I have two cents and Bill has $n$ cents. Bill wants to buy some pencils, which come in two different packages. One package of pencils costs 6 cents for 7 pencils, and the other package of pencils costs a [i]dime for a dozen[/i] pencils (i.e. 10 cents for 12 pencils). Bill notes that he can spend [b]all[/b] $n$ of his cents on some combination of pencil packages to get $P$ pencils. However, if I [i]give my two cents[/i] to Bill, he then notes that he can instead spend [b]all[/b] $n+2$ of his cents on some combination of pencil packages to get fewer than $P$ pencils. What is the smallest value of $n$ for which this is possible? Note: Both times Bill must spend [b]all[/b] of his cents on pencil packages, i.e. have zero cents after either purchase.

For given positive integers $n$ and $N$, let $P_n$ be the set of all polynomials $f(x)=a_0+a_1x+\cdots+a_nx^n$ with integer coefficients such that: [list] (a) $|a_j| \le N$ for $j = 0,1, \cdots ,n$; (b) The set $\{ j \mid a_j = N\}$ has at most two elements. [/list] Find the number of elements of the set $\{f(2N) \mid f(x) \in P_n\}$.

In a group of $2021$ people, $1400$ of them are $\emph{saboteurs}$. Sherlock wants to find one saboteur. There are some missions that each needs exactly $3$ people to be done. A mission fails if at least one of the three participants in that mission is a saboteur! In each round, Sherlock chooses $3$ people, sends them to a mission and sees whether it fails or not. What is the minimum number of rounds he needs to accomplish his goal?

Let $ABC$ be a triangle inscribed in the circle $\mathcal{C}(O,R)$ and circumscribed to the circle $\mathcal{C}(L,r)$. Denote $d=\dfrac{Rr}{R+r}$. Show that there exists a triangle $DEF$ such that for any interior point $M$ in $ABC$ there exists a point $X$ on the sides of $DEF$ such that $MX\le d$. [i]Dan Brânzei[/i]

Let $n$ be a positive integer and let $(a_1,a_2,\ldots ,a_{2n})$ be a permutation of $1,2,\ldots ,2n$ such that the numbers $|a_{i+1}-a_i|$ are pairwise distinct for $i=1,\ldots ,2n-1$. Prove that $\{a_2,a_4,\ldots ,a_{2n}\}=\{1,2,\ldots ,n\}$ if and only if $a_1-a_{2n}=n$.

Let $N$ be a positive integer. Prove that there exist three permutations $a_1,\dots,a_N$, $b_1,\dots,b_N$, and $c_1,\dots,c_N$ of $1,\dots,N$ such that \[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\] for every $k=1,2,\dots,N$.

We say that a function $f: \mathbb{N}\rightarrow\mathbb{N}$ has the $(\mathcal{P})$ property if, for any $y\in\mathbb{N}$, the equation $f(x)=y$ has exactly 3 solutions. a) Prove that there exist an infinity of functions with the $(\mathcal{P})$ property ; b) Find all monotonously functions with the $(\mathcal{P})$ property ; c) Do there exist monotonously functions $f: \mathbb{Q}\rightarrow\mathbb{Q}$ satisfying the $(\mathcal{P})$ property ?

A point $X$ is the origin of three rays such that the angle between any two of them equals $120^{\circ}$. Let $\omega$ be an arbitrary circle with radius $R$ such that $X$ lies inside it, and $A$, $B$, $C$ be the common points of the rays with this circle. Find $max(XA+XB+XC)$. Proposed by: F.Nilov

Let $n=p_1^{\alpha_1}\cdots p_s^{\alpha_s}\ge2$. If for any $\alpha\in\mathbb N$, $p_i-1\nmid\alpha$, where $i=1,2,\ldots,s$, prove that $n\mid\sum_{\alpha\in\mathbb Z^*_n}\alpha^{\alpha}$ where $\mathbb Z^*_n=\{a\in\mathbb Z_n:\gcd(a,n)=1\}$.

Suppose a function $f : \mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ satisfies $f(f(n)) + f(n+1) = n+2$ for all positive integer $n$. Prove that $f(f(n)+n) = n+1$ for all positive integer $n$.

Let $$\begin{array}{cccc}{a_{1,1}} & {a_{1,2}} & {a_{1,3}} & {\dots} \\ {a_{2,1}} & {a_{2,2}} & {a_{2,3}} & {\cdots} \\ {a_{3,1}} & {a_{3,2}} & {a_{3,3}} & {\cdots} \\ {\vdots} & {\vdots} & {\vdots} & {\ddots}\end{array}$$ be a doubly infinite array of positive integers, and suppose each positive integer appears exactly eight times in the array. Prove that $a_{m, n}>m n$ for some pair of positive integers $(m, n) .$

There are $n$ blocks placed on the unit squares of a $n \times n$ chessboard such that there is exactly one block in each row and each column. Find the maximum value $k$, in terms of $n$, such that however the blocks are arranged, we can place $k$ rooks on the board without any two of them threatening each other. (Two rooks are not threatening each other if there is a block lying between them.)

Let $ABC$ be a triangle in the coordinate plane with vertices on lattice points and with $AB = 1$. Suppose the perimeter of $ABC$ is less than $17$. Find the largest possible value of $1/r$, where $r$ is the inradius of $ABC$.

Let $F_n$ denote the $n$th Fibonacci number, with $F_0=0, F_1=1$ and $F_{n}=F_{n-1}+F_{n-2}$ for $n \geq 2$. There exists a unique two digit prime $p$ such that for all $n$, $p | F_{n+100} + F_n$. Find $p$. [i]Proposed by Sam Rosenstrauch[/i]

The sequence $a_1, a_2, ... , a_k, ...$ is constructed according to the rules: $$a_{2n} = a_n,a_{4n+1} = 1,a_{4n+3} = 0$$Prove that it is non-periodical sequence.

Let $P_1,P_2,P_3,P_4$ be the graphs of four quadratic polynomials drawn in the coordinate plane. Suppose that $P_1$ is tangent to $P_2$ at the point $q_2,P_2$ is tangent to $P_3$ at the point $q_3,P_3$ is tangent to $P_4$ at the point $q_4$, and $P_4$ is tangent to $P_1$ at the point $q_1$. Assume that all the points $q_1,q_2,q_3,q_4$ have distinct $x$-coordinates. Prove that $q_1,q_2,q_3,q_4$ lie on a graph of an at most quadratic polynomial.

In $\mathbb{R}^3$, for a rectangular box $\Delta$, let $10\Delta$ be the box with the same center as $\Delta$ but dilated by $10$. For example, if $\Delta$ is an $1 \times 1 \times 10$ box (hence with Lebesgue measure $10$), then $10\Delta$ is the $10 \times 10 \times 100$ box with the same center and orientation as $\Delta$. \medskip If two rectangular boxes $\Delta_1$ and $\Delta_2$ satisfy $\Delta_1 \subset 10\Delta_2$ and $\Delta_2 \subset 10 \Delta_1$, we say that they are [i]almost identical[/i]. Find the largest real number $a$ such that the following holds for some $C=C(a)>0$: For every positive integer $N$ and every collection $S$ of $1 \times 1 \times N$ boxes in $\mathbb{R}^3$, assuming that (i) $\vert S\vert=N$, (ii) every pair of boxes $(\Delta_1,\Delta_2)$ taken from $S$ are not almost identical, and (iii) the long edge of each box in $S$ forms an angle $\frac{\pi}{4}$ against the $xy$-plane. Then the volume \[\left\vert \bigcup_{\Delta \in S} \Delta\right\vert \ge CN^a.\]

Let $\omega$ and $\Omega$ be circles of radius $1$ and $R>1$ respectively that are internally tangent at a point $P$. Two tangent lines to $\omega$ are drawn such that they meet $\Omega$ at only three points $A$, $B$, and $C$, none of which are equal to $P$. If triangle $ABC$ has side lengths in a ratio of $3:4:5$, find the sum of all possible values of $R$. [i]Proposed by Connor Gordon[/i]

A positive rational number $x$ is called [i]banzai [/i] if the following conditions are met: $\bullet$ $x=\frac{p}{q}&gt;1$ where $p,q$ are comprime natural numbers $\bullet$ exist constants $\alpha,N$ such that for all integers $n\geq N$,$$\mid \left\{\,x^n\right\} -\alpha\mid \leq \dfrac{1}{2(p+q)}.$$ Find the total number of banzai numbers. Note:$\left\{\,x\right\}$ means fractional part of $x$ [i](tatari/nightmare)[/i]

We divide entire $ Z$ into $ n$ subsets such that difference of any two elements in a subset will not be a prime number. $ n$ is at least ? $\textbf{(A)}\ 6 \qquad\textbf{(B)}\ 5 \qquad\textbf{(C)}\ 4 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$

Find all positive integer solutions to $2x^2+5y^2=11(xy-11)$.

For any set $S$, let $|S|$ denote the number of elements in $S$, and let $n(S)$ be the number of subsets of $S$, including the empty set and the set $S$ itself. If $A$, $B$ and $C$ are sets for which \[n(A) + n(B) + n(C) = n(A \cup B \cup C)\quad\text{and}\quad |A| = |B| = 100,\] then what is the minimum possible value of $|A \cap B \cap C|$? $ \textbf{(A)}\ 96\qquad\textbf{(B)}\ 97\qquad\textbf{(C)}\ 98\qquad\textbf{(D)}\ 99\qquad\textbf{(E)}\ 100 $

In the figure, $[ABCD]$ is a square of side $1$. The points $E, F, G$ and $H$ are such that $[AFB], [BGC], [CHD]$ and $[DEA]$ are right-angled triangles. Knowing that the circles inscribed in each of these triangles and the circle inscribed in the square $[EFGH]$ has all the same radius, what is the measure of the radius of the circles? [img]https://1.bp.blogspot.com/-l37AEXa7_-c/X4KaJwe6HQI/AAAAAAAAMk4/14wvIipf26cRge_GqKSRwH32bp291vX4QCLcBGAsYHQ/s0/2018%2Bportugal%2Bp2.png[/img]

Let $d_1, d_2, ... , d_k$ be the positive divisors of $n = 1990!$. Show that $\sum \frac{d_i}{\sqrt{n}} = \sum \frac{\sqrt{n}}{d_i}$.

Ten mutually distinct non-zero reals are given such that for any two, either their sum or their product is rational. Prove that squares of all these numbers are rational.