Found problems: 85335
Pablo and Silvia play on a $2010 \times 2010$ board. To start the game, Pablo writes an integer in every cell. After he is done, Silvia repeats the following operation as many times as she wants: she chooses three cells that form an $L$, like in the figure below, and adds $1$ to each of the numbers in these three cells. Silvia wins if, after doing the operation many times, all of the numbers in the board are multiples of $10$.
Prove that Silvia can always win.
$\begin{array}{|c|c} \cline{1-1} \; & \; \\ \hline \; & \multicolumn{1}{|c|}{\;} \\ \hline \end{array} \qquad \begin{array}{c|c|} \cline{2-2} \; & \; \\ \hline \multicolumn{1}{|c|}{\;} & \; \\ \hline \end{array} \qquad \begin{array}{|c|c} \hline \; & \multicolumn{1}{|c|}{\;} \\ \hline \multicolumn{1}{|c|}{\;} & \; \\ \cline{1-1} \end{array} \qquad \begin{array}{c|c|} \hline \multicolumn{1}{|c|}{\;} & \; \\ \hline \; & \multicolumn{1}{|c|}{\;} \\ \cline{2-2} \end{array}$
Evaluate \[\sqrt{2013+276\sqrt{2027+278\sqrt{2041+280\sqrt{2055+\ldots}}}}\]
Find all functions $ f\in\mathcal{C}^1 [0,1] $ that satisfy $ f(1)=-1/6 $ and
$$ \int_0^1 \left( f'(x) \right)^2 dx\le 2\int_0^1 f(x)dx. $$
Prove that there exist infinitely many pairs $(a, b)$ of relatively prime positive integers such that \[\frac{a^{2}-5}{b}\;\; \text{and}\;\; \frac{b^{2}-5}{a}\] are both positive integers.
Given a convex polyhedron with an even number of edges.
Prove that we can attach an arrow to each edge, such that for every vertex of the polyhedron, the number of the arrows ending in this vertex is even.
The incircle of tangential quadrilateral $ABCD$ intersects diagonal $BD$ at $P$ and $Q$ $(BP<BQ).$ Let $UV$ be the diameter of the incircle perpendicular to $AC$ $(BU<BV).$ Show that the lines $AC,PV,$ and $QU$ pass through one point.
[i]Based on problem 2 of IOM 2018, Moscow[/i]
Let $a$, $b$ $c$ be the natural numbers, such that every digit occurs exactly the same number of times in each of the numbers $a$, $b$, $c$. Is it possible that $a + b + c = 10^{1001}$? Justify your answer.
Alice has 8 coins. She knows for sure only that 7 of these coins are genuine and weigh the same, while the remaining one is counterfeit and is either heavier or lighter than any of the other 7. Bob has a balance scale. The scale shows which plate is heavier but does not show by how much. For each measurement, Alice pays Bob beforehand a fee of one coin. If a genuine coin has been paid, Bob tells Alice the correct weighing outcome, but if a counterfeit coin has been paid, he gives a random answer. Alice wants to identify 5 genuine coins and not to give any of these coins to Bob. Can Alice achieve this result for sure?
Let $M,N\in M_2(\mathbb C)$ be two nonzero matrices such that
$$M^2=N^2=0_2\text{ and }MN+NM=I_2$$where $0_2$ is the $2\times2$ zero matrix and $I_2$ the $2\times2$ unit matrix. Prove that there is an invertible matrix $A\in M_2(\mathbb C)$ such that
$$M=A\begin{pmatrix}0&1\\0&0\end{pmatrix}A^{-1}\text{ and }N=A\begin{pmatrix}0&0\\1&0\end{pmatrix}A^{-1}.$$
Let $ABC$ be a triangle for which $AB+BC = 3AC$. Let $D$ and $E$ be the points of tangency of the incircle with the sides $AB$ and $BC$ respectively, and let $K$ and $L$ be the other endpoints of the diameters originating from $D$ and $E.$ Prove that $C , A, L$, and $K$ lie on a circle.
Albert and Brita play a game with a bar of $19$ adjacent squares. Initially, there is a button on the middle square of the bar. At every turn Albert mentions one positive integer less than $5$, and Brita moves button a number of squares in the direction of her choice - while doing so however, Brita must not move the button more than twice in one direction order. Prove that Albert can choose the numbers so that by the $19$th turn, Brita to be forced to move the button out of the bar.
Let $ABCD$ be an isosceles trapezium inscribed in circle $\omega$, such that $AB||CD$. Let $P$ be a point on the circle $\omega$. Let $H_1$ and $H_2$ be the orthocenters of triangles $PAD$ and $PBC$ respectively. Prove that the length of $H_1H_2$ remains constant, when $P$ varies on the circle.
Let $ M \ge 3$ be an integer and let $ S \equal{} \{3,4,5,\ldots,m\}$. Find the smallest value of $ m$ such that for every partition of $ S$ into two subsets, at least one of the subsets contains integers $ a$, $ b$, and $ c$ (not necessarily distinct) such that $ ab \equal{} c$.
[b]Note[/b]: a partition of $ S$ is a pair of sets $ A$, $ B$ such that $ A \cap B \equal{} \emptyset$, $ A \cup B \equal{} S$.
Is it possible to disassemble and reassemble a $4\times 4\times 4$ Rubik's Cuble in at least $577,800$ non-equivalent ways?
Notes:
1. When we reassemble the cube, a corner cube has to go to a corner cube, an edge cube must go to an edge cube and a central cube must go to a central cube.
2. Two positions of the cube are called equivalent if they can be obtained from one two another by rotating the faces or layers which are parallel to the faces.
In a convex quadrilateral $ABCD$ diagonals intersect at $E$ and $BE = \sqrt{2}\cdot ED, \: \angle BEC = 45^{\circ}.$ Let $F$ be the foot of the perpendicular from $A$ to $BC$ and $P$ be the second intersection point of the circumcircle of triangle $BFD$ and line segment $DC$. Find $\angle APD$.
Let $a\ge b$ and $c\ge d$ be real numbers. Prove that the equation
\[(x+a)(x+d)+(x+b)(x+c)=0\]
has real roots.
When counting from $3$ to $201$, $53$ is the $51^{\text{st}}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{\text{th}}$ number counted. What is $n$?
$\textbf{(A) }146\qquad \textbf{(B) } 147\qquad\textbf{(C) } 148\qquad\textbf{(D) }149\qquad\textbf{(E) }150$
The base $5$ number $32$ is equal to the base $7$ number $23$. There are two $3$-digit numbers in base $5$ which similarly have their digits reversed when expressed in base $7$. What is their sum, in base $5$? (You do not need to include the base $5$ subscript in your answer).
There are three positive integers written on a blackboard every minute. You can pick two written numbers $a$ and $b$ and replace them with $a \cdot b$ and $|a-b|$. Prove that it is always possible to make two of the numbers zero.
Find all positive integers n with the property that $n^3 - 1$ is a perfect square.
Given that $x$ and $y$ are nonzero real numbers such that $x+\frac{1}{y}=10$ and $y+\frac{1}{x}=\frac{5}{12}$, find all possible values of $x$.
Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]
Let $S$ be a subset of $\{1, 2, . . . , 500\}$ such that no two distinct elements of S have a
product that is a perfect square. Find, with proof, the maximum possible number of elements
in $S$.
In $\triangle ABC$ with $\angle BAC = 60^{\circ}$ and circumcircle $\omega$, the angle bisector of $\angle BAC$ intersects side $\overline{BC}$ at point $D$, and line $AD$ is extended past $D$ to a point $A'$. Let points $E$ and $F$ be the feet of the perpendiculars of $A'$ onto lines $AB$ and $AC$, respectively. Suppose that $\omega$ is tangent to line $EF$ at a point $P$ between $E$ and $F$ such that $\tfrac{EP}{FP} = \tfrac{1}{2}$. Given that $EF=6$, the area of $\triangle ABC$ can be written as $\tfrac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
[i]Proposed by Taiki Aiba[/i]
A ball with mass $m$ projected horizontally off the end of a table with an initial kinetic energy $K$. At a time $t$ after
it leaves the end of the table it has kinetic energy $3K$. What is $t$? Neglect air resistance.
(A) $(3/g)\sqrt{K/m}$
(B) $(2/g)\sqrt{K/m}$
(C) $(1/g)\sqrt{8K/m}$
(D) $(K/g)\sqrt{6/m}$
(E) $(2K/g)\sqrt{1/m}$