Part 1For a triangle $ABC$, we denote by $P$ the orthogonal projection of $A$ on $BC$ and by $D$ the reflection of $C$ in $AP$. Triangle $ABC$ is said to be pseudo-right at $A$ if $\left|\angle B-\angle C\right|=\frac\pi2$. Specifically, it is pseudo-right at $A$ and obtuse at $B$ if $\angle B-\angle C=\frac\pi2$. (a) Prove that triangle $ABC$ is pseudo-right at $A$ if and only if triangle $ABD$ is right at $A$. (b) Prove that $PA^2=PB\cdot PC$ if and only if $\triangle ABC$ is right at $A$ or pseudo-right at $A$. (c) Prove that triangle $ABC$ is pseudo-right at $A$ if and only if its orthocenter is symmetric to $A$ with respect to $BC$. (d) Let $R$ be the circumradius of $\triangle ABC$. Prove that $PB+PC=2R$ if and only if $\triangle ABC$ is right at $A$ or pseudo-right at $A$. (e) Prove that $\triangle ABC$ is pseudo-right at $A$ if and only if the line $AP$ is tangent to the circumcircle of $\triangle ABC$. (f) Let $\alpha,\beta,\gamma$ be the points in the complex plane corresponding to $A,B,C,$ respectively. i. Give a necessary and sufficient condition on $\frac{\alpha-\beta}{\alpha-\gamma}(\beta-\gamma)^2$ that $\triangle ABC$ is pseudo-right at $A$. ii. Set $\beta=-\gamma=e^{\frac{i\pi}4}$. Find the set $E_1$ of points $A$ in the plane for which $\triangle ABC$ is pseudo-right at $A$. iii. Set $\beta=-\gamma=1$. Find the set $E_2$ of points $A$ in the plane for which $\triangle ABC$ is pseudo-right at $A$. iv. Which geometric transformation takes $E_2$ to $E_1$? Part 2(a) Let $(a,b,c)$ be a triple of positive numbers. Prove that the following conditions are equivalent: (i) There is a pseudo-right at $A$ and obtuse at $B$ triangle $ABC$ with $AB=c$, $BC=a$, $CA=b$. (ii) $b^2-c^2=a\sqrt{b^2+c^2}$ (iii) There exist real numbers $\rho>0$ and $0<\theta<\frac\pi4$ such that $a=\rho\cos2\theta$, $b=\rho\cos\theta$, and $c=\rho\sin\theta$. If these conditions are satisfied, prove that $\theta$ is the measure of one of the angles of $\angle ABC$. Can you give a geometric interpretation for $\rho$? (b) Let $\triangle ABC$ be pseudo-right at $A$ and obtuse at $B$ and let its side lengths be rational. Define $\rho$ and $\theta$ as above. In this question you can use without proof that $\cos2\phi=\frac{1-\tan^2\phi}{1+\tan^2\phi}$ and $\sin2\phi=\frac{2\tan\phi}{1+\tan^2\phi}$. i. Prove that $\rho$ is rational and deduce that so in $\tan\frac\theta2$. Let $p,q$ be the coprime positive integers with $\tan\frac\theta2=\frac pq$. ii. Prove that $0<p<q\left(\sqrt2-1\right)$ and show the existence of a positive rational number $r$ such that $$a=r\left(p^4-6p^2q^2+q^4\right),\qquad b=r\left(q^4-r^4\right),\qquad c=2pqr\left(p^2+q^2\right).$$(c) Conversely, show that the formulas in 2b give side lengths of a triangle that is pseudo-right at $A$ and obtuse at $B$. (d) i. Let $p$ and $q$ be coprime positive integers. Find the greatest positive divisor of $p^4-6p^2q^2+q^4,q^4-p^4,2pq\left(p^2+q^2\right)$ in terms of parity of $p$ and $q$. ii. Describe all triples of integers $(a,b,c)$ for which there is a triangle $ABC$, pseudo-right at $A$ and obtuse at $B$, with $AB=c$, $BC=a$, $CA=b$. (e) Solve in $\mathbb N$ the equation $x^2\left(y^2+z^2\right)=\left(y^2-z^2\right)^2$. (f) Solve in $\mathbb Q^+$ the equation $x^2\left(y^2+z^2\right)=\left(y^2-z^2\right)^2$. (g) Solve in $\mathbb N$ the equation $x^2\left(y^2-z^2\right)^2=\left(y^2+z^2\right)^3$. Part 3Let $\mathcal H$ be the curve defined by $x\ge1$ and $y=\sqrt{x^2-1}$ and let $A=(r,s)$ be a point on $\mathcal H$. Denote by $\mathcal A$ the area of the set of points satisfying $1\le x\le r$ and $y^2\le x^2-1$. (a) Calculate $\mathcal A$ in terms of $r$ and $s$. (For example, you can rotate the image by $\frac\pi4$.) (b) (Based on a result by Pierre Fermat in 1658.) Let $u$ be a positive and $n$ be a natural number such that $u^n=r+s$. For each integer $k$, $1\le k\le n$, consider the right-angled trapezoid (possibly degenerated into a triangle) having a lateral side with endpoints at $\left(u^{k-1},0\right)$ and $\left(u^k,0\right)$, the bases with slope $-1$, and the top right angle at the point on $\mathcal H$ with abscissa $\frac{u^{k-1}+u^{1-k}}2$. i. Prove that the trapezoid $T_k$ is well-defined for each $k$ and draw a sketch. ii. Why can we conjecture that the sum of these areas of these trapezoids has the limit $\frac{\mathcal A+s^2}2$ when $u$ approaches $+\infty$? iii. Prove the conjecture using another sequence of trapezoids combined with the first. iv. Find the value of $\mathcal A$. (c) Let $B=(1,0)$ and $C=(-1,0)$ and let $A=(x,y)$ be a point with $x,y\ge0$ for which $\triangle ABC$ is a pseudo-rectangle at $A$. Denote by $S$ the area of $\triangle ABC$ and by $S’$ the area of the part of the triangle consisting of points $(X,Y)$ with $Y^2\le X^2-1$. Determine, if it exists, the limit of $\frac{S’}S$ when $x\to\infty$. Part 4In the plane $z=0$ in coordinate space, let $\mathfrak L$ be the circle with center $O$ and radius $1$ and let $T$ and $P$ be distinct points such that $TP$ is the tangent to $\mathfrak L$ at $T$. The line $OP$ meets $\mathfrak L$ at $B$ and $C$, and $\mathfrak D$ is the line through $P$ perpendicular to the plane $z=0$. (a) i. Show that there exist two points $A,A’$ on $\mathfrak D$ such that triangles $ABC$ and $A’BC$ are pseudo-right at $A$ and $A’$. Show how to construct these points. ii. Prove that the coordinates of these two points satisfy $x^2+y^2=z^2+1$. (b) Let $\mathcal H$ be the set of points $A$ and $A’$ when $T$ and $P$ vary. i. What is the intersection of $\mathcal H$ with a plane orthogonal to the $x$-axis? ii. What is the intersection of $\mathcal H$ with a plane containing the $x$-axis? iii. Prove that $\mathcal H$ is a union of lines and describe these lines. (c) We are now interested in points of set $\mathcal H$ with integer coordinates. i. Let $(x,y,z)$ be one such point. Prove that $x$ or $y$ is odd. Denote by $\mathfrak I$ the set of points $(x,y,z)$ with positive integer coordinates and with $x$ odd such that $x^2+y^2=z^2+1$. ii. Let $d$ be a fixed positive integer. Prove that the set of points $(x,y,z)\in\mathfrak I$ with $\gcd(x+1,y+z)=d$ is empty if $d$ is odd and infinite if $d$ is even. iii. Let $m\ge3$ be an integer. How many elements $(x,y,z)$ of $\mathfrak I$ with $x=m$ are there? Write down these elements for $m=3,5,7,9$.