Found problems: 1148
Let $p$ be a prime number such that $p\equiv 1\pmod{4}$. Determine $\sum_{k=1}^{\frac{p-1}{2}}\left \lbrace \frac{k^2}{p} \right \rbrace$, where $\{x\}=x-[x]$.
Prime number $p>3$ is congruent to $2$ modulo $3$. Let $a_k = k^2 + k +1$ for $k=1, 2, \ldots, p-1$. Prove that product $a_1a_2\ldots a_{p-1}$ is congruent to $3$ modulo $p$.
In right triangle $ ABC$ with right angle at $ C$, $ \angle BAC < 45$ degrees and $ AB \equal{} 4$. Point $ P$ on $ AB$ is chosen such that $ \angle APC \equal{} 2\angle ACP$ and $ CP \equal{} 1$. The ratio $ \frac{AP}{BP}$ can be represented in the form $ p \plus{} q\sqrt{r}$, where $ p,q,r$ are positive integers and $ r$ is not divisible by the square of any prime. Find $ p\plus{}q\plus{}r$.
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all $m,n\in\mathbb{N}$,
\[(2^m+1)f(n)f(2^mn)=2^mf(n)^2+f(2^mn)^2+(2^m-1)^2n. \]
Consider the quadratic equation \(x^2 + a_0 x + b_0\) for some real numbers \((a_0, b_0)\). Repeat the following procedure as many times as possible:
Let \(c_i = \min \{r_i, s_i\}\), with \(r_i, s_i\) being the roots of the equation \(x^2 + a_i x + b_i\). The new equation is written as \(x^2 + b_i x + c_i\). That is, for the next iteration of the procedure, \(a_{i+1} = b_i\) and \(b_{i+1} = c_i\).
We say that \((a_0, b_0)\) is an $\textit{interesting}$ pair if, after a finite number of steps, the equation we obtain after one step is the same, so that \((a_i, b_i) = (a_{i+1}, b_{i+1})\). Find all $\textit{interesting}$ pairs.
If $a_0$ is a positive real number, consider the sequence $\{a_n\}$ defined by:
\[ a_{n+1} = \frac{a^2_n - 1}{n+1}, n \geq 0. \]
Show that there exist a real number $a > 0$ such that:
[b]i.)[/b] for all $a_0 \geq a,$ the sequence $\{a_n\} \rightarrow \infty,$
[b]ii.)[/b] for all $a_0 < a,$ the sequence $\{a_n\} \rightarrow 0.$
If $ r$ and $ s$ are the roots of $ x^2\minus{}px\plus{}q\equal{}0$, then $ r^2\plus{}s^2$ equals:
$ \textbf{(A)}\ p^2\plus{}2q \qquad
\textbf{(B)}\ p^2\minus{}2q \qquad
\textbf{(C)}\ p^2\plus{}q^2 \qquad
\textbf{(D)}\ p^2\minus{}q^2 \qquad
\textbf{(E)}\ p^2$
Prove that the sequence $ \{y_{n}\}_{n \ge 1}$ defined by
\[ y_{0}=1, \; y_{n+1}= \frac{1}{2}\left( 3y_{n}+\sqrt{5y_{n}^{2}-4}\right) \]
consists only of integers.
An ordered pair $( b , c )$ of integers, each of which has absolute value less than or equal to five, is chosen at random, with each
such ordered pair having an equal likelihood of being chosen. What is the probability that the equation $x^ 2 + bx + c = 0$ will
[i]not[/i] have distinct positive real roots?
$\textbf{(A) }\frac{106}{121}\qquad\textbf{(B) }\frac{108}{121}\qquad\textbf{(C) }\frac{110}{121}\qquad\textbf{(D) }\frac{112}{121}\qquad\textbf{(E) }\text{none of these}$
Which one below cannot be expressed in the form $x^2+y^5$, where $x$ and $y$ are integers?
$
\textbf{(A)}\ 59170
\qquad\textbf{(B)}\ 59149
\qquad\textbf{(C)}\ 59130
\qquad\textbf{(D)}\ 59121
\qquad\textbf{(E)}\ 59012
$
Real numbers $x$ and $y$ satisfy the equation $x^2+y^2=10x-6y-34$. What is $x+y$?
$ \textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }6\qquad\textbf{(E) }8 $
Find all non-negative integers $m,n,p,q$ such that \[ p^mq^n = (p+q)^2 +1 . \]
The coefficients $a,b,c$ of a polynomial $f:\mathbb{R}\to\mathbb{R}, f(x)=x^3+ax^2+bx+c$ are mutually distinct integers and different from zero. Furthermore, $f(a)=a^3$ and $f(b)=b^3.$ Determine $a,b$ and $c$.
The $ 52$ cards in a deck are numbered $ 1, 2, \ldots, 52$. Alex, Blair, Corey, and Dylan each picks a card from the deck without replacement and with each card being equally likely to be picked, The two persons with lower numbered cards from a team, and the two persons with higher numbered cards form another team. Let $ p(a)$ be the probability that Alex and Dylan are on the same team, given that Alex picks one of the cards $ a$ and $ a\plus{}9$, and Dylan picks the other of these two cards. The minimum value of $ p(a)$ for which $ p(a)\ge\frac12$ can be written as $ \frac{m}{n}$. where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.
Calculate
\[\int\cdots\int \exp\left(-\sum_{1\le i\le j\le n}x_ix_j\right)dx_1\cdots dx_n\]
Let $ABCD$ be a square, and let $E$ and $F$ be points on $\overline{AB}$ and $\overline{BC}$, respectively. The line through $E$ parallel to $\overline{BC}$ and the line through $F$ parallel to $\overline{AB}$ divide $ABCD$ into two squares and two non square rectangles. The sum of the areas of the two squares is $\frac{9}{10}$ of the area of square $ABCD$. Find $\frac{AE}{EB} + \frac{EB}{AE}$.
Let $a$ be a positive real number. Then prove that the polynomial
\[ p(x)=a^3x^3+a^2x^2+ax+a \]
has integer roots if and only if $a=1$ and determine those roots.
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and
\[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\]
[i]Alex Zhu.[/i]
[hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]
Find all solutions in integers $m,n$ of the equation \[ (m-n)^2 = \frac{4mn}{ m+n-1}. \]
Prove that there are no integers $m$ and $n$ such that
\[19m^2+95mn+2000n^2=1995.\]
Let $P(x)$ be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers $x$ and $y$, we have the inequality $P(xy)^2 \leqslant P(x^2)P(y^2)$.
[i]E. Malinnikova[/i]
Show that from a set of $11$ square integers one can select six numbers $a^2,b^2,c^2,d^2,e^2,f^2$ such that $a^2+b^2+c^2 \equiv d^2+e^2+f^2\pmod{12}$.
The sequence $ \{a_n\}$ satisfies $ a_0 \equal{} 0, a_{n \plus{} 1} \equal{} ka_n \plus{} \sqrt {(k^2 \minus{} 1)a_n^2 \plus{} 1}, n \equal{} 0, 1, 2, \ldots$, where $ k$ is a fixed positive integer. Prove that all the terms of the sequence are integral and that $ 2k$ divides $ a_{2n}, n \equal{} 0, 1, 2, \ldots$.
Find all function pairs $(f,g)$ where each $f$ and $g$ is a function defined on the integers and with values, such that, for all integers $a$ and $b$,
\[f(a+b)=f(a)g(b)+g(a)f(b)\\
g(a+b)=g(a)g(b)-f(a)f(b).\]
A monic quadratic polynomial $f$ with integer coefficients attains prime values at three consecutive integer points.show that it attains a prime value at some other integer point as well.