Navvye/Olympiad-Problems-CSV huggingface dataset:
2005 Croatia National Olympiad
The incircle of a triangle $ABC$ touches $AC, BC$ , and $AB$ at $M , N$, and $R$, respectively. Let $S$ be a point on the smaller arc $MN$ and $t$ be the tangent to this arc at $S$ . The line $t$ meets $NC$ at $P$ and $MC$ at $Q$. Prove that the lines $AP, BQ, SR, MN$ have a common point.
2005 Croatia National Olympiad
Find the locus of points inside a trihedral angle such that the sum of their distances from the faces of the trihedral angle has a ﬁxed positive value $a$.
2005 Croatia National Olympiad
The vertices of a regular $2005$-gon are colored red, white and blue. Whenever two vertices of different colors stand next to each other, we are allowed to recolor them into the third color.

(a) Prove that there exists a ﬁnite sequence of allowed recolorings after which all the vertices are of the same color.

(b) Is that color uniquely determined by the initial coloring?
2005 Croatia National Olympiad
A sequence $(a_{n})$ is deﬁned by $a_{1}= 1$ and $a_{n}= a_{1}a_{2}...a_{n-1}+1$ for $n \geq 2.$ Find the smallest real number $M$ such that $\sum_{n=1}^{m}\frac{1}{a_{n}}<M\; \forall m\in\mathbb{N}$.
2005 Croatia National Olympiad
Let $P(x)$ be a monic polynomial of degree $n$ with nonnegative coefficients and the free term equal to $1$. Prove that if all the roots of $P(x)$ are real, then $P(x) \geq (x+1)^{n}$ holds for every $x \geq 0$.
2005 Croatia National Olympiad
Show that there is a unique positive integer which consists of the digits $2$ and $5$, having $2005$ digits and divisible by $2^{2005}$.
2005 Croatia National Olympiad
Let $P$ and $Q$ be points on the sides $BC$ and $CD$ of a convex quadrilateral $ABCD$, respectively, such that $\angle{BAP}=\angle{ DAQ}$. Prove that the triangles $ABP$ and $ADQ$ have equal area if and only if the line joining their orthocenters is perpendicular to $AC.$
2019 Czech and Slovak Olympiad III A
Find all triplets $(x,y,z)\in\mathbb{R}^3$ such that

\begin{align*}

x^2-yz &= |y-z|+1, \\ 

y^2-zx &= |z-x|+1, \\ 

z^2-xy &= |x-y|+1.

\end{align*}
2019 Czech and Slovak Olympiad III A
Let be $ABCD$ a rectangle with $|AB|=a\ge b=|BC|$. Find points $P,Q$ on the line $BD$ such that $|AP|=|PQ|=|QC|$. Discuss the solvability with respect to the lengths $a,b$.
2019 Czech and Slovak Olympiad III A
Let $a,b,c,n$ be positive integers such that the following conditions hold

(i) numbers $a,b,c,a+b+c$ are pairwise coprime,

(ii) number $(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)$ is a perfect $n$-th power.

Prove, that the product $abc$ can be expressed as a difference of two perfect $n$-th powers.
2019 Czech and Slovak Olympiad III A
Let be $ABC$ an acute-angled triangle. Consider point $P$ lying on the opposite ray to the ray $BC$ such that $|AB|=|BP|$. Similarly, consider point $Q$ on the opposite ray to the ray $CB$ such that $|AC|=|CQ|$. Denote $J$ the excenter of $ABC$ with respect to $A$ and $D,E$ tangent points of this excircle with the lines $AB$ and $AC$, respectively. Suppose that the opposite rays to $DP$ and $EQ$ intersect in $F\neq J$. Prove that $AF\perp FJ$.
2019 Czech and Slovak Olympiad III A
Prove that there are infinitely many integers which cannot be expressed as $2^a+3^b-5^c$ for non-negative integers $a,b,c$.
2019 Czech and Slovak Olympiad III A
Assume we can fill a table $n\times n$ with all numbers $1,2,\ldots,n^2-1,n^2$ in such way that arithmetic means of numbers in every row and every column is an integer. Determine all such positive integers $n$.
2018 Czech and Slovak Olympiad III A
In a group of people, there are some mutually friendly pairs. For positive integer $k\ge3$ we say the group is $k$-great, if every (unordered) $k$-tuple of people from the group can be seated around a round table it the way that all pairs of neighbors are mutually friendly. (Since this was the 67th year of CZE/SVK MO,) show that if the group is 6-great, then it is 7-great as well.

Bonus (not included in the competition): Determine all positive integers $k\ge3$ for which, if the group is $k$-great, then it is  $(k+1)$-great as well.
2018 Czech and Slovak Olympiad III A
Let $x,y,z$ be real numbers such that the numbers $$\frac{1}{|x^2+2yz|},\quad\frac{1}{|y^2+2zx|},\quad\frac{1}{|z^2+2xy|}$$are lengths of sides of a (non-degenerate) triangle. Determine all possible values of $xy+yz+zx$.
2018 Czech and Slovak Olympiad III A
In triangle $ABC$ let be $D$ an intersection of $BC$ and the $A$-angle bisector. Denote $E,F$ the circumcenters of $ABD$ and $ACD$ respectively. Assuming that the circumcenter of $AEF$ lies on the line $BC$ what is the possible size of the angle $BAC$ ?
2018 Czech and Slovak Olympiad III A
Let $a,b,c$ be integers which are lengths of sides of a triangle, $\gcd(a,b,c)=1$ and all the values $$\frac{a^2+b^2-c^2}{a+b-c},\quad\frac{b^2+c^2-a^2}{b+c-a},\quad\frac{c^2+a^2-b^2}{c+a-b}$$are integers as well. Show that $(a+b-c)(b+c-a)(c+a-b)$ or $2(a+b-c)(b+c-a)(c+a-b)$ is a perfect square.
2018 Czech and Slovak Olympiad III A
Let $ABCD$ an isosceles trapezoid with the longer base $AB$. Denote $I$ the incenter of $\Delta ABC$ and $J$ the excenter relative to the vertex $C$ of $\Delta ACD$. Show that the lines $IJ$ and $AB$ are parallel.
2018 Czech and Slovak Olympiad III A
Determine the least positive integer $n$ with the following property – for every 3-coloring of numbers $1,2,\ldots,n$ there are two (different) numbers $a,b$ of the same color such that $|a-b|$ is a perfect square.
2017 Czech And Slovak Olympiad III A
There are $100$ diamonds on the pile, $50$ of which are genuine and $50$ false. We invited a peculiar expert who alone can recognize which are which. Every time we show him some three diamonds, he would pick two and tell (truthfully) how many of them are genuine . Decide whether we can surely detect all genuine diamonds regardless how the expert chooses the pairs to be considered.
2017 Czech And Slovak Olympiad III A
Find all pairs of real numbers $k, l$ such that inequality $ka^2 + lb^2> c^2$ applies to the lengths of sides $a, b, c$ of any triangle.
2017 Czech And Slovak Olympiad III A
Find all functions $f: R \to  R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$
2017 Czech And Slovak Olympiad III A
For each sequence of $n$ zeros and $n$ units, we assign a number that is a number sections of the same digits in it. (For example, sequence $00111001$ has $4$ such sections $00, 111,00, 1$.) For a given $n$ we sum up all the numbers assigned to each such sequence. Prove that the sum total is equal to $(n+1){2n \choose n} $
2017 Czech And Slovak Olympiad III A
Given is the acute triangle $ABC$ with the intersection of altitudes $H$.  The angle bisector of angle $BHC$ intersects side $BC$ at point $D$. Mark $E$ and $F$ the symmetrics of the point $D$ wrt lines $AB$ and $AC$. Prove that the circle circumscribed around the triangle $AEF$ passes through the midpoint of the arc $BAC$
2017 Czech And Slovak Olympiad III A
Given is a nonzero integer $k$.

Prove that equation $k =\frac{x^2 - xy + 2y^2}{x + y}$ has an odd number of ordered integer pairs $(x, y)$ just when $k$ is  divisible by seven.
2016 Czech And Slovak Olympiad III A
Let $p> 3$ be a prime number. Determine the number of all ordered sixes $(a, b, c, d, e, f)$ of positive integers whose sum is $3p$ and all fractions $\frac{a + b}{c + d},\frac{b + c}{d + e},\frac{c + d}{e + f},\frac{d + e}{f + a},\frac{e + f}{a + b}$ have integer values.
2016 Czech And Slovak Olympiad III A
Let us denote successively $r$ and $r_a$ the radii of the inscribed circle and the exscribed circle wrt to side BC of triangle $ABC$. Prove that if it is true that $r+r_a=|BC|$ , then the triangle $ABC$ is a right one
2016 Czech And Slovak Olympiad III A
Mathematical clubs are popular among the inhabitants of the same city. Every two of them they have at least one member in common. Prove that we can give the people of the city compasses and rulers so that only one inhabitant gets both, while each club will to have both a ruler and a compass at the full participation of its members.
2016 Czech And Slovak Olympiad III A
For positive numbers $a, b, c$ holds $(a + c) (b^2 + a c) = 4a$.

Determine the maximum value of $b + c$ and find all triplets of numbers $(a, b, c)$ for which expression takes this value
2016 Czech And Slovak Olympiad III A
In the triangle $ABC$, $| BC | = 1$ and there is exactly one point $D$ on the  side  $BC$  such that $|DA|^2 =  |DB| \cdot |DC|$. Determine all possible values of the perimeter of  the triangle $ABC$.
2016 Czech And Slovak Olympiad III A
We put a figure of a king on some $6 \times  6$ chessboard. It can in one thrust jump either vertically or horizontally. The length of this jump is alternately one and two squares, whereby a jump of one (i.e. to the adjacent square) of the piece begins. Decide whether you can choose the starting position of the pieces so that after a suitable sequence

$35$ jumps visited each box of the chessboard just once.
2015 Czech and Slovak Olympiad III A
Find all 4-digit numbers $n$, such that $n=pqr$, where $p<q<r$ are distinct primes, such that $p+q=r-q$ and $p+q+r=s^2$, where $s$ is a prime number.
2015 Czech and Slovak Olympiad III A
Let $A=[0,0]$ and $B=[n,n]$. In how many ways can we go from $A$ to $B$, if we always want to go from lattice point to its neighbour (i.e. point with one coordinate the same and one smaller or bigger by one), we never want to visit the same point twice and we want our path to have length $2n+2$?

(For example, path $[0,0],[0,1],[-1,1],[-1,2],[0,2],[1,2],[2,2],[2,3],[3,3]$ is one of the paths for $n=3$)
2015 Czech and Slovak Olympiad III A
In triangle $\triangle ABC$ with median from $B$ not perpendicular to $AB$ nor $BC$, we call $X$ and $Y$ points on $AB$ and $BC$, which lie on the axis of the median from $B$. Find all such triangles, for which $A,C,X,Y$ lie on one circumrefference.
2015 Czech and Slovak Olympiad III A
Find all real triples $(a,b,c)$, for which $$a(b^2+c)=c(c+ab)$$ $$b(c^2+a)=a(a+bc)$$ $$c(a^2+b)=b(b+ca).$$
2015 Czech and Slovak Olympiad III A
In given triangle $\triangle ABC$, difference between sizes of each pair of sides is at least $d>0$. Let $G$ and $I$ be the centroid and incenter of $\triangle ABC$ and $r$ be its inradius. Show that $$[AIG]+[BIG]+[CIG]\ge\frac{2}{3}dr,$$ where $[XYZ]$ is (nonnegative) area of triangle $\triangle XYZ$.
2015 Czech and Slovak Olympiad III A
Integer $n>2$ is given. Find the biggest integer $d$, for which holds, that from any set $S$ consisting of $n$ integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by $d$.
2014 Czech and Slovak Olympiad III A
Let be $n$ a positive integer. Denote all its (positive) divisors as $1=d_1<d_2<\cdots<d_{k-1}<d_k=n$.

Find all values of $n$ satisfying $d_5-d_3=50$ and $11d_5+8d_7=3n$.



(Day 1, 1st problem

author: Matúš Harminc)
2014 Czech and Slovak Olympiad III A
A segment $AB$ is given in (Euclidean) plane. Consider all triangles $XYZ$ such, that $X$ is an inner point of $AB$, triangles $XBY$ and $XZA$ are similar (in this order of vertices) and points $A, B, Y, Z$ lie on a circle in this order. Find a set of midpoints of all such segments $YZ$.



(Day 1, 2nd problem

authors: Michal Rolínek, Jaroslav Švrček)
2014 Czech and Slovak Olympiad III A
Suppose we have a $8\times8$ chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into $1\times2$ domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.



(Day 1, 3rd problem

author: Michal Rolínek)
2014 Czech and Slovak Olympiad III A
$234$ viewers came to the cinema. Determine for which$ n \ge 4$ the viewers could be can be arranged in $n$ rows so that every viewer in $i$-th row gets to know just $j$ viewers in $j$-th row for any $i, j \in \{1, 2,... , n\}, i\ne j$. (The relationship of acquaintance is mutual.)



(Tomáš Jurík)
2014 Czech and Slovak Olympiad III A
Given is the acute triangle $ABC$. Let us denote $k$ a circle with diameter $AB$. Another circle, tangent to $AB$ at point $A$ and passing through point $C$ intersects the circle $k$ at point $P, P \ne A$. Another  circle which touches AB at point $B$ and passes point $C$, intersects the circle $k$ at point $Q, Q \ne B$. Prove that the intersection of the line $AQ$ and $BP$ lies on one of the sides of angle $ACB$.



(Peter Novotný)
2014 Czech and Slovak Olympiad III A
For arbitrary non-negative numbers $a$ and $b$ prove inequality

$\frac{a}{\sqrt{b^2+1}}+\frac{b}{\sqrt{a^2+1}}\ge\frac{a+b}{\sqrt{ab+1}}$,

and find, where equality occurs.



(Day 2, 6th problem

authors: Tomáš Jurík, Jaromír Šimša)
2013 Czech And Slovak Olympiad IIIA
Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$
2013 Czech And Slovak Olympiad IIIA
Each of the thieves in the $n$-member party ($n \ge 3$) charged a certain number of coins. All the coins were $100n$. Thieves decided to share their prey as follows: at each step, one of the bandits puts one coin to the other two. Find them all natural numbers $n \ge 3$ for which after a finite number of steps each outlaw can have $100$ coins no matter how many coins each thug has charged.
2013 Czech And Slovak Olympiad IIIA
In the parallelolgram A$BCD$ with the center $S$, let $O$ be the center of the circle of the inscribed triangle $ABD$ and let $T$  be the touch point with the diagonal $BD$. Prove that the lines $OS$ and $CT$ are parallel.
2013 Czech And Slovak Olympiad IIIA
On the board is written in decimal the integer positive number $N$. If it is not a single digit number, wipe its last digit $c$ and replace the number $m$ that remains on the board with a number $m -3c$. (For example, if $N = 1,204$ on the board, $120 - 3 \cdot  4 = 108$.) Find all the natural numbers $N$, by repeating the adjustment described eventually we get the number $0$.
2013 Czech And Slovak Olympiad IIIA
Given the parallelogram $ABCD$ such that the feet $K, L$ of the perpendiculars from point $D$ on the sides $AB, BC$ respectively are internal points. Prove that $KL \parallel AC$ when $|\angle BCA| + |\angle ABD| = |\angle BDA| + |\angle ACD|$.
2013 Czech And Slovak Olympiad IIIA
Find all positive real numbers $p$ such that $\sqrt{a^2 + pb^2} +\sqrt{b^2 + pa^2} \ge a + b + (p - 1) \sqrt{ab}$ holds for any pair of positive real numbers $a, b$.
2012 Czech And Slovak Olympiad IIIA
Find all integers for which $n$ is $n^4 -3n^2 + 9$ prime
2012 Czech And Slovak Olympiad IIIA
Find out the maximum possible area of the triangle $ABC$ whose medians have lengths satisfying inequalities $m_a \le 2, m_b \le 3, m_c \le  4$.
2012 Czech And Slovak Olympiad IIIA
Prove that there are two numbers $u$ and $v$, between any $101$ real numbers that apply $100 |u - v| \cdot |1 - uv|  \le (1 + u^2)(1 + v^2)$
2012 Czech And Slovak Olympiad IIIA
Inside the parallelogram $ABCD$ is a point $X$. Make a line that passes through point $X$ and divides the parallelogram into two parts whose areas differ from each other the most.
2012 Czech And Slovak Olympiad IIIA
In a group of $90$ children each has at least $30$ friends (friendship is mutual). Prove that they can be divided into three $30$-member groups so that each child has its own a group of at least one friend.
2012 Czech And Slovak Olympiad IIIA
In the set of real numbers solve the system of equations

$x^4+y^2+4=5yz$

$y^4+z^2+4=5zx$

$z^4+x^2+4=5xy$
2011 Czech and Slovak Olympiad III A
In a certain triangle $ABC$, there are points $K$ and $M$ on sides $AB$ and $AC$, respectively, such that if $L$ is the intersection of $MB$ and $KC$, then both $AKLM$ and $KBCM$ are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle $ABC$.
2011 Czech and Slovak Olympiad III A
Find all triples $(p, q, r)$ of prime numbers for which $$(p+1)(q+2)(r+3)=4pqr. $$
2011 Czech and Slovak Olympiad III A
Suppose that $x$, $y$, $z$ are real numbers  satisfying $$x+y+z=12,\qquad\text{and}\qquad x^2+y^2+z^2=54.$$ Prove that:



(a) Each of the numbers

$xy$, $yz$, $zx$ is at least $9$, but at most $25$.

(b) One of the numbers $x$, $y$, $z$ is at most $3$, and another one is at least $5$.
2011 Czech and Slovak Olympiad III A
Consider a quadratic polynomial $ax^2+bx+c$ with real coefficients satisfying $a\ge 2$, $b\ge 2$, $c\ge 2$. Adam and Boris play the following game. They alternately take turns with Adam first. On Adam’s turn, he can choose one of the polynomial’s coefficients and replace it with the sum of the other two coefficients. On Boris’s turn, he can choose one of the polynomial’s coefficients and replace it with the product of the other two coefficients. The winner is the player who first produces a polynomial with two distinct real roots. Depending on the values of $a$, $b$ and $c$, determine who has a winning strategy.
2011 Czech and Slovak Olympiad III A
In acute triangle ABC, which is not equilateral, let $P$ denote the foot of the altitude from $C$ to side $AB$; let $H$ denote the orthocenter; let $O$ denote the circumcenter; let $D$ denote the intersection of line $CO$ with $AB$; and let $E$ denote the midpoint of $CD$. Prove that line $EP$ passes through the midpoint of $OH$.
2011 Czech and Slovak Olympiad III A
Let $\mathbb{R}^+$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that for any $x,y\in\mathbb{R}^+$, we have $$ f(x)f(y)=f(y)f\Big(xf(y)\Big)+\frac{1}{xy}.$$
2010 Czech And Slovak Olympiad III A
Determine all pairs of integers $a, b$ for which they apply $4^a + 4a^2 + 4 = b^2$ .
2010 Czech And Slovak Olympiad III A
A circular target with a radius of $12$ cm was hit by $19$ shots. Prove that the distance between two hits is less than $7$ cm.
2010 Czech And Slovak Olympiad III A
Rumburak kidnapped $31$ members of party $A$ , $28$ members of party $B$, $23$ members of party $C$, $19$ members of Party $D$ and each of them in a separate cell. After work out occasionally they could walk in the yard and talk. Once three people started to talk to each other members of three different parties, Rumburak re-registered them to the fourth party as a punishment.(They never talked to each other more than three kidnapped.)

a) Could it be that after some time all were abducted by members of one party? Which?

b) Determine all four positive integers of which the sum is $101$ and which as the numbers of kidnapped members of the four parties allow the Rumburaks all of them became members of one party over time.
2010 Czech And Slovak Olympiad III A
A circle $k$ is given with a non-diameter chord $AC$.  On the tangent at point $A$ select point $X \ne A$ and mark $D$ the intersection of the circle $k$ with the interior of the line $XC$ (if any). Let $B$ a point in circle $k$ such that quadrilateral $ABCD$ is a trapezoid . Determine the set of intersections of lines $BC$ and $AD$ belonging to all such trapezoids.
2010 Czech And Slovak Olympiad III A
On the board are written numbers $1, 2,. . . , 33$. In one step we select two numbers written on the product of which is the square of the natural number, we wipe off the two chosen numbers and write the square root of their product on the board. This way we continue to the board only the numbers remain so that the product of neither of them is a square. (In one we can also wipe out two identical numbers and replace them with the same number.) Prove that at least $16$ numbers remain on the board.
2010 Czech And Slovak Olympiad III A
Find the minimum of the expression  $\frac{a + b + c}{2} -\frac{[a, b] + [b, c] + [c, a]}{a + b + c}$ where the variables $a, b, c$ are any integers greater than $1$ and $[x, y]$ denotes the least common multiple of numbers $x, y$.
2009 Czech and Slovak Olympiad III A
Knowing that the numbers $p, 3p+2, 5p+4, 7p+6, 9p+8$, and $11p+10$ are all primes, prove that $6p+11$ is a composite number.
2009 Czech and Slovak Olympiad III A
Rectangle $ABCD$ is inscribed in circle $O$. Let the projections of a point $P$ on minor arc $CD$ onto $AB,AC,BD$ be $K,L,M$, respectively. Prove that $\angle LKM=45$if and only if $ABCD$ is a square.
2009 Czech and Slovak Olympiad III A
Find the least value of $x>0$ such that for all positive real numbers $a,b,c,d$ satisfying $abcd=1$, the inequality

$a^x+b^x+c^x+d^x\ge\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}$ is true.
2009 Czech and Slovak Olympiad III A
A positive integer $n$ is called good if and only if there exist exactly $4$ positive integers $k_1, k_2, k_3, k_4$ such that $n+k_i|n+k_i^2$ ($1 \leq k \leq 4$). Prove that:



$58$ is good;



$2p$ is good if and only if $p$ and $2p+1$ are both primes ($p>2$).
2009 Czech and Slovak Olympiad III A
At every vertex $A_k(1\le k\le n)$ of a regular $n$-gon, $k$ coins are placed. We can do the following operation: in each step, one can choose two arbitrarily coins and move them to their adjacent vertices respectively, one clockwise and one anticlockwise. Find all positive integers $n$ such that after a finite number of operations, we can reach the following configuration: there are $n+1-k$ coins at vertex $A_k$ for all $1\le k\le n$.
2009 Czech and Slovak Olympiad III A
Given two fixed points $O$ and $G$ in the plane. Find the locus of the vertices of triangles whose circumcenters and centroids are $O$ and $G$ respectively.
2008 Czech and Slovak Olympiad III A
Find all pairs of real numbers $(x,y)$ satisfying:

$$x+y^2=y^3,$$$$y+x^2=x^3.$$
2008 Czech and Slovak Olympiad III A
Two disjoint circles $W_1(S_1,r_1)$ and $W_2(S_2,r_2)$ are given in the plane. Point $A$ is on circle $W_1$ and $AB,AC$ touch the circle $W_2$ at $B,C$ respectively. Find the loci of the incenter and orthocenter of triangle $ABC$.
2008 Czech and Slovak Olympiad III A
Find all pairs of integers $(a,b)$ such that $a^2+ab+1\mid b^2+ab+a+b-1$.
2008 Czech and Slovak Olympiad III A
In decimal representation, we call an integer $k$-carboxylic if and only if it can be represented as a sum of $k$ distinct integers, all of them greater than $9$, whose digits are the same. For instance, $2008$ is $5$-carboxylic because $2008=1111+666+99+88+44$. Find, with an example, the smallest integer $k$ such that $8002$ is $k$-carboxylic.
2008 Czech and Slovak Olympiad III A
At one moment, a kid noticed that the end of the hour hand, the end of the minute hand and one of the twelve numbers (regarded as a point) of his watch formed an equilateral triangle. He also calculated that $t$ hours would elapse for the next similar case. Suppose that the ratio of the lengths of the minute hand (whose length is equal to the distance from the center of the watch plate to any of the twelve numbers) and the hour hand is $k>1$. Find the maximal value of $t$.
2008 Czech and Slovak Olympiad III A
Find the greatest value of $p$ and the smallest value of $q$ such that for any triangle in the plane, the inequality

$$p<\frac{a+m}{b+n}<q$$

holds, where $a,b$ are it's two sides and $m,n$ their corresponding medians.
2007 Czech and Slovak Olympiad III A
A stone is placed in a square of a chessboard with $n$ rows and $n$ columns. We can alternately undertake two operations:

(a) move the stone to a square that shares a common side with the square in which it stands;

(b) move it to a square sharing only one common vertex with the square in which it stands.



In addition, we are required that the first step must be (b). Find all integers $n$ such that the stone can go through a certain path visiting every square exactly once.
2007 Czech and Slovak Olympiad III A
In a cyclic quadrilateral $ABCD$, let $L$ and $M$ be the incenters of $ABC$ and $BCD$ respectively. Let $R$ be a point on the plane such that $LR\bot AC$ and $MR\bot BD$.Prove that triangle $LMR$ is isosceles.
2007 Czech and Slovak Olympiad III A
Consider a function $f:\mathbb N\rightarrow \mathbb N$ such that for any two positive integers $x,y$, the equation $f(xf(y))=yf(x)$ holds. Find the smallest possible value of $f(2007)$.
2007 Czech and Slovak Olympiad III A
The set $M=\{1,2,\ldots,2007\}$ has the following property: If $n$ is an element of $M$, then all terms in the arithmetic progression with its first term $n$ and common difference $n+1$, are in $M$. Does there exist an integer $m$ such that all integers greater than $m$ are elements of $M$?