Given a prime number $p$ congruent to $1$ modulo $5$ such that $2p+1$ is also prime, show that there exists a matrix of $0$s and $1$s containing exactly $4p$ (respectively, $4p+2$) $1$s no sub-matrix of which contains exactly $2p$ (respectively, $2p+1$) $1$s.

Let $m$ and $n$ be positive integers. Prove that the number $2n-1$ is divisible by $(2^m-1)^2$ if and only if $n$ is divisible by $m(2^m-1)$.

Find the sum of the $23$ smallest positive integers that are $4$ more than a multiple of $23$ and whose last two digits are $23.$

Find all integer polynomials $f$ such that there are infinitely many pairs of relatively prime natural numbers $(a,b)$ so that $a+b \mid f(a)+f(b)$.

Prove that for every positive integer $n$ there exists an $n$-digit number divisible by $5^n$ all of whose digits are odd.

For all integers $n\ge 2$ with the following property: [list] [*] for each pair of positive divisors $k,~\ell <n$, at least one of the numbers $2k-\ell$ and $2\ell-k$ is a (not necessarily positive) divisor of $n$ as well.[/list]

Let the irrational number \[\alpha =1-\cfrac{1}{2a_1-\cfrac{1}{2a_2-\cfrac{1}{2a_3-\cdots}}}\] where coefficients $a_1, a_2, \ldots$ are positive integers, infinitely many of which are greater than $1$. Prove that for every positive integer $N$ at least half of the numbers $\lfloor \alpha\rfloor, \lfloor 2\alpha\rfloor, \ldots, \lfloor N\alpha\rfloor$ are even. [i]Proposed by Géza Kós, Budapest[/i]

[b]p1.[/b] We say that six positive integers form a magic triangle if they are arranged in a triangular array as in the figure below in such a way that each number in the top two rows is equal to the sum of its two neighbors in the row directly below it. The triangle shown is magic because $4 = 1 + 3$, $5 = 3 + 2$, and $9 = 4 + 5$. $$9$$ $$4\,\,\,\,5$$ $$1\,\,\,\,3\,\,\,\,2$$ (a) Find a magic triangle such that the numbers at the three corners are $10$, $20$, and $2010$, with $2010$ at the top. (b) Find a magic triangle such that the numbers at the three corners are $20$, $201$, and $2010$, with $2010$ at the top, or prove that no such triangle exists. [b]p2.[/b] (a) The equalities $\frac12+\frac13+\frac16= 1$ and $\frac12+\frac13+\frac17+\frac{1}{42}= 1$ express $1$ as a sum of the reciprocals of three (respectively four) distinct positive integers. Find five positive integers $a < b < c <d < e$ such that $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d}+\frac{1}{e}= 1.$$ (b) Prove that for any integer $m \ge 3$, there exist $m$ positive integers $d_1 < d_2 <... < d_m$ such that $$\frac{1}{d_1}+\frac{1}{d_2}+ ... +\frac{1}{d_m}= 1.$$ [b]p3.[/b] Suppose that $P(x) = a_nx^n +... + a_1x + a_0$ is a polynomial of degree n with real coefficients. Say that the real number $b$ is a balance point of $P$ if for every pair of real numbers $a$ and $c$ such that $b$ is the average of $a$ and $c$, we have that $P(b)$ is the average of $P(a)$ and $P(c)$. Assume that $P$ has two distinct balance points. Prove that $n$ is at most $1$, i.e., that $P$ is a linear function. [b]p4.[/b] A roller coaster at an amusement park has a train consisting of $30$ cars, each seating two people next to each other. $60$ math students want to take as many rides as they can, but are told that there are two rules that cannot be broken. First, all $60$ students must ride each time, and second, no two students are ever allowed to sit next to each other more than once. What is the maximal number of roller coaster rides that these students can take? Justify your answer. [b]p5.[/b] Let $ABCD$ be a convex quadrilateral such that the lengths of all four sides and the two diagonals of $ABCD$ are rational numbers. If the two diagonals $AC$ and $BD$ intersect at a point $M$, prove that the length of $AM$ is also a rational number. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that $gcd (a + b, lcm(a, b)) = gcd (a, b)$ for any $a, b$.

Let $a$ and $b$ be coprime integers, greater than or equal to $1$. Prove that all integers $n$ greater than or equal to $(a - 1)(b - 1)$ can be written in the form: \[n = ua + vb, \qquad \text{with} (u, v) \in \mathbb N \times \mathbb N.\]

Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are a) natural numbers, b) integers

Find all natural numbers a, b such that $ a^{n}\plus{} b^{n} \equal{} c^{n\plus{}1}$ where c and n are naturals.

Find all four-digit numbers such that when decomposed into prime factors, each number has the sum of its prime factors equal to the sum of the exponents.

Let $N$ be a positive integer. There are $N$ tasks, numbered $1, 2, 3, \ldots, N$, to be completed. Each task takes one minute to complete and the tasks must be completed subjected to the following conditions: [list] [*] Any number of tasks can be performed at the same time. [*] For any positive integer $k$, task $k$ begins immediately after all tasks whose numbers are divisors of $k$, not including $k$ itself, are completed. [*] Task 1 is the first task to begin, and it begins by itself. [/list] Suppose $N = 2017$. How many minutes does it take for all of the tasks to complete? Which tasks are the last ones to complete?

We arranged all the prime numbers in the ascending order: $p_1=2<p_2<p_3<\cdots$. Also assume that $n_1<n_2<\cdots$ is a sequence of positive integers that for all $i=1,2,3,\cdots$ the equation $x^{n_i} \equiv 2 \pmod {p_i}$ has a solution for $x$. Is there always a number $x$ that satisfies all the equations? [i]Proposed by Mahyar Sefidgaran , Yahya Motevasel[/i]

[b]p1.[/b] Find the area of a right triangle with legs of lengths $20$ and $18$. [b]p2.[/b] How many $4$-digit numbers (without leading zeros) contain only $2,0,1,8$ as digits? Digits can be used more than once. [b]p3.[/b] A rectangle has perimeter $24$. Compute the largest possible area of the rectangle. [b]p4.[/b] Find the smallest positive integer with $12$ positive factors, including one and itself. [b]p5.[/b] Sammy can buy $3$ pencils and $6$ shoes for $9$ dollars, and Ben can buy $4$ pencils and $4$ shoes for $10$ dollars at the same store. How much more money does a pencil cost than a shoe? [b]p6.[/b] What is the radius of the circle inscribed in a right triangle with legs of length $3$ and $4$? [b]p7.[/b] Find the angle between the minute and hour hands of a clock at $12 : 30$. [b]p8.[/b] Three distinct numbers are selected at random fromthe set $\{1,2,3, ... ,101\}$. Find the probability that $20$ and $18$ are two of those numbers. [b]p9.[/b] If it takes $6$ builders $4$ days to build $6$ houses, find the number of houses $8$ builders can build in $9$ days. [b]p10.[/b] A six sided die is rolled three times. Find the probability that each consecutive roll is less than the roll before it. [b]p11.[/b] Find the positive integer $n$ so that $\frac{8-6\sqrt{n}}{n}$ is the reciprocal of $\frac{80+6\sqrt{n}}{n}$. [b]p12.[/b] Find the number of all positive integers less than $511$ whose binary representations differ from that of $511$ in exactly two places. [b]p13.[/b] Find the largest number of diagonals that can be drawn within a regular $2018$-gon so that no two intersect. [b]p14.[/b] Let $a$ and $b$ be positive real numbers with $a > b $ such that $ab = a +b = 2018$. Find $\lfloor 1000a \rfloor$. Here $\lfloor x \rfloor$ is equal to the greatest integer less than or equal to $x$. [b]p15.[/b] Let $r_1$ and $r_2$ be the roots of $x^2 +4x +5 = 0$. Find $r^2_1+r^2_2$ . [b]p16.[/b] Let $\vartriangle ABC$ with $AB = 5$, $BC = 4$, $C A = 3$ be inscribed in a circle $\Omega$. Let the tangent to $\Omega$ at $A$ intersect $BC$ at $D$ and let the tangent to $\Omega$ at $B$ intersect $AC$ at $E$. Let $AB$ intersect $DE$ at $F$. Find the length $BF$. [b]p17.[/b] A standard $6$-sided die and a $4$-sided die numbered $1, 2, 3$, and $4$ are rolled and summed. What is the probability that the sum is $5$? [b]p18.[/b] Let $A$ and $B$ be the points $(2,0)$ and $(4,1)$ respectively. The point $P$ is on the line $y = 2x +1$ such that $AP +BP$ is minimized. Find the coordinates of $P$. [b]p19.[/b] Rectangle $ABCD$ has points $E$ and $F$ on sides $AB$ and $BC$, respectively. Given that $\frac{AE}{BE}=\frac{BF}{FC}= \frac12$, $\angle ADE = 30^o$, and $[DEF] = 25$, find the area of rectangle $ABCD$. [b]p20.[/b] Find the sum of the coefficients in the expansion of $(x^2 -x +1)^{2018}$. [b]p21.[/b] If $p,q$ and $r$ are primes with $pqr = 19(p+q+r)$, find $p +q +r$ . [b]p22.[/b] Let $\vartriangle ABC$ be the triangle such that $\angle B$ is acute and $AB < AC$. Let $D$ be the foot of altitude from $A$ to $BC$ and $F$ be the foot of altitude from $E$, the midpoint of $BC$, to $AB$. If $AD = 16$, $BD = 12$, $AF = 5$, find the value of $AC^2$. [b]p23.[/b] Let $a,b,c$ be positive real numbers such that (i) $c > a$ (ii) $10c = 7a +4b +2024$ (iii) $2024 = \frac{(a+c)^2}{a}+ \frac{(c+a)^2}{b}$. Find $a +b +c$. [b]p24.[/b] Let $f^1(x) = x^2 -2x +2$, and for $n > 1$ define $f^n(x) = f ( f^{n-1}(x))$. Find the greatest prime factor of $f^{2018}(2019)-1$. [b]p25.[/b] Let $I$ be the incenter of $\vartriangle ABC$ and $D$ be the intersection of line that passes through $I$ that is perpendicular to $AI$ and $BC$. If $AB = 60$, $C A =120$, and $CD = 100$, find the length of $BC$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

For each integer $n > 1$, find a set of $n$ integers $\{a_1, a_2,..., a_n\}$ such that the set of numbers $\{a_1+a_j | 1 \le i \le j \le n\}$ leave distinct remainders when divided by $n(n + 1)/2$. If such a set of integers does not exist, give a proof.

Let $ \mathbb{Z}$ be the set of all integers. Define the set $ \mathbb{H}$ as follows: (1). $ \dfrac{1}{2} \in \mathbb{H}$, (2). if $ x \in \mathbb{H}$, then $ \dfrac{1}{1\plus{}x} \in \mathbb{H}$ and also $ \dfrac{x}{1\plus{}x} \in \mathbb{H}$. Prove that there exists a bijective function $ f: \mathbb{Z} \rightarrow \mathbb{H}$.

A finite sequence of integers $ a_0, a_1, \ldots, a_n$ is called quadratic if for each $ i$ in the set $ \{1,2 \ldots, n\}$ we have the equality $ |a_i \minus{} a_{i\minus{}1}| \equal{} i^2.$ a.) Prove that any two integers $ b$ and $ c,$ there exists a natural number $ n$ and a quadratic sequence with $ a_0 \equal{} b$ and $ a_n \equal{} c.$ b.) Find the smallest natural number $ n$ for which there exists a quadratic sequence with $ a_0 \equal{} 0$ and $ a_n \equal{} 1996.$

The least common multiple of $a$ and $b$ is $12$, and the least common multiple of $b$ and $c$ is $15$. What is the least possible value of the least common multiple of $a$ and $c$? $\textbf{(A) }20\qquad\textbf{(B) }30\qquad\textbf{(C) }60\qquad\textbf{(D) }120\qquad \textbf{(E) }180$

Given an integer $ a$. Let $ p$ is prime number such that $ p|a$ and $ p \equiv \pm 3 (mod8)$. Define a sequence $ \{a_n\}_{n \equal{} 0}^\infty$ such that $ a_n \equal{} 2^n \plus{} a$. Prove that the sequence $ \{a_n\}_{n \equal{} 0}^\infty$ has finitely number of square of integer.

Let $P(x)$ and $Q(x)$ be polynomials of degree $p$ and $q$ respectively such that every coefficient is $1$ or $2023$. If $P(x)$ divides $Q(x)$, prove that $p+1$ divides $q+1$.

The smallest positive integer that does not divide $1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8 \times 9$ is:

Prove that the natural numbers can be divided into two groups in a way that both conditions are fulfilled: 1) For every prime number $p$ and every natural number $n$, the numbers $p^n,p^{n+1}$ and $p^{n+2}$ do not have the same colour. 2) There does not exist an infinite geometric sequence of natural numbers of the same colour.

Let $n{}$ be a positive integer and let $a{}$ and $b{}$ be positive integers congruent to 1 modulo 4. Prove that there exists a positive integer $k{}$ such that at least one of the numbers $a^k-b$ and $b^k-a$ is divisible by $2^n.$ [i]Cătălin Liviu Gherghe[/i]