A lock has $16$ keys arranged in a $4\times 4$ array, each key oriented either horizontally or vertically. In order to open it, all the keys must be vertically oriented. When a key is switched to another position, all the other keys in the same row and column automatically switch their positions too. Show that no matter what the starting positions are, it is always possible to open this lock. (Only one key at a time can be switched.)

[b]a)[/b] Prove that $ 0=\inf\{ |x\sqrt 2+y\sqrt 3+y\sqrt 5|\big| x,y,z\in\mathbb{Z} ,x^2+y^2+z^2>0 \} $ [b]b)[/b] Prove that there exist three positive rational numbers $ a,b,c $ such that the expression $ E(x,y,z):=xa+yb+zc $ vanishes for infinitely many integer triples $ (x,y,z), $ but it doesn´t get arbitrarily close to $ 0. $

Sabrina has a fair tetrahedral die whose faces are numbered 1, 2, 3, and 4, respectively. She creates a sequence by rolling the die and recording the number on its bottom face. However, she discards (without recording) any roll such that appending its number to the sequence would result in two consecutive terms that sum to 5. Sabrina stops the moment that all four numbers appear in the sequence. Find the expected (average) number of terms in Sabrina's sequence.

Andrea and Bruno play a game on a table with $11$ rows and $9$ columns. First Andrea divides the table in $33$ zones. Each zone is formed by $3$ contiguous cells aligned vertically or horizontally, as shown in the figure. [code] ._ |_| |_| _ _ _ |_| |_|_|_| [/code] Then, Bruno writes one of the numbers $0, 1, 2, 3, 4, 5$ in each cell in such a way that the sum of the numbers in each zone is equal to $5$. Bruno wins if the sum of the numbers written in each of the $9$ columns of the table is a prime number. Otherwise, Andrea wins. Show that Bruno always has a winning strategy.

Let $a_1,a_2,\dots$ be a sequence of integers satisfying $a_1=2$ and: $$a_n=\begin{cases}a_{n-1}+1, & \text{ if }n\ne a_k \text{ for some }k=1,2,\dots,n-1; \\ a_{n-1}+2, & \text{ if } n=a_k \text{ for some }k=1,2,\dots,n-1. \end{cases}$$ Find the value of $a_{2022!}$.

Let $a, b, c$ be real numbers such that $a+b+c > 0$, $ab+bc+ca > 0$, $abc > 0$. Show that $a, b, c$ are all positive.

We are given a cyclic quadrilateral $ABCD \quad AB\not=CD$. Quadrilaterals $AKDL$ and $CMBN$ are rhombuses with equal sides. Prove, that $KLMN$ is cyclic

We calculate the sum of $7$ natural consecutive pairs (e.g. $2+4+6+8+10+12+14$) and we will call the result $A$, then the sum of the next $7$ consecutive pairs (in the example, $16+ 18+...$) and its result we will call $B$, and finally we calculate the sum of the following $7$ consecutive pairs and its result we will call $C$. Can the product $ABC$ be $2002^3$?

Circles $\mathcal{C}_{1}$ and $\mathcal{C}_{2}$ intersect at two points, one of which is $(9,6),$ and the product of the radii is $68.$ The x-axis and the line $y=mx$, where $m>0,$ are tangent to both circles. It is given that $m$ can be written in the form $a\sqrt{b}/c,$ where $a,$ $b,$ and $c$ are positive integers, $b$ is not divisible by the square of any prime, and $a$ and $c$ are relatively prime. Find $a+b+c.$

Let $ a$ be positive constant number. Evaluate $ \int_{ \minus{} a}^a \frac {x^2\cos x \plus{} e^{x}}{e^{x} \plus{} 1}\ dx.$

Points $A$, $V_1$, $V_2$, $B$, $U_2$, $U_1$ lie fixed on a circle $\Gamma$, in that order, and such that $BU_2 > AU_1 > BV_2 > AV_1$. Let $X$ be a variable point on the arc $V_1 V_2$ of $\Gamma$ not containing $A$ or $B$. Line $XA$ meets line $U_1 V_1$ at $C$, while line $XB$ meets line $U_2 V_2$ at $D$. Let $O$ and $\rho$ denote the circumcenter and circumradius of $\triangle XCD$, respectively. Prove there exists a fixed point $K$ and a real number $c$, independent of $X$, for which $OK^2 - \rho^2 = c$ always holds regardless of the choice of $X$. [i]Proposed by Andrew Gu and Frank Han[/i]

Determine all positive integers $M$ such that the sequence $a_0, a_1, a_2, \cdots$ defined by \[ a_0 = M + \frac{1}{2} \qquad \textrm{and} \qquad a_{k+1} = a_k\lfloor a_k \rfloor \quad \textrm{for} \, k = 0, 1, 2, \cdots \] contains at least one integer term.

Given two circles $k_1$ and $k_2$ which intersect at two different points $A$ and $B$. The tangent to the circle $k_2$ at the point $A$ meets the circle $k_1$ again at the point $C_1$. The tangent to the circle $k_1$ at the point $A$ meets the circle $k_2$ again at the point $C_2$. Finally, let the line $C_1C_2$ meet the circle $k_1$ in a point $D$ different from $C_1$ and $B$. Prove that the line $BD$ bisects the chord $AC_2$.

In a parallelogram $ABCD$ with the side ratio $AB : BC = 2 : \sqrt 3$ the normal through $D$ to $AC$ and the normal through $C$ to $AB$ intersects in the point $E$ on the line $AB$. What is the relationship between the lengths of the diagonals $AC$ and $BD$?

On square $ABCD$, point $E$ lies on side $\overline{AD}$ and point $F$ lies on side $\overline{BC}$, so that $BE=EF=FD=30$. Find the area of square $ABCD$.

Business is a little slow at Lou's Fine Shoes, so Lou decides to have a sale. On Friday, Lou increases all of Thursday's prices by $10$ percent. Over the weekend, Lou advertises the sale: "Ten percent off the listed price. Sale starts Monday." How much does a pair of shoes cost on Monday that cost $40$ dollars on Thursday? $\textbf{(A)}\ 36 \qquad \textbf{(B)}\ 39.60 \qquad \textbf{(C)}\ 40 \qquad \textbf{(D)}\ 40.40 \qquad \textbf{(E)}\ 44$

Given sequences $a_n=\frac{1}{n}{\sqrt[n] {_{2n}P_n}},\ b_n=\frac{1}{n^2}{\sqrt[n] {_{4n}P_{2n}}}$ and $c_n=\sqrt[n]{\frac{_{8n}P_{4n}}{_{6n}P_{4n}}}$, find $\lim_{n\to\infty} a_n,\ \lim_{n\to\infty} b_n$and $\lim_{n\to\infty} c_n.$

Let $k\geq 2$ be a natural number. Suppose that $a_1, a_2, \dots a_{2021}$ is a monotone decreasing sequence of non-negative numbers such that \[\sum_{i=n}^{2021}a_i\leq ka_n\] for all $n=1,2,\dots 2021$. Prove that $a_{2021}\leq 4(1-\frac{1}{k})^{2021}a_1$.

A right circular cone has a base with radius 600 and height $200\sqrt{7}$. A fly starts at a point on the surface of the cone whose distance from the vertex of the cone is 125, and crawls along the surface of the cone to a point on the exact opposite side of the cone whose distance from the vertex is $375\sqrt{2}$. Find the least distance that the fly could have crawled.

For any real number $a$ and positive integer $k$, define \[ {a \choose k} = \frac{a(a-1)(a-2)\cdots(a-(k-1))}{k(k-1)(k-2)\cdots(2)(1)}. \]What is \[{-\frac{1}{2} \choose 100} \div {\frac{1}{2} \choose 100}?\] $ \textbf{(A)}\ -199\qquad\textbf{(B)}\ -197\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ 197\qquad\textbf{(E)}\ 199 $

The probability that event $A$ occurs is $\frac{3}{4}$; the probability that event $B$ occurs is $\frac{2}{3}$. Let $p$ be the probability that both $A$ and $B$ occur. The smallest interval necessarily containing $p$ is the interval $ \textbf{(A)}\ \Big[\frac{1}{12},\frac{1}{2}\Big]\qquad\textbf{(B)}\ \Big[\frac{5}{12},\frac{1}{2}\Big]\qquad\textbf{(C)}\ \Big[\frac{1}{2},\frac{2}{3}\Big]\qquad\textbf{(D)}\ \Big[\frac{5}{12},\frac{2}{3}\Big]\qquad\textbf{(E)}\ \Big[\frac{1}{12},\frac{2}{3}\Big]$

The ``Manhattan distance" between two cells is the shortest distance between those cells when traveling up, down, left, or right, as if one were traveling along city blocks rather than as the crow flies. Place numbers from $1$-$6$ in some cells so the following criteria are satisfied: $1.$ A cell contains at most one number. Cells can be left empty. $2.$ For each cell containing a number $N$ in the grid, exactly two other cells containing $N$ are at a Manhattan distance of $N.$ $3.$ For each cell containing a number $N$ in the grid, no other cells containing $N$ are at a Manhattan distance less than $N.$ [asy] //credits to fruitmonster97 for the diagram unitsize(1.25cm); //The gridlines for(int i=-3;i<4;++i){ draw((i,3)--(i,-3),lightgray+linewidth(1)); } for(int j=-3;j<4;++j){ if (j==0){ draw((4,j)--(-4,j),lightgray+linewidth(1)); }else{ draw((3,j)--(-3,j),lightgray+linewidth(1)); } } //The outline draw((-4,-1)--(-4,1)--(-3,1)--(-3,3)--(3,3)--(3,1)--(4,1)--(4,-1)--(3,-1)--(3,-3)--(-3,-3)--(-3,-1)--cycle); //The numbers label("$1$",(-0.5,0.5)); label("$1$",(0.5,-0.5)); label("$2$",(-1.5,1.5)); label("$2$",(-2.5,-1.5)); label("$3$",(2.5,1.5)); label("$3$",(-3.5,0.5)); label("$4$",(3.5,-0.5)); label("$4$",(1.5,-2.5)); label("$4$",(-1.5,2.5)); label("$5$",(-2.5,1.5)); label("$5$",(2.5,-1.5)); label("$6$",(1.5,-1.5)); [/asy]

Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers $$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$

We have a blue triangle. In every move, we divide the blue triangle by angle bisector to $2$ triangles and color one triangle in red. Prove, that after some moves we color more than half of the original triangle in red.

Given a positive integer $n \geq 2$, whose canonical prime factorization is $n = p_1^{\alpha_1}p_2^{\alpha_2} \ldots p_k^{\alpha_k}$, we define the following functions: $$\varphi(n) = n\bigg(1 -\frac{1}{p_1}\bigg) \bigg(1 -\frac{1}{p_2}\bigg) \ldots \bigg(1 -\frac {1}{p_k}\bigg) ; \overline{\varphi}(n) = n\bigg(1 +\frac{1}{p_1}\bigg) \bigg(1 +\frac{1}{p_2}\bigg) \ldots \bigg(1 + \frac{1}{p_k}\bigg)$$ Consider all positive integers $n$ such that $\overline{\varphi}(n)$ is a multiple of $n + \varphi(n) $. (a) Prove that $n$ is even. (b) Determine all positive integers $n$ that satisfy this property.