John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary. John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well. What was Mary's Number?

Let $ n \equal{} 2k \minus{} 1$ where $ k \geq 6$ is an integer. Let $ T$ be the set of all $ n\minus{}$tuples $ (x_1, x_2, \ldots, x_n)$ where $ x_i \in \{0,1\}$ $ \forall i \equal{} \{1,2, \ldots, n\}$ For $ x \equal{} (x_1, x_2, \ldots, x_n) \in T$ and $ y \equal{} (y_1, y_2, \ldots, y_n) \in T$ let $ d(x,y)$ denote the number of integers $ j$ with $ 1 \leq j \leq n$ such that $ x_i \neq x_j$, in particular $ d(x,x) \equal{} 0.$ Suppose that there exists a subset $ S$ of $ T$ with $ 2^k$ elements that has the following property: Given any element $ x \in T,$ there is a unique element $ y \in S$ with $ d(x, y) \leq 3.$ Prove that $ n \equal{} 23.$

If $x, y, z$ are positive real numbers find the minimum value of $$2\sqrt{(x+y+z) \left( \frac{1}{x}+ \frac{1}{y} + \frac{1}{z} \right)} - \sqrt{ \left( 1+ \frac{x}{y} \right) \left( 1+ \frac{y}{z} \right)}$$

Two people $A$ and $B$ start from the same place at the same time to travel around a circular track of length $100$ m in opposite directions. First $B$ goes more slowly than $A$ until they meet then by doubling his rate he next meets $A$ at the starting point. Let $d$ m be the distance travelled by $B$ before he met $A$ for the first time after leaving the starting point. Find the integer closest to $d$.

$\Delta ABC$ is right-angled at $B$, with $AB=1$ and $BC=3$. $E$ is the foot of perpendicular from $B$ to $AC$. $BA$ and $BE$ are produced to $D$ and $F$ respectively such that $D$, $F$, $C$ are collinear and $\angle DAF=\angle BAC$. Find the length of $AD$.

The least number is $m$ and the greatest number is $M$ among $ a_1 ,a_2 ,\ldots,a_n$ satisfying $ a_1 \plus{}a_2 \plus{}...\plus{}a_n \equal{}0$. Prove that \[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]

Show that there exists a sequence of positive integers $x_1, x_2,…x_n,…$ that satisfies the following two conditions: (i) Every positive integer appears exactly once, (ii) For every $n=1,2,…$ the partial sum $x_1+x_2+…+x_n$ is divisible by $n^n$.

Let $ ABC$ be a triangle, let $ AB > AC$. Its incircle touches side $ BC$ at point $ E$. Point $ D$ is the second intersection of the incircle with segment $ AE$ (different from $ E$). Point $ F$ (different from $ E$) is taken on segment $ AE$ such that $ CE \equal{} CF$. The ray $ CF$ meets $ BD$ at point $ G$. Show that $ CF \equal{} FG$.

Let $\mathcal{C}$ be a family of subsets of $A=\{1,2,\dots,100\}$ satisfying the following two conditions: 1) Every $99$ element subset of $A$ is in $\mathcal{C}.$ 2) For any non empty subset $C\in\mathcal{C}$ there is $c\in C$ such that $C\setminus\{c\}\in \mathcal{C}.$ What is the least possible value of $|\mathcal{C}|$?

Paul and his pet octahedron like to play games together. For this game, the octahedron randomly draws an arrow on each of its faces pointing to one of its three edges. Paul then randomly chooses a face and progresses from face to adjacent face, as determined by the arrows on each face, and he wins if he reaches every face of the octahedron. What is the probability that Paul wins?

\[\int_0^1 (x^6+6x^5+15x^4+15x^2+6x+1)\mathrm dx\] [i]Proposed by Robert Trosten[/i]

The function $f$ sends sequences to sequences in the following way: given a sequence $\{a_n\}_{n=0}^{\infty}$ of real numbers, $f$ sends $\{a_n\}_{n=0}^{\infty}$ to the sequence $\{b_n\}_{n=0}^{\infty},$ where $b_n=\sum_{k=0}^n a_k \tbinom{n}{k}$ for all $n \ge 0.$ Let $\{F_n\}_{n=0}^{\infty}$ be the Fibonacci sequence, defined by $F_0=0, F_1=1,$ and $F_{n+2}=F_{n+1}+F_n$ for all $n \ge 0.$ Let $\{c_n\}_{n=0}^{\infty}$ denote the sequence obtained by applying the function $f$ to the sequence $\{F_n\}_{n=0}^{\infty}$ $2022$ times. Find $c_5 \pmod{1000}.$

Consider the sequence of integer such that: $ a_1 = 2$ $ a_2 = 5$ $ a_{n + 1} = (2 - n^2)a_n + (2 + n^2)a_{n - 1}, \forall n\ge 2$ Find all triplies $ (x,y,z) \in \mathbb{N}^3$ such that $ a_xa_y = a_z$.

For any function $f(x)$, in the same rectangular coordinates, figures of function $y=f(x-1)$ and $y=f(-x+1)$ $\text{(A)}$ are symmetrical about $x$-axis $\text{(B)}$ are symmetrical about line $x=1$ $\text{(C)}$ are symmetrical about line $x=-1$ $\text{(D)}$ are symmetrical about $y$-axis

If $\angle A = 60^\circ $, $\angle E = 40^\circ $ and $\angle C = 30^\circ $, then $\angle BDC = $ [asy] pair A,B,C,D,EE; A = origin; B = (2,0); C = (5,0); EE = (1.5,3); D = (1.75,1.5); draw(A--C--D); draw(B--EE--A); dot(A); dot(B); dot(C); dot(D); dot(EE); label("$A$",A,SW); label("$B$",B,S); label("$C$",C,SE); label("$D$",D,NE); label("$E$",EE,N); [/asy] $\text{(A)}\ 40^\circ \qquad \text{(B)}\ 50^\circ \qquad \text{(C)}\ 60^\circ \qquad \text{(D)}\ 70^\circ \qquad \text{(E)}\ 80^\circ$

Let $ABCDEF$ be an equiangular hexagon. Let $P$ be the point that is a distance of 6 from $BC$, $DE$, and $FA$. If the distances from $P$ to $AB$, $CD$, and $EF$ are $8$, $11$, and $5$ respectively, find $(DE-AB)^2$. [i]Proposed by Andy Xu[/i]

Let $k$ be a positive integer and $u, v$ be real numbers. Consider $P(x) = (x - u^k) (x - uv) (x -v^k) = x^3 + ax^2 + bx + c$. (a) For $k = 2$ prove that if $a, b, c$ are rational then so is $uv$. (b) Is that also true for $k = 3$?

Let $ S$ be a finite set of points in the plane such that no three of them are on a line. For each convex polygon $ P$ whose vertices are in $ S$, let $ a(P)$ be the number of vertices of $ P$, and let $ b(P)$ be the number of points of $ S$ which are outside $ P$. A line segment, a point, and the empty set are considered as convex polygons of $ 2$, $ 1$, and $ 0$ vertices respectively. Prove that for every real number $ x$ \[\sum_{P}{x^{a(P)}(1 \minus{} x)^{b(P)}} \equal{} 1,\] where the sum is taken over all convex polygons with vertices in $ S$. [i]Alternative formulation[/i]: Let $ M$ be a finite point set in the plane and no three points are collinear. A subset $ A$ of $ M$ will be called round if its elements is the set of vertices of a convex $ A \minus{}$gon $ V(A).$ For each round subset let $ r(A)$ be the number of points from $ M$ which are exterior from the convex $ A \minus{}$gon $ V(A).$ Subsets with $ 0,1$ and 2 elements are always round, its corresponding polygons are the empty set, a point or a segment, respectively (for which all other points that are not vertices of the polygon are exterior). For each round subset $ A$ of $ M$ construct the polynomial \[ P_A(x) \equal{} x^{|A|}(1 \minus{} x)^{r(A)}. \] Show that the sum of polynomials for all round subsets is exactly the polynomial $ P(x) \equal{} 1.$ [i]Proposed by Federico Ardila, Colombia[/i]

Let $m$ be a positive integer and let $A$, respectively $B$, be two alphabets with $m$, respectively $2m$ letters. Let also $n$ be an even integer which is at least $2m$. Let $a_n$ be the number of words of length $n$, formed with letters from $A$, in which appear all the letters from $A$, each an even number of times. Let $b_n$ be the number of words of length $n$, formed with letters from $B$, in which appear all the letters from $B$, each an odd number of times. Compute $\frac{b_n}{a_n}$.

Determine all triples $(a, b, c)$ of positive integers such that $$a! +b! = 2^{c!} .$$

For each positive integer $n$, let $k(n)$ be the number of ones in the binary representation of $2023 \cdot n$. What is the minimum value of $k(n)$?

Let $\triangle ABC$ be a right-angled triangle with $\angle BAC = 90^{\circ}$ and let $E$ be the foot of the perpendicular from $A$ to $BC$. Let $Z \ne A$ be a point on the line $AB$ with $AB = BZ$. Let $(c)$ be the circumcircle of the triangle $\triangle AEZ$. Let $D$ be the second point of intersection of $(c)$ with $ZC$ and let $F$ be the antidiametric point of $D$ with respect to $(c)$. Let $P$ be the point of intersection of the lines $FE$ and $CZ$. If the tangent to $(c)$ at $Z$ meets $PA$ at $T$, prove that the points $T$, $E$, $B$, $Z$ are concyclic. Proposed by [i]Theoklitos Parayiou, Cyprus[/i]

Make a graphical representation of the function $y=\vert \vert \vert x-1 \vert -2 \vert -3 \vert$ on the interval $-8 \leq x \leq 8$.

Sides $AB,~ BC, ~CD$ and $DA$, respectively, of convex quadrilateral $ABCD$ are extended past $B,~ C ,~ D$ and $A$ to points $B',~C',~ D'$ and $A'$. Also, $AB = BB' = 6,~ BC = CC' = 7, ~CD = DD' = 8$ and $DA = AA' = 9$; and the area of $ABCD$ is 10. The area of $A 'B 'C'D'$ is $\textbf{(A) }20\qquad\textbf{(B) }40\qquad\textbf{(C) }45\qquad\textbf{(D) }50\qquad \textbf{(E) }60$

Is it possible to place a triangle with area $1999$ and perimeter $19992$ in the interior of a triangle with area $2000$ and perimeter $20002$?