Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 Sharygin Geometry Olympiad
Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le  \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.
2005 Sharygin Geometry Olympiad
Given a circle centered at the origin.

Prove that there is a circle of smaller radius that has no less points with integer coordinates.
2005 Sharygin Geometry Olympiad
We took a non-equilateral acute-angled triangle and marked $4$ wonderful points in it: the centers of the inscribed and circumscribed circles, the center of gravity (the point of intersection of the medians) and the point of intersection of heights. Then the triangle itself was erased. It turned out that it was impossible to establish which of the centers corresponds to each of the marked points. Find the angles of the triangle
2005 Sharygin Geometry Olympiad
A circle is inscribed in the triangle $ ABC$ and it's center $I$ and the points of tangency $P, Q, R$ with the sides $BC$, $C A$ and $AB$ are marked, respectively. With a single ruler, build a point $K$ at which the circle passing through the vertices B and $C$ touches (internally) the inscribed circle.
2005 Sharygin Geometry Olympiad
On the plane are three straight lines $\ell_1, \ell_2,\ell_3$, forming a triangle, and the point $O$ is marked,  the center of the circumscribed circle of this triangle. For an arbitrary point X of the plane, we denote by $X_i$ the point symmetric to the point X with respect to the line $\ell_i, i = 1,2,3$.

a) Prove that for an arbitrary point $M$ the straight lines connecting the midpoints of the segments $O_1O_2$ and $M_1M_2, O_2O_3$ and $M_2M_3, O_3O_1$ and $M_3M_1$ intersect at one point,

b) where can this intersection point lie?
2005 Sharygin Geometry Olympiad
As you know, the moon revolves around the earth. We assume that the Earth and the Moon are points, and the Moon rotates around the Earth in a circular orbit with a period of one revolution per month.

The flying saucer is in the plane of the lunar orbit. It can be jumped through the Moon and the Earth - from the old place (point $A$), it instantly appears in the new (at point $A '$) so that either the Moon or the Earth is in the middle of segment $AA'$. Between the jumps, the flying saucer hangs motionless in outer space.

1) Determine the minimum number of jumps a flying saucer will need to jump from any point inside the lunar orbit to any other point inside the lunar orbit.

2) Prove that a flying saucer, using an unlimited number of jumps, can jump from any point inside the lunar orbit to any other point inside the lunar orbit for any period of time, for example, in a second.
2005 Sharygin Geometry Olympiad
Let $I$ be the center of the sphere inscribed in the tetrahedron $ABCD, A ', B', C ', D'$ be the centers of the spheres circumscribed around the tetrahedra $IBCD, ICDA, IDAB, IABC$, respectively. Prove that the sphere circumscribed around $ABCD$ lies entirely inside the circumscribed around $A'B'C'D '$.
2005 Sharygin Geometry Olympiad
The planet  Tetraincognito covered by  ocean has the shape of a regular tetrahedron with an edge of $900$ km. What area of the ocean will the tsunami' cover $2$ hours after the earthquake with the epicenter in

a) the center of the face,

b) the middle of the edge,

if the tsunami propagation speed is $300$ km / h?
2005 Sharygin Geometry Olympiad
Perpendiculars at their centers of gravity (points of intersection of medians) are restored to the faces of the tetrahedron. Prove that the projections of the three perpendiculars to the fourth face intersect at one point.
2005 Sharygin Geometry Olympiad
Envelop the cube in one layer with five convex pentagons of equal areas.
2005 Sharygin Geometry Olympiad
A triangle is given, all the angles of which are smaller than $\phi$, where $\phi <2\pi / 3$. Prove that in space there is a point from which all sides of the triangle are visible at an angle $\phi$.
2005 Sharygin Geometry Olympiad
The quadrangle $ABCD$ is inscribed in a circle whose center $O$ lies inside it.

Prove that if $\angle BAO = \angle DAC$, then the diagonals of the quadrilateral are perpendicular.
2005 Sharygin Geometry Olympiad
Find all isosceles triangles that cannot be cut into three isosceles triangles with the same sides.
2005 Sharygin Geometry Olympiad
Given a circle and points $A, B$ on it.  Draw the set of midpoints of the segments, one of the ends of which lies on one of the arcs $AB$, and the other on the second.
2005 Sharygin Geometry Olympiad
Let $P$ be the intersection point of the diagonals of the quadrangle $ABCD$, $M$ the intersection point of the lines connecting the midpoints of its opposite sides, $O$ the intersection point of the perpendicular bisectors of the diagonals, $H$ the intersection point of the lines connecting the orthocenters of the triangles $APD$ and $BCP$, $APB$ and $CPD$. Prove that $M$ is the midpoint of $OH$.
2005 Sharygin Geometry Olympiad
It is given that for no side of the triangle from the height drawn to it, the bisector and the median it is impossible to make a triangle. Prove that one of the angles of the triangle is greater than $135^o$
2005 Sharygin Geometry Olympiad
A convex quadrangle without parallel sides is given. For each triple of its vertices, a point is constructed that supplements this triple to a parallelogram, one of the diagonals of which coincides with the diagonal of the quadrangle. Prove that of the four points constructed, exactly one lies inside the original quadrangle.
2005 Sharygin Geometry Olympiad
A triangle can be cut into three similar triangles.

Prove that it can be cut into any number of triangles similar to each other.
2005 Sharygin Geometry Olympiad
Two parallel chords $AB$ and $CD$ are drawn in a circle with center $O$.

Circles with diameters $AB$ and $CD$ intersect at point $P$.

Prove that the midpoint of the segment $OP$ is equidistant from lines $AB$ and $CD$.
2005 Sharygin Geometry Olympiad
Two segments $A_1B_1$ and $A_2B_2$ are given on the plane, with $\frac{A_2B_2}{A_1B_1} = k < 1$. On segment $A_1A_2$, point $A_3$ is taken, and on the extension of this segment beyond point $A_2$, point $A_4$ is taken, so $\frac{A_3A_2}{A_3A_1} =\frac{A_4A_2}{A_4A_1}= k$. Similarly, point $B_3$ is taken on segment $B_1B_2$ , and on the extension of this the segment beyond point $B_2$ is point $B_4$, so $\frac{B_3B_2}{B_3B_1} =\frac{B_4B_2}{B_4B_1}= k$. Find the angle between lines $A_3B_3$ and $A_4B_4$.



(Netherlands)
2005 Sharygin Geometry Olympiad
Two circles of radius $1$ intersect at points $X, Y$, the distance between which is also equal to $1$. From point $C$ of one circle, tangents $CA, CB$ are drawn to the other. Line $CB$ will cross the first circle a second time at point $A'$. Find the distance $AA'$.
2005 Sharygin Geometry Olympiad
Let $H$ be the orthocenter of triangle $ABC$, $X$ be an arbitrary point. A circle with a diameter of $XH$ intersects lines $AH, BH, CH$ at points $A_1, B_1, C_1$ for the second time, and lines $AX BX, CX$ at points $A_2, B_2, C_2$. Prove that lines A$_1A_2, B_1B_2, C_1C_2$ intersect at one point.
2005 Sharygin Geometry Olympiad
$A_1, B_1, C_1$ are the midpoints of the sides $BC,CA,BA$ respectively of an equilateral triangle $ABC$. Three parallel lines, passing through $A_1, B_1, C_1$  intersect, respectively, lines $B_1C_1, C_1A_1, A_1B_1$ at points $A_2, B_2, C_2$. Prove that the lines $AA_2, BB_2, CC_2$ intersect at one point lying on the circle circumscribed around the triangle $ABC$.
2005 Sharygin Geometry Olympiad
Convex quadrilateral $ABCD$ is given. Lines $BC$ and $AD$ intersect at point $O$, with $B$ lying on the segment $OC$, and $A$ on the segment $OD$. $I$ is the center of the circle inscribed in the $OAB$ triangle, $J$ is the center of the circle exscribed in the triangle $OCD$  touching the side of $CD$ and the extensions of the other two sides. The perpendicular  from the midpoint of the segment $IJ$ on the lines $BC$ and $AD$ intersect the corresponding sides of the quadrilateral (not the extension) at points $X$ and $Y$. Prove that the segment $XY$ divides the perimeter of the quadrilateral$ABCD$ in half, and from all segments with this property and ends on $BC$ and $AD$, segment $XY$ has the smallest length.
2005 Sharygin Geometry Olympiad
Inside the inscribed quadrilateral $ABCD$ there is a point $K$, the distances from which to the sides $ABCD$ are proportional to these sides. Prove that $K$ is the intersection point of the diagonals of $ABCD$.
2005 Sharygin Geometry Olympiad
In the triangle $ABC , \angle A = \alpha,  BC = a$. The inscribed circle touches the lines $AB$ and $AC$ at points $M$ and $P$. Find the length of the chord cut by the line $MP$ in a circle with diameter $BC$.
2005 Sharygin Geometry Olympiad
The angle and the point $K$ inside it are given on the plane.  Prove that there is a point $M$ with the following property:

if an arbitrary line passing through intersects the sides of the angle at points $A$ and $B$, then $MK$ is the bisector of the angle $AMB$.
2005 Sharygin Geometry Olympiad
The sphere inscribed in the tetrahedron $ABCD$  touches its faces at points $A',B',C',D'$. The segments $AA'$ and $BB'$ intersect, and the point of their intersection lies on the inscribed sphere. Prove that the segments $CC'$ and $DD'$ also intersect on the inscribed sphere.
2019 Singapore MO Open
In the acute-angled triangle $ABC$ with circumcircle $\omega$ and orthocenter $H$, points $D$ and $E$ are the feet of the perpendiculars from $A$ onto $BC$ and from $B$ onto $AC$ respecively. Let $P$ be a point on the minor arc $BC$ of $\omega$ . Points $M$ and $N$ are the  feet of the perpendiculars from $P$ onto lines $BC$ and $AC$ respectively. Let $PH$ and $MN$ intersect at $R$. Prove that $\angle DMR=\angle MDR$.
2019 Singapore MO Open
find all functions $f : \mathbb{Z} \to \mathbb{Z}$ such that



$f(-f(x)-f(y)) = 1-x-y$ $\quad \forall x,y \in \mathbb{Z}$
2019 Singapore MO Open
A robot is placed at point $P$ on the $x$-axis but different from $(0,0)$ and $(1,0)$ and can only move along the axis either to the left or to the right. Two players play the following game. Player $A$ gives a distance and $B$ gives a direction and the robot will move the indicated distance along the indicated direction. Player $A$ aims to move the robot to either $(0,0)$ or $(1,0)$. Player $B$'s aim is to stop $A$ from achieving his aim. For which $P$ can $A$ win?
2019 Singapore MO Open
Let $p \equiv 2 \pmod 3$ be a prime, $k$ a positive integer and $P(x) = 3x^{\frac{2p-1}{3}}+3x^{\frac{p+1}{3}}+x+1$. For any integer $n$, let $R(n)$ denote the remainder when $n$ is divided by $p$ and let $S = \{0,1,\cdots,p-1\}$. At each step, you can either (a) replaced every element $i$ of $S$ with $R(P(i))$ or (b) replaced every element $i$ of $S$ with $R(i^k)$. Determine all $k$ such that there exists a finite sequence of steps that reduces $S$ to $\{0\}$.



Proposed by fattypiggy123
2019 Singapore MO Open
In a $m\times n$ chessboard ($m,n\ge 2$), some dominoes are placed (without overlap) with each domino covering exactly two adjacent cells. Show that if no more dominoes can be added to the grid, then at least $2/3$ of the chessboard is covered by dominoes.



Proposed by DVDthe1st, mzy and jjax
2018 Singapore MO Open
Consider a regular cube with side length $2$.  Let $A$ and $B$ be $2$ vertices that are furthest apart.  Construct a sequence of points on the surface of the cube $A_1$, $A_2$, $\ldots$, $A_k$ so that $A_1=A$, $A_k=B$ and for any $i = 1,\ldots, k-1$, the distance from $A_i$ to $A_{i+1}$ is $3$.  Find the minimum value of $k$.
2018 Singapore MO Open
Let O be a point inside triangle ABC such that $\angle BOC$ is $90^\circ$ and $\angle BAO = \angle BCO$.  Prove that $\angle OMN$ is $90$ degrees, where $M$ and $N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively.
2018 Singapore MO Open
Let $n$ be a positive integer.  Show that there exists an integer $m$ such that

$$

2018m^2+20182017m+2017

$$is divisible by $2^n$.
2018 Singapore MO Open
each of the squares in a 2 x 2018 grid of squares is to be coloured black or white such that in any 2 x 2 block , at least one of the 4 squares is white. let P be the number of ways of colouring the grid. find the largest k so that $3^k$ divides P.
2018 Singapore MO Open
consider a polynomial P(x,y,z) in three variables with integer coefficients such that for any real numbers a,b,c



P(a,b,c)=0 <=> a=b=c



find the largest integer r such that for all such polynomials P(x,y,z) and integers n,m



m^r | P(n,n+m,n+2m)
2017 Singapore MO Open
The incircle of $\vartriangle ABC$ touches the sides $BC,CA,AB$ at $D,E,F$ respectively. A circle through $A$ and $B$ encloses $\vartriangle ABC$ and intersects the line $DE$ at points $P$ and $Q$. Prove that the midpoint of $AB$ lies on the circumircle of $\vartriangle PQF$.
2017 Singapore MO Open
Let $a_1,a_2,...,a_n,b_1,b_2,...,b_n,p$  be real numbers with $p >- 1$. Prove that

$$\sum_{i=1}^{n}(a_i-b_i)\left(a_i (a_1^2+a_2^2+...+a_n^2)^{p/2}-b_i (b_1^2+b_2^2+...+b_n^2)^{p/2}\right)\ge 0$$
2017 Singapore MO Open
Find the smallest positive integer $n$ so that $\sqrt{\frac{1^2+2^2+...+n^2}{n}}$ is an integer.
2017 Singapore MO Open
Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that  $x_1<y_1<x_2<y_2<...<x_n<y_n$
2017 Singapore MO Open
Let $A$ and $B$ be two $n \times n$ square arrays. The cells of $A$ are labelled by the numbers from $1$ to $n^2$ from left to right starting from the top row, whereas the cells of $B$ are labelled by the numbers from $1$ to $n^2$ along rising north-easterly diagonals starting with the upper left-hand corner. Stack the array $B$ on top of the array $A$. If two overlapping cells have the same number, they are coloured red. Determine those $n$ for which there is at least one red cell other than the cells at top left corner, bottom right corner and the centre (when $n$ is odd). Below shows the arrays for $n=4$.
2016 Singapore MO Open
Let $D$ be a point in the interior of $\triangle{ABC}$ such that $AB=ab$, $AC=ac$, $BC=bc$, $AD=ad$, $BD=bd$, $CD=cd$. Show that $\angle{ABD}+\angle{ACD}=60^{\circ}$.



Source: 2016 Singapore Mathematical Olympiad (Open) Round 2, Problem 1
2016 Singapore MO Open
Let $a, b, c$ be real numbers such that $0 < a, b, c < 1/2$ and $a + b + c= 1$. Prove that for all real numbers $x,y,z$,

$$abc(x + y + z)^2 \ge ayz( 1- 2a) + bxz( 1 - 2b) + cxy( 1 - 2c)$$. When does equality hold?
2016 Singapore MO Open
Let $n$ be a prime number. Show that there is a permutation $a_1,a_2,...,a_n$ of $1,2,...,n$ so that $a_1,a_1a_2,...,a_1a_2...a_n$ leave distinct remainders when divided by $n$
2016 Singapore MO Open
Let $b$  be a number with $-2 < b < 0$. Prove that there exists a positive integer $n$ such that all the coefficients of the polynomial $(x + 1)^n(x^2 + bx + 1)$ are positive.
2016 Singapore MO Open
A total of $731$ objects are put into $n$ nonempty bags where $n$ is a positive integer. These bags can be distributed into $17$ red boxes and also into $43$ blue boxes so that each red and each blue box contain $43$ and $17$ objects, respectively. Find the minimum value of $n$.
2015 Singapore MO Open
In an acute-angled triangle $\triangle ABC$, D is the point on BC such that AD bisects ∠BAC,

E and F are the feet of the perpendiculars from D onto AB and AC respectively. The

segments BF and CE intersect at K. Prove that AK is perpendicular to BC.
2015 Singapore MO Open
A boy lives in a small island in which there are three roads at every junction. He starts

from his home and walks along the roads. At each junction he would choose to turn

to the road on his right or left alternatively, i.e., his choices would be . . ., left, right,

left,... Prove that he will eventually return to his home.
2015 Singapore MO Open
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$, where $\mathbb{R}$ is the set of real numbers, such that

$f(x)f(yf(x) - 1) = x^2 f(y) - f(x) \quad\forall x,y \in \mathbb{R}$
2015 Singapore MO Open
Let $f_0, f_1,...$ be the Fibonacci sequence: $f_0 = f_1 = 1, f_n = f_{n-1} + f_{n-2}$ if $n \geq 2$.

Determine all possible positive integers $n$ so that there is a positive integer $a$ such that

$f_n \leq a \leq f_{n+1}$ and that $a(  \frac{1}{f_1}+\frac{1}{f_1f_2}+\cdots+\frac{1}{f_1f_2...f_n}  )$

is an integer.
2015 Singapore MO Open
Let n > 3 be a given integer. Find the largest integer d (in terms of n) such that for

any set S of n integers, there are four distinct (but not necessarily disjoint) nonempty

subsets, the sum of the elements of each of which is divisible by d.
2014 Singapore MO Open
The quadrilateral ABCD is inscribed in a circle which has diameter BD. Points A’ and B’ are symmetric to A and B with respect to the line BD and AC respectively. If the lines A’C, BD intersect at P and AC, B’D intersect at Q, prove that PQ is perpendicular to AC.
2014 Singapore MO Open
Find all functions from the reals to the reals satisfying

$$f(xf(y) + x) = xy + f(x)$$
2014 Singapore MO Open
Let $0<a_1<a_2<\cdots <a_n$ be real numbers. Prove that

$$\left (\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}\right )^2 \leq \frac{1}{a_1}+\frac{1}{a_2-a_1}+\cdots +\frac{1}{a_n-a_{n-1}}.$$
2014 Singapore MO Open
Fill up each square of a $50$ by $50$ grid with an integer. Let $G$ be the configuration of $8$ squares obtained by taking a $3$ by $3$ grid and removing the central square. Given that for any such $G$ in the $50$ by $50$ grid, the sum of integers in its squares is positive, show there exist a $2$ by $2$ square such that the sum of its entries is also positive.
2014 Singapore MO Open
Determine the largest odd positive integer $n$ such that every odd integer $k$ with $1<k<n$ and $\gcd(k, n)=1$ is a prime.
2013 Singapore MO Open
Let $a_1$, $a_2$, ... be a sequence of integers defined recursively by $a_1=2013$ and for $n \ge 1$, $a_{n+1}$ is the sum of the $2013$-th powers of the digits of $a_n$. Do there exist distinct positive integers $i$, $j$ such that $a_i=a_j$?
2013 Singapore MO Open
Let $ABC$ be an acute-angled triangle and let $D$, $E$, and $F$ be the midpoints of $BC$, $CA$, and $AB$ respectively. Construct a circle, centered at the orthocenter of triangle $ABC$, such that triangle $ABC$ lies in the interior of the circle. Extend $EF$ to intersect the circle at $P$, $FD$ to intersect the circle at $Q$ and $DE$ to intersect the circle at $R$. Show that $AP=BQ=CR$.
2013 Singapore MO Open
Let n be a positve integer. prove there exists a positive integer n st $n^{2013}-n^{20}+n^{13}-2013$ has at least N distinct prime factors.
2013 Singapore MO Open
Let $F$ be a finite non-empty set of integers and let $n$ be a positive integer. Suppose that



$\bullet$ Any $x \in F$ may be written as $x=y+z$ for some $y$, $z \in F$;

$\bullet$ If $1 \leq k \leq n$ and $x_1$, ..., $x_k \in F$, then $x_1+\cdots+x_k \neq 0$.



Show that $F$ has at least $2n+2$ elements.
2013 Singapore MO Open
Let $ABC$ be a triangle with integral side lengths such that $\angle A=3\angle B$. Find the minimum value of its perimeter.
2012 Singapore MO Open
The incircle with centre $I$ of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ at $D, E, F$ respectively. The line $ID$ intersects the segment $EF$ at $K$. Proof that $A, K, M$ collinear, where $M$ is the midpoint of $BC$.
2012 Singapore MO Open
Find all functions $f:\mathbb{R}\to\mathbb{R}$ so that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y$ that belongs to $\mathbb{R}$.
2012 Singapore MO Open
For each $i=1,2,..N$, let $a_i,b_i,c_i$ be integers such that at least one of them is odd. Show that one can find integers $x,y,z$ such that $xa_i+yb_i+zc_i$ is odd for at least $\frac{4}{7}N$ different values of $i$.
2012 Singapore MO Open
Let $p$ be an odd prime. Prove that

$$1^{p-2}+2^{p-2}+\cdots+\left(\frac{p-1}{2}\right)^{p-2}\equiv\frac{2-2^p}{p}\pmod p.$$
2012 Singapore MO Open
There are $2012$ distinct points in the plane, each of which is to be coloured using one of $n$ colours, so that the numbers of points of each colour are distinct. A set of $n$ points is said to be multi-coloured if their colours are distinct. Determine $n$ that maximizes the number of multi-coloured sets.
2011 Singapore MO Open
In the acute-angled non-isosceles triangle $ABC$, $O$ is its circumcenter, $H$ is its orthocenter and $AB>AC$. Let $Q$ be a point on $AC$ such that the extension of $HQ$ meets the extension of $BC$ at the point $P$. Suppose $BD=DP$, where $D$ is the foot of the perpendicular from $A$ onto $BC$. Prove that $\angle ODQ=90^{\circ}$.
2011 Singapore MO Open
If 46 squares are colored red in a $9\times 9$ board, show that there is a $2\times 2$ block on the board in which at least 3 of the squares are colored red.
2011 Singapore MO Open
Let $x,y,z>0$ such that $\frac1x+\frac1y+\frac1z<\frac{1}{xyz}$. Show that

$$\frac{2x}{\sqrt{1+x^2}}+\frac{2y}{\sqrt{1+y^2}}+\frac{2z}{\sqrt{1+z^2}}<3.$$
2011 Singapore MO Open
Find all polynomials $P(x)$ with real coefficients such that

$$P(a)\in\mathbb{Z}\ \ \ \text{implies that}\ \ \ a\in\mathbb{Z}.$$
2011 Singapore MO Open
Find all pairs of positive integers $(m,n)$ such that

$$m+n-\frac{3mn}{m+n}=\frac{2011}{3}.$$
2010 Singapore MO Open
Let $CD$ be a chord of a circle $\Gamma_1$ and $AB$ a diameter of $\Gamma_1$ perpendicular to $CD$ at $N$ with $AN > NB$. A circle $\Gamma_2$ centered at $C$ with radius $CN$ intersects $\Gamma_1$ at points $P$ and $Q$. The line $PQ$ intersects $CD$ at $M$ and $AC$ at $K$; and the extension of $NK$ meets $\Gamma_2$ at $L$. Prove that $PQ$ is perpendicular to $AL$
2010 Singapore MO Open
Let $(a_n), (b_n)$, $n = 1,2,...$ be two sequences of integers defined by $a_1 = 1, b_1 = 0$ and for $n \geq 1$



$a_{n+1} = 7a_n + 12b_n + 6$

$b_{n+1} = 4a_n + 7b_n + 3$



Prove that $a_n^2$ is the difference of two consecutive cubes.
2010 Singapore MO Open
Suppose that $a_1,...,a_{15}$ are prime numbers forming an arithmetic progression with common difference $d > 0$ if $a_1 > 15$ show that $d > 30000$
2010 Singapore MO Open
Let $n$ be a positive integer. Find the smallest positive integer $k$ with the property that for any colouring nof the squares of a $2n$ by $k$ chessboard with $n$ colours, there are $2$ columns and $2$ rows such that the $4$ squares in their intersections have the same colour.
2009 Singapore MO Open
let $O$ be the center of the circle inscribed in a rhombus ABCD. points E,F,G,H are chosen on sides AB, BC, CD, DA respectively so that EF and GH are tangent to inscribed circle. show that EH and FG are parallel.
2009 Singapore MO Open
a palindromic number is a number which is unchanged when order of its digits is reversed.

prove that the arithmetic progression 18, 37,.. contains infinitely many palindromic numbers.
2009 Singapore MO Open
for $k\in\mathbb{N}$ , define $A_n$ for $n=1,2,...$ by

$A_{n+1} = \frac{ nA_n+2(n+1)^{2k}  }{n+2}  , A_1=1$

Prove $A_n$ is integer for all $n\geq 1$, and $A_n$ is odd if and only if $n\equiv$1 or 2(mod 4)
2009 Singapore MO Open
find largest constant C st

$\sum_{i=1}^{4} (x_i+1/x_i)^3 \geq C$

for all positive real numbers $x_1,..,x_4$ st

$x_1^3+x_3^3+3x_1x_3=x_2+x_4=1$
2009 Singapore MO Open
Find all integers x,y,z with $2\leq x\leq y\leq z$ st

$xy \equiv 1 $(mod z) $xz\equiv 1$(mod y) $yz \equiv 1$ (mod x)
2008 Singapore MO Open
Find all pairs of positive integers $ (n,k)$ so that $ (n+1)^k-1=n!$.
2008 Singapore MO Open
in the acute triangle $\triangle ABC$.

M is a point in the interior of the segment AC  and N is a point on the extension of segment AC such that MN=AC.

let D,E be the feet of perpendiculars from M,N onto lines BC,AB respectively

prove that the orthocentre of $\triangle ABC$ lies on circumcircle of $\triangle BED$