Let $b \geqslant 2$ be a positive integer. Anu has an infinite collection of notes with exactly $b-1$ copies of a note worth $b^k-1$ rupees, for every integer $k\geqslant 1$. A positive integer $n$ is called payable if Anu can pay exactly $n^2+1$ rupees by using some collection of her notes. Prove that if there is a payable number, there are infinitely many payable numbers. [i]Proposed by Shantanu Nene[/i]

Consider the numbers $ a_n=1-\binom{n}{3} +\binom{n}{6} -\cdots, b_n= -\binom{n}{1} +\binom{n}{4}-\binom{n}{7} +\cdots $ and $ c_n=\binom{n}{2} -\binom{n}{5} +\binom{n}{8} -\cdots , $ for a natural number $ n\ge 2. $ Prove that $$ a_n^2+b_n^2+c_n^2-a_nb_n-b_nc_n-c_na_n =3^{n-1}. $$

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

A permutation $\pi : \{1,2,\ldots,N\} \rightarrow \{1,2, \ldots,N\}$ is [i]very odd[/i] if the smallest positive integer $k$ such that $\pi^k(a) = a$ for all $1 \le a \le N$ is odd, where $\pi^k$ denotes $\pi$ composed with itself $k$ times. Let $X_0 = 1,$ and for $i \ge 1,$ let $X_i$ be the fraction of all permutations of $\{1,2,\ldots,i\}$ that are very odd. Let $S$ denote the set of all ordered $4$-tuples $(A,B,C,D)$ of nonnegative integers such that $A+B +C +D =2023.$ Find the last three digits of the integer $$2023\sum_{(A,B,C,D) \in S} X_AX_BX_CX_D.$$

On a table, there is an empty bag and a chessboard containing exactly one token on each square. Next to the table is a large pile that contains an unlimited supply of tokens. Using only the following types of moves what is the maximum possible number of tokens that can be in the bag? $\bullet$ Type 1: Choose a non-empty square on the chessboard that is not in the rightmost column. Take a token from this square and place it, along with one token from the pile, on the square immediately to its right. $\bullet$ Type 2: Choose a non-empty square on the chessboard that is not in the bottommost row. Take a token from this square and place it, along with one token from the pile, on the square immediately below it. $\bullet$ Type 3: Choose two adjacent non-empty squares. Remove a token from each and put them both into the bag.

Consider the function \[ f(x,n) = \frac{\binom n0 + \binom n2 x + \binom n4x^2 + \cdots}{\binom n1 + \binom n3 x + \binom n5 x^2 + \cdots}, \] where $n$ is a positive integer. Express $f(x,n+1)$ rationally in terms of $f(x,n)$ and $x$. Hence, or otherwise, evaluate $\textstyle\lim_{n\to\infty}f(x,n)$ for suitable fixed values of $x$. (The symbols $\textstyle\binom nr$ represent the binomial coefficients.)

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

Let $ a,b,c,d$ be the positive integers such that $ a > b > c > d$ and $ (a \plus{} b \minus{} c \plus{} d) | (ac \plus{} bd)$ . Prove that if $ m$ is arbitrary positive integer , $ n$ is arbitrary odd positive integer, then $ a^n b^m \plus{} c^m d^n$ is composite number

Real numbers between 0 and 1, inclusive, are chosen in the following manner. A fair coin is flipped. If it lands heads, then it is flipped again and the chosen number is 0 if the second flip is heads and 1 if the second flip is tails. On the other hand, if the first coin flip is tails, then the number is chosen uniformly at random from the closed interval $[0,1]$. Two random numbers $x$ and $y$ are chosen independently in this manner. What is the probability that $|x-y| > \tfrac{1}{2}$? $\textbf{(A)} \frac{1}{3} \qquad \textbf{(B)} \frac{7}{16} \qquad \textbf{(C)} \frac{1}{2} \qquad \textbf{(D)} \frac{9}{16} \qquad \textbf{(E)} \frac{2}{3}$

Is it possible to tile $2003 \times 2003$ board by $1 \times 2$ dominoes placed horizontally and $1 \times 3$ rectangles placed vertically?

Suppose we choose two numbers $x,y\in[0,1]$ uniformly at random. If the probability that the circle with center $(x,y)$ and radius $|x-y|$ lies entirely within the unit square $[0,1]\times [0,1]$ is written as $\tfrac{p}{q}$ with $p$ and $q$ relatively prime nonnegative integers, then what is $p^2+q^2$?

Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions: (i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ; (ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$ (iii) $\bigcup_{X \in F} X = R$

Every square of a $3\times3$ board is assigned a sign $+$ or $-$. In every move, one square is selected and the signs are changed in the selected square and all the neighboring squares (two squares are neighboring if they have a common side). Is it true that, no matter how the signs were initially distributed, one can obtain a table in which all signs are $-$ after finitely many moves?

The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?

Prove that if the integers $a_{1}$, $a_{2}$, $\cdots$, $a_{n}$ are all distinct, then the polynomial \[(x-a_{1})^{2}(x-a_{2})^{2}\cdots (x-a_{n})^{2}+1\] cannot be expressed as the product of two nonconstant polynomials with integer coefficients.

Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$ Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$

Let $A$ and $B$ be sets of positive integers with $|A|\ge 2$ and $|B|\ge 2$. Let $S$ be a set consisting of $|A|+|B|-1$ numbers of the form $ab$ where $a\in A$ and $b\in B$. Prove that there exist pairwise distinct $x,y,z\in S$ such that $x$ is a divisor of $yz$.

Alice has an orange $\text{3-by-3-by-3}$ cube, which is comprised of $27$ distinguishable, $\text{1-by-1-by-1}$ cubes. Each small cube was initially orange, but Alice painted $10$ of the small cubes completely black. In how many ways could she have chosen $10$ of these smaller cubes to paint black such that every one of the $27$ $\text{3-by-1-by-1}$ sub-blocks of the $\text{3-by-3-by-3}$ cube contains at least one small black cube?

Let $P$ be a point inside triangle $ABC$ such that $\angle CPA = 90^o$ and $\angle CBP = \angle CAP$. Prove that $\angle P XY = 90^o$, where $X$ and $Y$ are the midpoints of $AB$ and $AC$ respectively.

A polynomial $p(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$ with integer coefficients is said to be divisible by an integer $m$ if $p(x)$ is divisible by m for all integers $x$. Prove that if $p(x)$ is divisible by $m$, then $k!a_0$ is also divisible by $m$. Also prove that if $a_0, k,m$ are non-negative integers for which $k!a_0$ is divisible by $m$, there exists a polynomial $p(x) = a_0x^k+\cdots+ a_k$ divisible by $m.$

Rose and Brunno play the game on a board shaped like a regular 1001-gon. Initially, all vertices of the board are white, and there is a chip at one of them. On each turn, Rose chooses an arbitrary positive integer \( k \), then Brunno chooses a direction: clockwise or counterclockwise, and moves the chip in the chosen direction by \( k \) vertices. If at the end of the turn the chip stands at a white vertex, this vertex is painted red. Find the greatest number of vertices that Rose can make red regardless of Brunno's actions, if the number of turns is not limited.

Given a triangle $ABC$ where $AC > BC$, $D$ is located on the circumcircle of $ABC$ such that $D$ is the midpoint of the arc $AB$ that contains $C$. $E$ is a point on $AC$ such that $DE$ is perpendicular to $AC$. Prove that $AE = EC + CB$.

There are $n \ge 2$ points in a square side $1$. Show that one can label the points $P_1, P_2, ... , P_n$ such that $\sum_{i=1}^n |P_{i-1} - P_i|^2 \le 4$, where we use cyclic subscripts, so that $P_0$ means $P_n$.

A large cube of size \(4 \times 4 \times 4\) is made up of 64 small unit cubes. Exactly 16 of these small cubes must be colored red, subject to the following condition: In each block of \(1 \times 1 \times 4\), \(1 \times 4 \times 1\), and \(4 \times 1 \times 1\) cubes, there must be exactly one red cube. Determine how many different ways it is possible to choose the 16 small cubes to be colored red. Note: Two colorings are considered different even if one can be obtained from the other by rotations or symmetries of the cube.

Suppose $a, b, c$ are complex numbers with $a + b + c = 0$, $a^2 + b^2 + c^2 = 0$, and $|a|,|b|,|c| \le 5$. Suppose further at least one of $a, b, c$ have real and imaginary parts that are both integers. Find the number of possibilities for such ordered triples $(a, b, c)$.