Find all integers $n$ such that $n^4-8n+15$ is product of two consecutive integers

Find a sequence of positive integer $f(n)$, $n \in \mathbb{N}$ such that $(1)$ $f(n) \leq n^8$ for any $n \geq 2$, $(2)$ for any pairwisely distinct natural numbers $a_1,a_2,\cdots, a_k$ and $n$, we have that $$f(n) \neq f(a_1)+f(a_2)+ \cdots + f(a_k)$$

Let $\angle A$ in a triangle $ABC$ be $60^\circ$. Let point $N$ be the intersection of $AC$ and perpendicular bisector to the side $AB$ while point $M$ be the intersection of $AB$ and perpendicular bisector to the side $AC$. Prove that $CB = MN$. [i](3 points)[/i]

p1. Find all real numbers that satisfy the equation $$(1 + x^2 + x^4 + .... + x^{2014})(x^{2016} + 1) = 2016x^{2015}$$ p2. Let $A$ be an integer and $A = 2 + 20 + 201 + 2016 + 20162 + ... + \underbrace{20162016...2016}_{40\,\, digits}$ Find the last seven digits of $A$, in order from millions to units. p3. In triangle $ABC$, points $P$ and $Q$ are on sides of $BC$ so that the length of $BP$ is equal to $CQ$, $\angle BAP = \angle CAQ$ and $\angle APB$ is acute. Is triangle $ABC$ isosceles? Write down your reasons. p4. Ayu is about to open the suitcase but she forgets the key. The suitcase code consists of nine digits, namely four $0$s (zero) and five $1$s. Ayu remembers that no four consecutive numbers are the same. How many codes might have to try to make sure the suitcase is open? p5. Fulan keeps $100$ turkeys with the weight of the $i$-th turkey, being $x_i$ for $i\in\{1, 2, 3, ... , 100\}$. The weight of the $i$-th turkey in grams is assumed to follow the function $x_i(t) = S_it + 200 - i$ where $t$ represents the time in days and $S_i$ is the $i$-th term of an arithmetic sequence where the first term is a positive number $a$ with a difference of $b =\frac15$. It is known that the average data on the weight of the hundred turkeys at $t = a$ is $150.5$ grams. Calculate the median weight of the turkey at time $t = 20$ days.

Let $ABC$ be a triangle with centroid $G$, and let $E$ and $F$ be points on side $BC$ such that $BE = EF = F C$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, so that $X$, $Y$ , and $G$ are not collinear. If the line through $E$ parallel to $XG$ and the line through $F$ parallel to $Y G$ intersect at $P\ne G$, prove that $GP$ passes through the midpoint of $XY$.

Given a positive integer $n \geq 2$ and real numbers $a_1, a_2, \ldots, a_n \in [0,1]$. Prove that there exist real numbers $b_1, b_2, \ldots, b_n \in \{0,1\}$, such that for all $1\leq k\leq l \leq n$ we have $$\left| \sum_{i=k}^l (a_i-b_i)\right| \leq \frac{n}{n+1}.$$

Let $p> 5$ be a prime such that none of its digits is divisible by $3$ or $7$. Prove that the equation $x^4 + p = 3y^4$ does not have integer solutions.

Through a point inside a triangle, three lines are drawn from the vertices to the opposite sides forming six triangular sections. Then: $ \textbf{(A)}\ \text{the triangles are similar in opposite pairs}$ $ \textbf{(B)}\ \text{the triangles are congruent in opposite pairs}$ $ \textbf{(C)}\ \text{the triangles are equal in area in opposite pairs}$ $ \textbf{(D)}\ \text{three similar quadrilaterals are formed}$ $ \textbf{(E)}\ \text{none of the above relations are true}$

Joe bikes $x$ miles East at $20$ mph to his friend’s house. He then turns South and bikes $x$ miles at $20$ mph to the store. Then, Joe turns East again and goes to his grandma’s house at $14$ mph. On this last leg, he has to carry flour he bought for her at the store. Her house is $2$ more miles from the store than Joe’s friend’s house is from the store. Joe spends a total of 1 hour on the bike to get to his grandma’s house. If Joe then rides straight home in his grandma’s helicopter at $78$ mph, how many minutes does it take Joe to get home from his grandma’s house

Let $P$ be an arbitrary point of the side line $AB$ of the triangle $ABC$. Mark the perpendicular projection of $P$ on the side lines $AC$ and $BC$ as $A_1$ and $B_1$ respectively. Denote $C_1$ he foot of the alttiude starting from $C$. Prove that the points $A_1$, $B_1$, $C_1$, $C$ and $P$ lie on a circle.

Let $P(x)=x^n +a_1x^{n-1}+a_2x^{n-2}+\ldots+a_{n-1}x+a_n$ be a polynomial of degree $n$ and $n$ real roots, all of them in the interval $(0,1)$. Prove that for all $k=\overline{1,n}$ the following inequality holds: \[(-1)^k(a_k+a_{k+1}+\ldots+a_n)>0.\] [i]Proposed by N. Safaei (Iran)[/i]

The function f is defined on the set of integers and satisfies $\bullet$ $f(n) = n - 2$, if $n \ge 2005$ $\bullet$ $f(n) = f(f(n+7))$, if $n < 2005$. Find $f(3)$.

$(a)$ Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality. $(b)$ Determine the points in a triangle that minimize the sum of the distances to the sides.

For any natural $n$, $f(n)$ is the number of labeled digraphs with $n$ vertices such that for any vertex the number if in-edges is equal to the number of out-edges and the total of (in+out) edges is even. Let $g(n)$ be the odd-analogous of $f(n)$. Find $g(n)-f(n)$ with proof . [hide=original formulation] Dado $n$ natural, seja $f(n)$ o número de grafos rotulados direcionados com $n$ vértices de modo que em cada vértice o número de arestas que chegam é igual ao número de arestas que saem e o número de arestas total do grafo é par . Defina $g(n)$ analogamente trocando "par" por "ímpar" na definição acima. Calcule $f(n) - g (n)$. (Observação: Um grafo rotulado direcionado é um par $G = (V, E)$ onde $V = \{1, 2, …, n\}$ e $E$ é um subconjunto de $V^2 -\{(i, i); 0 < i < n + 1\}$).[/hide]

A hunter and an invisible rabbit play a game in the Euclidean plane. The rabbit's starting point, $A_0,$ and the hunter's starting point, $B_0$ are the same. After $n-1$ rounds of the game, the rabbit is at point $A_{n-1}$ and the hunter is at point $B_{n-1}.$ In the $n^{\text{th}}$ round of the game, three things occur in order: [list=i] [*]The rabbit moves invisibly to a point $A_n$ such that the distance between $A_{n-1}$ and $A_n$ is exactly $1.$ [*]A tracking device reports a point $P_n$ to the hunter. The only guarantee provided by the tracking device to the hunter is that the distance between $P_n$ and $A_n$ is at most $1.$ [*]The hunter moves visibly to a point $B_n$ such that the distance between $B_{n-1}$ and $B_n$ is exactly $1.$ [/list] Is it always possible, no matter how the rabbit moves, and no matter what points are reported by the tracking device, for the hunter to choose her moves so that after $10^9$ rounds, she can ensure that the distance between her and the rabbit is at most $100?$ [i]Proposed by Gerhard Woeginger, Austria[/i]

Upon sides $AB$ and $BC$ of triangle $ABC$ are constructed squares $ABB_{1}A_{1}$ and $BCC_{1}B_{2}$. Prove that lines $AC_{1}$, $CA_{1}$ and altitude from $B$ to side $AC$ are concurrent.

Consider numbers of the form $1a1$, where $a$ is a digit. How many pairs of such numbers are there such that their sum is also a palindrome? [i]Note: A palindrome is a number which reads the same from left to right and from right to left. Examples: $353$, $91719$.[/i]

For a positive integer \( n \), let \( \tau(n) \) denote the number of positive divisors of \( n \). Determine whether there exists a positive integer triple \( a, b, c \) such that there are exactly $1012$ positive integers \( K \) not greater than $2024$ that satisfies the following: the equation \[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \] holds for some positive integers $x,y,z$.

Let $p\ge 3$ be a prime number. Show that there exist $p$ positive integers $a_1,a_2,\ldots ,a_p$ not exceeding $2p^2$ such that the $\frac{p(p-1)}{2}$ sums $a_i+a_j\ (i<j)$ are all distinct.

Let $p$ be an odd prime. Let $A(n)$ be the number of subsets of $\{1,2,...,n\}$ such that the sum of elements of the subset is a multiple of $p$. Prove that if $2^{p-1}-1$ is not a multiple of $p^2$, there exists infinitely many positive integer $m$ for any integer $k$ that satisfies the following. (The sum of elements of the empty set is 0.) $$\frac{A(m)-k}{p}\in\mathbb{Z}$$

Let $ABCD$ be a cyclic quadrilateral with $\angle ABC = 60^o$ and $| BC | = | CD |$. Prove that $|CD| + |DA| = |AB|$

Determine all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfy \[f(f(x)+y)=2x+f(f(y)-x)\quad\text{for all real}\ x,y. \]

Vernonia High School has 85 seniors, each of whom plays on at least one of the school’s three varsity sports teams: football, baseball, and lacrosse. It so happens that $74$ are on the football team; $26$ are on the baseball team; $17$ are on both the football and lacrosse teams; $18$ are on both the baseball and football teams; and $13$ are on both the baseball and lacrosse teams. Compute the number of seniors playing all three sports, given that twice this number are members of the lacrosse team.

Let $n$ be a positive integer, $n\geq 2$. Let $M=\{0,1,2,\ldots n-1\}$. For an integer nonzero number $a$ we define the function $f_{a}: M\longrightarrow M$, such that $f_{a}(x)$ is the remainder when dividing $ax$ at $n$. Find a necessary and sufficient condition such that $f_{a}$ is bijective. And if $f_{a}$ is bijective and $n$ is a prime number, prove that $a^{n(n-1)}-1$ is divisible by $n^{2}$.

Can a tetrahedron scan turn out to be a triangle with sides $3, 4$ and $5$ (a tetrahedron can be cut only along the edges)?