We will consider odd natural numbers $n$ such that$$n|2023^n-1$$ $\textbf{a.}$ Find the smallest two such numbers. $\textbf{b.}$ Prove that there exists infinitely many such $n$

The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

Let $S=\{1,2,\cdots, n\}$ and let $T$ be the set of all ordered triples of subsets of $S$, say $(A_1, A_2, A_3)$, such that $A_1\cup A_2\cup A_3=S$. Determine, in terms of $n$, \[ \sum_{(A_1,A_2,A_3)\in T}|A_1\cap A_2\cap A_3|\]

Show that $ x^3 + x+ a^2 = y^2 $ has at least one pair of positive integer solution $ (x,y) $ for each positive integer $ a $.

In the acute-angled triangle $ABC$ is drawn the altitude $CH$. A ray beginning at point $C$ that lies inside the $\angle BCA$ and intersects for second time the circles circumscribed circles of $\vartriangle BCH$ and $\vartriangle ABC$ at points $X$ and $Y$ respectively. It turned out that $2CX = CY$. Prove that the line $HX$ bisects the segment $AC$. (Hilko Danilo)

Consider the set $A = \{1, 2, ..., n\}$, where $n \in N, n \ge 6$. Show that $A$ is the union of three pairwise disjoint sets, with the same cardinality and the same sum of their elements, if and only if $n$ is a multiple of $3$.

Let $P_1, P_2, \ldots, P_n$ be distinct $2$-element subsets of $\{1, 2, \ldots, n\}$. Suppose that for every $1 \le i < j \le n$, if $P_i \cap P_j \neq \emptyset$, then there is some $k$ such that $P_k = \{i, j\}$. Prove that if $a \in P_i$ for some $i$, then $a \in P_j$ for exactly one value of $j$ not equal to $i$.

In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right? [asy] size(340); int i, j; for(i = 0; i<10; i = i+1) { for(j = 0; j<5; j = j+1) { if(10*j + i == 11 || 10*j + i == 12 || 10*j + i == 14 || 10*j + i == 15 || 10*j + i == 18 || 10*j + i == 32 || 10*j + i == 35 || 10*j + i == 38 ) { } else{ label("$*$", (i,j));} }} label("$\leftarrow$"+"Dec. 31", (10.3,0)); label("Jan. 1"+"$\rightarrow$", (-1.3,4));[/asy]

We have a square of side $1$ and a number $\ell$ such that $0 <\ell <\sqrt2$. Two players $A$ and $B$, in turn, draw in the square an open segment (without its two ends) of length $\ell $, starts A. Each segment after the first cannot have points in common with the previously drawn segments. He loses the player who cannot make his play. Determine if either player has a winning strategy.

If you pick a random $3$-digit number, what is the probability that its hundreds digit is triple the ones digit?

Given that $(1 + \tan 1^{\circ})(1 + \tan 2^{\circ}) \ldots (1 + \tan 45^{\circ}) = 2^n$, find $n$.

Let $\sum_{i=1}^n a_i x_i =0$, $a_i,x_i\in \mathbb{Z}$. It is known that however we color $\mathbb{Z}$ with finite number of colors, then the given equation has a monochromatic (of one color) solution. Prove that there is some non-empty sum of its coefficients equal to 0.

For a finite sequence $A = (a_1, a_2,\ldots,a_n)$ of numbers, the [i]Cesaro sum[/i] of $A$ is defined to be \[\frac{S_1 + S_2 + \cdots + S_n}{n}\] where $S_k = a_1 + a_2 + \cdots + a_k\ \ \ \ (1 \le k \le n)$. If the Cesaro sum of the 99-term sequence $(a_1, a_2, \ldots, a_{99})$ is $1000$, what is the Cesaro sum of the 100-term sequence $(1,a_1,a_2,\ldots,a_{99})$? $ \textbf{(A)}\ 991\qquad\textbf{(B)}\ 999\qquad\textbf{(C)}\ 1000\qquad\textbf{(D)}\ 1001\qquad\textbf{(E)}\ 1009 $

Given an $8\times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squares and four white squares so that no two of the eight chosen squares are in the same row or column? [i]Proposed by James Lin.[/i]

Krut and Vert go by car from point $A$ to point $B$. The car leaves $A$ in the direction of $B$, but every $3$ km of the road Krut turns $90^o$ to the left, and every $7$ km of the road Vert turns $90^o$ to the right ( if they try to turn at the same time, the car continues to go in the same direction). Will Krut and Vert be able to get to $B$ if the distance between $A$ and $B$ is $100$ km?

Let $\mathcal{L}$ be the set of all lines in the plane and let $\mathcal{P}$ be the set of all points in the plane. Determine whether there exists a function $g : \mathcal{L} \to \mathcal{P}$ such that for any two distinct non-parallel lines $\ell_1, \ell_2 \in \mathcal{L}$, we have $(a)$ $g(\ell_1) \neq g(\ell_2)$, and $(b)$ if $\ell_3$ is the line passing through $g(\ell_1)$ and $g(\ell_2)$, then $g(\ell_3)$ is the intersection of $\ell_1$ and $\ell_2$.

Let $x_1,x_2,\ldots,x_n$ be distinct positive integers. Prove that \[ x_1^2+x_2^2 + \cdots + x_n^2 \geq \frac {2n+1}3 ( x_1+x_2+\cdots + x_n). \] [i]Laurentiu Panaitopol[/i]

Suppose that $ P\equal{}2^m$ and $ Q\equal{}3^n$. Which of the following is equal to $ 12^{mn}$ for every pair of integers $ (m,n)$? $ \textbf{(A)}\ P^2Q \qquad \textbf{(B)}\ P^nQ^m \qquad \textbf{(C)}\ P^nQ^{2m} \qquad \textbf{(D)}\ P^{2m}Q^n \qquad \textbf{(E)}\ P^{2n}Q^m$

About the equation $ ax^2 \minus{} 2x\sqrt {2} \plus{} c \equal{} 0$, with $ a$ and $ c$ real constants, we are told that the discriminant is zero. The roots are necessarily: $ \textbf{(A)}\ \text{equal and integral} \qquad\textbf{(B)}\ \text{equal and rational} \qquad\textbf{(C)}\ \text{equal and real}$ $ \textbf{(D)}\ \text{equal and irrational} \qquad\textbf{(E)}\ \text{equal and imaginary}$

Three speed skaters have a friendly "race" on a skating oval. They all start from the same point and skate in the same direction, but with different speeds that they maintain throughout the race. The slowest skater does $1$ lap per minute, the fastest one does $3.14$ laps per minute, and the middle one does $L$ laps a minute for some $1 < L < 3.14$. The race ends at the moment when all three skaters again come together to the same point on the oval (which may differ from the starting point.) Determine the number of different choices for $L$ such that exactly $117$ passings occur before the end of the race. Note: A passing is defined as when one skater passes another one. The beginning and the end of the race when all three skaters are together are not counted as passings.

A company that sells keychains has to pay $\mathdollar500$ in maintenance fees each day and then it pays each work $\mathdollar15$ an hour. Each worker makes $5$ keychains per hour, which are sold at $\mathdollar3.10$ each. What is the least number of workers the company has to hire in order to make a profit in an $8$-hour workday?

Starting from 37, adding 5 before each previous term, forms the following sequence: \[37,537,5537,55537,555537,...\] How many prime numbers are there in this sequence?

The odd positive integers $1,3,5,7,\cdots,$ are arranged into in five columns continuing with the pattern shown on the right. Counting from the left, the column in which $ 1985$ appears in is the [asy] int i,j; for(i=0; i<4; i=i+1) { label(string(16*i+1), (2*1,-2*i)); label(string(16*i+3), (2*2,-2*i)); label(string(16*i+5), (2*3,-2*i)); label(string(16*i+7), (2*4,-2*i)); } for(i=0; i<3; i=i+1) { for(j=0; j<4; j=j+1) { label(string(16*i+15-2*j), (2*j,-2*i-1)); }} dot((0,-7)^^(0,-9)^^(2*4,-8)^^(2*4,-10)); for(i=-10; i<-6; i=i+1) { for(j=1; j<4; j=j+1) { dot((2*j,i)); }} [/asy] $ \textbf{(A)} \text{ first} \qquad \textbf{(B)} \text{ second} \qquad \textbf{(C)} \text{ third} \qquad \textbf{(D)} \text{ fourth} \qquad \textbf{(E)} \text{ fifth}$

$E$ is the midpoint of the side $AB$ of the square $ABCD$, and $F, G$ are any points on the sides $BC$, $CD$ such that $EF$ is parallel to $AG$. Show that $FG$ touches the inscribed circle of the square.

There are $m$ identical rectangular chocolate bars and $n$ people. Each chocolate bar may be cut into two (possibly unequal) pieces at most once. For which $m$ and $n$ is it possible to split the chocolate evenly among all the people? [i]Selected from the Kvant Magazine (D. Bugaenko and N. Konstantinov)[/i]