Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 Slovenia National Olympiad
The incircle of a right triangle $ABC$ touches the hypotenuse $AB$ at a point $D$. Show that the area of $\triangle ABC$ equals $AD\cdot DB$.
1999 Slovenia National Olympiad
On a mountain, three shepherds cyclically alternate shearing the same herd of sheep. The shepherds agreed to obey the following rules:

(i) Every day a sheep can be shorn* on one side only;

(ii) Every day at least one sheep must be shorn;

(iii) No two days the same group of sheep can be shorn.

The shepherd who first breaks the agreement will have to accompany the herd in the valley next fall. Can anyone of the shepherds shear the sheep in such a way to make sure that he will avoid this punishment?



*shorn is the past tense of shear
1999 Slovenia National Olympiad
Prove that the product of three consecutive positive integers is never a perfect square.
1999 Slovenia National Olympiad
Three unit vectors $a,b,c$ are given on the plane. Prove that one can choose the signs in the expression $x=\pm a\pm b\pm c$ so as to obtain a vector $x$ with $|x|\le\sqrt2$.
1999 Slovenia National Olympiad
A semicircle with diameter $AB$ is given. Two non-intersecting circles $k_1$ and $k_2$ with different radii touch the diameter $AB$ and touch the semicircle internally at $C$ and $D$, respectively. An interior common tangent $t$ of $k_1$ and $k_2$ touches $k_1$ at $E$ and $k_2$ at $F$. Prove that the lines $CE$ and $DF$ intersect on the semicircle.
1999 Slovenia National Olympiad
Three integers are written on a blackboard. At every step one of them is erased and the sum of the other two decreased by $1$ is written instead. Is it possible to obtain the numbers $17,75,91$ if the three initial numbers were:

$\textbf{(a)}~2,2,2$;

$\textbf{(b)}~3,3,3$?
1999 Slovenia National Olympiad
What is the smallest possible value of $\left|12^m-5^n\right|$, where $m$ and $n$ are positive integers?
1999 Slovenia National Olympiad
Consider the polynomial $p(x)=x^{1999}+2x^{1998}+3x^{1997}+\ldots+2000$. Find a nonzero polynomial whose roots are the reciprocal values of the roots of $p(x)$.
1999 Slovenia National Olympiad
Let $O$ be the circumcenter of a triangle $ABC$, $P$ be the midpoint of $AO$, and $Q$ be the midpoint of $BC$. If $\angle ABC=4\angle OPQ$ and $\angle ACB=6\angle OPQ$, compute $\angle OPQ$.
1999 Slovenia National Olympiad
A pawn is put on each of $2n$ arbitrary selected cells of an $n\times n$ board ($n>1$). Prove that there are four cells that are marked with pawns and whose centers form a parallelogram.
1999 Slovenia National Olympiad
Let $r_1,r_2,\ldots,r_m$ be positive rational numbers with a sum of $1$. Find the maximum values of the function $f:\mathbb N\to\mathbb Z$ defined by

$$f(n)=n-\lfloor r_1n\rfloor-\lfloor r_2n\rfloor-\ldots-\lfloor r_mn\rfloor$$
1999 Slovenia National Olympiad
The numbers $1,\frac12,\frac13,\ldots,\frac1{1999}$ are written on a blackboard. In every step, we choose two of them, say $a$ and $b$, erase them, and write the number $ab+a+b$ instead. This step is repeated until only one number remains. Can the last remaining number be equal to $2000$?
1999 Slovenia National Olympiad
A section of a rectangular parallelepiped by a plane is a regular hexagon. Prove that this parallelepiped is a cube.
1999 Slovenia National Olympiad
Let be given three-element subsets $A_1,A_2,\ldots,A_6$ of a six-element set $X$. Prove that the elements of $X$ can be colored with two colors in such a way that none of the given subsets are monochromatic.
2000 Slovenia National Olympiad
In the expression $4\cdot\text{RAKEC}=\text{CEKAR}$, each letter represents a (decimal) digit. Replace the letters so that the equality is true.
2000 Slovenia National Olympiad
Find all real numbers $a$ for which the following equation has a unique real solution:

$$|x-1|+|x-2|+\ldots+|x-99|=a.$$
2000 Slovenia National Olympiad
Let $ABC$ be a triangle such that the altitude $CD$ is equal to $AB$. The squares $DBEF$ and $ADGH$ are constructed with $F,G$ on $CD$. Show that the segments $CD,AE$ and $BH$ are concurrent.
2000 Slovenia National Olympiad
All vertices of a convex $n$-gon ($n\ge3$) in the plane have integer coordinates. Show that its area is at least $\frac{n-2}2$.
2000 Slovenia National Olympiad
Let $n$ be the number of ordered $5$-tuples $(a_1,a_2,\ldots,a_5)$ of positive integers such that $\frac1{a_1}+\frac1{a_2}+\ldots+\frac1{a_5}=1$. Is $n$ an even number?
2000 Slovenia National Olympiad
Three students start walking with constant speeds at the same time, each along a straight line in the plane. Prove that if the students are not on the same line at the beginning, then they will be on the same line at most twice during their journey.
2000 Slovenia National Olympiad
A point $D$ is taken inside an isosceles triangle $ABC$ with base $AB$ and $\angle C=80^\circ$ such that $\angle DAB=10^\circ$ and $\angle DBA=20^\circ$. Compute $\angle ACD$.
2000 Slovenia National Olympiad
Alex and Jack have $1000$ sheets each. Each of them writes the numbers $1,\ldots,2000$ on his sheets in an arbitrary order, with one number on each side of a sheet. The sheets are to be placed on the floor so that one side of each sheet is visible. Prove that they can do so in such a way that each of the numbers from $1$ to $2000$ is visible.
2000 Slovenia National Olympiad
Find all prime numbers whose base $b$ representations (for some $b$) contain each of the digits $0,1,\ldots,b-1$ exactly once. (Digit $0$ may appear as the first digit.)
2000 Slovenia National Olympiad
Consider the polynomial $p(x)=a_nx^n+\ldots+a_1x+a_0$ with real coefficients such that $0\le a_i\le a_0$ for each $i=1,2,\ldots,n$. If $a$ is the coefficient of $x^{n+1}$ in the polynomial $q(x)=p(x)^2$, prove that $2a\le p(1)^2$.
2000 Slovenia National Olympiad
Let $H$ be the orthocenter of an acute-angled triangle $ABC$ with $AC\ne BC$. The line through the midpoints of the segments $AB$ and $HC$ intersects the bisector of $\angle ACB$ at $D$. Suppose that the line $HD$ contains the circumcenter of $\triangle ABC$. Determine $\angle ACB$.
2000 Slovenia National Olympiad
A pile of $2000$ coins is given on a table. In each step, we choose a pile with at least three coins, remove one coin from it, and divide the rest of this pile into two piles (not necessarily of the same size). Is it possible that after several steps each pile on the table has exactly three coins?
2000 Slovenia National Olympiad
The sequence $(a_n)$ is given by $a_1=2$, $a_2=500$, $a_3=2000$ and

$$\frac{a_{n+2}+a_{n+1}}{a_{n+1}+a_{n-1}}=\frac{a_{n+1}}{a_{n-1}}\qquad\text{for }n\ge2$$Prove that all terms of this sequence are positive integers and that $a_{2000}$ is divisible by $2^{2000}$.
2000 Slovenia National Olympiad
Find all functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$,

$$f(x-f(y))=1-x-y.$$
2000 Slovenia National Olympiad
The diagonals of a cyclic quadrilateral $ABCD$ intersect at $E$. Let $F$ and $G$ be the midpoints of $AB$ and $CD$ respectively. Prove that the lines through $E,F$ and $G$ perpendicular to $AD,BD$ and $AC$, respectively, intersect in a single point.
2000 Slovenia National Olympiad
Three boxes with at least one marble in each are given. In each step we double the number of marbles in one of the boxes, taking the required number of boxes from one of the other two boxes. Is it always possible to have one of the boxes empty after several steps?
2001 Slovenia National Olympiad
None of the positive integers $k,m,n$ are divisible by $5$. Prove that at least one of the numbers $k^2-m^2,m^2-n^2,n^2-k^2$ is divisible by $5$.
2001 Slovenia National Olympiad
Tina wrote a positive number on each of five pieces of paper. She did not say which numbers she wrote, but revealed their pairwise sums instead: $17,20,28,14,42,36,28,39,25,31$. Which numbers did she write?
2001 Slovenia National Olympiad
For an arbitrary point $P$ on a given segment $AB$, two isosceles right triangles $APQ$ and $PBR$ with the right angles at $Q$ and $R$ are constructed on the same side of the line $AB$. Prove that the distance from the midpoint $M$ of $QR$ to the line $AB$ does not depend on the choice of $P$.
2001 Slovenia National Olympiad
Andrej and Barbara play the following game with two strips of newspaper of length $a$ and $b$. They alternately cut from any end of any of the strips a piece of length $d$. The player who cannot cut such a piece loses the game. Andrej allows Barbara to start the game. Find out how the lengths of the strips determine the winner.
2001 Slovenia National Olympiad
Determine all positive integers $a,b,c$ such that $ab + ac + bc$ is a prime number and

$$\frac{a+b}{a+c}=\frac{b+c}{b+a}.$$
2001 Slovenia National Olympiad
Let $p(n)$ denote the product of decimal digits of a positive integer $n$. Computer the sum $p(1)+p(2)+\ldots+p(2001)$.
2001 Slovenia National Olympiad
Let $E$ and $F$ be points on the side $AB$ of a rectangle $ABCD$ such that $AE = EF$. The line through $E$ perpendicular to $AB$ intersects the diagonal $AC$ at $G$, and the segments $FD$ and $BG$ intersect at $H$. Prove that the areas of the triangles $FBH$ and $GHD$ are equal.
2001 Slovenia National Olympiad
Find the smallest number of squares on an $8\times8$ board that should be colored so that every $L$-tromino on the board contains at least one colored square.
2001 Slovenia National Olympiad
(a) Prove that $\sqrt{n+1}-\sqrt n<\frac1{2\sqrt n}<\sqrt n-\sqrt{n-1}$ for all $n\in\mathbb N$.

(b) Prove that the integer part of the sum $1+\frac1{\sqrt2}+\frac1{\sqrt3}+\ldots+\frac1{\sqrt{m^2}}$, where $m\in\mathbb N$, is either $2m-2$ or $2m-1$.
2001 Slovenia National Olympiad
Find all rational numbers $r$ such that the equation $rx^2 + (r + 1)x + r = 1$ has integer solutions.
2001 Slovenia National Olympiad
A point $D$ is taken on the side $BC$ of an acute-angled triangle $ABC$ such that $AB = AD$. Point $E$ on the altitude from $C$ of the triangle is such that the circle $k_1$ with center $E$ is tangent to the line $AD$ at $D$. Let $k_2$ be the circle through $C$ that is tangent to $AB$ at $B$. Prove that $A$ lies on the line determined by the common chord of $k_1$ and $k_2$.
2001 Slovenia National Olympiad
Cross-shaped tiles are to be placed on a $8\times8$ square grid without overlapping. Find the largest possible number of tiles that can be placed.
2001 Slovenia National Olympiad
Let $a,b,c,d,e,f$ be positive numbers such that $a,b,c,d$ is an arithmetic progression, and $a,e,f,d$ is a geometric progression. Prove that $bc\ge ef$.
2001 Slovenia National Olympiad
Find all prime numbers $p$ for which $3^p-(p+2)^2$ is also prime.
2001 Slovenia National Olympiad
Let $D$ be the foot of the altitude from $A$ in a triangle $ABC$. The angle bisector at $C$ intersects $AB$ at a point $E$. Given that $\angle CEA=\frac\pi4$, compute $\angle EDB$.
2001 Slovenia National Olympiad
Let $n\ge4$ points on a circle be denoted by $1$ through $n$. A pair of two nonadjacent points denoted by $a$ and $b$ is called regular if all numbers on one of the arcs determined by $a$ and $b$ are less than $a$ and $b$. Prove that there are exactly $n-3$ regular pairs.
2005 Slovenia National Olympiad
If $x,y,z$ are real numbers such that $xyz=1$, evaluate

$$\frac{x+1}{xy+x+1}+\frac{y+1}{yz+y+1}+\frac{z+1}{zx+z+1}.$$
2005 Slovenia National Olympiad
Find all prime numbers $p$ for which the number $p^2+11$ has less than $11$ divisors.
2005 Slovenia National Olympiad
Suppose that a triangle $ABC$ with incenter $I$ satisfies $CA+AI=BC$. Find the ratio between the measures of the angles $\angle BAC$ and $\angle CBA$.
2005 Slovenia National Olympiad
The friends Alex, Ben, and Charles prepared a lot of labels and wrote one of the numbers $2,3,4,5,6,7,8$ on each label. Then Mary joined them and glued one label onto the forehead of each friend. Of course, each of the friends can see the labels on the others’ foreheads, but not the one on his own forehead. Mary told them: ”The numbers on your foreheads are not all distinct, and their product is a perfect square.” Can any of the friends find out the number on his forehead?
2005 Slovenia National Olympiad
Find all real numbers $x,y$ such that $x^3-y^3=7(x-y)$ and $x^3+y^3=5(x+y)$.
2005 Slovenia National Olympiad
For which prime numbers $p$ and $q$ is $(p+1)^q$ a perfect square?
2005 Slovenia National Olympiad
Let $T$ be a point inside a square $ABCD$. The lines $TA,TB,TC,TD$ meet the circumcircle of $ABCD$ again at $A',B',C',D'$, respectively. Prove that $A'B'\cdot C'D'=A'D'\cdot B'C'$.
2005 Slovenia National Olympiad
The village chatterboxes are exchanging their gossip by phone every day so that any two of them talk to each other exactly once. A certain day, every chatterbox called up at least one of the other chatterboxes. Show that there exist three chatterboxes such that the first called up the second, the second called up the third, and the third called up the first.
2005 Slovenia National Olympiad
Evaluate the sum $\left\lfloor\log_21\right\rfloor+\left\lfloor\log_22\right\rfloor+\left\lfloor\log_23\right\rfloor+\ldots+\left\lfloor\log_2256\right\rfloor$.
2005 Slovenia National Olympiad
Find the smallest prime number $p$ for which the number $p^3+2p^2+p$ has exactly $42$ divisors.
2005 Slovenia National Olympiad
In an isosceles triangle $ABC$ with $AB = AC$, $D$ is the midpoint of $AC$ and $E$ is the projection of $D$ onto $BC$. Let $F$ be the midpoint of $DE$. Prove that the lines $BF$ and $AE$ are perpendicular if and only if the triangle $ABC$ is equilateral.
2005 Slovenia National Olympiad
Several teams from Littletown and Bigtown took part on a tournament. There were nine more teams from Bigtown than those from Littletown. Any two teams played exactly one match, and the winner and loser got 1 and 0 points respectively (no ties). The teams from Bigtown in total gained nine times more points than those from Littletown. What is the maximum possible number of wins of the best team from Littletown?
2005 Slovenia National Olympiad
Find all positive numbers $x$ such that $20\{x\}+0.5\lfloor x\rfloor = 2005$.
2005 Slovenia National Olympiad
Let $(a_n)$ be a geometrical progression with positive terms. Define $S_n=\log a_1+\log a_2+\ldots+\log a_n$. Prove that if $S_n=S_m$ for some $m\ne n$, then $S_{n+m}=0$.
2005 Slovenia National Olympiad
The tangent lines from a point $P$ meet a circle $k$ at $A$ and $B$. Let $X$ be an arbitrary point on the shorter arc $AB$, and $C$ and $D$ be the orthogonal projections of $P$ onto the lines $AX$ and $BX$, respectively. Prove that the line $CD$ passes through a fixed point $Y$ as $X$ moves along the arc $AB$.
2005 Slovenia National Olympiad
William was bored at the math lesson, so he drew a circle and $n\ge3$ empty cells around the circumference. In every cell he wrote a positive number. Later on he erased the numbers and in every cell wrote the geometric mean of the numbers previously written in the two neighboring cells. Show that there exists a cell whose number was not replaced by a larger number.
2010 Slovenia National Olympiad
For a real number $t$ and positive real numbers $a,b$ we have

$$2a^2-3abt+b^2=2a^2+abt-b^2=0$$

Find $t.$
2010 Slovenia National Olympiad
Find all prime numbers $p, q, r$ such that

$$15p+7pq+qr=pqr.$$
2010 Slovenia National Olympiad
Let $ABC$ be an isosceles triangle with apex at $C.$ Let $D$ and $E$ be two points on the sides $AC$ and $BC$ such that the angle bisectors $\angle DEB$ and $\angle ADE$ meet at $F,$ which lies on segment $AB.$ Prove that $F$ is the midpoint of $AB.$
2010 Slovenia National Olympiad
Find the smallest three-digit number such that the following holds:

If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits.
2010 Slovenia National Olympiad
Let $ABCD$ be a square with the side of $20$ units. Amir divides this square into $400$ unit squares. Reza then picks $4$ of the vertices of these unit squares. These vertices lie inside the square $ABCD$ and define a rectangle with the sides parallel to the sides of the square $ABCD.$ There are exactly $24$ unit squares which have at least one point in common with the sides of this rectangle. Find all possible values for the area of a rectangle with these properties.



NoteNote: Vid changed to Amir, and Eva change to Reza!
2010 Slovenia National Olympiad
Let $a,b$ be real numbers such that $|a| \neq |b|$ and $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6.$ Find the value of the expression $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}.$
2010 Slovenia National Olympiad
Let $a, b$ and $c$ be nonzero digits. Let $p$ be a prime number which divides the three digit numbers $\overline{abc}$ and $\overline{cba}.$ Show that $p$ divides at least one of the numbers $a+b+c, a-b+c$ and $a-c.$
2010 Slovenia National Olympiad
Let $ABC$ be an acute triangle. A line parallel to $BC$ intersects the sides $AB$ and $AC$ at $D$ and $E$, respectively. The circumcircle of the triangle $ADE$ intersects the segment $CD$ at $F  \ (F \neq D).$ Prove that the triangles $AFE$ and $CBD$ are similar.
2010 Slovenia National Olympiad
Let $x,y$ and $z$ be real numbers such that $0 \leq x,y,z \leq 1.$ Prove that

$$xyz+(1-x)(1-y)(1-z) \leq 1.$$

When does equality hold?
2010 Slovenia National Olympiad
Let $ABC$ be an equilateral triangle with the side of $20$ units. Amir divides this triangle into $400$ smaller equilateral triangles with the sides of $1$ unit. Reza then picks $4$ of the vertices of these smaller triangles. The vertices lie inside the triangle $ABC$ and form a parallelogram with sides parallel to the sides of the triangle $ABC.$ There are exactly $46$ smaller triangles that have at least one point in common with the sides of this parallelogram. Find all possible values for the area of this parallelogram.



[asy]

unitsize(150);

defaultpen(linewidth(0.7));

int n = 20; /* # of vertical lines, including BC */

pair A = (0,0), B = dir(-30), C = dir(30);

draw(A--B--C--cycle,linewidth(1)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0));

label("$A$",A,W); label("$C$",C,NE); label("$B$",B,SE);

for(int i = 1; i < n; ++i) {

draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n);

draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n);

draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n);

}[/asy]



[Thanks azjps for drawing the diagram.]



NoteNote: Vid changed to Amir, and Eva change to Reza!
2010 Slovenia National Olympiad
Let $a,b,c$ be positive integers. Prove that $a^2+b^2+c^2$ is divisible by $4$, if and only if $a,b,c$ are even.
2010 Slovenia National Olympiad
Find all real $x$ in the interval $[0, 2\pi)$ such that

$$27 \cdot 3^{3 \sin x} = 9^{\cos^2 x}.$$
2010 Slovenia National Olympiad
Let $ABC$ be an acute triangle with $|AB| > |AC|.$ Let $D$ be a point on the side $AB$, such that the angles $\angle ACD$ and $\angle CBD$ are equal. Let $E$ denote the midpoint of $BD,$ and let $S$ be the circumcenter of the triangle $BCD.$ Prove that the points $A, E, S$ and $C$ lie on the same circle.
2010 Slovenia National Olympiad
Find all non-zero real numbers $x$ such that

$$\min \left\{ 4, x+ \frac 4x \right\} \geq 8 \min \left\{ x,\frac 1x\right\} .$$
2010 Slovenia National Olympiad
Ten pirates find a chest filled with golden and silver coins. There are twice as many silver coins in the chest as there are golden. They divide the golden coins in such a way that the difference of the numbers of coins given to any two of the pirates is not divisible by $10.$ Prove that they cannot divide the silver coins in the same way.
2010 Slovenia National Olympiad
Find all prime numbers $p, q$ and $r$ such that $p>q>r$ and the numbers $p-q, p-r$ and $q-r$ are also prime.
2010 Slovenia National Olympiad
Let $\mathfrak K_1$ and $\mathfrak K_2$ be circles centered at $O_1$ and $O_2,$ respectively, meeting at the points $A$ and $B.$ Let $p$ be the line through the point $A$ meeting the circles $\mathfrak K_1$ and $\mathfrak K_2$ again at $C_1$ and $C_2.$ Assume that $A$ lies between $C_1$ and $C_2.$ Denote the intersection of the lines $C_1O_1$ and $C_2O_2$ by $D.$ Prove that the points $C_1, C_2, B$ and $D$ lie on the same circle.
2010 Slovenia National Olympiad
Find all functions $f: [0, +\infty) \to [0, +\infty)$ satisfying the equation

$$(y+1)f(x+y) = f\left(xf(y)\right)$$

For all non-negative real numbers $x$ and $y.$
2010 Slovenia National Olympiad
For real numbers $a, b$ and $c$ we have

$$(2b-a)^2 + (2b-c)^2 = 2(2b^2-ac).$$

Prove that the numbers $a, b$ and $c$ are three consecutive terms in some arithmetic sequence.
2010 Slovenia National Olympiad
For what positive integers $n \geq 3$ does there exist a polygon with $n$ vertices (not necessarily convex) with property that each of its sides is parallel to another one of its sides?
2021 South Africa National Olympiad
Find the smallest and largest integers with decimal representation of the form $ababa$ ($a \neq 0$) that are divisible by $11$.
2021 South Africa National Olympiad
Let $PAB$ and $PBC$ be two similar right-angled triangles (in the same plane) with $\angle PAB = \angle PBC = 90^\circ$ such that $A$ and $C$ lie on opposite sides of the line $PB$. If $PC = AC$, calculate the ratio $\frac{PA}{AB}$.
2021 South Africa National Olympiad
Determine the smallest integer $k > 1$ such that there exist $k$ distinct primes whose squares sum to a power of $2$.
2021 South Africa National Olympiad
Let $ABC$ be a triangle with $\angle ABC \neq 90^\circ$ and $AB$ its shortest side. Denote by $H$ the intersection of the altitudes of triangle $ABC$. Let $K$ be the circle through $A$ with centre $B$. Let $D$ be the other intersection of $K$ and $AC$. Let $K$ intersect the circumcircle of $BCD$ again at $E$. If $F$ is the intersection of $DE$ and $BH$, show that $BD$ is tangent to the circle through $D$, $F$, and $H$.
2021 South Africa National Olympiad
Determine all polynomials $a(x)$, $b(x)$, $c(x)$, $d(x)$ with real coefficients satisfying the simultaneous equations

\begin{align*}

  b(x) c(x) + a(x) d(x) & = 0 \\
  a(x) c(x) + (1 - x^2) b(x) d(x) & = x + 1.

\end{align*}
2021 South Africa National Olympiad
Jacob and Laban take turns playing a game. Each of them starts with the list of square numbers $1, 4, 9, \dots, 2021^2$, and there is a whiteboard in front of them with the number $0$ on it. Jacob chooses a number $x^2$ from his list, removes it from his list, and replaces the number $W$ on the whiteboard with $W + x^2$. Laban then does the same with a number from his list, and the repeat back and forth until both of them have no more numbers in their list. Now every time that the number on the whiteboard is divisible by $4$ after a player has taken his turn, Jacob gets a sheep. Jacob wants to have as many sheep as possible. What is the greatest number $K$ such that Jacob can guarantee to get at least $K$ sheep by the end of the game, no matter how Laban plays?
2020 South Africa National Olympiad
Find the smallest positive multiple of $20$ with exactly $20$ positive divisors.
2020 South Africa National Olympiad
Let $S$ be a square with sides of length $2$ and $R$ be a rhombus with sides of length $2$ and angles measuring $60^\circ$ and $120^\circ$. These quadrilaterals are arranged to have the same centre and the diagonals of the rhombus are parallel to the sides of the square. Calculate the area of the region on which the figures overlap.
2020 South Africa National Olympiad
If $x$, $y$, $z$ are real numbers satisfying

\begin{align*}

(x + 1)(y + 1)(z + 1) & = 3 \\
(x + 2)(y + 2)(z + 2) & = -2 \\
(x + 3)(y + 3)(z + 3) & = -1, 

\end{align*}find the value of

$$ (x + 20)(y + 20)(z + 20). $$
2020 South Africa National Olympiad
A positive integer $k$ is said to be visionary if there are integers $a > 0$ and $b \geq 0$ such that $a \cdot k + b \cdot (k + 1) = 2020.$ How many visionary integers are there?
2020 South Africa National Olympiad
Let $ABC$ be a triangle, and let $T$ be a point on the extension of $AB$ beyond $B$, and $U$ a point on the extension of $AC$ beyond $C$, such that $BT = CU$. Moreover, let $R$ and $S$ be points on the extensions of $AB$ and $AC$ beyond $A$ such that $AS = AT$ and $AR = AU$. Prove that $R$, $S$, $T$, $U$ lie on a circle whose centre lies on the circumcircle of $ABC$.