Navvye/Olympiad-Problems-CSV huggingface dataset:
2020 South Africa National Olympiad
Marjorie is the drum major of the world's largest marching band, with more than one million members. She would like the band members to stand in a square formation. To this end, she determines the smallest integer $n$ such that the band would fit in an $n \times n$ square, and lets the members form rows of $n$ people. However, she is dissatisfied with the result, since some empty positions remain. Therefore, she tells the entire first row of $n$ members to go home and repeats the process with the remaining members. Her aim is to continue it until the band forms a perfect square, but as it happens, she does not succeed until the last members are sent home. Determine the smallest possible number of members in this marching band.
2019 South Africa National Olympiad
Determine all positive integers $a$ for which $a^a$ is divisible by $20^{19}$.
2019 South Africa National Olympiad
We have a deck of $90$ cards that are numbered from $10$ to $99$ (all two-digit numbers). How many sets of three or more different cards in this deck are there such that the number on one of them is the sum of the other numbers, and those other numbers are consecutive?
2019 South Africa National Olympiad
Let $A$, $B$, $C$ be points on a circle whose centre is $O$ and whose radius is $1$, such that $\angle BAC = 45^\circ$. Lines $AC$ and $BO$ (possibly extended) intersect at $D$, and lines $AB$ and $CO$ (possibly extended) intersect at $E$. Prove that $BD \cdot CE = 2$.
2019 South Africa National Olympiad
The squares of an $8 \times 8$ board are coloured alternatingly black and white. A rectangle consisting of some of the squares of the board is called important if its sides are parallel to the sides of the board and all its corner squares are coloured black. The side lengths can be anything from $1$ to $8$ squares. On each of the $64$ squares of the board, we write the number of important rectangles in which it is contained. The sum of the numbers on the black squares is $B$, and the sum of the numbers on the white squares is $W$. Determine the difference $B - W$.
2019 South Africa National Olympiad
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that

$$

  f(a^3) + f(b^3) + f(c^3) + 3f(a + b)f(b + c)f(c + a) = {(f(a + b + c))}^3

$$for all integers $a, b, c$.
2019 South Africa National Olympiad
Determine all pairs $(m, n)$ of non-negative integers that satisfy the equation

$$

  20^m - 10m^2 + 1 = 19^n.

$$
2018 South Africa National Olympiad
One hundred empty glasses are arranged in a $10 \times 10$ array. Now we pick $a$ of the rows and pour blue liquid into all glasses in these rows, so that they are half full. The remaining rows are filled halfway with yellow liquid. Afterwards, we pick $b$ of the columns and fill them up with blue liquid. The remaining columns are filled up with yellow liquid. The mixture of blue and yellow liquid turns green. If both halves have the same colour, then that colour remains as it is.



Determine all possible combinations of values for $a$ and $b$ so that exactly half of the glasses contain green liquid at the end.

Is it possible that precisely one quarter of the glasses contain green liquid at the end?
2018 South Africa National Olympiad
In a triangle $ABC$, $AB = AC$, and $D$ is on $BC$. A point $E$ is chosen on $AC$, and a point $F$ is chosen on $AB$, such that $DE = DC$ and $DF = DB$. It is given that $\frac{DC}{BD} = 2$ and $\frac{AF}{AE} = 5$. Determine that value of $\frac{AB}{BC}$.
2018 South Africa National Olympiad
Determine the smallest positive integer $n$ whose prime factors are all greater than $18$, and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$.
2018 South Africa National Olympiad
Let $ABC$ be a triangle with circumradius $R$, and let $\ell_A, \ell_B, \ell_C$ be the altitudes through $A, B, C$ respectively. The altitudes meet at $H$. Let $P$ be an arbitrary point in the same plane as $ABC$. The feet of the perpendicular lines through $P$ onto $\ell_A, \ell_B, \ell_C$ are $D, E, F$ respectively. Prove that the areas of $DEF$ and $ABC$ satisfy the following equation:

$$

  \operatorname{area}(DEF) = \frac{{PH}^2}{4R^2} \cdot \operatorname{area}(ABC).

$$
2018 South Africa National Olympiad
Determine all sequences $a_1, a_2, a_3, \dots$ of nonnegative integers such that $a_1 < a_2 < a_3 < \dots$ and $a_n$ divides $a_{n - 1} + n$ for all $n \geq 2$.
2018 South Africa National Olympiad
Let $n$ be a positive integer, and let $x_1, x_2, \dots, x_n$ be distinct positive integers with $x_1 = 1$. Construct an $n \times 3$ table where the entries of the $k$-th row are $x_k, 2x_k, 3x_k$ for $k = 1, 2, \dots, n$. Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table.



Prove that at least three entries remain at the end of the procedure.

Prove that there are infinitely many possible choices for $n$ and $x_1, x_2, \dots, x_n$ such that only three entries remain.
2017 South Africa National Olympiad
Together, the two positive integers $a$ and $b$ have $9$ digits and contain each of the digits $1, 2, 3, 4, 5, 6, 7, 8, 9$ exactly once. For which possible values of $a$ and $b$ is the fraction $a/b$ closest to $1$?
2017 South Africa National Olympiad
Let $ABCD$ be a rectangle with side lengths $AB = CD = 5$ and $BC = AD = 10$. $W, X, Y, Z$ are points on $AB, BC, CD$ and $DA$ respectively chosen in such a way that $WXYZ$ is a kite, where $\angle ZWX$ is a right angle. Given that $WX = WZ = \sqrt{13}$ and $XY = ZY$, determine the length of $XY$.
2017 South Africa National Olympiad
A representation of $\frac{17}{20}$ as a sum of reciprocals

$$

  \frac{17}{20} = \frac{1}{a_1} + \frac{1}{a_2} + \cdots + \frac{1}{a_k}

$$is called a calm representation with $k$ terms if the $a_i$ are distinct positive integers and at most one of them is not a power of two.



(a) Find the smallest value of $k$ for which $\frac{17}{20}$ has a calm representation with $k$ terms.

(b) Prove that there are infinitely many calm representations of $\frac{17}{20}$.
2017 South Africa National Olympiad
Andile and Zandre play a game on a $2017 \times 2017$ board. At the beginning, Andile declares some of the squares forbidden, meaning the nothing may be placed on such a square. After that, they take turns to place coins on the board, with Zandre placing the first coin. It is not allowed to place a coin on a forbidden square or in the same row or column where another coin has already been placed. The player who places the last coin wins the game.



What is the least number of squares Andile needs to declare as forbidden at the beginning to ensure a win? (Assume that both players use an optimal strategy.)
2017 South Africa National Olympiad
Let $ABC$ be a triangle with circumcircle $\Gamma$. Let $D$ be a point on segment $BC$ such that $\angle BAD = \angle DAC$, and let $M$ and $N$ be points on segments $BD$ and $CD$, respectively, such that $\angle MAD = \angle DAN$. Let $S, P$ and $Q$ (all different from $A$) be the intersections of the rays $AD$, $AM$ and $AN$ with $\Gamma$, respectively.



Show that the intersection of $SM$ and $QD$ lies on $\Gamma$.
2017 South Africa National Olympiad
Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.
2016 South African National Olympiad
At the start of the Mighty Mathematicians Football Team's first game of the season, their coach noticed that the jersey numbers of the 22 players on the field were all the numbers from 1 to 22. At halftime, the coach substituted her goal-keeper, with jersey number 1, for a reserve player. No other substitutions were made by either team at or before halftime. The coach noticed that after the substitution, no two players on the field had the same jersey number and that the sums of the jersey numbers of each of the teams were exactly equal.  Determine

* the greatest possible jersey number of the reserve player,

* the smallest possible (positive) jersey number of the reserve player.
2016 South African National Olympiad
Determine all pairs of real numbers $a$ and $b$, $b > 0$, such that the solutions to the two equations

$$x^2 + ax + a = b \qquad \text{and} \qquad x^2 + ax + a = -b$$are four consecutive integers.
2016 South African National Olympiad
The inscribed circle of triangle $ABC$, with centre $I$, touches sides $BC$, $CA$ and $AB$ at $D$, $E$ and

$F$, respectively. Let $P$ be a point, on the same side of $FE$ as $A$, for which $\angle PFE = \angle BCA$

and $\angle PEF = \angle ABC$. Prove that $P$, $I$ and $D$ lie on a straight line.
2016 South African National Olympiad
For which integers $n \geq 2$ is it possible to draw $n$ straight lines in the plane in such a way that there are at least $n - 2$ points where exactly three of the lines meet?
2016 South African National Olympiad
For every positive integer $n$, determine the greatest possible value of the quotient

$$\frac{1-x^{n}-(1-x)^{n}}{x(1-x)^n+(1-x)x^n}$$where $0 < x < 1$.
2016 South African National Olympiad
Let $k$ and $m$ be integers with $1 < k < m$. For a positive integer $i$, let $L_i$ be the least common multiple of $1,2,\ldots,i$.

Prove that $k$ is a divisor of $L_i \cdot [\binom{m}{i} - \binom{m-k}{i}]$ for all $i \geq 1$. [Here, $\binom{n}{i} = \frac{n!}{i!(n-i)!}$ denotes a binomial coefficient. Note that $\binom{n}{i} = 0$ if $n < i$.]
2015 South Africa National Olympiad
Points $E$ and $F$ lie inside a square $ABCD$ such that the two triangles $ABF$ and $BCE$ are equilateral. Show that $DEF$ is an equilateral triangle.
2015 South Africa National Olympiad
Determine all pairs of real numbers $a$ and $x$ that satisfy the simultaneous equations $$5x^3 + ax^2 + 8 = 0$$ and $$5x^3 + 8x^2 + a = 0.$$
2015 South Africa National Olympiad
We call a divisor $d$ of a positive integer $n$ special if $d + 1$ is also a divisor of $n$. Prove: at most half the positive divisors of a positive integer can be special. Determine all positive integers for which exactly half the positive divisors are special.
2015 South Africa National Olympiad
Let $ABC$ be an acute-angled triangle with $AB < AC$, and let points $D$ and $E$ be chosen on the side $AC$ and $BC$ respectively in such a way that $AD = AE = AB$. The circumcircle of $ABE$ intersects the line $AC$ at $A$ and $F$ and the line $DE$ at $E$ and $P$. Prove that $P$ is the circumcentre of $BDF$.
2015 South Africa National Olympiad
Several small villages are situated on the banks of a straight river. On one side, there are 20 villages in a row, and on the other there are 15 villages in a row. We would like to build bridges, each of which connects a village on the one side with a village on the other side. The bridges must not cross, and it should be possible to get from any village to any other village using only those bridges (and not any roads that might exist between villages on the same side of the river). How many different ways are there to build the bridges.
2015 South Africa National Olympiad
Suppose that $a$ is an integer and that $n! + a$ divides $(2n)!$ for infinitely many positive integers $n$. Prove that $a = 0$.
2014 South africa National Olympiad
Determine the last two digits of the product of the squares of all positive odd integers less than $2014$.
2014 South africa National Olympiad
Given that

$$\frac{a-b}{c-d}=2\quad\text{and}\quad\frac{a-c}{b-d}=3$$

for certain real numbers $a,b,c,d$, determine the value of

$$\frac{a-d}{b-c}.$$
2014 South africa National Olympiad
In obtuse triangle $ABC$, with the obtuse angle at $A$, let $D$, $E$, $F$ be the feet of the altitudes through $A$, $B$, $C$ respectively. $DE$ is parallel to $CF$, and $DF$ is parallel to the angle bisector of $\angle BAC$. Find the angles of the triangle.
2014 South africa National Olympiad
(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also

$$ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).$$



(b) Show that there are no two positive integers $a$ and $b$ such that

$$ab+\gcd(a,b)+\text{lcm}(a,b)=2014.$$
2014 South africa National Olympiad
Let $n > 1$ be an integer. An $n \times n$-square is divided into $n^2$ unit squares. Of these unit squares, $n$ are coloured green and $n$ are coloured blue, and all remaining ones are coloured white. Are there more such colourings for which there is exactly one green square in each row and exactly one blue square in each column; or colourings for which there is exactly one green square and exactly one blue square in each row?
2014 South africa National Olympiad
Let $O$ be the centre of a two-dimensional coordinate system, and let $A_1, A_2,  \ldots ,A_n$ be points in the first quadrant and $B_1, B_2, \ldots , B_m$ points in the second quadrant. We associate numbers $a_1, a_2, \ldots , a_n$ to the points $A_1, A_2, \ldots ,A_n$ and numbers $b_1, b_2, \ldots, b_m$ to the points $B_1, B_2, \ldots , B_m$, respectively. It turns out that the area of triangle $OA_jB_k$ is always equal to the product $a_jb_k$, for any $j$ and $k$. Show that either all the $A_j$ or all the $B_k$ lie on a single line through $O$.
2013 South africa National Olympiad
2013 is the first year since the Middle Ages that consists of four consecutive digits. How many such years are there still to come after 2013 (and before 10000)?
2013 South africa National Olympiad
A is a two-digit number and B is a three-digit number such that A increased by B% equals B reduced by A%. Find all possible pairs (A, B).
2013 South africa National Olympiad
Let ABC be an acute-angled triangle and AD one of its altitudes (D on BC). The line through D parallel to AB is denoted by $l$, and t is the tangent to the circumcircle of ABC at A. Finally, let E be the intersection of $l$ and t. Show that CE and t are perpendicular to each other.
2013 South africa National Olympiad
Determine all pairs of polynomials $f$ and $g$ with real coefficients such that $$ x^2 \cdot g(x) = f(g(x)). $$
2013 South africa National Olympiad
Some coins are placed on a $20 \times 13$ board. Two coins are called neighbours if they are in the same row or column and no other coins lie between them. What is the largest number of coins that can be placed on the board if no coin is allowed to have more than two neighbours?
2013 South africa National Olympiad
Let $ABC$ be an acute-angled triangle with $AC \neq BC$, and let $O$ be the circumcentre and $F$ the foot of the altitude through $C$. Furthermore, let $X$ and $Y$ be the feet of the perpendiculars dropped from $A$ and $B$ respectively to (the extension of) $CO$. The line $FO$ intersects the circumcircle of $FXY$ a second time at $P$. Prove that $OP<OF$.
2012 South africa National Olympiad
Given that

$\frac{1+3+5+\cdots+(2n-1)}{2+4+6+\cdots+(2n)}=\frac{2011}{2012}$,

determine n.
2012 South africa National Olympiad
Let $ABCD$ be a square and $X$ a point such that $A$ and $X$ are on opposite sides of $CD$. The lines $AX$ and $BX$ intersect $CD$ in $Y$ and $Z$ respectively. If the area of $ABCD$ is $1$ and the area of $XYZ$ is $\frac{2}{3}$, determine the length of $YZ$
2012 South africa National Olympiad
Sixty points, of which thirty are coloured red, twenty are coloured blue and ten are coloured green, are marked on a circle. These points divide the circle into sixty arcs. Each of these arcs is assigned a number according to the colours of its endpoints: an arc between a red and a green point is assigned a number $1$, an arc between a red and a blue point is assigned a number $2$, and an arc between a blue and a green point is assigned a number $3$. The arcs between two points of the same colour are assigned a number $0$. What is the greatest possible sum of all the numbers assigned to the arcs?
2012 South africa National Olympiad
Let $p$ and $k$ be positive integers such that $p$ is prime and $k>1$. Prove that there is at most one pair $(x,y)$ of positive integers such that

$x^k+px=y^k$.
2012 South africa National Olympiad
Let $ABC$ be a triangle such that $AB\neq AC$. We denote its orthocentre by $H$, its circumcentre by $O$ and the midpoint of $BC$ by $D$. The extensions of $HD$ and $AO$ meet in $P$. Prove that triangles $AHP$ and $ABC$ have the same centroid.
2012 South africa National Olympiad
Find all functions $f:\mathbb{N}\to\mathbb{R}$ such that

$f(km)+f(kn)-f(k)f(mn)\ge 1$

for all $k,m,n\in\mathbb{N}$.
2011 South africa National Olympiad
Consider the sequence $2, 3, 5, 6, 7, 8, 10, ...$ of all positive integers that are not perfect squares.

Determine the $2011^{th}$ term of the sequence.
2011 South africa National Olympiad
Suppose that $x$ and $y$ are real numbers that satisfy the system of equations



$2^x-2^y=1$

$4^x-4^y=\frac{5}{3}$



Determine $x-y$
2011 South africa National Olympiad
We call a sequence of $m$ consecutive integers a friendly sequence if its first term is divisible by $1$, the second by $2$, ..., the $(m-1)^{th}$ by $m-1$, and in addition, the last term is divisible by $m^2$



Does a friendly sequence exist for (a) $m=20$ and (b) $m=11$?
2011 South africa National Olympiad
An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network feasible if it satisfies the following conditions:



All connections operate in both directions

If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.





Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?
2011 South africa National Olympiad
Let $\mathbb{N}_0$ denote the set of all nonnegative integers. Determine all functions $f:\mathbb{N}_0\to\mathbb{N}_0$ with the following two properties:



$0\le f(x)\le x^2$ for all $x\in\mathbb{N}_0$



$x-y$ divides $f(x)-f(y)$ for all $x,y\in\mathbb{N}_0$ with $x>y$
2011 South africa National Olympiad
In triangle $ABC$, the incircle touches $BC$ in $D$, $CA$ in $E$ and $AB$ in $F$. The bisector of $\angle BAC$ intersects $BC$ in $G$. The lines $BE$ and $CF$ intersect in $J$. The line through $J$ perpendicular to $EF$ intersects $BC$ in $K$. Prove that



$\frac{GK}{DK}=\frac{AE}{CE}+\frac{AF}{BF}$
2010 South africa National Olympiad
For a positive integer $n$, $S(n)$ denotes the sum of its digits and $U(n)$ its unit digit. Determine all positive integers $n$ with the property that

$$n = S(n) + U(n)^2.$$
2010 South africa National Olympiad
Consider a triangle $ABC$ with $BC = 3$. Choose a point $D$ on $BC$ such that $BD = 2$. Find the value of

$$AB^2 + 2AC^2 - 3AD^2.$$
2010 South africa National Olympiad
Determine all positive integers $n$ such that $5^n - 1$ can be written as a product of an even number of consecutive integers.
2010 South africa National Olympiad
Given $n$ positive real numbers satisfying $x_1 \ge x_2 \ge \cdots \ge x_n \ge 0$ and $x_1^2+x_2^2+\cdots+x_n^2=1$, prove that

$$\frac{x_1}{\sqrt{1}}+\frac{x_2}{\sqrt{2}}+\cdots+\frac{x_n}{\sqrt{n}}\ge 1.$$
2010 South africa National Olympiad
(a) A set of lines is drawn in the plane in such a way that they create more than 2010 intersections at a particular angle $\alpha$. Determine the smallest number of lines for which this is possible.



(b) Determine the smallest number of lines for which it is possible to obtain exactly 2010 such intersections.
2010 South africa National Olympiad
Write either $1$ or $-1$ in each of the cells of a $(2n) \times (2n)$-table, in such a way that there are exactly $2n^2$ entries of each kind. Let the minimum of the absolute values of all row sums and all column sums be $M$. Determine the largest possible value of $M$.
2009 South africa National Olympiad
Determine the smallest integer $n > 1$ with the property that $n^2(n - 1)$ is divisible by 2009.
2009 South africa National Olympiad
Let $ABCD$ be a rectangle and $E$ the reflection of $A$ with respect to the diagonal $BD$. If $EB = EC$, what is the ratio $\frac{AD}{AB}$ ?
2009 South africa National Olympiad
Ten girls, numbered from 1 to 10, sit at a round table, in a random order. Each girl then receives a new number, namely the sum of her own number and those of her two neighbours. Prove that some girl receives a new number greater than 17.
2009 South africa National Olympiad
Let $x_1,x_2,\dots,x_n$ be a finite sequence of real numbersm mwhere $0<x_i<1$ for all $i=1,2,\dots,n$. Put $P=x_1x_2\cdots x_n$, $S=x_1+x_2+\cdots+x_n$ and $T=\frac{1}{x_1}+\frac{1}{x_2}+\cdots+\frac{1}{x_n}$. Prove that

$$\frac{T-S}{1-P}>2.$$
2009 South africa National Olympiad
A game is played on a board with an infinite row of holes labelled $0, 1, 2, \dots$. Initially, $2009$ pebbles are put into hole $1$; the other holes are left empty. Now steps are performed according to the following scheme:



(i) At each step, two pebbles are removed from one of the holes (if possible), and one pebble is put into each of the neighbouring holes.



(ii) No pebbles are ever removed from hole $0$.



(iii) The game ends if there is no hole with a positive label that contains at least two pebbles.



Show that the game always terminates, and that the number of pebbles in hole $0$ at the end of the game is independent of the specific sequence of steps. Determine this number.
2009 South africa National Olympiad
Let $A$ denote the set of real numbers $x$ such that $0\le x<1$. A function $f:A\to \mathbb{R}$ has the properties:



(i) $f(x)=2f(\frac{x}{2})$ for all $x\in A$;



(ii) $f(x)=1-f(x-\frac{1}{2})$ if $\frac{1}{2}\le x<1$.



Prove that



(a) $f(x)+f(1-x)\ge \frac{2}{3}$ if $x$ is rational and $0<x<1$.



(b) There are infinitely many odd positive integers $q$ such that equality holds in (a) when $x=\frac{1}{q}$.
2008 South africa National Olympiad
Determine the number of positive divisors of $2008^8$ that are less than $2008^4$.
2008 South africa National Olympiad
Let $ABCD$ be a convex quadrilateral with the property that $AB$ extended and $CD$ extended intersect at a right angle. Prove that $AC\cdot BD>AD\cdot BC$.
2008 South africa National Olympiad
Let $a,b,c$ be positive real numbers. Prove that

$$(a+b)(b+c)(c+a)\ge 8(a+b-c)(b+c-a)(c+a-b)$$

and determine when equality occurs.
2008 South africa National Olympiad
A pack of $2008$ cards, numbered from $1$ to $2008$, is shuffled in order to play a game in which each move has two steps:



(i) the top card is placed at the bottom;



(ii) the new top card is removed.



It turns out that the cards are removed in the order $1,2,\dots,2008$. Which card was at the top before the game started?
2008 South africa National Olympiad
Triangle $ABC$ has orthocentre $H$. The feet of the perpendiculars from $H$ to the internal and external bisectors of $\hat{A}$ are $P$ and $Q$ respectively. Prove that $P$ is on the line that passes through $Q$ and the midpoint of $BC$. (Note: The ortohcentre of a triangle is the point where the three altitudes intersect.)
2008 South africa National Olympiad
Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$,

$$f(a+b)=f(a)g(b)+g(a)f(b)\\
g(a+b)=g(a)g(b)-f(a)f(b).$$
2007 South africa National Olympiad
Determine whether $ \frac{1}{\sqrt{2}} - \frac{1}{\sqrt{6}}$ is less than or greater than $ \frac{3}{10}$.
2007 South africa National Olympiad
Consider the equation $ x^4 = ax^3 + bx^2 + cx + 2007$, where $ a,b,c$ are real numbers. Determine the largest value of $ b$ for which this equation has exactly three distinct solutions, all of which are integers.
2007 South africa National Olympiad
In acute-angled triangle $ ABC$, the points $ D,E,F$ are on sides $ BC,CA,AB$, respectively such that $ \angle AFE = \angle BFD, \angle FDB = \angle EDC, \angle DEC = \angle FEA$. Prove that $ AD$ is perpendicular to $ BC$.