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pseudo-right triangles, locus x^2+y^2=z^2+1

Source: French MO 2002

April 8, 2021
number theorygeometry

Problem Statement

[hide=Part 1] For a triangle ABCABC, we denote by PP the orthogonal projection of AA on BCBC and by DD the reflection of CC in APAP.
Triangle ABCABC is said to be pseudo-right at AA if ∣∠B−∠C∣=π2\left|\angle B-\angle C\right|=\frac\pi2. Specifically, it is pseudo-right at AA and obtuse at BB if ∠B−∠C=π2\angle B-\angle C=\frac\pi2.
(a) Prove that triangle ABCABC is pseudo-right at AA if and only if triangle ABDABD is right at AA. (b) Prove that PA2=PB⋅PCPA^2=PB\cdot PC if and only if △ABC\triangle ABC is right at AA or pseudo-right at AA. (c) Prove that triangle ABCABC is pseudo-right at AA if and only if its orthocenter is symmetric to AA with respect to BCBC. (d) Let RR be the circumradius of △ABC\triangle ABC. Prove that PB+PC=2RPB+PC=2R if and only if △ABC\triangle ABC is right at AA or pseudo-right at AA. (e) Prove that △ABC\triangle ABC is pseudo-right at AA if and only if the line APAP is tangent to the circumcircle of △ABC\triangle ABC. (f) Let α,β,γ\alpha,\beta,\gamma be the points in the complex plane corresponding to A,B,C,A,B,C, respectively. i. Give a necessary and sufficient condition on α−βα−γ(β−γ)2\frac{\alpha-\beta}{\alpha-\gamma}(\beta-\gamma)^2 that △ABC\triangle ABC is pseudo-right at AA. ii. Set β=−γ=eiπ4\beta=-\gamma=e^{\frac{i\pi}4}. Find the set E1E_1 of points AA in the plane for which △ABC\triangle ABC is pseudo-right at AA. iii. Set β=−γ=1\beta=-\gamma=1. Find the set E2E_2 of points AA in the plane for which △ABC\triangle ABC is pseudo-right at AA. iv. Which geometric transformation takes E2E_2 to E1E_1?
[hide=Part 2] (a) Let (a,b,c)(a,b,c) be a triple of positive numbers. Prove that the following conditions are equivalent:
(i) There is a pseudo-right at AA and obtuse at BB triangle ABCABC with AB=cAB=c, BC=aBC=a, CA=bCA=b. (ii) b2−c2=ab2+c2b^2-c^2=a\sqrt{b^2+c^2} (iii) There exist real numbers ρ>0\rho>0 and 0<θ<π40<\theta<\frac\pi4 such that a=ρcos⁡2θa=\rho\cos2\theta, b=ρcos⁡θb=\rho\cos\theta, and c=ρsin⁡θc=\rho\sin\theta.
If these conditions are satisfied, prove that θ\theta is the measure of one of the angles of ∠ABC\angle ABC. Can you give a geometric interpretation for ρ\rho? (b) Let △ABC\triangle ABC be pseudo-right at AA and obtuse at BB and let its side lengths be rational. Define ρ\rho and θ\theta as above. In this question you can use without proof that cos⁡2ϕ=1−tan⁡2ϕ1+tan⁡2ϕ\cos2\phi=\frac{1-\tan^2\phi}{1+\tan^2\phi} and sin⁡2ϕ=2tan⁡ϕ1+tan⁡2ϕ\sin2\phi=\frac{2\tan\phi}{1+\tan^2\phi}. i. Prove that ρ\rho is rational and deduce that so in tan⁡θ2\tan\frac\theta2. Let p,qp,q be the coprime positive integers with tan⁡θ2=pq\tan\frac\theta2=\frac pq. ii. Prove that 0<p<q(2−1)0<p<q\left(\sqrt2-1\right) and show the existence of a positive rational number rr such that a=r(p4−6p2q2+q4),b=r(q4−r4),c=2pqr(p2+q2).a=r\left(p^4-6p^2q^2+q^4\right),\qquad b=r\left(q^4-r^4\right),\qquad c=2pqr\left(p^2+q^2\right). (c) Conversely, show that the formulas in 2b give side lengths of a triangle that is pseudo-right at AA and obtuse at BB. (d) i. Let pp and qq be coprime positive integers. Find the greatest positive divisor of p4−6p2q2+q4,q4−p4,2pq(p2+q2)p^4-6p^2q^2+q^4,q^4-p^4,2pq\left(p^2+q^2\right) in terms of parity of pp and qq. ii. Describe all triples of integers (a,b,c)(a,b,c) for which there is a triangle ABCABC, pseudo-right at AA and obtuse at BB, with AB=cAB=c, BC=aBC=a, CA=bCA=b. (e) Solve in N\mathbb N the equation x2(y2+z2)=(y2−z2)2x^2\left(y^2+z^2\right)=\left(y^2-z^2\right)^2. (f) Solve in Q+\mathbb Q^+ the equation x2(y2+z2)=(y2−z2)2x^2\left(y^2+z^2\right)=\left(y^2-z^2\right)^2. (g) Solve in N\mathbb N the equation x2(y2−z2)2=(y2+z2)3x^2\left(y^2-z^2\right)^2=\left(y^2+z^2\right)^3.
[hide=Part 3] Let H\mathcal H be the curve defined by x≥1x\ge1 and y=x2−1y=\sqrt{x^2-1} and let A=(r,s)A=(r,s) be a point on H\mathcal H. Denote by A\mathcal A the area of the set of points satisfying 1≤x≤r1\le x\le r and y2≤x2−1y^2\le x^2-1.
(a) Calculate A\mathcal A in terms of rr and ss. (For example, you can rotate the image by π4\frac\pi4.) (b) (Based on a result by Pierre Fermat in 1658.) Let uu be a positive and nn be a natural number such that un=r+su^n=r+s. For each integer kk, 1≤k≤n1\le k\le n, consider the right-angled trapezoid (possibly degenerated into a triangle) having a lateral side with endpoints at (uk−1,0)\left(u^{k-1},0\right) and (uk,0)\left(u^k,0\right), the bases with slope −1-1, and the top right angle at the point on H\mathcal H with abscissa uk−1+u1−k2\frac{u^{k-1}+u^{1-k}}2.
i. Prove that the trapezoid TkT_k is well-defined for each kk and draw a sketch. ii. Why can we conjecture that the sum of these areas of these trapezoids has the limit A+s22\frac{\mathcal A+s^2}2 when uu approaches +∞+\infty? iii. Prove the conjecture using another sequence of trapezoids combined with the first. iv. Find the value of A\mathcal A. (c) Let B=(1,0)B=(1,0) and C=(−1,0)C=(-1,0) and let A=(x,y)A=(x,y) be a point with x,y≥0x,y\ge0 for which △ABC\triangle ABC is a pseudo-rectangle at AA. Denote by SS the area of △ABC\triangle ABC and by S’S’ the area of the part of the triangle consisting of points (X,Y)(X,Y) with Y2≤X2−1Y^2\le X^2-1. Determine, if it exists, the limit of S’S\frac{S’}S when x→∞x\to\infty.
[hide=Part 4] In the plane z=0z=0 in coordinate space, let L\mathfrak L be the circle with center OO and radius 11 and let TT and PP be distinct points such that TPTP is the tangent to L\mathfrak L at TT. The line OPOP meets L\mathfrak L at BB and CC, and D\mathfrak D is the line through PP perpendicular to the plane z=0z=0.
(a) i. Show that there exist two points A,A’A,A’ on D\mathfrak D such that triangles ABCABC and A’BCA’BC are pseudo-right at AA and A’A’. Show how to construct these points. ii. Prove that the coordinates of these two points satisfy x2+y2=z2+1x^2+y^2=z^2+1. (b) Let H\mathcal H be the set of points AA and A’A’ when TT and PP vary. i. What is the intersection of H\mathcal H with a plane orthogonal to the xx-axis? ii. What is the intersection of H\mathcal H with a plane containing the xx-axis? iii. Prove that H\mathcal H is a union of lines and describe these lines. (c) We are now interested in points of set H\mathcal H with integer coordinates. i. Let (x,y,z)(x,y,z) be one such point. Prove that xx or yy is odd. Denote by I\mathfrak I the set of points (x,y,z)(x,y,z) with positive integer coordinates and with xx odd such that x2+y2=z2+1x^2+y^2=z^2+1. ii. Let dd be a fixed positive integer. Prove that the set of points (x,y,z)∈I(x,y,z)\in\mathfrak I with gcd⁡(x+1,y+z)=d\gcd(x+1,y+z)=d is empty if dd is odd and infinite if dd is even. iii. Let m≥3m\ge3 be an integer. How many elements (x,y,z)(x,y,z) of I\mathfrak I with x=mx=m are there? Write down these elements for m=3,5,7,9m=3,5,7,9.
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