Find all quadruples $(a,b,c,d)$ of positive real numbers such that $abcd=1,a^{2012}+2012b=2012c+d^{2012}$ and $2012a+b^{2012}=c^{2012}+2012d$.

Prove that we can find an infinite set of positive integers of the from $2^n-3$ (where $n$ is a positive integer) every pair of which are relatively prime.

Square $ABCD$ has side length $5$. Draw E on $BC$ and $F$ on $AD$ such that $BE < AF$. Next, flip $ABCD$ across $EF$ to a square $A'B'C'D'$ such that $C'$ lies in the interior of $ABCD$ and $C$ lies in the interior of $A'B'C'D'$. Suppose that $CC' = 4$ and $DD' = 2$. Compute $AA'$.

A square board with a size of $2020 \times 2020$ is divided into $2020^2$ small squares of size $1 \times 1$. Each of these small squares will be coloured black or white. Determine the number of ways to colour the board such that for every $2\times 2$ square, which consists of $4$ small squares, contains $2$ black small squares and $2$ white small squares.

Consider all pairs of points $(a,b,c)$ and $(d,e, f )$ in the $3$-D coordinate system with $ad +be +c f = -2023$. What is the least positive integer that can be the distance between such a pair of points? [i]Proposed by William Hua[/i]

Find all prime numbers p such that $2^p+p^2 $ is also a prime number.

Consider a polynomial in $n$ variables with real coefficients. We know that if every variable is $\pm1$, the value of the polynomial is positive, or negative if the number of $-1$'s is even, or odd, respectively. Prove that the degree of this polynomial is at least $n$.

Let $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Find $x$ such that $x\lfloor x\lfloor x\lfloor x\rfloor\rfloor \rfloor = 88$ [list=1] [*] $\pi$ [*] 3.14 [*] $\frac{22}{7}$ [*] All of these [/list]

a) What can be the number of symmetry axes of a checked polygon, that is, of a polygon whose sides lie on lines of a list of checked paper? (Indicate all possible values.) b) What can be the number of symmetry axes of a checked polyhedron, that is, of a polyhedron consisting of equal cubes which border one to another by plane facets?

Two regular polygons, a $7$-gon and a $17$-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)

Let $R$ be a noncommutative finite ring with multiplicative identity element $1$. Show that if the subring generated by $I \cup \{1\}$ is $R$ for each nonzero ideal $I$ then $R$ is simple.

Which of the following statements is (are) equivalent to the statement "If the pink elephant on planet alpha has purple eyes, then the wild pig on planet beta does not have a long nose"? $\textbf{I. }$ "If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha has purple eyes." $\textbf{II. }$ "If the pink elephant on planet alpha does not have purple eyes, then the wild pig on planet beta does not have a long nose. $\textbf{III. }$ "If the wild pig on planet beta has a long nose, then the pink elephant on planet alpha does not have purple eyes." $ \textbf{IV. }$ "The pink elephant on planet alpha does not have purple eyes, or the wild pig on planet beta does not have a long nose." $\textbf{(A) }\textbf{I. }\text{and }\textbf{II. }\text{only}\qquad\textbf{(B) }\textbf{III. }\text{and }\textbf{IV. }\text{only}\qquad\textbf{(C) }\textbf{II. }\text{and }\textbf{IV. }\text{only}\qquad\textbf{(D) }\textbf{II. }\text{and }\textbf{III. }\text{only}\qquad \textbf{(E) }\text{and }\textbf{III. }\text{only}$

Let $ABC$ be a triangle and $D$ be the foot of the altitude from $A$. Let $l$ be the line that passes through the midpoints of $BC$ and $AC$. $E$ is the reflection of $D$ over $l$. Prove that the circumcentre of $\triangle ABC$ lies on the line $AE$.

Let $T$ be a non-empty finite subset of positive integers $\ge 1$. A subset $S$ of $T$ is called [b]good [/b] if for every integer $t\in T$ there exists an $s$ in $S$ such that $gcd(t,s) >1$. Let \[A={(X,Y)\mid X\subseteq T,Y\subseteq T,gcd(x,y)=1 \text{for all} x\in X, y\in Y}\] Prove that : $a)$ If $X_0$ is not [b]good[/b] then the number of pairs $(X_0,Y)$ in $A$ is [b]even[/b]. $b)$ the number of good subsets of $T$ is [b]odd[/b].

Let $\left(a_{1},\ a_{2},\ a_{3},\ ...\right)$ be a sequence of reals such that $a_{n}\geq\frac{\left(n-1\right)a_{n-1}+\left(n-2\right)a_{n-2}+...+2a_{2}+1a_{1}}{\left(n-1\right)+\left(n-2\right)+...+2+1}$ for every integer $n\geq 2$. Prove that $a_{n}\geq\frac{a_{n-1}+a_{n-2}+...+a_{2}+a_{1}}{n-1}$ for every integer $n\geq 2$. [i]Generalization.[/i] Let $\left(b_{1},\ b_{2},\ b_{3},\ ...\right)$ be a monotonically increasing sequence of positive reals, and let $\left(a_{1},\ a_{2},\ a_{3},\ ...\right)$ be a sequence of reals such that $a_{n}\geq\frac{b_{n-1}a_{n-1}+b_{n-2}a_{n-2}+...+b_{2}a_{2}+b_{1}a_{1}}{b_{n-1}+b_{n-2}+...+b_{2}+b_{1}}$ for every integer $n\geq 2$. Prove that $a_{n}\geq\frac{a_{n-1}+a_{n-2}+...+a_{2}+a_{1}}{n-1}$ for every integer $n\geq 2$. darij

Given $ 12$ points in a plane no three of which are collinear, the number of lines they determine is: $ \textbf{(A)}\ 24 \qquad\textbf{(B)}\ 54 \qquad\textbf{(C)}\ 120 \qquad\textbf{(D)}\ 66 \qquad\textbf{(E)}\ \text{none of these}$