Found problems: 22
2004 Germany Team Selection Test, 2
Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties:
(a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$.
(b) We have $f\left(2\right) = 0$.
(c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$.
[b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.
1986 China Team Selection Test, 4
Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points.
[b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$.
[b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.
2021 Thailand TST, 1
For a positive integer $n$, consider a square cake which is divided into $n \times n$ pieces with at most one strawberry on each piece. We say that such a cake is [i]delicious[/i] if both diagonals are fully occupied, and each row and each column has an odd number of strawberries.
Find all positive integers $n$ such that there is an $n \times n$ delicious cake with exactly $\left\lceil\frac{n^2}{2}\right\rceil$ strawberries on it.
2004 Germany Team Selection Test, 1
A function $f$ satisfies the equation
\[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\]
for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.
2021 Thailand TST, 2
Prove that, for all positive integers $m$ and $n$, we have $$\left\lfloor m\sqrt{2} \right\rfloor\cdot\left\lfloor n\sqrt{7} \right\rfloor<\left\lfloor mn\sqrt{14} \right\rfloor.$$
2019 Slovenia Team Selection Test, 3
Let $ABC$ be a non-right triangle and let $M$ be the midpoint of $BC$. Let $D$ be a point on $AM$ (D≠A, D≠M). Let ω1 be a circle through $D$ that intersects $BC$ at $B$ and let ω2 be a circle through $D$ that intersects $BC$ at $C$. Let $AB$ intersect ω1 at $B$ and $E$, and let $AC$ intersect ω2 at $C$ and $F$.
Prove, that the tangent on ω1 at $E$ and the tangent on ω2 at $F$ intersect on $AM$.
2005 Germany Team Selection Test, 2
Let $M$ be a set of points in the Cartesian plane, and let $\left(S\right)$ be a set of segments (whose endpoints not necessarily have to belong to $M$) such that one can walk from any point of $M$ to any other point of $M$ by travelling along segments which are in $\left(S\right)$. Find the smallest total length of the segments of $\left(S\right)$ in the cases
[b]a.)[/b] $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$.
[b]b.)[/b] $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$.
In other words, find the Steiner trees of the set $M$ in the above two cases.
2004 Germany Team Selection Test, 2
In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$.
Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.
2004 Germany Team Selection Test, 4
Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation
\[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\]
Show that at least two of these integers are equal to each other.
2005 Italy TST, 1
A stage course is attended by $n \ge 4$ students. The day before the final exam, each group of three students conspire against another student to throw him/her out of the exam. Prove that there is a student against whom there are at least $\sqrt[3]{(n-1)(n- 2)} $conspirators.
2005 Germany Team Selection Test, 2
Let $M$ be a set of points in the Cartesian plane, and let $\left(S\right)$ be a set of segments (whose endpoints not necessarily have to belong to $M$) such that one can walk from any point of $M$ to any other point of $M$ by travelling along segments which are in $\left(S\right)$. Find the smallest total length of the segments of $\left(S\right)$ in the cases
[b]a.)[/b] $M = \left\{\left(-1,0\right),\left(0,0\right),\left(1,0\right),\left(0,-1\right),\left(0,1\right)\right\}$.
[b]b.)[/b] $M = \left\{\left(-1,-1\right),\left(-1,0\right),\left(-1,1\right),\left(0,-1\right),\left(0,0\right),\left(0,1\right),\left(1,-1\right),\left(1,0\right),\left(1,1\right)\right\}$.
In other words, find the Steiner trees of the set $M$ in the above two cases.
2004 Germany Team Selection Test, 2
Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$.
Find all points $B$ on the diameter $d$ in the interior of $k$ such that
\[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\]
(i. e. give an explicit description of these points without using the points $M$ and $N$).
1987 China Team Selection Test, 1
a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying:
[b]i.[/b] each has $k$ elements;
[b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$)
[b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements.
b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.
2004 Germany Team Selection Test, 1
A function $f$ satisfies the equation
\[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\]
for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.
2005 Italy TST, 1
A stage course is attended by $n \ge 4$ students. The day before the final exam, each group of three students conspire against another student to throw him/her out of the exam. Prove that there is a student against whom there are at least $\sqrt[3]{(n-1)(n- 2)} $conspirators.
1986 China Team Selection Test, 4
Mark $4 \cdot k$ points in a circle and number them arbitrarily with numbers from $1$ to $4 \cdot k$. The chords cannot share common endpoints, also, the endpoints of these chords should be among the $4 \cdot k$ points.
[b]i.[/b] Prove that $2 \cdot k$ pairwisely non-intersecting chords can be drawn for each of whom its endpoints differ in at most $3 \cdot k - 1$.
[b]ii.[/b] Prove that the $3 \cdot k - 1$ cannot be improved.
2004 Germany Team Selection Test, 2
In a triangle $ABC$, let $D$ be the midpoint of the side $BC$, and let $E$ be a point on the side $AC$. The lines $BE$ and $AD$ meet at a point $F$.
Prove: If $\frac{BF}{FE}=\frac{BC}{AB}+1$, then the line $BE$ bisects the angle $ABC$.
2004 Germany Team Selection Test, 4
Let the positive integers $x_1$, $x_2$, $...$, $x_{100}$ satisfy the equation
\[\frac{1}{\sqrt{x_1}}+\frac{1}{\sqrt{x_2}}+...+\frac{1}{\sqrt{x_{100}}}=20.\]
Show that at least two of these integers are equal to each other.
2021 Thailand TST, 2
Let $\mathcal{A}$ be the set of all $n\in\mathbb{N}$ for which there exist $k\in\mathbb{N}$ and $a_0,a_1,\dots,a_{k-1}\in \{1,2,\dots,9\}$ such that $a_0 \geq a_1 \geq \cdots \geq a_{k-1}$ and $n = a_0 +a_1 \cdot 10^1 +\cdots +a_{k-1}\cdot 10^{k-1}$. Let $\mathcal{B}$ be the set of all $m \in\mathbb{N}$ for which there exist $l \in\mathbb{N}$ and $b_0,b_1,\dots,b_{l-1} \in \{1,2,\dots,9\}$ such that $b_0 \leq b_1 \leq \cdots\leq b_{l-1}$ and $m = b_0 + b_1 \cdot 10^1 + \cdots+ b_{l-1}\cdot 10^{l-1}$.
[list=a]
[*] Are there infinitely many $n\in \mathcal{A}$ such that $n^2-3\in\mathcal{A} \ ?$
[*] Are there infinitely many $m\in \mathcal{B}$ such that $m^2-3\in\mathcal{B} \ ?$
[/list]
[i]Proposed by Pakawut Jiradilok and Wijit Yangjit[/i]
2004 Germany Team Selection Test, 2
Find all functions $f: \Bbb{R}_{0}^{+}\rightarrow \Bbb{R}_{0}^{+}$ with the following properties:
(a) We have $f\left( xf\left( y\right) \right) \cdot f\left( y\right) =f\left( x+y\right)$ for all $x$ and $y$.
(b) We have $f\left(2\right) = 0$.
(c) For every $x$ with $0 < x < 2$, the value $f\left(x\right)$ doesn't equal $0$.
[b]NOTE.[/b] We denote by $\Bbb{R}_{0}^{+}$ the set of all non-negative real numbers.
2004 Germany Team Selection Test, 2
Let $d$ be a diameter of a circle $k$, and let $A$ be an arbitrary point on this diameter $d$ in the interior of $k$. Further, let $P$ be a point in the exterior of $k$. The circle with diameter $PA$ meets the circle $k$ at the points $M$ and $N$.
Find all points $B$ on the diameter $d$ in the interior of $k$ such that
\[\measuredangle MPA = \measuredangle BPN \quad \text{and} \quad PA \leq PB.\]
(i. e. give an explicit description of these points without using the points $M$ and $N$).
1987 China Team Selection Test, 1
a.) For all positive integer $k$ find the smallest positive integer $f(k)$ such that $5$ sets $s_1,s_2, \ldots , s_5$ exist satisfying:
[b]i.[/b] each has $k$ elements;
[b]ii.[/b] $s_i$ and $s_{i+1}$ are disjoint for $i=1,2,...,5$ ($s_6=s_1$)
[b]iii.[/b] the union of the $5$ sets has exactly $f(k)$ elements.
b.) Generalisation: Consider $n \geq 3$ sets instead of $5$.