ABC is a right triangle, right at B, in which the height BD is drawn. E is a point on side BC such that AE = EC = 8. If BD is 6 and DE = k , find k

There is a lack of trigonometric bash practice, and I want to see techniques to do these problems. So here are 10 kinds of problems that are usually out in the wild. How do you tackle these problems? (I had ChatGPT write a code for this.). Please give me some general techniques to solve these kinds of problems, especially set 2b. I’ll add more later.

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problem set 1a

problem set 2a

problem set 2b
answers 2b

General techniques so far:

Trick 1: one thing to keep in mind is

Many of these seem to be reducible to this. The half can be written as , and , . This is proven in solution 1a-1. We will refer to this as Trick 1.

Let be positive real numbers such that . Find the minimum value of the expression


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Hi! I'm not comfortable with latex yet so bear with me please.
Q. in triABC, BD and CE are the bisectors of angles B,C cutting CA, AB at D,E respectively. if angle BDE= 24deg and angle CED= 18deg, find the angles of triABC.
I did find out angle A which comes out to be Click to reveal hidden text
but i'm stuck on the other two. help would be appreciated.
thanks!

Title says most of it. I've been meaning to post a problem set on HSM since at least a few months ago, but since I proposed the most recent problems I made to the 2025 SSMO, I had to wait for that happen. (Hence, most of these problems will probably be familiar if you participated in that contest, though numbers and wording may be changed.) The problems are very roughly arranged by difficulty. Enjoy!

Problem 1: An increasing sequence of positive integers has the property that the sum of its first terms is divisible by for every positive integer . Let be the number of such sequences satisfying . Compute the remainder when is divided by .

Problem 2: Rhombus has side length . Point lies on segment such that line is perpendicular to line . Given that , the area of can be expressed as , where and are relatively prime positive integers. Compute .

Problem 3: Positive integers and satisfy , , and . If the number of possible ordered pairs is equal to , compute the remainder when is divided by .

Problem 4: Let be a triangle. Point lies on side , point lies on side , and point lies on side such that , , and . Let be the foot of the altitude from to . Given that , , and , the value of can be expressed as , where and are relatively prime positive integers. Compute .

Problem 5: Anna has a three-term arithmetic sequence of integers. She divides each term of her sequence by a positive integer and tells Bob that the three resulting remainders are , , and , in some order. For how many values of is it possible for Bob to uniquely determine ?

Problem 6: There is a unique ordered triple of positive reals satisfying the system of equations The value of can be expressed as , where and are positive integers such that is square-free. Compute .

Problem 7: Let be the set of all monotonically increasing six-term sequences whose terms are all integers between and inclusive. We say a sequence in is symmetric if for every integer , the number of terms of that are at least is . The probability that a randomly chosen element of is symmetric is , where and are relatively prime positive integers. Compute .

Problem 8: For a positive integer , let denote the value of the binary number obtained by reading the binary representation of from right to left. Find the smallest positive integer such that the equation has at least ten positive integer solutions .

Problem 9: Let be a quadratic polynomial with a positive leading coefficient. There exists a positive real number such that and for . Compute .

Problem 10: Find the number of ordered triples of positive integers such that and is a multiple of .