MathDB

2022 Dutch IMO TST

Part of Dutch IMO TST

Subcontests

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2 tangents on different circles concurrent with 1 common tangent

Two circles Γ1\Gamma_1 and Γ2\Gamma_2are given with centres O1O_1 and O2O_2 and common exterior tangents ℓ1\ell_1 and ℓ2\ell_2. The line ℓ1\ell_1 intersects Γ1\Gamma_1 in AA and Γ2\Gamma_2 in BB. Let XX be a point on segment O1O2O_1O_2, not lying on Γ1\Gamma_1 or Γ2\Gamma_2. The segment AXAX intersects Γ1\Gamma_1 in Y≠AY \ne A and the segment BXBX intersects Γ2\Gamma_2 in Z≠BZ \ne B. Prove that the line through YY tangent to Γ1\Gamma_1 and the line through ZZ tangent to Γ2\Gamma_2 intersect each other on ℓ2\ell_2.

n boxes in a row, n + 1 identical stones

Let n>1n > 1 be an integer. There are nn boxes in a row, and there are n+1n + 1 identical stones. A distribution is a way to distribute the stones over the boxes, in which every stone is in exactly one of the boxes. We say that two of such distributions are a stone’s throw away from each other if we can obtain one distribution from the other by moving exactly one stone from one box to another. The cosiness of a distribution aa is defined as the number of distributions that are a stone’s throw away from aa. Determine the average cosiness of all possible distributions.
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n = d^3 + m^3 , m | n, d is the smallest divisor of n greater than 1

Determine all positive integers n≥2n \ge 2 which have a positive divisor m∣nm | n satisfying n=d3+m3.n = d^3 + m^3. where dd is the smallest divisor of nn which is greater than 11.

diophantine ab =2(1 + cd), exists a triangle with sides a - c, b - d and c + d

Find all quadruples (a,b,c,d)(a, b, c, d) of non-negative integers such that ab=2(1+cd)ab =2(1 + cd) and there exists a non-degenerate triangle with sides of length a−ca - c, b−db - d, and c+dc + d.

F is circumcenter of DMN if B is circumcenter of FMN

Consider an acute triangle ABCABC with ∣AB∣>∣CA∣>∣BC∣|AB| > |CA| > |BC|. The vertices D,ED, E, and FF are the base points of the altitudes from A,BA, B, and CC, respectively. The line through F parallel to DEDE intersects BCBC in MM. The angular bisector of ∠MFE\angle MF E intersects DEDE in NN. Prove that FF is the circumcentre of △DMN\vartriangle DMN if and only if BB is the circumcentre of △FMN\vartriangle FMN.
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min of max {xy, (x- 1)(y - 1), x + y - 2xy} for 0 <= x, y <= 1

For real numbers xx and yy we define M(x,y)M(x, y) to be the maximum of the three numbers xyxy, (x−1)(y−1)(x- 1)(y - 1), and x+y−2xyx + y - 2xy. Determine the smallest possible value of M(x,y)M(x, y) where xx and yy range over all real numbers satisfying 0≤x,y≤10 \le x, y \le 1.

divisors on the exponent

Let nn be a natural number. An integer a>2a>2 is called nn-decomposable, if an−2na^n-2^n is divisible by all the numbers of the form ad+2da^d+2^d, where d≠nd\neq n is a natural divisor of nn. Find all composite n∈Nn\in \mathbb{N}, for which there's an nn-decomposable number.

15 numbered lights on the ceiling of a room, 15 switches each for 2 lights

There are 1515 lights on the ceiling of a room, numbered from 11 to 1515. All lights are turned off. In another room, there are 1515 switches: a switch for lights 11 and 22, a switch for lights 22 and 33, a switch for lights 33 en 44, etcetera, including a sqitch for lights 1515 and 11. When the switch for such a pair of lights is turned, both of the lights change their state (from on to off, or vice versa). The switches are put in a random order and all look identical. Raymond wants to find out which switch belongs which pair of lights. From the room with the switches, he cannot see the lights. He can, however, flip a number of switches, and then go to the other room to see which lights are turned on. He can do this multiple times. What is the minimum number of visits to the other room that he has to take to determine for each switch with certainty which pair of lights it corresponds to?