The number of integers $n \geq 10$ such that the product $\binom { n } { 10 } \cdot \binom { n + 1 } { 10 }$ is a perfect square is
(A) 0
(B) 1
(C) 2
(D) 3

The number of all integer solutions of the equation $x ^ { 2 } + y ^ { 2 } + x - y = 2021$ is
(A) 5 .
(B) 7 .
(C) 1 .
(D) 0 .

For a positive integer $n$, the equation $$x ^ { 2 } = n + y ^ { 2 } , \quad x , y \text { integers} ,$$ does not have a solution if and only if
(A) $n = 2$.
(B) $n$ is a prime number.
(C) $n$ is an odd number.
(D) $n$ is an even number not divisible by 4 .

Let $p < q$ be prime numbers such that $p^2 + q^2 + 7pq$ is a perfect square. Then, the largest possible value of $q$ is:
(A) 7
(B) 11
(C) 23
(D) 29

Let $\mathbb { N }$ denote the set of natural numbers, and let $\left( a _ { i } , b _ { i } \right)$, $1 \leq i \leq 9$, be nine distinct tuples in $\mathbb { N } \times \mathbb { N }$. Show that there are three distinct elements in the set $\left\{ 2 ^ { a _ { i } } 3 ^ { b _ { i } } : 1 \leq i \leq 9 \right\}$ whose product is a perfect cube.

If $n ^ { 2 } + 19 n + 92$ is a perfect square, then the possible values of $n$ may be
(A) - 19
(B) - 8
(C) - 4
(D) - 11

Let $n$ be a positive integer, and $x$ and $y$ be non-negative integers. We are to examine the solutions of the following equation in $x$ and $y$
$$x ^ { 2 } - y ^ { 2 } = n . \tag{1}$$
First of all, by transforming (1), we obtain
$$( x + y ) ( x - y ) = n . \tag{2}$$
(1) When we find the solutions $( x , y )$ of (1) in the cases where $n = 8$ and $n = 9$, we have that if $n = 8$, then $( x , y ) = ( \mathbf { A } , \mathbf { B } )$, and if $n = 9$, then $( x , y ) = ( \mathbf { C } , \mathbf { D } ) , ( \mathbf { E } , \mathbf { F } )$. Note that you should write the solutions in the order such that $\mathbf{C} \leq \mathbf{E}$.
(2) For each of $\mathbf { G }$ $\sim$ $\mathbf { R }$ in the following sentences, choose the correct answer from among (0) $\sim$ (9) given below.
The following is a proof that (3) given below is the necessary and sufficient condition for (1) to have a solution.
Proof: First, suppose that $( x , y )$ satisfies (1). If $x$ and $y$ are both even or both odd, then both $x + y$ and $x - y$ are $\mathbf { G }$. Hence, by (2) we see that $n$ is a multiple of $\mathbf { H }$.
Next, if one of $x$ and $y$ is even and the other is odd, then both $x + y$ and $x - y$ are $\mathbf{I}$, and hence $n$ is $\mathbf{J}$.
Thus we see that
$$\text{``} n \text{ is a multiple of } \mathbf{H} \text{, or } n \text{ is } \mathbf{J} \text{''} \quad \ldots\ldots (3)$$
is a necessary condition for (1) to have a solution.
Conversely, suppose that $n$ satisfies the condition (3). If $n$ is a multiple of $\mathbf{H}$, then $n$ can be represented as $n = \mathbf{H} \cdot k$, where $k$ is a positive integer. So, if for example we take $x + y = \mathbf { K } \cdot k$ and $x - y = 2$, then $( x , y ) = ( k + \mathbf { L } , k - \mathbf { M } )$, which shows that (1) has a solution.
On the other hand, if $n$ is $\mathbf{J}$, then $n$ can be represented as $n = \mathbf { N } \ell + \mathbf { O }$, where $\ell$ is a non-negative integer. So, if for example we take $x + y = \mathbf { P } \ell + \mathbf { Q }$ and $x - y = 1$, then $( x , y ) = ( \ell + \mathbf { R } , \ell )$, which shows that (1) has a solution.
From the above, we see that the necessary and sufficient condition for (1) to have a solution is (3).
(0) 0 (1) 1 (2) 2 (3) 3 (4) 4 (5) 5 (6) 6 (7) even (8) odd (9) prime

2. For ALL APPLICANTS.

(i) Suppose $x , y$, and $z$ are whole numbers such that $x ^ { 2 } - 19 y ^ { 2 } = z$. Show that for any such $x , y$ and $z$, it is true that
$$\left( x ^ { 2 } + N y ^ { 2 } \right) ^ { 2 } - 19 ( 2 x y ) ^ { 2 } = z ^ { 2 }$$
where $N$ is a particular whole number which you should determine.
(ii) Find $z$ if $x = 13$ and $y = 3$. Hence find a pair of whole numbers ( $x , y$ ) with $x ^ { 2 } - 19 y ^ { 2 } = 4$ and with $x > 2$.
(iii) Hence find a pair of positive whole numbers $( x , y )$ with $x ^ { 2 } - 19 y ^ { 2 } = 1$ and with $x > 1$.
Is your solution the only such pair of positive whole numbers $( x , y )$ ? Justify your answer.
(iv) Prove that there are no whole number solutions $( x , y )$ to $x ^ { 2 } - 25 y ^ { 2 } = 1$ with $x > 1$.
(v) Find a pair of positive whole numbers $( x , y )$ with $x ^ { 2 } - 17 y ^ { 2 } = 1$ and with $x > 1$.
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Let a, b and c be three consecutive even integers arranged from smallest to largest such that the geometric mean of b and c is $\sqrt { 2 }$ times the geometric mean of a and b.
Accordingly, what is the sum $\mathrm { a } + \mathrm { b } + \mathrm { c }$?
A) 12
B) 18
C) 24
D) 30
E) 36