Found problems: 85335
Let $ S$ be a finite family of squares on a plane such that every point on that plane is contained in at most $ k$ squares in $ S$. Prove that $ P$ can be divided into $ 4(k\minus{}1)\plus{}1$ sub-family such that in each sub-family, each pair of squares are disjoint.
There are a) $26$; b) $30$ identical-looking coins in a circle. It is known that exactly two of them are fake. Real coins weigh the same, fake ones too, but they are lighter than the real ones. How can you determine in three weighings on a cup scale without weights whether there are fake coins lying nearby or not??
[i]Proposed by A. Gribalko[/i]
Each integer should be colored by one of three colors, including red. Each number which can be represent as a sum of two numbers of different colors should be red. Each color should be used. Is this coloring possible?
Let $AA_1$ and $BB_1$ be the altitudes of an acute-angled, non-isosceles triangle $ABC$. Also, let $A_0$ and $B_0$ be the midpoints of its sides $BC$ and $CA$, respectively. The line $A_1B_1$ intersects the line $A_0B_0$ at a point $C'$. Prove that the line $CC'$ is perpendicular to the Euler line of the triangle $ABC$ (this is the line that joins the orthocenter and the circumcenter of the triangle $ABC$).
There are $100$ people in a queue waiting to enter a hall. The hall has exactly $100$ seats numbered from $1$ to $100$. The first person in the queue enters the hall, chooses any seat and sits there. The $n$-th person in the queue, where $n$ can be $2, . . . , 100$, enters the hall after $(n-1)$-th person is seated. He sits in seat number $n$ if he finds it vacant; otherwise he takes any unoccupied seat. Find the total number of ways in which $100$ seats can be filled up, provided the $100$-th person occupies seat number $100$.
Rațiu and Horațiu are playing a game on a $100\times 100$ grid. They make moves alternatively, starting with Rațiu. At a move, a player places a token on an empty cell of the grid. If a player places a token on a cell which is adjacent to another cell with a token, he loses. Determine who has a winning strategy.
If $ y \equal{} \sqrt{x^2 \minus{} 2x \plus{} 1} \plus{} \sqrt{x^2 \plus{} 2x \plus{} 1}$, then $ y$ is:
$ \textbf{(A)}\ 2x\qquad
\textbf{(B)}\ 2(x \plus{} 1)\qquad
\textbf{(C)}\ 0\qquad
\textbf{(D)}\ |x \minus{} 1| \plus{} |x \plus{} 1|\qquad
\textbf{(E)}\ \text{none of these}$
We are given an $(n^2-1)\times(n^2-1)$ checkered board. A set of $n{}$ cells is called [i]progressive[/i] if the centers of the cells lie on a straight line and form $n-1$ equal intervals. Find the number of progressive sets.
[i]Proposed by P. Kozhevnikov[/i]
A country with $n$ cities has some two-way roads connecting certain pairs of cities. Someone notices that if the country is split into two parts in any way, then there would be at most $kn$ roads between the two parts (where $k$ is a fixed positive integer). What is the largest integer $m$ (in terms of $n$ and $k$) such that there is guaranteed to be a set of $m$ cities, no two of which are directly connected by a road?
Let $n=(10^{2020}+2020)^2$. Find the sum of all the digits of $n$.
Let $ABC$ be a triangle and $D$ be the midpoint of the segment $BC$. The circle that passes through $D$ and tangent to $AB$ at $B$, and the circle that passes through $D$ and tangent to $AC$ at $C$ intersect at $M\neq D$. Let $M'$ be the reflection of $M$ with respect to $BC$. Prove that $M'$ is on $AD$.
Marisa has a collection of $2^8-1=255$ distinct nonempty subsets of $\{1, 2, 3, 4, 5, 6, 7, 8\}$. For each step she takes two subsets chosen uniformly at random from the collection, and replaces them with either their union or their intersection, chosen randomly with equal probability. (The collection is allowed to contain repeated sets.) She repeats this process $2^8-2=254$ times until there is only one set left in the collection. What is the expected size of this set?
You are given three lists A, B, and C. List A contains the numbers of the form $10^{k}$ in base 10, with $k$ any integer greater than or equal to 1. Lists B and C contain the same numbers translated into base 2 and 5 respectively: \[\begin{array}{lll}A & B & C \\ 10 & 1010 & 20 \\ 100 & 1100100 & 400 \\ 1000 & 1111101000 & 13000 \\ \vdots & \vdots & \vdots \end{array}.\] Prove that for every integer $n > 1$, there is exactly one number in exactly one of the lists B or C that has exactly $n$ digits.
There are $2024$ points on a circle. A purple elephant labels the points $P_1,P_2,\ldots,P_{2024}$ in some order, and walks along the points from $P_1$ to $P_{2024}$ in this order, while laying some eggs. To ensure the elephant does not step on the eggs it laid, the chords $P_1P_2, P_2P_3, \ldots, P_{2023}P_{2024}$ must not intersect each other except possibly at their endpoints. How many labellings are there? (Note: Two labellings are the same if one is a rotation of the other.)
[i](Proposed by Ho Janson)[/i]
The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.
Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\textsl{monotonic decreasing}$ if when $X \subseteq Y\subseteq S$, then $f(X) \geq f(Y)$ holds. Prove that: $f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)$ for $X, Y \subseteq S$ if and only if $g(X)=f(X \cup \{ a \}) - f(X)$ is a $\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \setminus \{a\}}$ of set $S \setminus \{a\}$ for any $a \in S$.
In a country named Fillip, there are three major cities called Alenda, Breda, Chenida. This country uses the unit of "FP". The distance between Alenda and Chenida is $100$ FP. Breda is $70$ FP from Alenda and $30$ FP from Chenida. Let us say that we take a road trip from Alenda to Chenida. After $2$ hours of driving, we are currently at $50$ FP away from Alenda and $50$ FP away from Chenida. How many FP are we away from Breda?
If $a$ and $b$ are digits for which
\begin{tabular}{ccc} & 2 & a\\ $\times$ & b & 3\\ \hline & 6 & 9\\ 9 & 2\\ \hline 9 & 8 & 9\end{tabular}
Then $a+b =$
A. 3
B. 4
C. 7
D. 9
E. 12
Find all positive integer solutions for the equation $(3x+1)(3y+1)(3z+1)=34xyz$
Thx
Find the number of triples of sets $(A, B, C)$ such that $A \cup B \cup C = \{1, 2, 3, ... , 2549\}$
Inside the triangle $ABC$ choose the point $M$, and on the side $BC$ - the point $K$ in such a way that $MK || AB$. The circle passing through the points $M, \, \, K, \, \, C,$ crosses the side $AC$ for the second time at the point $N$, a circle passing through the points $M, \, \, N, \, \, A, $ crosses the side $AB$ for the second time at the point $Q$. Prove that $BM = KQ$.
(Nagel Igor)
Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$.
[i]Jakob Jurij Snoj, Slovenia[/i]
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]
If $x,y,z,p,q,r$ are real numbers such that \[\frac{1}{x+p}+\frac{1}{y+p}+\frac{1}{z+p}=\frac{1}{p}\]\[\frac{1}{x+q}+\frac{1}{y+q}+\frac{1}{z+q}=\frac{1}{q}\]\[\frac{1}{x+r}+\frac{1}{y+r}+\frac{1}{z+r}=\frac{1}{r}.\]Find the numerical value of $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}$.
An unfair coin has probability $p$ of coming up heads on a single toss. Let $w$ be the probability that, in $5$ independent toss of this coin, heads come up exactly $3$ times. If $w = 144 / 625$, then
$ \textbf{(A)}\ p\text{ must be }2/5$
$ \textbf{(B)}\ p\text{ must be }3/5$
$ \textbf{(C)}\ p\text{ must be greater than }3/5$
$ \textbf{(D)}\ p\text{ is not uniquely determined}$
$ \textbf{(E)}\ \text{there is no value of }p\text{ for which }w = 144/625$