MathDB

2020 Dutch IMO TST

Part of Dutch IMO TST

Subcontests

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1/r is an integer when r min of (a/b - c/d) when gcd (a, b) = 1, c<=a, d <= b

Let a,b≥2a, b \ge 2 be positive integers with gcd(a,b)=1gcd (a, b) = 1. Let rr be the smallest positive value that ab−cd\frac{a}{b}- \frac{c}{d} can take, where cc and dd are positive integers satisfying c≤ac \le a and d≤bd \le b. Prove that 1r\frac{1}{r} is an integer.

perpendicular wanted, tangents on circumcircle related

Let ABCABC be an acute-angled triangle and let PP be the intersection of the tangents at BB and CC of the circumscribed circle of △ABC\vartriangle ABC. The line through AA perpendicular on ABAB and cuts the line perpendicular on ACAC through CC at XX. The line through AA perpendicular on ACAC cuts the line perpendicular on ABAB through BB at YY. Show that AP⊥XYAP \perp XY.

2 player game of m tiles of kx1 on a nxn board k <= n <= 2k - 1

Given are two positive integers kk and nn with k≤n≤2k−1k \le n \le 2k - 1. Julian has a large stack of rectangular k×1k \times 1 tiles. Merlin calls a positive integer mm and receives mm tiles from Julian to place on an n×nn \times n board. Julian first writes on every tile whether it should be a horizontal or a vertical tile. Tiles may be used the board should not overlap or protrude. What is the largest number mm that Merlin can call if he wants to make sure that he has all tiles according to the rule of Julian can put on the plate?
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1x1,1x2, ..., 1xn tiles in a nxn board, red n (n + 1)/2 cells

For a positive integer nn, we consider an n×nn \times n board and tiles with dimensions 1×1,1×2,...,1×n1 \times 1, 1 \times 2, ..., 1 \times n. In how many ways exactly can 12n(n+1)\frac12 n (n + 1) cells of the board are colored red, so that the red squares can all be covered by placing the nn tiles all horizontally, but also by placing all nn tiles vertically?
Two colorings that are not identical, but by rotation or reflection from the board into each other count as different.

a + b = \phi (a) + \phi (b) + gcd (a, b)

Find all pairs (a,b)(a, b) of positive integers for which a+b=ϕ(a)+ϕ(b)+gcd(a,b)a + b = \phi (a) + \phi (b) + gcd (a, b).
Here ϕ(n) \phi (n) is the number of numbers kk from {1,2,...,n}\{1, 2,. . . , n\} with gcd(n,k)=1gcd (n, k) = 1.

f( - f (x) - f (y))= 1 -x - y , in Zxz

Find all functions f:Z→Zf: Z \to Z that satisfy f(−f(x)−f(y))=1−x−yf(-f (x) - f (y))= 1 -x - y for all x,y∈Zx, y \in Z
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P (x^2) + 2P (x) = P (x)^2 + 2

Determine all polynomials P(x)P (x) with real coefficients that apply P(x2)+2P(x)=P(x)2+2P (x^2) + 2P (x) = P (x)^2 + 2.

999 integers 2 player game

Ward and Gabrielle are playing a game on a large sheet of paper. At the start of the game, there are 999999 ones on the sheet of paper. Ward and Gabrielle each take turns alternatingly, and Ward has the first turn. During their turn, a player must pick two numbers a and b on the sheet such that gcd(a,b)=1gcd(a, b) = 1, erase these numbers from the sheet, and write the number a+ba + b on the sheet. The first player who is not able to do so, loses. Determine which player can always win this game.

fixed point wanted, point on circumcircle, reflection wrt angle bisector

Given is a triangle ABCABC with its circumscribed circle and ∣AC∣<∣AB∣| AC | <| AB |. On the short arc ACAC, there is a variable point D≠AD\ne A. Let EE be the reflection of AA wrt the inner bisector of ∠BDC\angle BDC. Prove that the line DEDE passes through a fixed point, regardless of point DD.
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&lt;BAC = 2 &lt;ABC wanted, AC + AI = BC given , incenter I

In acute-angled triangle ABC,IABC, I is the center of the inscribed circle and holds ∣AC∣+∣AI∣=∣BC∣| AC | + | AI | = | BC |. Prove that ∠BAC=2∠ABC\angle BAC = 2 \angle ABC.

P_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n

Given are real numbers a1,a2,...,a2020a_1, a_2,..., a_{2020}, not necessarily different. For every n≥2020n \ge 2020, define an+1a_{n + 1} as the smallest real zero of the polynomial Pn(x)=x2n+a1x2n−2+a2x2n−4+...+an−1x2+anP_n (x) = x^{2n} + a_1x^{2n - 2} + a_2x^{2n - 4} +... + a_{n -1}x^2 + a_n, if it exists. Assume that an+1a_{n + 1} exists for all n≥2020n \ge 2020. Prove that an+1≤ana_{n + 1} \le a_n for all n≥2021n \ge 2021.

k = d (a) = d (b) = d (2a + 3b), no of positive divisors

For a positive number nn, we write d(n)d (n) for the number of positive divisors of nn. Determine all positive integers kk for which exist positive integers aa and bb with the property k=d(a)=d(b)=d(2a+3b)k = d (a) = d (b) = d (2a + 3b).