Found problems: 137
KoMaL A Problems 2021/2022, A. 809
Let the lengths of the sides of triangle $ABC$ be denoted by $a,b,$ and $c,$ using the standard notations. Let $G$ denote the centroid of triangle $ABC.$ Prove that for an arbitrary point $P$ in the plane of the triangle the following inequality is true: \[a\cdot PA^3+b\cdot PB^3+c\cdot PC^3\geq 3abc\cdot PG.\][i]Proposed by János Schultz, Szeged[/i]
KoMaL A Problems 2017/2018, A. 707
$100$ betyárs stand on the Hortobágy plains. Every betyár's field of vision is a $100$ degree angle. After each of them announces the number of other betyárs they see, we compute the sum of these $100$ numbers. What is the largest value this sum can attain?
KoMaL A Problems 2019/2020, A. 762
In a forest, there are $n$ different trees (considered as points), no three of which lie on the same line. John takes photographs of the forest such that all trees are visible (and no two trees are behind each other). What is the largest number of orders of in which the trees that can appear on the photos?
[i]Proposed by Gábor Mészáros, Sunnyvale, Kalifornia[/i]
KoMaL A Problems 2019/2020, A. 764
We call a diagonal of a polygon [i]nice[/i], if it is entirely inside the polygon or entirely outside the polygon. Let $P$ be an $n$–gon with no three of its vertices being on the same line. Prove that $P$ has at least $3(n-3)/2$ nice diagonals.
[i]Proposed by Bálint Hujter, Budapest and Gábor Szűcs, Szikszó[/i]
KoMaL A Problems 2023/2024, A. 859
Path graph $U$ is given, and a blindfolded player is standing on one of its vertices. The vertices of $U$ are labeled with positive integers between 1 and $n$, not necessarily in the natural order. In each step of the game, the game master tells the player whether he is in a vertex with degree 1 or with degree 2. If he is in a vertex with degree 1, he has to move to its only neighbour, if he is in a vertex with degree 2, he can decide whether he wants to move to the adjacent vertex with the lower or with the higher number. All the information the player has during the game after $k$ steps are the $k$ degrees of the vertices he visited and his choice for each step. Is there a strategy for the player with which he can decide in finitely many steps how many vertices the path has?
KoMaL A Problems 2020/2021, A. 784
Let $n,s,$ and $t$ be positive integers and $0<\lambda<1.$ A simple graph on $n$ vertices with at least $\lambda n^2$ edges is given. We say that $(x_1,\ldots,x_s,y_1,\ldots,y_t)$ is a [i]good intersection[/i] if letters $x_i$ and $y_j$ denote not necessarily distinct vertices and every $x_iy_j$ is an edge of the graph $(1\leq i\leq s,$ $1\leq j\leq t).$ Prove that the number of good insertions is at least $\lambda^{st}n^{s+t}.$
[i]Proposed by Kada Williams, Cambridge[/i]
KoMaL A Problems 2019/2020, A. 774
Let $O$ be the circumcenter of triangle $ABC,$ and $D$ be an arbitrary point on the circumcircle of $ABC.$ Let points $X, Y$ and $Z$ be the orthogonal projections of point $D$ onto lines $OA, OB$ and $OC,$ respectively. Prove that the incenter of triangle $XYZ$ is on the Simson-Wallace line of triangle $ABC$ corresponding to point $D.$
KoMaL A Problems 2023/2024, A. 881
We visit all squares exactly once on a $n\times n$ chessboard (colored in the usual way) with a king. Find the smallest number of times we had to switch colors during our walk.
[i]Proposed by Dömötör Pálvölgyi, Budapest[/i]
KoMaL A Problems 2023/2024, A. 858
Prove that the only integer solution of the following system of equations is $u=v=x=y=z=0$: $$uv=x^2-5y^2, (u+v)(u+2v)=x^2-5z^2$$
KoMaL A Problems 2024/2025, A. 900
In a room, there are $n$ lights numbered with positive integers $1, 2, \ldots, n$. At the beginning of the game subsets $S_1, S_2,\ldots,S_k$ of $\{1,\ldots, n\}$ can be chosen. For every integer $1\le i\le k$, there is a button that turns on the lights corresponding to the elements of $S_i$ and also a button that turns off all the lights corresponding to the elements of $S_i$. For any positive integer $n$, determine the smallest $k$ for which it is possible to choose the sets $S_1, S_2, \ldots, S_n$ in such a way that allows any combination of the $n$ lights to be turned on, starting from the state where all the lights are off.
[i]Proposed by Kristóf Zólomy, Budapest[/i]
KoMaL A Problems 2021/2022, A. 821
[b]a)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ and positive integer $M$ there exists $n\in\mathbb N$ such that set $\left\{k\in \mathbb N : f(n,k)=g(k)\right\}$ has at least $M$ elements?
[b]b)[/b] Is it possible to find a function $f:\mathbb N^2\to\mathbb N$ such that for every function $g:\mathbb N\to\mathbb N$ there exists $n\in \mathbb N$ such that set $\left\{k\in\mathbb N : f(n,k)=g(k)\right\}$ has an infinite number of elements?
KoMaL A Problems 2021/2022, A. 819
Let $G$ be an arbitrarily chosen finite simple graph. We write non-negative integers on the vertices of the graph such that for each vertex $v$ in $G,$ the number written on $v$ is equal to the number of vertices adjacent to $v$ where an even number is written. Prove that the number of ways to achieve this is a power of $2$.
KoMaL A Problems 2018/2019, A. 744
Show that for every odd integer $N>5$ there exist vectors $\bf u,v,w$ in (three-dimensional) space which are pairwise perpendicular, not parallel with any of the coordinate axes, have integer coordinates, and satisfy $N\bf =|u|=|v|=|w|.$
[i]Based on problem 2 of the 2018 Kürschák contest[/i]
2016 Indonesia TST, 4
The Hawking Space Agency operates $n-1$ space flights between the $n$ habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet.
In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely $1,2, ... , \binom{n}{2}$ monetary units in some order. prove that $n$ or $n-2$ is a square number.
KoMaL A Problems 2023/2024, A. 882
Let $H_1, H_2,\ldots, H_m$ be non-empty subsets of the positive integers, and let $S$ denote their union. Prove that
\[\sum_{i=1}^m \sum_{(a,b)\in H_i^2}\gcd(a,b)\ge\frac1m \sum_{(a,b)\in S^2}\gcd(a,b).\]
[i]Proposed by Dávid Matolcsi, Berkeley[/i]
KoMaL A Problems 2019/2020, A. 763
Let $k\geq 2$ be an integer. We want to determine the weight of $n$ balls. One try consists of choosing two balls, and we are given the sum of the weights of the two chosen balls. We know that at most $k$ of the answers can be wrong. Let $f_k(n)$ denote the smallest number for which it is true that we can always find the weights of the balls with $f_k(n)$ tries (the tries don't have to be decided in advance). Prove that there exist numbers $a_k$ and $b_k$ for which $|f_k(n)-a_kn|\leq b_k$ holds.
[i]Proposed by Surányi László, Budapest and Bálint Virág, Toronto[/i]
KoMaL A Problems 2017/2018, A. 722
The Hawking Space Agency operates $n-1$ space flights between the $n$ habitable planets of the Local Galaxy Cluster. Each flight has a fixed price which is the same in both directions, and we know that using these flights, we can travel from any habitable planet to any habitable planet.
In the headquarters of the Agency, there is a clearly visible board on a wall, with a portrait, containing all the pairs of different habitable planets with the total price of the cheapest possible sequence of flights connecting them. Suppose that these prices are precisely $1,2, ... , \binom{n}{2}$ monetary units in some order. prove that $n$ or $n-2$ is a square number.
KoMaL A Problems 2024/2025, A. 903
Let the irrational number
\[\alpha =1-\cfrac{1}{2a_1-\cfrac{1}{2a_2-\cfrac{1}{2a_3-\cdots}}}\]
where coefficients $a_1, a_2, \ldots$ are positive integers, infinitely many of which are greater than $1$. Prove that for every positive integer $N$ at least half of the numbers $\lfloor \alpha\rfloor, \lfloor 2\alpha\rfloor, \ldots, \lfloor N\alpha\rfloor$ are even.
[i]Proposed by Géza Kós, Budapest[/i]
KoMaL A Problems 2022/2023, A.837
Let all the edges of tetrahedron \(A_1A_2A_3A_4\) be tangent to sphere \(S\). Let \(\displaystyle a_i\) denote the length of the tangent from \(A_i\) to \(S\). Prove that
\[\bigg(\sum_{i=1}^4 \frac 1{a_i}\bigg)^{\!\!2}> 2\bigg(\sum_{i=1}^4 \frac1{a_i^2}\bigg). \]
[i]Submitted by Viktor Vígh, Szeged[/i]
KoMaL A Problems 2021/2022, A. 806
Four distinct lines are given in the plane, which are not concurrent and no three of which are parallel. Prove that it is possible to find four points in the plane, $A,B,C,$ and $D$ with the following properties:
[list=1]
[*]$A,B,C,$ and $D$ are collinear in this order;
[*]$AB=BC=CD$;
[*]with an appropriate order of the four given lines, $A$ is on the first, $B$ is on the second, $C$ is on the third and $D$ is on the fourth line.
[/list]
[i]Proposed by Kada Williams, Cambridge[/i]
KoMaL A Problems 2023/2024, A. 868
A set of points in the plane is called disharmonic, if the ratio of any two distances between the points is between $100/101$ and $101/100$, or at least $100$ or at most $1/100$.
Is it true that for any distinct points $A_1,A_2,\ldots,A_n$ in the plane it is always possible to find distinct points $A_1',A_2',\ldots, A_n'$ that form a disharmonic set of points, and moreover $A_i, A_j$ and $A_k$ are collinear in this order if and only if $A_i', A_j'$ and $A_k'$ are collinear in this order (for all distinct $1 \le i,j,k\le n$?
[i]Submitted by Dömötör Pálvölgyi and Balázs Keszegh, Budapest[/i]
KoMaL A Problems 2020/2021, A. 801
For which values of positive integer $m$ is it possible to find polynomials $P, Q\in\mathbb{C} [x]$, with degrees at least two, such that \[x(x+1)\cdots(x+m-1)=P(Q(x)).\][i]Proposed by Navid Safaei, Tehran[/i]
KoMaL A Problems 2024/2025, A. 902
In triangle $ABC$, interior point $D$ is chosen such that triangle $BCD$ is equilateral. Let $E$ be the isogonal conjugate of point $D$ with respect to triangle $ABC$. Define point $P$ on the ray $AB$ such that $AP=BE$. Similarly, define point $Q$ on the ray $AC$ such that $AQ=CE$. Prove that line $AD$ bisects segment $PQ$.
[i]Proposed by Áron Bán-Szabó, Budapest[/i]
KoMaL A Problems 2022/2023, A. 846
Let $n$ be a positive integer and let vectors $v_1$, $v_2$, $\ldots$, $v_n$ be given in the plain. A flea originally sitting in the origin moves according to the following rule: in the $i$th minute (for $i=1,2,\ldots,n$) it will stay where it is with probability $1/2$, moves with vector $v_i$ with probability $1/4$, and moves with vector $-v_i$ with probability $1/4$. Prove that after the $n$th minute there exists no point which is occupied by the flea with greater probability than the origin.
[i]Proposed by Péter Pál Pach, Budapest[/i]
KoMaL A Problems 2023/2024, A.860
A 0-1 sequence of length $2^k$ is given. Alice can pick a member from the sequence, and reveal it (its place and its value) to Bob. Find the largest number $s$ for which Bob can always pick $s$ members of the sequence, and guess all their values correctly.
Alice and Bob can discuss a strategy before the game with the aim of maximizing the number of correct guesses of Bob. The only information Bob has is the length of the sequence and the member of the sequence picked by Alice.