Practice (115)

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A bee starts flying from point $P_0$. She flies $1$ inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a \sqrt{b} + c \sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$ ?

Let positive real number $x$, $y$, and $z$ satisfy $x+y+z=1$. Find the minimal value of $u=\sqrt{x^2 + y^2 + xy} + \sqrt{y^2 +z^2 +yz} +\sqrt{z^2 +x^2 + xz}$

Let $a_n=\binom{2020}{3n-1}$. Find the vale of $\displaystyle\sum_{n=1}^{673}a_n$.

Let $A=x\cos^2{\theta} + y\sin^2{\theta}$, $B=x\sin^2{\theta}+y\sin^2{\theta}$, where $x$, $y$, $A$, and $B$ are all real numbers. Prove $x^2 + y^2 \ge A^2 + B^2$

Let $S_n$ be the minimal value of $\displaystyle\sum_{k=1}^n\sqrt{(2k-1)^2+a_k^2}$, where $n\in\mathbb{N}$, $a_1, a_2, \cdots, a_n\in\mathbb{R}^+$, and $a_1+a_2+\cdots a_n = 17$. If there exists a unique $n$ such that $S_n$ is also an integer, find $n$.

Let sequences {$a_n$} and {$b_n$} satisfy: $a_n=a_{n-1}\cos{\theta} - b_{n-1}\sin{\theta}$ and $b_n=a_{n-1}\sin{\theta}+b_{n-1}\cos{\theta}$. If $a_1=1$ and $b_1=\tan{\theta}$, where $\theta$ is a known real number, find the general formula for {$a_n$} and {$b_n$}.

Let polynomials $P(x)$, $Q(x)$, $R(x)$, and $S(x)$ satisfy: $$P(x^5) + xQ(x^5)+x^2R(x^5)=(x^4+x^3+x^2+x+1)S(x)$$ Prove: $(x-1) | P(x)$

Let $f(z) = z^2 + az + b$, where both $a$ and $b$ are complex numbers. If for all $|z|=1$, find the values of $a$ and $b$.

Let $ABCD$ be an cyclic quadrilateral and let $HA, HB, HC, HD$ be the orthocentres of triangles $BCD, CDA, DAB$, and ABC respectively. Prove that the quadrilaterals $ABCD$ and $H_AH_BH_CH_D$ are congruent.

Consider triangle ABC and its circumcircle S. Reflect the circle with respect to AB, AC, BC to get three new circles SAB, SBC, and SBC. Show that these three circles intersect at a common point and identify this point.

The line $T$ is tangent to the circumcircle of acute triangle $ABC$ at $B$. Let $K$ be the projection of the orthocenter of triangle $ABC$ onto line $T$ ($K$ is the root of the perpendicular from the orthocenter to $S$). Let $L$ be the midpoint of side $AC$. Show that the triangle $BKL$ is isosceles.

The squares $BCDE$, $CAFG$, and $ABHI$ are constructed outside the triangle $ABC$. Let $GCDQ$ and $EBHP$ be parallelograms. Prove that $APQ$ is isosceles and $\angle PAQ =\frac{\pi}{2} $.

Given triangle $ABC$, construct equilateral triangle $ABC_1$, $BCA_1$, $CAB_1$ on the outside of $ABC$. Let $P, Q$ denote the midpoints of $C_1A_1$ and $C_1B_1$ respectively. Let $R$ be the midpoint of $AB$. Prove that triangle $PQR$ is isosceles.

(Napolean's Triangle) Given triangle $ABC$, construct an equilateral triangle on the outside of each of the sides. Let $P, Q, R$ be the centroids of these equilateral triangles, prove that triangle $PQR$ is equilateral.

(Simson Line) Let $Z$ be a point on the circumcircle of triangle $ABC$ and $P, Q, R$ be the feet of perpendiclars from $Z$ to $BC, AC, AB$ respectively. Prove that $P, Q, R$ are collinear. (This line is called the Simson line of triangle $ABC$ from $Z$.)

Given cyclic quadrilateral $ABCD$, let $P$ and $Q$ be the reflection of $C$ across lies $AB$ and $AD$ respectively. Prove that PQ passes through the orthocentre of triangle $ABD$.

Let $W_1W_2W_3$ be a triangle with circumcircle $S$, and let $A_1, A_2, A_3$ be the midpoints of $W_2W_3, W_1W_3, W_1W_2$ respectively. From Ai drop a perpendicular to the line tangent to $S$ at $W_i$. Prove that these perpendicular lines are concurrent and identify this point of concurrency.

Let $A_0A_1A_2A_3A_4A_5A_6$ be a regular 7-gon. Prove that $$\frac{1}{A_0A_1} = \frac{1}{A_0A_2}+\frac{1}{A_0A_3}$$

Given point $P_0$ in the plane of triangle $A_1A_2A_3$. Denote $A_s = A_{s-3}$, for $s > 3$. Construct points $P_1; P_2; \cdots$ sequentially such that point $P_{k+1}$ is $P_k$ rotated $120^\circ$ counter-clockwise around $A_{k+1}$. Prove that if $P_{1986} = P_0$ then triangle $A_1A_2A_3$ is isosceles.

Point $H$ is the orthocenter of triangle $ABC$. Points $D, E$ and $F$ lie on the circumcircle of triangle $ABC$ such that $AD\parallel BE\parallel CF$. Points $S, T,$ and $U$ are the respective reflections of $D, E, F$ across the lines $BC, CA$ and $AB$. Prove that $S, T, U, H$ are cyclic.

Let $ABCD$ be a cyclic quadrilateral. Let $P, Q, R$ be the feet of the perpendiculars from $D$ to the lines $BC, CA$ and $AB$ respectively. Show that $PQ = QR$ iff the bisectors of $\angle ABC$ and $\angle ADC$ meet on $AC$.

Let $O$ be the circumcentre of triangle $ABC$. A line through $O$ intersects sides $AB$ and $AC$ at $M$ and $N$ respectively. Let $S$ and $R$ be the midpoints of $BN$ and $CM$, respectively. Prove that $\angle ROS = \angle BAC$.

Let $ABCD$ be a convex quadrilateral for which $AC = BD$. Equilateral triangles are constructed on the sides of the quadrilateral and pointing outward. Let $O_1, O_2, O_3, O_4$ be the centres of the triangles constructed on $AB, BC, CD,$ and $DA$ respectively. Prove that lines $O_1O_3$ and $O_2O_4$ are perpendicular.

Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle BAP = \angle CAQ$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcentre of triangle $BCR$. Prove that $AO \perp PQ$.

Show that $$\sin\frac{\pi}{2n+1}\cdot\sin\frac{2\pi}{2n+1}\cdots\sin\frac{n\pi}{2n+1}=\frac{\sqrt{2n+1}}{2^n}$$