF$ and $\triangle ABC$ equals

$$\frac{2pqr}{(p+q)(q+r)(r+p)}.$$
1995 South africa National Olympiad
The convex quadrilateral $ABCD$ has area $1$, and $AB$ is produced to $E$, $BC$ to $F$, $CD$ to $G$ and $DA$ to $H$, such that $AB=BE$, $BC=CF$, $CD=DG$ and $DA=AH$. Find the area of the quadrilateral $EFGH$.
1995 South africa National Olympiad
Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$.
1995 South africa National Olympiad
The circumcircle of $\triangle ABC$ has radius $1$ and centre $O$ and $P$ is a point inside the triangle such that $OP=x$. Prove that

$$AP\cdot BP\cdot CP\le(1+x)^2(1-x),$$

with equality if and only if $P=O$.
2021 South East Mathematical Olympiad
A sequence $\{a_n\}$ is defined recursively by $a_1=\frac{1}{2}, $ and for $n\ge 2,$ $0<a_n\leq a_{n-1}$ and

$$a_n^2(a_{n-1}+1)+a_{n-1}^2(a_n+1)-2a_na_{n-1}(a_na_{n-1}+a_n+1)=0.$$$(1)$ Determine the general formula of the sequence $\{a_n\};$

$(2)$ Let $S_n=a_1+\cdots+a_n.$ Prove that for $n\ge 1,$ $\ln\left(\frac{n}{2}+1\right)<S_n<\ln(n+1).$
2021 South East Mathematical Olympiad
In $\triangle ABC$,$AB=AC>BC$, point $O,H$ are the circumcenter and orthocenter of $\triangle ABC$ respectively,$G $ is the midpoint of segment $AH$ , $BE$ is the altitude on $AC$ . Prove that if $OE\parallel BC$, then $H$ is the incenter of $\triangle GBC$.
2021 South East Mathematical Olympiad
Let $p$ be an odd prime and $\{u_i\}_{i\ge 0}$be an integer sequence.

Let $v_n=\sum_{i=0}^{n} C_{n}^{i} p^iu_i$ where $C_n^i$ denotes the binomial coefficients.

If $v_n=0$ holds for infinitely many $n$ , prove that it holds for every positive integer $n$.
2021 South East Mathematical Olympiad
Suppose there are $n\geq{5}$ different points arbitrarily arranged on a circle, the labels are $1, 2,\dots $, and $n$, and the permutation is $S$. For a permutation , a “descending chain” refers to several consecutive points on the circle , and its labels is a clockwise descending sequence (the length of sequence is at least $2$), and the descending chain cannot be extended to longer .The point with the largest label in the chain is called the "starting point of descent", and the other  points in the chain are called the “non-starting point of descent” . For example: there are two descending chains $5, 2$and $4, 1$ in $5, 2, 4, 1, 3$ arranged in a clockwise direction, and $5$ and $4$ are their  starting points of descent respectively, and $2, 1$ is the non-starting point of descent . Consider the following operations: in the first round, find all descending chains in the permutation $S$, delete all non-starting points of descent , and then repeat the first round of operations for the arrangement of the remaining points, until no more descending chains can be found. Let $G(S)$ be the number of all descending chains that permutation $S$ has appeared in the operations, $A(S)$ be the average value of $G(S)$of all possible n-point permutations $S$.

(1) Find $A(5)$.

(2)For $n\ge{6}$ , prove that $\frac{83}{120}n-\frac{1}{2} \le A(S) \le \frac{101}{120}n-\frac{1}{2}.$
2021 South East Mathematical Olympiad
To commemorate the $43rd$ anniversary of the restoration of mathematics competitions, a mathematics enthusiast

arranges the first $2021$ integers $1,2,\dots,2021$ into a sequence $\{a_n\}$ in a certain order, so that the sum of any consecutive $43$ items in the sequence is a multiple of $43$.

(1) If the sequence of numbers is connected end to end into a circle, prove that the sum of any consecutive $43$ items on the circle is also a multiple of $43$;

(2) Determine the number of sequences $\{a_n\}$ that meets the conditions of the question.
2021 South East Mathematical Olympiad
Let $ABCD$ be a cyclic quadrilateral. Let $E$ be a point on side $BC,$ $F$ be a point on side $AE,$ $G$ be a point on the exterior angle bisector of $\angle BCD,$ such that $EG=FG,$ $\angle EAG=\dfrac12\angle BAD.$ Prove that $AB\cdot AF=AD\cdot AE.$
2021 South East Mathematical Olympiad
Let $a,b,c$ be pairwise distinct positive real, Prove that$$\dfrac{ab+bc+ca}{(a+b)(b+c)(c+a)}<\dfrac17(\dfrac{1}{|a-b|}+\dfrac{1}{|b-c|}+\dfrac{1}{|c-a|}).$$
2021 South East Mathematical Olympiad
Determine all the pairs of positive integers $(a,b),$ such that $$14\varphi^2(a)-\varphi(ab)+22\varphi^2(b)=a^2+b^2,$$where $\varphi(n)$ is Euler's totient function.
2021 South East Mathematical Olympiad
Let $p\geq 5$ be a prime number, and set $M=\{1,2,\cdots,p-1\}.$ Define $$T=\{(n,x_n):p|nx_n-1\ \textup{and}\ n,x_n\in M\}.$$If $\sum_{(n,x_n)\in T}n\left[\dfrac{nx_n}{p}\right]\equiv k \pmod {p},$ with $0\leq k\leq p-1,$ where $\left[\alpha\right]$ denotes the largest integer that does not exceed $\alpha,$ determine the value of $k.$
2021 South East Mathematical Olympiad
Let $a,b,c\geq 0$ and $a^2+b^2+c^2\leq 1.$ Prove that$$\frac{a}{a^2+bc+1}+\frac{b}{b^2+ca+1}+\frac{c}{c^2+ab+1}+3abc<\sqrt 3$$
2021 South East Mathematical Olympiad
For positive integer $k,$ we say that it is a Taurus integer if  we can delete one element from the set $M_k=\{1,2,\cdots,k\},$ such that the sum of remaining $k-1$ elements is a positive perfect square. For example, $7$ is a Taurus integer, because if we delete $3$ from $M_7=\{1,2,3,4,5,6,7\},$ the sum of remaining $6$ elements is $25,$ which is a positive perfect square.

$(1)$ Determine whether $2021$ is a Taurus integer.

$(2)$ For positive integer $n,$ determine the number of Taurus integers in $\{1,2,\cdots,n\}.$