Let \[N = 69^{5} + 5\cdot 69^{4} + 10\cdot 69^{3} + 10\cdot 69^{2} + 5\cdot 69 + 1.\] How many positive integers are factors of $N$? $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 69\qquad\textbf{(D)}\ 125\qquad\textbf{(E)}\ 216 $

A function $f$ satisfies $f(\cos x) = \cos (17x)$ for every real $x$. Show that $f(\sin x) =\sin (17x)$ for every $x \in \mathbb{R}.$

Let $k\ge 2$, $n\ge 1$, $a_1, a_2,\dots, a_k$ and $b_1, b_2, \dots, b_n$ be integers such that $1<a_1<a_2<\dots <a_k<b_1<b_2<\dots <b_n$. Prove that if $a_1+a_2+\dots +a_k>b_1+b_2+\dots + b_n$, then $a_1\cdot a_2\cdot \ldots \cdot a_k>b_1\cdot b_2 \cdot \ldots \cdot b_n$.

a) Find at least two functions $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$ for all $x \in \mathbb{R}^+.$ b) Let $f: \mathbb{R}^+ \rightarrow \mathbb{R}^+$ be a function such that $$\displaystyle{2f(x^2)\geq xf(x) + x,}$$ for all $x \in \mathbb{R}^+.$ Show that $ f(x^3)\geq x^2,$ for all $x \in \mathbb{R}^+.$ Can we find the best constant $a\in \Bbb{R}$ such that $f(x)\geq x^a,$ for all $x \in \mathbb{R}^+?$

Let $a,b,c$ be real numbers with $a,c\ne0$. Prove that if $r$ is a real root of $ax^2+bx+c=0$ and $s$ a real root of $-ax^2+bx+c=0$, then there is a root of a $\frac a2x^2+bx+c=0$ between $r$ and $s$.

Let $a$ and $b$ be non-negative integers. Consider a sequence $s_1$, $s_2$, $s_3$, $. . .$ such that $s_1 = a$, $s_2 = b$, and $s_{i+1} = |s_i - s_{i-1}|$ for $i \ge 2$. Prove that there is some $i$ for which $s_i = 0$.

Suppose that $a,b,c,d$ are positive real numbers satisfying $(a+c)(b+d)=ac+bd$. Find the smallest possible value of $$\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}.$$ [i]Israel[/i]

Determine all functions $f : R^+ \to R^+$, so that for all $x, y > 0$: $$f(xy) \le \frac{xf(y) + yf(x)}{2}$$

Are there four real numbers $a, b, c, d$ for every three positive real numbers $x, y, z$ with the property $ad + bc = x$, $ac + bd = y$, $ab + cd = z$ and one of the numbers $a, b, c, d$ is equal to the sum of the other three?

Given real numbers $x$ and $y$, such that $$x^4 y^2 + y^4 + 2 x^3 y + 6 x^2 y + x^2 + 8 \le 0 .$$ Prove that $x \ge - \frac16$

Let the sequence $\{a_n\}_{n \geq 1}$ be defined by \[ a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N} \] Prove that \[ a_n^{2025} >n^{2024} \] for all positive integers $n \geq 2$. $\textbf{Proposed by Prajit Adhikari, Nepal.}$

Let the sequence $a_1,a_2,\ldots,a_n,\ldots$ is defined by the conditions: $a_1=2$ and $a_{n+1}=a_n^2-a_n+1$ $(n=1,2,\ldots)$. Prove that: (a) $a_m$ and $a_n$ are relatively prime numbers when $m\ne n$. (b) $\lim_{n\to\infty}\sum_{k=1}^n\frac1{a_k}=1$ [i]I. Tonov[/i]

Find all quadruplets $(a, b, c, d)$ of real numbers satisfying the system $(a + b)(a^2 + b^2) = (c + d)(c^2 + d^2)$ $(a + c)(a^2 + c^2) = (b + d)(b^2 + d^2)$ $(a + d)(a^2 + d^2) = (b + c)(b^2 + c^2)$ (Slovakia)

Real numbers $(x,y)$ satisfy the following equations: $$(x + 3)(y + 1) + y^2 = 3y$$ $$-x + x(y + x) = - 2x - 3.$$ Find the sum of all possible values of $x$. [i]Team #1[/i]

[b]p1.[/b] A boy has as many sisters as brothers. How ever, his sister has twice as many brothers as sisters. How many boys and girls are there in the family? [b]p2.[/b] Solve each of the following problems. (1) Find a pair of numbers with a sum of $11$ and a product of $24$. (2) Find a pair of numbers with a sum of $40$ and a product of $400$. (3) Find three consecutive numbers with a sum of $333$. (4) Find two consecutive numbers with a product of $182$. [b]p3.[/b] $2008$ integers are written on a piece of paper. It is known that the sum of any $100$ numbers is positive. Show that the sum of all numbers is positive. [b]p4.[/b] Let $p$ and $q$ be prime numbers greater than $3$. Prove that $p^2 - q^2$ is divisible by $24$. [b]p5.[/b] Four villages $A,B,C$, and $D$ are connected by trails as shown on the map. [img]https://cdn.artofproblemsolving.com/attachments/4/9/33ecc416792dacba65930caa61adbae09b8296.png[/img] On each path $A \to B \to C$ and $B \to C \to D$ there are $10$ hills, on the path $A \to B \to D$ there are $22$ hills, on the path $A \to D \to B$ there are $45$ hills. A group of tourists starts from $A$ and wants to reach $D$. They choose the path with the minimal number of hills. What is the best path for them? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

Determine whether $ \frac{1}{\sqrt{2}} \minus{} \frac{1}{\sqrt{6}}$ is less than or greater than $ \frac{3}{10}$.

positive reals $a_1, a_2, . . . $ satisfying (i) $a_{n+1}=a_1^2\cdot a_2^2 \cdot . . . \cdot a_n^2-3$(all positive integers $n$) (ii) $\frac{1}{2}(a_1+\sqrt{a_2-1})$ is positive integer. prove that $\frac{1}{2}(a_1 \cdot a_2 \cdot . . . \cdot a_n + \sqrt{a_{n+1}-1})$ is positive integer

Find the value of the expression $$f\left( \frac{1}{2000} \right)+f\left( \frac{2}{2000} \right)+...+ f\left( \frac{1999}{2000} \right)+f\left( \frac{2000}{2000} \right)+f\left( \frac{2000}{1999} \right)+...+f\left( \frac{2000}{1} \right)$$ assuming $f(x) =\frac{x^2}{1 + x^2}$ .

Consider the sums of the form $\sum_{k=1}^{n} \epsilon_k k^3,$ where $\epsilon_k \in \{-1, 1\}.$ Is any of these sums equal to $0$ if [b](a)[/b] $n=2000;$ [b](b)[/b] $n=2001 \ ?$

Let $ P(x) \in \mathbb{Z}[x]$ be a polynomial of degree $ \text{deg} P \equal{} n > 1$. Determine the largest number of consecutive integers to be found in $ P(\mathbb{Z})$. [i]B. Berceanu[/i]

10. Let $n \ge 5$ be an odd number and let $r$ be an integer such that $1\le r \le (n-1)/2$. IN a sports tournament, $n$ players take part in a series of contests. In each contest, $2r+1$ players participate, and the scores obtained by the players are the numbers $$-r, -(r-1),\cdots, -1, 0, 1 \cdots, r-1, r$$ in some order. Each possible subset of $2r+1$ players takes part together in exactly one contest. let the final score of player $i$ be $S_i$, for each $i=1, 2,\cdots,n$. Define $N$ to be the smallest difference between the final scores of two players, i.e., $$N = \min_{i<j}|S_i - S_j|.$$ Determine, with proof, the maximum possible value of $N$.

Two players play a game as follows. The first player chooses two non-zero integers A and B. The second player forms a quadratic with A, B and 1998 as coefficients (in any order). The first player wins iff the equation has two distinct rational roots. Show that the first player can always win.

If $z_1$ , $z_2$ are the roots of the equation with real coefficients $z^2+az+b = 0$, prove that $ z^n_1 + z^n_2$ is a real number for any natural value of $n$. If particular of the equation $z^2 - 2z + 2 = 0$, express, as a function of $n$, the said sum.

Solve: $$ \left\{ \begin{matrix} x+y=\sqrt{4z -1} \\ y+z=\sqrt{4x -1} \\ z+x=\sqrt{4y -1}\end{matrix}\right. . $$