MathDB

2022 Cyprus JBMO TST

Part of Cyprus JBMO TST

Subcontests

(4)
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Find number in 2022-th position out of several four-digit numbers

Consider the digits 1,2,3,4,5,6,71, 2, 3, 4, 5, 6, 7. (a) Determine the number of seven-digit numbers with distinct digits that can be constructed using the digits above. (b) If we place all of these seven-digit numbers in increasing order, find the seven-digit number which appears in the 2022th2022^{\text{th}} position.

Maximal subset with |1/x-1/y| >= 1/1000

Let AA be a subset of {1,2,3,…,50}\{1, 2, 3, \ldots, 50\} with the property: for every x,y∈Ax,y\in A with x≠yx\neq y, it holds that ∣1x−1y∣>11000.\left| \frac{1}{x}- \frac{1}{y}\right|>\frac{1}{1000}. Determine the largest possible number of elements that the set AA can have.

Replace a and b with a-2b

The numbers 1,2,3,…,101, 2, 3, \ldots , 10 are written on the blackboard. In each step, Andrew chooses two numbers a,ba, b which are written on the blackboard such that a⩾2ba\geqslant 2b, he erases them, and in their place writes the number a−2ba-2b.
Find all numbers nn, such that after a sequence of steps as above, at the end only the number nn will remain on the blackboard.
3
3

Minimum value of x^5+y^5-x^4y-xy^4+x^2+4x+7 with x+y non-negative

If x,yx,y are real numbers with x+y⩾0x+y\geqslant 0, determine the minimum value of the expression K=x5+y5−x4y−xy4+x2+4x+7K=x^5+y^5-x^4y-xy^4+x^2+4x+7 For which values of x,yx,y does KK take its minimum value?

Midpoint of foot of perpendicular and circumcenter

Let ABCABC be an acute-angled triangle, and let D,ED, E and KK be the midpoints of its sides AB,ACAB, AC and BCBC respectively. Let OO be the circumcentre of triangle ABCABC, and let MM be the foot of the perpendicular from AA on the line BCBC. From the midpoint PP of OMOM we draw a line parallel to AMAM, which meets the lines DEDE and OAOA at the points TT and ZZ respectively. Prove that:
(a) the triangle DZEDZE is isosceles (b) the area of the triangle DZEDZE is given by the formula EDZE=BC⋅OK8E_{DZE}=\frac{BC\cdot OK}{8}

Two similar looking inequalities with the constraint abc=1

If a,b,ca,b,c are positive real numbers with abc=1abc=1, prove that (a) 2(aba+b+bcb+c+cac+a)⩾9ab+bc+ca2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{ab+bc+ca} (b)2(aba+b+bcb+c+cac+a)⩾9a2b+b2c+c2a2\left(\frac{ab}{a+b}+\frac{bc}{b+c}+\frac{ca}{c+a}\right) \geqslant \frac{9}{a^2 b+b^2 c+c^2 a}
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Square divided into two triangles and a parellelogram of equal areas

Let ABCDABCD be a square. Let E,ZE, Z be points on the sides AB,CDAB, CD of the square respectively, such that DE∥BZDE\parallel BZ. Assume that the triangles △EAD,△ZCB\triangle EAD, \triangle ZCB and the parallelogram BEDZBEDZ have the same area.
If the distance between the parallel lines DEDE and BZBZ is equal to 11, determine the area of the square.

Solve p^3+q^3+1=p^2q^2 in prime numbers

Determine all pairs of prime numbers (p,q)(p, q) which satisfy the equation p3+q3+1=p2q2 p^3+q^3+1=p^2q^2

Find the measure of the angle

In a triangle ABCABC with A^=80∘\widehat{A}=80^{\circ} and B^=60∘\widehat{B}=60^{\circ}, the internal angle bisector of C^\widehat{C} meets the side ABAB at the point DD. The parallel from DD to the side ACAC, meets the side BCBC at the point EE. Find the measure of the angle ∠EAB\angle EAB.
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x^2+6x+33 is perfect square

Find all integer values of xx for which the value of the expression x2+6x+33x^2+6x+33 is a perfect square.

Solve [x/2]+[x/3] = x-2022

Determine all real numbers x∈Rx\in\mathbb{R} for which ⌊x2⌋+⌊x3⌋=x−2022. \left\lfloor \frac{x}{2} \right\rfloor + \left\lfloor \frac{x}{3} \right\rfloor=x-2022. The notation ⌊z⌋\lfloor z \rfloor, for z∈Rz\in\mathbb{R}, denotes the largest integer which is less than or equal to zz. For example: \lfloor 3.98 \rfloor =3   \text{and}   \lfloor 0.14 \rfloor =0.

At least one of 2k-1, 5k-1, 13k-1 is not a perfect square

Prove that for every natural number kk, at least one of the integers 2k-1,   5k-1   \text{and}   13k-1 is not a perfect square.