MathDB

1

Part of BIMO 2022

Problems(7)

f(xf(x)+2y)=f(x)^2+x+2f(y)

Source: Own. IMO 2022 Malaysian Training Camp 1

1/29/2022
Find all functions f:R→Rf:\mathbb{R}\rightarrow \mathbb{R} such that for all real numbers x,yx,y, we have f(xf(x)+2y)=f(x)2+x+2f(y)f(xf(x)+2y)=f(x)^2+x+2f(y)
algebrafunctional equation
Intersection lies on fixed circle

Source: Own. IMO 2022 Malaysian Training Camp 1

2/27/2022
A pentagon ABCDEABCDE is such that ABCDABCD is cyclic, BE∥CDBE\parallel CD, and DB=DEDB=DE. Let us fix the points B,C,D,EB,C,D,E and vary AA on the circumcircle of BCDBCD. Let P=AC∩BEP=AC\cap BE, and Q=BC∩DEQ=BC\cap DE.
Prove that the second intersection of circles (ABE)(ABE) and (PQE)(PQE) lie on a fixed circle.
geometry
assignment of vertices and edges?

Source: IMO 2022 Malaysian Training Camp 1

2/26/2022
Given a graph GG, consider the following two quantities,
∙\bullet Assign to each vertex a number in {0,1,2}\{0,1,2\} such that for every edge e=uve=uv, the numbers assigned to uu and vv have sum at least 22. Let A(G)A(G) be the minimum possible sum of the numbers written to each vertex satisfying this condition.
∙\bullet Assign to each edge a number in {0,1,2}\{0,1,2\} such that for every vertex vv, the sum of numbers on all edges containing vv is at most 22. Let B(G)B(G) be the maximum possible sum of the numbers written to each edge satisfying this condition.
Prove that A(G)=B(G)A(G)=B(G) for every graph GG.
[Note: This question is not original]
[Extra: Show that this statement is still true if we replace 22 to nn, if and only if nn is even (where we replace {0,1,2}\{0,1,2\} to {0,1,⋯ ,n}\{0,1,\cdots, n\})]
combinatorics
(APF) intersect (AQE)

Source: Own. IMO 2022 Malaysian Training Camp 2

3/13/2022
Let ABCABC be a triangle, and let BE,CFBE, CF be the altitudes. Let ℓ\ell be a line passing through AA. Suppose ℓ\ell intersect BEBE at PP, and ℓ\ell intersect CFCF at QQ. Prove that:
i) If ℓ\ell is the AA-median, then circles (APF)(APF) and (AQE)(AQE) are tangent.
ii) If ℓ\ell is the inner AA-angle bisector, suppose (APF)(APF) intersect (AQE)(AQE) again at RR, then ARAR is perpendicular to ℓ\ell.
geometry
largest possible k?

Source: Own. IMO 2022 Malaysian Training Camp 2

3/14/2022
Let a,b,c,a, b, c, be nonnegative reals with a+b+c=3 a+b+c=3 , find the largest positive real k k so that for all a,b,c,a,b,c, we have a2+b2+c2+k(abc−1)≥3 a^2+b^2+c^2+k(abc-1)\ge 3
algebrainequalitiesInequality
n divides x^n-y^n implies n^2 does?

Source: Own. IMO 2022 Malaysian Training Camp 3

4/17/2022
Find all positive integer nn such that for all positive integers x x , y y , n∣xn−yn⇒n2∣xn−yn n \mid x^n-y^n \Rightarrow n^2 \mid x^n-y^n .
number theory
Projection inequality

Source: Own. Malaysian IMO TST 2022 P1

5/7/2022
Given an acute triangle ABCABC, mark 33 points X,Y,ZX, Y, Z in the interior of the triangle. Let X1,X2,X3X_1, X_2, X_3 be the projections of XX to BC,CA,ABBC, CA, AB respectively, and define the points Yi,ZiY_i, Z_i similarly for i=1,2,3i=1, 2, 3.
a) Suppose that XiYi<XiZiX_iY_i<X_iZ_i for all i=1,2,3i=1,2,3, prove that XY<XZXY<XZ.
b) Prove that this is not neccesarily true, if triangle ABCABC is allowed to be obtuse.
Proposed by Ivan Chan Kai Chin
geometryinequalities