Formulas

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64   
Consider all $1000$-element subsets of the set $\{1, 2, 3, ... , 2015\}$. From each such subset choose the least element. Find the arithmetic mean of all of these least elements.

156   
Solve in positive integers $x^2 - 4xy + 5y^2 = 169$.

169   
Let the lengths of two sides of a right triangle be $l$ and $m$, respectively, and the length of the hypotenuse be $n$. If both $m$ and $n$ are positive integers, and $l$ is a prime number, show that $2(l+m+n)$ must be a perfect square.

1602   
A right triangle has perimeter $32$ and area $20$. What is the length of its hypotenuse?

1624   
How many right triangles have integer leg lengths $a$ and $b$ and a hypotenuse of length $b+1$, where $b<100$?

1835   
Let $\triangle ABC$ be a right triangle whose three sides' lengths are all integers. Prove among its three sides' lengths, at lease one is a multiple of $3$, one is a multiple of $4$, and one is a multiple of $5$. (Note: they can be the same side. For example, in the $5-12-13$, $12$ is both a multiple of $3$ and $4$.)

2298   
Show that for any right triangle whose sides' lengths are all integers, - one side's length must be a multiple of 3, and - one side's length must be a multiple of 4, and - one side's length must be a multiple of 5 Please note these sides may not be distinct. For example, in a 5-12-13 triangle, 12 is a multiple of both 3 and 4.

2311   
Let integers $a$, $b$ and $c$ be the lengths of a right triangle's three sides, where $c > b > a$. Show that $\frac{(c-a)(c-b)}{2}$ must be a square number.

2321   
Show that $x^4 + y^4 = z^2$ is not solvable in positive integers.

2331   
Show that there exists an infinite sequence of positive integers $a_1, a_2, \cdots$ such that $$S_n=a_1^2 + a_2^2 + \cdots + a_n^2$$ is square for any positive integer $n$.

2332   
Show that the sides of a Pythagorean triangle in which the hypotenuse exceeds the larger leg by 1 are given by $\frac{n^2-1}{2}$, $n$ and $\frac{n^2+1}{2}$

2333   
Show that if the lengths of all the three sides in a right triangle are whole numbers, then radius of its incircle is always a whole number too.

2347   
Find all the Pythagorean triangles whose two sides are consecutive integers.

2349   
Solve in positive integers the equation $x^2 + y^2 = z^4$, where $\gcd(x,y)=1$ and $x$ is even.

2461   
Let $a, b, c$ be respectively the lengths of three sides of a triangle, and $r$ be the triangle's inradius. Show that $$r = \frac{1}{2}\sqrt{\frac{(b+c-a)(c+a-b)(b+a-c)}{a+b+c}}$$

2537   
The sides of a right triangle all have lengths that are whole numbers. The sum of the length of one leg and the hypotenuse is 49. Find the sum of all the possible lengths of the other leg. (A) 7 (B) 49 (C) 63 (D) 71 (E) 96

2690   
Find the remainder when $1\times 2 + 2\times 3 + 3\times 4 + \cdots + 2018\times 2019$ is divided by $2020$.

2765   
A grid point is defined as a point whose $x$ and $y$ coordinates are both integers. How many grid points are there on the circle which is centered at (199, 0) with a radius of 199?

2829   
Find all right triangles whose sides' lengths are all integers, and areas equal circumstance numerically.

2831   
Solve in positive integers the equation $3^x + 4^y = 5^z$ .

2860   
Let $a, b$, and $c$ be three positive integers such that $\frac{1}{a^2}+\frac{1}{b^2}=\frac{1}{c^2}$. Find the sum of all possible $a$ where $a \le 100$.

2861   
Note that $1 + 2 + 3 + 45 + 6 + 78+9=144$. How many different ways are there to make a total of 144 using only digits of 1, 2, 3, 4, $\cdots$, 7, 8, 9, in that order, with some addition signs.

3178   

Assuming a small packet of mm’s can contain anywhere from $20$ to $40$ mm’s in $6$ different colours. How many different mm packets are possible?


4124   

Octagon $ABCDEFGH$ with side lengths $AB = CD = EF = GH = 10$ and $BC = DE = FG = HA = 11$ is formed by removing 6-8-10 triangles from the corners of a $23$ $\times$ $27$ rectangle with side $\overline{AH}$ on a short side of the rectangle, as shown. Let $J$ be the midpoint of $\overline{AH}$, and partition the octagon into 7 triangles by drawing segments $\overline{JB}$, $\overline{JC}$, $\overline{JD}$, $\overline{JE}$, $\overline{JF}$, and $\overline{JG}$. Find the area of the convex polygon whose vertices are the centroids of these 7 triangles.


4127   

Let $ABCD$ be a convex quadrilateral with $AB = CD = 10$, $BC = 14$, and $AD = 2\sqrt{65}$. Assume that the diagonals of $ABCD$ intersect at point $P$, and that the sum of the areas of triangles $APB$ and $CPD$ equals the sum of the areas of triangles $BPC$ and $APD$. Find the area of quadrilateral $ABCD$.


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