High $R^2$ squared and high $p$-value for simple linear regression Let's assume that we have simple linear regression:
      $\hat{y} = bx + \text{intercept}$.
Is it possible to have a high p-value and high $R^2$ (or low p-value and low $R^2$)? I've been looking for examples of this. When the linear regression has multiple parameters, I saw some examples where p-value for some parameters are low, but overall $R^2$ is low as well, but I was wondering if it's possible for the linear regression of a single parameter. 
 A: This looks like a self-study, so I'll offer a hint: Is either or both of these measures (R-square and p-value) related to the sample size?
A: Yes, it is possible. The $R^2$ and the $t$ statistic (used to compute the p-value) are related exactly by:
$
|t| = \sqrt{\frac{R^2}{(1- R^2)}(n -2)}
$
Therefore, you can have a high $R^2$ with a high p-value (a low $|t|$) if you have a small sample.
For instance, take $n = 3$. For this sample size to give you a (two-sided) p-value less then 10% you would need an $R^2$ greater than 85% -- anything less than that would give you "non-significant" p-value.
As a concrete example, the simulation below produces an $R^2$ close to 0.5 with a p-value of $0.516$.
set.seed(10)
n <- 3
x <- rnorm(n, 0, 1)
y <- 1 + x + rnorm(n, 0, 1)
summary(m1 <- lm(y ~ x))

Call:
lm(formula = y ~ x)

Residuals:
       1        2        3 
-0.36552  0.42802 -0.06251 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)
(Intercept)   0.7756     0.4261    1.82    0.320
x             0.5065     0.5333    0.95    0.516

Residual standard error: 0.5663 on 1 degrees of freedom
Multiple R-squared:  0.4743,    Adjusted R-squared:  -0.05148 
F-statistic: 0.9021 on 1 and 1 DF,  p-value: 0.5164

For the opposite case (low p-value with low $R^2$), you can trivially obtain that by setting a regression where $x$ has a low explanatory power and let $n \to \infty$ to get a p-value as small as you want.
A: Here is another example:
$y_1 = c + \epsilon,y_2 = c,\ y_3 = \epsilon,$
where $c$ is a constant and $\epsilon \sim \mathcal{N}(0, \sigma^2)$ is a Gaussian noise.
Consider the two regression problems:
(1) $y_1 = \hat{\beta}_2 y_2 +\epsilon_2$
(2) $y_1 = \hat{\beta}_3 y_3 +\epsilon_3$
Could you tell in which case, we have a high $R^2$ and a high $p$-value; and in which case, we have a low $R^2$ and a low $p$-value?
p.s. $\frac{R^2}{1-R^2}$ in the formula of Carlos' answer is signal-to-noise ratio of the regression.
