Exploiting Symmetries or Asymmetries (Imbalances)
Read this topicExplanations 4
- Choose the Middle Term! When looking at arithmetic sequences, why look at them as aaa, a+da+da+d, a+2da+2da+2d, a+3da+3da+3d, a+4da+4da+4d? Why not look at them as a−2da-2da−2d,…
By Cody Johnson, Ahaan Rungta
- Exploiting symmetry in polynomial coefficients (Note: I believe the title refers to one-variable polynomials, since I clicked on that category to get here. If someone wants to p…
By Vincent Huang, Victor Wang
- Elementary Symmetric Sums The elementary symmetric sums of a set of numbers, S={a,b,c,d,⋯ }S=\{a, b, c, d, \cdots\}S={a,b,c,d,⋯}, is defined as follows: S1=a+b+c+d+⋯ ,S_1=a+b+c+…
By Kevin Jiang
- Symmetry is often an extremely useful tool. For example, suppose you have a rectangle, and you draw a diagonal splitting the rectangle into two triangles. The two triangles are …
By Victor Wang
Problems 8
- Pairwise distinct points P1,P2,…,P16P_1,P_2,\ldots, P_{16}P1,P2,…,P16 lie on the perimeter of a square with side length 444 centered at OOO such that ∣PiPi+1∣=1\lvert P_iP_{i…
- How many ordered pairs of integers (x,y)(x, y)(x,y) are there such that 0<∣xy∣<36?0 < \left\vert xy \right\vert < 36?0<∣xy∣<36? Source: Math Prize for Girls
- Given that x+1x=3,x+\frac{1}{x}=3,x+x1=3, evaluate x4+1x4.x^4+\frac{1}{x^4}.x4+x41.
- Given that x+y=9x + y = 9x+y=9 and xy=1,xy = 1,xy=1, compute (x−y)2.(x-y)^2.(x−y)2.
- A shipping clerk wishes to determine the weight of each of the 5 boxes. Each box weighs a different integer amount less than 100 lbs. The only scale available measures weight in…
- Calculate the sum of all the digits of the numbers in the sequence 1,2,3,...,999.1, 2, 3, ..., 999.1,2,3,...,999.
- You might have noticed that on a standard die, the sides with 1 and 6 are opposite, the sides with 2 and 5 are opposite, and the sides with 3 and 4 are opposite. This configurat…
- You have two standard dice. On each, the sides with 111 and 666 are opposite, the sides with 222 and 555 are opposite, and the sides with 333 and 444 are opposite. You are playi…