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The Altarians use an interesting number system.
In Altarian, the natural numbers 1 to 5 are written r, s, sr, s/s, and ss. For what it's worth, the Altarians use z for 0.
Thus, s/s+ss=sr/sr and ssr+sr/sr=s/s/s/s. Obviously, s/s*ss=s/s/ss.
Give the Altarian equivalents of the Earth numbers 21 and 36.
Too easy? How about 37 and 38?
Let triangle ABC and a point D on AC such that BD = DC = 3. If AD = 6 and ∠ ACB = 30◦, calculate ∠ ABD.
I have a clock app that shows not only the time and date, but also shows a polar projection map (specifically azimuthal equidistant centered on the north pole) that includes all of the world that's north of 25° South. The portions of the world that are lit at that time by the sun are shown in light versions of its colors, while nighttime is shown in darker shades.
That app was written so that the sunlight is assumed to come directly from the left so the center of the arc defining the left edge of daylight is at would normally be called the "9 o'clock" position, though this is obviously where noon is. The 25° South limit allows the location of where the sun is directly overhead to be shown, but it also is near enough to the equator to allow every horizontal line through the map to be all daytime pixels, all nighttime pixels, or a straight run of daylight pixels followed by nighttime pixels.
If the map were to have been made to extend all the way to the south pole, not only would there be areas of great distortion, but the programming would also have to consider horizontal rows of pixels that went from day to night and back to day, or even the reverse at certain times of year: night, day and night again. When the map goes only as far out as 25° S this doesn't happen. Such would happen though before the outer limit got all the way to the south pole.
The question is: To what south latitude could the map be extended without this happening? ... so any line of pixels would still have at most two parts, not three.
Assume the Sun can be anywhere from 23.4° S to 23.4° N, and that daylight is where the geographic location is less than 90° from where the sun is directly overhead.
A cube of size 4 × 4 × 4 is divided into 16 equal squares per face, with numbers from 1 to 96 randomly assigned to these squares. An operation consists of taking two squares that share a vertex, summing their numbers, and rewriting this sum in one of the squares while leaving the other blank. After performing several such operations, only one number remains. Prove that regardless of the order of operations, the final remaining number is always the same. Also find this number.
On a circumference, points A and B are on opposite arcs of diameter CD. Line segments CE and DF are perpendicular to AB such that A, E, F and B are collinear in this order. Given that AE=1, find the length of BF.
Positive numbers x, y satisfy equality x 3 + y 3 = x - y. Prove that x 2 + y 2 < 1.
Let P be a variable point on the circumference of a quarter-circle with radii OA, OB and ∠AOB = 90◦. H is the projection of P on OA. Find the locus of the incenter of the
right-angled triangle HPO.
You have N sticks with lengths 1, 2, 3, ..., N.
Provide a closed form function in terms of N for the number of distinct valid triangles can you make from these N sticks.
note: reflections and rotations are considered to be the same, and valid triangles must have positive area.
Inspired by 5, 6, Pick Up Sticks
Show that any triangle has two sides whose lengths a and b satisfy (√5-1)/2<a/b<(√5+1)/2.
At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining 1/2 points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players with the same number of points the younger received the better ranking. After the competition Jan realizes that the best three players together got as many points as the last 9 players obtained points together. And Joerg noted that the number of ties was maximal. Determine the number of ties.
Find the product of the factors of 1980n for n ∈ N
Can the graphs of a polynomial of degree 20 and the function
y=1/x40 have exactly 30 points of intersection?
M is the midpoint of the side AB in an equilateral triangle ABC.
The point D on the side BC is such that BD:DC=3:1.
On the line passing through C and parallel to MD there is a point T
inside the triangle ABC such that ∠ CTA=150.
Find the ∠ MTD.
Consider triangle ABC an isosceles triangle such that AB = BC. Let P be a point satisfying
∠ ABP = 80, ∠ CBP = 20, and AC = BP
Find all possible values of ∠ BCP.
Alice writes down all 7-digit numbers using the digits 1, 2, 3, 4, 5, 6, and 7 exactly once. Prove that there are no two numbers among them where one is a multiple of the other.
Compute the smallest integer n for which it is possible to draw an n-gon whose vertex angles all measure 167◦
or 174◦.
Find the angles of the triangle which satisfies R(b + c) = a√(bc) where a,
b, c, R are the sides and the circumradius of the triangle.
Compute the area of the circle circumscribed about an equiangular hexagon with sides of lengths
20, 16, 20, 16, 20, 16.
Let a,b,c be positive real numbers with a+b>c. Prove that ax + sin(bx) + cos(cx) > 1 for all x ∈ (0, pi/(a+b+c)).
Let ABC be a triangle with incenter I and incircle ω. Let l be the tangent to ω parallel to BC and distinct from BC. Let D be the intersection of l and AC, and let M be the midpoint of ID. Prove that ∠AMD = ∠DBC.
On a 4 × 4 board, each cell is colored either black or white such that each row and each column have an even number of black cells. How many ways can the board be colored?
For how many non-empty subsets S of {1, 2, 3, 4, 5, 6, 7, 8, 9} is it true that the number (max S)/(min S) is an element of S?
Let's consider all possible quadratic trinomials of the form x2 + ax + b, where a and b are positive integers not exceeding some positive integer N. Prove that the number of pairs of such trinomials having a common root does not exceed N2.
There are 1001 stacks of coins S1, S2, ..., S1001. Initially, stack Sk has k coins for each k = 1,2, ...,1001. In an operation, one selects an ordered pair (i,j) of indices i and j satisfying 1 ≤ i < j ≤ 1001 subject to two conditions:
The stacks Si and Sj must each have at least 1 coin.
The ordered pair (i,j) must not have been selected before.
Then, if Si and Sj have a coins and b coins respectively, one removes gcd(a,b) coins from each stack.
What is the maximum number of times this operation could be performed?
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