p1. Given a set of $n$ the first natural number. If one of the numbers is removed, then the average number remaining is $21\frac14$ . Specify the number which is deleted. p2. Ipin and Upin play a game of Tic Tac Toe with a board measuring $3 \times 3$. Ipin gets first turn by playing $X$. Upin plays $O$. They must fill in the $X$ or $O$ mark on the board chess in turn. The winner of this game was the first person to successfully compose a sign horizontally, vertically, or diagonally. Determine as many final positions as possible, if Ipin wins in the $4$th step. For example, one of the positions the end is like the picture on the side. [img]https://cdn.artofproblemsolving.com/attachments/6/a/a8946f24f583ca5e7c3d4ce32c9aa347c7e083.png[/img] p3. Numbers $ 1$ to $10$ are arranged in pentagons so that the sum of three numbers on each side is the same. For example, in the picture next to the number the three numbers are $16$. For all possible arrangements, determine the largest and smallest values ​​of the sum of the three numbers. [img]https://cdn.artofproblemsolving.com/attachments/2/8/3dd629361715b4edebc7803e2734e4f91ca3dc.png[/img] p4. Define $$S(n)=\sum_{k=1}^{n}(-1)^{k+1}\,\, , \,\, k=(-1)^{1+1}1+(-1)^{2+1}2+...+(-1)^{n+1}n$$ Investigate whether there are positive integers $m$ and $n$ that satisfy $S(m) + S(n) + S(m + n) = 2011$ p5. Consider the cube $ABCD.EFGH$ with side length $2$ units. Point $A, B, C$, and $D$ lie in the lower side plane. Point $I$ is intersection point of the diagonal lines on the plane of the upper side. Next, make a pyramid $I.ABCD$. If the pyramid $I.ABCD$ is cut by a diagonal plane connecting the points $A, B, G$, and $H$, determine the volume of the truncated pyramid low part.

Determine all positive integers $n$ for which there exist positive integers $a_1,a_2, ..., a_n$ with $a_1 + 2a_2 + 3a_3 +... + na_n = 6n$ and $\frac{1}{a_1}+\frac{2}{a_2}+\frac{3}{a_3}+ ... +\frac{n}{a_n}= 2 + \frac1n$

Determine the primes $p$ for which the numbers $2\lfloor p/k\rfloor - 1, \ k = 1,2,\ldots, p,$ are all quadratic residues modulo $p.$ [i]Vlad Matei[/i]

Find all integers $a, b, c, d$ such that $$\begin{cases} ab - 2cd = 3 \\ ac + bd = 1\end{cases}$$

Each integer has been colored in one of three colors. Prove that exist two different numbers of the same color, whose difference is a perfect square.

Let $N$ be a natural number and $x_1, \ldots , x_n$ further natural numbers less than $N$ and such that the least common multiple of any two of these $n$ numbers is greater than $N$. Prove that the sum of the reciprocals of these $n$ numbers is always less than $2$: $\sum^n_{i=1} \frac{1}{x_i} < 2.$

An integer $n\ge2$ is called [i]resistant[/i], if it is coprime to the sum of all its divisors (including $1$ and $n$). Determine the maximum number of consecutive resistant numbers. For instance: * $n=5$ has sum of divisors $S=6$ and hence is resistant. * $n=6$ has sum of divisors $S=12$ and hence is not resistant. * $n=8$ has sum of divisors $S=15$ and hence is resistant. * $n=18$ has sum of divisors $S=39$ and hence is not resistant.

[b]p1.[/b] The Evergreen School booked buses for a field trip. Altogether, $138$ people went to West Lake, while $115$ people went to East Lake. The buses all had the same number of seats, and every bus has more than one seat. All seats were occupied and everybody had a seat. How many seats were there in each bus? [b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $50$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a 6 cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $150$ coins? [b]p3.[/b] Pinocchio multiplied two $2$ digit numbers. But witch Masha erased some of the digits. The erased digits are the ones marked with a $*$. Could you help Pinocchio to restore all the erased digits? $\begin{tabular}{ccccc} & & & 9 & 5 \\ x & & & * & * \\ \hline & & & * & * \\ + & 1 & * & * & \\ \hline & * & * & * & * \\ \end{tabular}$ Find all solutions. [b]p4.[/b] There are $50$ senators and $435$ members of House of Representatives. On Friday all of them voted a very important issue. Each senator and each representative was required to vote either "yes" or "no". The announced results showed that the number of "yes" votes was greater than the number of "no" votes by $24$. Prove that there was an error in counting the votes. [b]p5.[/b] Was there a year in the last millennium (from $1000$ to $2000$) such that the sum of the digits of that year is equal to the product of the digits? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.

Prove that there exists no innite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$: $p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.

Find the smallest prime $q$ such that $$q = a_1^2 + b_1^2 = a_2^2 + 2b_2^2 = a_3^2 + 3b_3^2 = ... = a_{10}^ 2 + 10b_{10}^2$$ where $a_i, b_i(i = 1, 2, ...,10)$ are positive integers

Consider the sequence $a_1, a_2, \dots$ of positive integers such that $a_1=2$ and $a_{n+1}=a_n^4+a_n^3-3a_n^2-a_n+2$, for all $n\geqslant 1$. Prove that there exist infinitely many prime numbers that don't divide any term of the sequence. [i]Proposed by Pavel Ciurea[/i]

Prove that the equation $7^x=1+y^2+z^2$ has no solutions over positive integers.

The sequence of integers $(a_{n})_{n \ge 1}$ is defined by $a_{1}= 2$, $a_{n+1}= 2a_{n}^{2}-1$. Prove that for each positive integer n, $n$ and $a_{n}$ are coprime.

Find all natural numbers $ n $ for which there exists two natural numbers $ a,b $ such that $$ n=S(a)=S(b)=S(a+b) , $$ where $ S(k) $ denotes the sum of the digits of $ k $ in base $ 10, $ for any natural number $ k. $ [i]Vasile Zidaru[/i] and [i]Mircea Lascu[/i]

A “number triangle” $(t_{n, k}) (0 \le k \le n)$ is defined by $t_{n,0} = t_{n,n} = 1 (n \ge 0),$ \[t_{n+1,m} =(2 -\sqrt{3})^mt_{n,m} +(2 +\sqrt{3})^{n-m+1}t_{n,m-1} \quad (1 \le m \le n)\] Prove that all $t_{n,m}$ are integers.

A positive integer $n\geqslant 3$ is [i]almost squarefree[/i] if there exists a prime number $p\equiv 1\bmod 3$ such that $p^2\mid n$ and $n/p$ is squarefree. Prove that for any almost squarefree positive integer $n$ the ratio $2\sigma(n)/d(n)$ is an integer.

Solve the following system of equations in natural numbers $$\begin{cases} a^4+14ab+1=n^4 \\ b^4+14bc+1=m^4 \\ c^4+14ca+1=k^4 \end{cases}$$

Let $D$ be a non-empty subset of positive integers and let $d$ be the greatest common divisor of $D$, and let $d\mathbb{Z}=[dn: n \in \mathbb{Z} ]$. Prove that there exists a bijection $f: \mathbb{Z} \rightarrow d\mathbb{Z} $ such that $| f(n+1)-f(n)|$ is member of $D$ for every integer $n$.

There are $n$ mathematicians seated around a circular table with $n$ seats numbered $1,2,3,\cdots,n$ in clockwise order. After a break they again sit around the table. The mathematicians note that there is a positive integer $a$ such that (1) for each $k$, the mathematician who was seated in seat $k$ before the break is seated in seat $ka$ after the break (where seat $i+n$ is seat $i$); (2) for every pair of mathematicians, the number of mathematicians sitting between them after the break, counting in both the clockwise and the counterclockwise directions, is different from either of the number of mathematicians sitting between them before the break. Find the number of possible values of $n$ with $1<n<1000$.

In parallelogram $ABCD$, points $E$ and $F$ on segments $AD$ and $CD$ are such that $\angle BCE=\angle BAF$. Points $K$ and $L$ on segments $AD$ and $CD$ are such that $AK=ED$ and $CL=FD$. Prove that $\angle BKD=\angle BLD$.

Find all positive integers n such that $n^5 + n + 1$ is prime.

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions: [list] [*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square. [*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$. [/list] Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$. And determine the value of $\sum_{k=N}^{N+2021}a_k$.