Found problems: 85335
Let $n, b$ and $c$ be positive integers. A group of $n$ pirates wants to fairly split their treasure. The treasure consists of $c \cdot n$ identical coins distributed over $b \cdot n$ bags, of which at least $n-1$ bags are initially empty. Captain Jack inspects the contents of each bag and then performs a sequence of moves. In one move, he can take any number of coins from a single bag and put them into one empty bag. Prove that no matter how the coins are initially distributed, Jack can perform at most $n-1$ moves and then split the bags among the pirates such that each pirate gets $b$ bags and $c$ coins.
Find the number of ways to partition $S = \{1, 2, 3, \dots, 2020\}$ into two disjoint sets $A$ and $B$ with $A \cup B = S$ so that if you choose an element $a$ from $A$ and an element $b$ from $B$, $a+b$ is never a multiple of $20$. $A$ or $B$ can be the empty set, and the order of $A$ and $B$ doesn't matter. In other words, the pair of sets $(A,B)$ is indistinguishable from the pair of sets $(B,A)$.
[i]Proposed by Timothy Qian[/i]
Let $AC$ and $BD$ be two chords of a circle with center $O$ such that they intersect at right angles inside the circle at the point $M$. Suppose $K$ and $L$ are midpoints of the chords $AB$ and $CD$ respectively. Prove that $OKML$ is a parallelogram.
There are given $2^{500}$ points on a circle labeled $1,2,\ldots ,2^{500}$ in some order. Prove that one can choose $100$ pairwise disjoint chords joining some of theses points so that the $100$ sums of the pairs of numbers at the endpoints of the chosen chord are equal.
$ABCD$ is cyclic quadrilateral and $O$ is the center of its circumcircle. Suppose that $AD \cap BC = E$ and $AC \cap BD = F$. Circle $\omega$ is tanget to line $AC$ and $BD$. $PQ$ is a diameter of $\omega$ that $F$ is orthocenter of $EPQ$. Prove that line $OE$ is passing through center of $\omega$
[i]Proposed by Mahdi Etesami Fard [/i]
We will say that a plane is [i]well-colored[/i] in several colors if it is divided into convex polygons with an area of at least $1/1000$ and each polygon is colored in one color. Points lying on the border of several polygons can be colored in any of their colors. Are there convex is a $9$-gon $R$ and a good coloring of the plane in $7$ colors such that in any polygon obtained from $R$ by a translate to any vector, all colors occupy the same area ($1/7$ of the area of $R$)?
A regular octagon is inscribed in a circle of radius 2. Alice and Bob play a game in which they take turns claiming vertices of the octagon, with Alice going first. A player wins as soon as they have selected three points that form a right angle. If all points are selected without either player winning, the game ends in a draw. Given that both players play optimally, find all positive areas of the convex polygon formed by Alice's points at the end of the game.
Find the smallest positive integer $n$ with the following property:
After painting black exactly $n$ cells of a $7\times 7$ board there always exists a $2\times 2$ square with at least three black cells.
A convex quadrilateral is inscribed in a circle of radius $1$. Prove that the difference between its perimeter and the sum of the lengths of its diagonals is greater than zero and less than $2.$
An acute-angled triangle $ABC$ is given, through the vertices $B$ and $C$ of which a circle $\Omega$, $A \notin \Omega$, is drawn. We consider all points $P \in \Omega$, that do not lie on none of the lines $AB$ and $AC$ and for which the common tangents of the circumscribed circles of triangles $APB$ and $APC$ are not parallel. Let $X_P$ be the point of intersection of such two common tangents.
a) Prove that the locus of points $X_P$ lies to some two lines.
b) Prove that if the circle $\Omega$ passes through the orthocenter of the triangle $ABC$, then one of these lines is the line $BC$.
Let $\mathcal{S}$ be the set of points $(x, y)$ in the Cartesian coordinate plane such that $xy > 0$ and $x^2 + y^2 + 2x + 4y \leq 2021.$ The total area of $\mathcal{S}$ can be written in the form $a\pi + b,$ where $a$ and $b$ are integers. Compute $a + b.$
Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.
On an infinite square grid we place finitely many [i]cars[/i], which each occupy a single cell and face in one of the four cardinal directions. Cars may never occupy the same cell. It is given that the cell immediately in front of each car is empty, and moreover no two cars face towards each other (no right-facing car is to the left of a left-facing car within a row, etc.). In a [i]move[/i], one chooses a car and shifts it one cell forward to a vacant cell. Prove that there exists an infinite sequence of valid moves using each car infinitely many times.
[i]Nikolai Beluhov[/i]
A pair of standard 6-sided fair dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference?
$ \textbf{(A)}\ \frac{1}{36} \qquad
\textbf{(B)}\ \frac{1}{12} \qquad
\textbf{(C)}\ \frac{1}{6} \qquad
\textbf{(D)}\ \frac{1}{4} \qquad
\textbf{(E)}\ \frac{5}{18}
$
How many moles of oxygen gas are produced by the decomposition of $245$ g of potassium chlorate?
\[\ce{2KClO3(s)} \rightarrow \ce{2KCl(s)} + \ce{3O2(g)}\]
Given:
Molar Mass/ $\text{g} \cdot \text{mol}^{-1}$
$\ce{KClO3}$: $122.6$
$ \textbf{(A)}\hspace{.05in}1.50 \qquad\textbf{(B)}\hspace{.05in}2.00 \qquad\textbf{(C)}\hspace{.05in}2.50 \qquad\textbf{(D)}\hspace{.05in}3.00 \qquad $
Find the maximum possible number of edges of a simple graph with $8$ vertices and without any quadrilateral. (a simple graph is an undirected graph that has no loops (edges connected at both ends to the same vertex) and no more than one edge between any two different vertices.)
For positive real numbers $x,y,z$ that satisfy $ xy + yz + zx + xyz=4$, prove that
$$\frac{x+y+z}{xy+yz+zx}\le 1+\frac{5}{247}\cdot \left( (x-y)^2+(y-z)^2+(z-x)^2\right)$$
Suppose that there are $16$ variables $\{a_{i,j}\}_{0\leq i,j\leq 3}$, each of which may be $0$ or $1$. For how many settings of the variables $a_{i,j}$ do there exist positive reals $c_{i,j}$ such that the polynomial \[f(x,y)=\sum_{0\leq i,j\leq 3}a_{i,j}c_{i,j}x^iy^j\] $(x,y\in\mathbb{R})$ is bounded below?
Triangle $ABC$ is such that $AB < AC$. The perpendicular bisector of side $BC$ intersects lines $AB$ and $AC$ at points $P$ and $Q$, respectively. Let $H$ be the orthocentre of triangle $ABC$, and let $M$ and $N$ be the midpoints of segments $BC$ and $PQ$, respectively. Prove that lines $HM$ and $AN$ meet on the circumcircle of $ABC$.
Fredek runs a private hotel. He claims that whenever $ n \ge 3$ guests visit the hotel, it is possible to select two guests that have equally many acquaintances among the other guests, and that also have a common acquaintance or a common unknown among the guests. For which values of $ n$ is Fredek right? (Acquaintance is a symmetric relation.)
Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ$. Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ$. Find the number of degrees in $\angle CMB$.
Simple graph $G$ has $19998$ vertices. For any subgraph $\bar G$ of $G$ with $9999$ vertices, $\bar G$ has at least $9999$ edges. Find the minimum number of edges in $G$
Let $x_1, x_2, \dots ,x_n$ ($n\ge 2$) be positive real numbers satisfying $\sum^{n}_{i=1}x_i=1$. Prove that:\[\sum^{n}_{i=1}\dfrac{x_i}{\sqrt{1-x_i}}\ge \dfrac{\sum_{i=1}^{n}\sqrt{x_i}}{\sqrt{n-1}}.\]
When Phil and Shelley stand on a scale together, the scale reads $151$ pounds. When Shelley and Ryan stand on the same scale together, the scale reads $132$ pounds. When Phil and Ryan stand on the same scale together, the scale reads $115$ pounds. Find the number of pounds Shelley weighs.
Show that the edges of a strongly connected bipolar graph can be oriented in such a way that for any edge $ e$ there is a simple directed path from pole $ p$ to pole $ q$ containing $ e$. (A strongly connected bipolar graph is a finite connected graph with two special vertices $ p$ and $ q$ having the property that there are no points $ x,y,x \not \equal{} y$, such that all paths from $ x$ to $ p$ as well as all paths from $ x$ to $ q$ contain $ y$.)
[i]A. Adam[/i]