Found problems: 1148
Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$
For how many rational numbers $p$ is the area of the triangle formed by the intercepts and vertex of $f(x) = -x^2+4px-p+1$ an integer?
Determine all pairs of functions $f,g:\mathbb{R}\rightarrow\mathbb{R}$ such that for any $x,y\in \mathbb{R}$,
\[f(x)f(y)=g(x)g(y)+g(x)+g(y).\]
Call admissible a set $A$ of integers that has the following property:
If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$.
Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers.
[i]Proposed by Warut Suksompong, Thailand[/i]
Positive integer $n$ cannot be divided by $2$ and $3$, there are no nonnegative integers $a$ and $b$ such that $|2^a-3^b|=n$. Find the minimum value of $n$.
Prove that for any positive integer $ n$, there exists only $ n$ degree polynomial $ f(x),$ satisfying $ f(0) \equal{} 1$ and $ (x \plus{} 1)[f(x)]^2 \minus{} 1$ is an odd function.
Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point.
EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet.
Note: fresh translation
Discriminant of a second degree polynomial with integer coefficients cannot be
$ \textbf{(A)}\ 23
\qquad\textbf{(B)}\ 24
\qquad\textbf{(C)}\ 25
\qquad\textbf{(D)}\ 28
\qquad\textbf{(E)}\ 33
$
Prove that: there exists a positive constant $K$, and an integer series $\{a_n\}$, satisfying:
$(1)$ $0<a_1<a_2<\cdots <a_n<\cdots $;
$(2)$ For any positive integer $n$, $a_n<1.01^n K$;
$(3)$ For any finite number of distinct terms in $\{a_n\}$, their sum is not a perfect square.
Show that there are no integers $a,b,c$ for which $a^2+b^2-8c=6$.
A line segment is divided so that the lesser part is to the greater part as the greater part is to the whole. If $ R$ is the ratio of the lesser part to the greater part, then the value of \[ R^{[R^{(R^2\plus{}R^{\minus{}1})}\plus{}R^{\minus{}1}]}\plus{}R^{\minus{}1}\] is
$ \textbf{(A)}\ 2
\qquad \textbf{(B)}\ 2R
\qquad \textbf{(C)}\ R^{\minus{}1}
\qquad \textbf{(D)}\ 2\plus{}R^{\minus{}1}
\qquad \textbf{(E)}\ 2\plus{}R$
If $ x^{2} \plus{}y^{2} \plus{}z\equal{}15$, $ x\plus{}y\plus{}z^{2} \equal{}27$ and $ xy\plus{}yz\plus{}zx\equal{}7$, then
$\textbf{(A)}\ 3\le \left|x\plus{}y\plus{}z\right|\le 4 \\ \textbf{(B)}\ 5\le \left|x\plus{}y\plus{}z\right|\le 6 \\ \textbf{(C)}\ 7\le \left|x\plus{}y\plus{}z\right|\le 8 \\ \textbf{(D)}\ 9\le \left|x\plus{}y\plus{}z\right|\le 10 \\ \textbf{(E)}\ \text{None}$
Let $\langle a_n\rangle $ and $ \langle b_n\rangle$ be two arithmetic sequences of numbers, and let $m$ be an integer greater than $2.$ Define $P_k(x)=x^2+a_kx+b_k,\ k=1,2,\cdots, m.$ Prove that if the quadratic expressions $P_1(x), P_m(x)$ do not have any real roots, then all the remaining polynomials also don't have real roots.
Define a quadratic trinomial to be "good", if it has two distinct real roots and all of its coefficients are distinct. Do there exist 10 positive integers such that there exist 500 good quadratic trinomials coefficients of which are among these numbers?
[I]Proposed by F. Petrov[/i]
Let $(A,+,\cdot)$ a 9 elements ring. Prove that the following assertions are equivalent:
(a) For any $x\in A\backslash\{0\}$ there are two numbers $a\in \{-1,0,1\}$ and $b\in \{-1,1\}$ such that $x^2+ax+b=0$.
(b) $(A,+,\cdot)$ is a field.
Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that
(1) For all integer $m$, $f(m) \ne 0$.
(2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.
Let $S$ be the set of all triangles $ABC$ for which \[ 5 \left( \dfrac{1}{AP} + \dfrac{1}{BQ} + \dfrac{1}{CR} \right) - \dfrac{3}{\min\{ AP, BQ, CR \}} = \dfrac{6}{r}, \] where $r$ is the inradius and $P, Q, R$ are the points of tangency of the incircle with sides $AB, BC, CA,$ respectively. Prove that all triangles in $S$ are isosceles and similar to one another.
Find all real numbers $ x$ such that $ \lfloor x^3 \rfloor \equal{} 4x \plus{} 3$.
Let $ S$ be a set of rational numbers such that
(a) $ 0\in S;$
(b) If $ x\in S$ then $ x\plus{}1\in S$ and $ x\minus{}1\in S;$ and
(c) If $ x\in S$ and $ x\notin\{0,1\},$ then $ \frac{1}{x(x\minus{}1)}\in S.$
Must $ S$ contain all rational numbers?
Let $P(n)$ be a polynomial of degree $m$ with integer coefficients, where $m \le 10$. Suppose that $P(0)=0$, $P(n)$ has $m$ distinct integer roots, and $P(n)+1$ can be factored as the product of two nonconstant polynomials with integer coefficients. Find the sum of all possible values of $P(2)$.
[i]Proposed by Evan Chen[/i]
Let $ n>4$ be a positive integer such that $ n$ is composite (not a prime) and divides $ \varphi (n) \sigma (n) \plus{}1$, where $ \varphi (n)$ is the Euler's totient function of $ n$ and $ \sigma (n)$ is the sum of the positive divisors of $ n$. Prove that $ n$ has at least three distinct prime factors.
Today was the 5th Kettering Olympiad - and here are the problems, which are very good intermediate problems.
1. Find all real $x$ so that $(1+x^2)(1+x^4)=4x^3$
2. Mark and John play a game. They have $100$ pebbles on a table. They take turns taking at least one at at most eight pebbles away. The person to claim the last pebble wins. Mark goes first. Can you find a way for Mark to always win? What about John?
3. Prove that
$\sin x + \sin 3x + \sin 5x + ... + \sin 11 x = (1-\cos 12 x)/(2 \sin x)$
4. Mark has $7$ pieces of paper. He takes some of them and splits each into $7$ pieces of paper. He repeats this process some number of times. He then tells John he has $2000$ pieces of paper. John tells him he is wrong. Why is John right?
5. In a triangle $ABC$, the altitude, angle bisector, and median split angle $A$ into four equal angles. Find the angles of $ABC.$
6. There are $100$ cities. There exist airlines connecting pairs of cities.
a) Find the minimal number of airlines such that with at most $k$ plane changes, one can go from any city to any other city.
b) Given that there are $4852$ airlines, show that, given any schematic, one can go from any city to any other city.
If $ \left|BC\right| \equal{} a$, $ \left|AC\right| \equal{} b$, $ \left|AB\right| \equal{} c$, $ 3\angle A \plus{} \angle B \equal{} 180{}^\circ$ and $ 3a \equal{} 2c$, then find $ b$ in terms of $ a$.
$\textbf{(A)}\ \frac {3a}{2} \qquad\textbf{(B)}\ \frac {5a}{4} \qquad\textbf{(C)}\ a\sqrt {2} \qquad\textbf{(D)}\ a\sqrt {3} \qquad\textbf{(E)}\ \frac {2a\sqrt {3} }{3}$
The quadratic equation $ x^2 \plus{} mx \plus{} n \equal{} 0$ has roots that are twice those of $ x^2 \plus{} px \plus{} m \equal{} 0$, and none of $ m,n,$ and $ p$ is zero. What is the value of $ n/p$?
$ \textbf{(A)}\ 1\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 4\qquad
\textbf{(D)}\ 8\qquad
\textbf{(E)}\ 16$
How many solutions does the equation $ \frac{[x]}{\{ x\}} =\frac{2007x}{2008} $ have?
[i]Mihail Bălună[/i]