Let $ n \geq 2, n \in \mathbb{N}$ and let $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i \equal{} 1, \ldots, n,$ and \[ \sum^n_{i\equal{}1} a_i \equal{} \sum^n_{i\equal{}1} b_i.\] Prove the inequality: \[ \sum^n_{i\equal{}1} b_i \prod^n_{j \equal{} 1, j \neq i} a_j \leq \frac{p}{(n\minus{}1)^{n\minus{}1}}.\]

Compute $\sqrt{(31)(30)(29)(28)+1}$.

[b]a)[/b] Let be three natural numbers, $ a>b\ge 3\le 3n, $ such that $ b^n|a^n-1. $ Prove that $ a^b>2^n. $ [b]b)[/b] Does there exist positive real numbers $ m $ which have the property that $ \log_8 (1+3\sqrt x) =\log_{27} (mx) $ if and only if $ 2^{x} +2^{1/x}\le 4? $

Let $ a_{ij}$ be nonnegative integers such that $ a_{ij}\equal{}0$ if and only if $ i>j$ and that $ \displaystyle\sum_{j\equal{}1}^{1988}a_{ij}\equal{}1988$ holds for all $ i\equal{}1,...,1988$. Find all real solutions of the system of equations: $ \displaystyle\sum_{j\equal{}1}^{1988} (1\plus{}a_{ij})x_j\equal{}i\plus{}1, 1 \le i \le 1988$.

Let $\mathbb{R}$ be the set of real numbers. Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that \[f(x+y)f(x-y)\geqslant f(x)^2-f(y)^2\] for every $x,y\in\mathbb{R}$. Assume that the inequality is strict for some $x_0,y_0\in\mathbb{R}$. Prove that either $f(x)\geqslant 0$ for every $x\in\mathbb{R}$ or $f(x)\leqslant 0$ for every $x\in\mathbb{R}$.

Let $a,b$ be real numbers such that the equation $$x^3+\sqrt{3}(a-1)x^2-6ax+b=0$$has three real roots. Prove that $|b|\leq |a+1|^3$. >Clarification: $|x|$ indicates the absolute value of $x$. For example, $|5|=5$; $|-1.23|=1.23$; etc

Let $n$ be a positive integer and let $p$ be a prime number. Prove that if $a$, $b$, $c$ are integers (not necessarily positive) satisfying the equations \[ a^n + pb = b^n + pc = c^n + pa\] then $a = b = c$. [i]Proposed by Angelo Di Pasquale, Australia[/i]

Determine all integers $a$ for which the equation \[x^{2}+axy+y^{2}=1\] has infinitely many distinct integer solutions $x, \;y$.

Find the values of $p$ for which the equation $x^5 - px-1 = 0$ has two roots $r$ and $s$ which are the roots of equation $x^2-ax+b= 0$ for some integers $a,b$.

Find all functions $ f: \mathbb{R}\mapsto\mathbb{R}$ such that for all $ x$, $ y$ $ \in\mathbb{R}$, \[ f(f(x))\plus{}f(f(y))\equal{}2y\plus{}f(x\minus{}y)\] holds.

Let $a$ and $b$ be integers. Find all polynomial with integer coefficients sucht that $P(n)$ divides $P(an+b)$ for infinitely many positive integer $n$

Let $ n$ be a positive integer, and let $ x$ and $ y$ be a positive real number such that $ x^n \plus{} y^n \equal{} 1.$ Prove that \[ \left(\sum^n_{k \equal{} 1} \frac {1 \plus{} x^{2k}}{1 \plus{} x^{4k}} \right) \cdot \left( \sum^n_{k \equal{} 1} \frac {1 \plus{} y^{2k}}{1 \plus{} y^{4k}} \right) < \frac {1}{(1 \minus{} x) \cdot (1 \minus{} y)}. \] [i]Author: Juhan Aru, Estonia[/i]

We define a sequence of numbers $a_n$ such that $a_0=1$ and for all $n\ge0$: \[2a_{n+1} ^3 + 2a_n ^3 = 3 a_{n +1} ^2 a_n + 3a_{n+1}a_n^2\] Find the sum of all $a_{2023}$'s possible values.

Let $f(X) = a_nX^n + a_{n-1}X^{n-1} + ...+ a_1X + p$ be a polynomial of integer coefficients where $p$ is a prime number. Assume that $p >\sum_{i=1}^n |a_i|$. Prove that $f(X)$ is irreducible.

Let $P(x)$ be a polynomial with real coefficients. Prove that there exist positive integers $n$ and $k$ such that $k$ has $n$ digits and more than $P(n)$ positive divisors.

[b]p1.[/b] Suppose that $a \oplus b = ab - a - b$. Find the value of $$((1 \oplus 2) \oplus (3 \oplus 4)) \oplus 5.$$ [b]p2.[/b] Neethin scores a $59$ on his number theory test. He proceeds to score a $17$, $23$, and $34$ on the next three tests. What score must he achieve on his next test to earn an overall average of $60$ across all five tests? [b]p3.[/b] Consider a triangle with side lengths $28$ and $39$. Find the number of possible integer lengths of the third side. [b]p4.[/b] Nithin is thinking of a number. He says that it is an odd two digit number where both of its digits are prime, and that the number is divisible by the sum of its digits. What is the sum of all possible numbers Nithin might be thinking of? [b]p5.[/b] Dora sees a fire burning on the dance floor. She calls her friends to warn them to stay away. During the first pminute Dora calls Poonam and Serena. During the second minute, Poonam and Serena call two more friends each, and so does Dora. This process continues, with each person calling two new friends every minute. How many total people would know of the fire after $6$ minutes? [b]p6.[/b] Charlotte writes all the positive integers $n$ that leave a remainder of $2$ when $2018$ is divided by $n$. What is the sum of the numbers that she writes? [b]p7.[/b] Consider the following grid. Stefan the bug starts from the origin, and can move either to the right, diagonally in the positive direction, or upwards. In how many ways can he reach $(5, 5)$? [img]https://cdn.artofproblemsolving.com/attachments/9/9/b9fdfdf604762ec529a1b90d663e289b36b3f2.png[/img] [b]p8.[/b] Let $a, b, c$ be positive numbers where $a^2 + b^2 + c^2 = 63$ and $2a + 3b + 6c = 21\sqrt7$. Find $\left( \frac{a}{c}\right)^{\frac{a}{b}} $. [b]p9.[/b] What is the sum of the distinct prime factors of $12^5 + 12^4 + 1$? [b]p10.[/b] Allen starts writing all permutations of the numbers $1$, $2$, $3$, $4$, $5$, $6$ $7$, $8$, $9$, $10$ on a blackboard. At one point he writes the permutation $9$, $4$, $3$, $1$, $2$, $5$, $6$, $7$, $8$, $10$. David points at the permutation and observes that for any two consecutive integers $i$ and $i+1$, all integers that appear in between these two integers in the permutation are all less than $i$. For example, $4$ and $5$ have only the numbers $3$, $1$, $2$ in between them. How many of the $10!$ permutations on the board satisfy this property that David observes? [b]p11.[/b] (Estimation) How many positive integers less than $2018$ can be expressed as the sum of $3$ square numbers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $k$ be a natural number. For each function $f : \mathbb{N}\to \mathbb{N}$ define the sequence of functions $(f_{m})_{m\geq 1}$ by $f_{1}= f$ and $f_{m+1}= f \circ f_{m}$ for $m \geq 1$ . Function $f$ is called $k$-[i]nice[/i] if for each $n \in\mathbb{N}: f_{k}(n) = f (n)^{k}$. (a) For which $k$ does there exist an injective $k$-nice function $f$ ? (b) For which $k$ does there exist a surjective $k$-nice function $f$ ?

Real numbers $x_1,x_2,...x_{2004},y_1,y_2,...y_{2004}$ differ from $1$ and are such that $x_ky_k=1$ for every $k=1,2,...,2004$. Calculate the sum $$S=\frac{1}{1-x_1^3}+\frac{1}{1-x_2^3}+...+\frac{1}{1-x_{2004}^3}+\frac{1}{1-y_1^3}+\frac{1}{1-y_2^3}+...+\frac{1}{1-y_{2004}^3}$$

i) Show there exist (not necessarily distinct) non-negative real numbers $a_1,a_2,\ldots,a_{10}$; $b_1,b_2,\ldots,b_{10}$, with $a_k+b_k \leq 4$ for all $1\leq k \leq 10$, such that $\max\{|a_i-a_j|, |b_i-b_j|\} \geq \dfrac{4}{3} > 1$ for all $1\leq i < j \leq 10$. ii) Prove for any (not necessarily distinct) non-negative real numbers $a_1,a_2,\ldots,a_{11}$; $b_1,b_2,\ldots,b_{11}$, with $a_k+b_k \leq 4$ for all $1\leq k \leq 11$, there exist $1\leq i < j \leq 11$ such that $\max\{|a_i-a_j|, |b_i-b_j|\} \leq 1$. ([i]Dan Schwarz[/i])

Compute $1 \times 4 - 2 \times 3 + 2 \times 5 - 3 \times 4 + 3 \times 6 - 4 \times 5 + 4 \times 7 - 5 \times 6 + 5 \times 8 - 6 \times 7.$

Evaluate the sum $$1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 1 0 + . . . - 2010$$ , where each three consecutive signs $+$ are followed by two signs $-$.

[b]p1.[/b] Compute the value of $N$, where $$N = 818^3 - 6 \cdot 818^2 \cdot 209 + 12 \cdot 818 \cdot 209^2 - 8 \cdot 209^3$$ [b]p2.[/b] Suppose $x \le 2019$ is a positive integer that is divisible by $2$ and $5$, but not $3$. If $7$ is one of the digits in $x$, how many possible values of $x$ are there? [b]p3.[/b] Find all non-negative integer solutions $(a,b)$ to the equation $$b^2 + b + 1 = a^2.$$ [b]p4.[/b] Compute the remainder when $\sum^{2019}_{n=1} n^4$ is divided by $53$. [b]p5.[/b] Let $ABC$ be an equilateral triangle and $CDEF$ a square such that $E$ lies on segment $AB$ and $F$ on segment $BC$. If the perimeter of the square is equal to $4$, what is the area of triangle $ABC$? [img]https://cdn.artofproblemsolving.com/attachments/1/6/52d9ef7032c2fadd4f97d7c0ea051b3766b584.png[/img] [b]p6.[/b] $$S = \frac{4}{1\times 2\times 3}+\frac{5}{2\times 3\times 4} +\frac{6}{3\times 4\times 5}+ ... +\frac{101}{98\times 99\times 100}$$ Let $T = \frac54 - S$. If $T = \frac{m}{n}$ , where $m$ and $n$ are relatively prime integers, find the value of $m + n$. [b]p7.[/b] Find the sum of $$\sum^{2019}_{i=0}\frac{2^i}{2^i + 2^{2019-i}}$$ [b]p8.[/b] Let $A$ and $B$ be two points in the Cartesian plane such that $A$ lies on the line $y = 12$, and $B$ lies on the line $y = 3$. Let $C_1$, $C_2$ be two distinct circles that intersect both $A$ and $B$ and are tangent to the $x$-axis at $P$ and $Q$, respectively. If $PQ = 420$, determine the length of $AB$. [b]p9.[/b] Zion has an average $2$ out of $3$ hit rate for $2$-pointers and $1$ out of $3$ hit rate for $3$-pointers. In a recent basketball match, Zion scored $18$ points without missing a shot, and all the points came from $2$ or $3$-pointers. What is the probability that all his shots were $3$-pointers? [b]p10.[/b] Let $S = \{1,2, 3,..., 2019\}$. Find the number of non-constant functions $f : S \to S$ such that $$f(k) = f(f(k + 1)) \le f(k + 1) \,\,\,\, for \,\,\,\, all \,\,\,\, 1 \le k \le 2018.$$ Express your answer in the form ${m \choose n}$, where $m$ and $n$ are integers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

What is the largest integer that must divide $n^5-5n^3+4n$ for all integers $n$? [i]2016 CCA Math Bonanza Lightning #2.4[/i]

It is known that the equation$ |x - 1| + |x - 2| +... + |x - 2001| = a$ has exactly one solution. Find $a$.

Find all positive integers $n \geqslant 2$ for which there exist $n$ real numbers $a_1<\cdots<a_n$ and a real number $r>0$ such that the $\tfrac{1}{2}n(n-1)$ differences $a_j-a_i$ for $1 \leqslant i<j \leqslant n$ are equal, in some order, to the numbers $r^1,r^2,\ldots,r^{\frac{1}{2}n(n-1)}$.