Chapter 5

Compute the sum $$ \sum_{k=0}^{3 n-1}(-1)^{k}\left(\begin{array}{c} 6 n \\ 2 k+1 \end{array}\right) 3^{k} . $$

Two unit squares \(K_{1}, K_{2}\) with centers \(M, N\) are situated in the plane so that \(M N=4 .\) Two sides of \(K_{1}\) are parallel to the line \(M N\), and one of the diagonals of \(K_{2}\) lies on \(M N\). Find the locus of the midpoint of \(X Y\) as \(X, Y\) vary over the interior of \(K_{1}, K_{2}\) respectivelv.

Let \(z_{1}, z_{2}, \ldots, z_{n}\) be distinct complex numbers such that \(\left|z_{1}\right|=\) \(\left|z_{2}\right|=\cdots=\left|z_{n}\right| .\) Prove that $$ \sum_{1 \leq i

$$ \text { Let } n \geq 3 \text { and } k \geq 2 \text { be positive integers and consider the complex } $$ numbers \(z=\cos \frac{2 \pi}{n}+i \sin \frac{2 \pi}{n}\) and \(\theta=1-z+z^{2}-z^{3}+\cdots+(-1)^{k-1} z^{k-1}\) a) If \(k\) is even, prove that \(\theta^{n}=1\) if and only if \(n\) is even and \(\frac{n}{2}\) divides \(k-1\) or \(k+1\). b) If \(k\) is odd, prove that \(\theta^{n}=1\) if and only if \(n\) divide \(k-1\) or \(k+1\).

Prove that $$ \cos \frac{\pi}{11}+\cos \frac{3 \pi}{11}+\cos \frac{5 \pi}{11}+\cos \frac{7 \pi}{11}+\cos \frac{9 \pi}{11}=\frac{1}{2} $$

On each side of a parallelogram a square is drawn external to the figure. Prove that the centers of the squares are the vertices of another square.

Consider equilateral triangles \(A B C\) and \(A^{\prime} B^{\prime} C^{\prime}\), both in the same plane and having the same orientation. Show that the segments \(\left[A A^{\prime}\right],\left[B B^{\prime}\right],\left[C C^{\prime}\right]\) can be the sides of a triangle.

Consider the quadratic equation $$ a^{2} z^{2}+a b z+c^{2}=0 $$ $$ \begin{aligned} &\text { where } a, b, c \in \mathbb{C}^{*} \text { and denote by } z_{1}, z_{2} \text { its roots. Prove that if } \frac{b}{c} \text { is a real number then }\\\ &\left|z_{1}\right|=\left|z_{2}\right| \text { or } \frac{z_{1}}{z_{2}} \in \mathbb{R} \end{aligned} $$

Let \(z_{1}, z_{2}, z_{3}\) be complex numbers such that $$ \left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=r>0 $$ and \(z_{1}+z_{2}+z_{3} \neq 0\). Prove that $$ \left|\frac{z_{1} z_{2}+z_{2} z_{3}+z_{3} z_{1}}{z_{1}+z_{2}+z_{3}}\right|=r $$

Curves \(A, B, C\) and \(D\) are defined in the plane as follows: $$ \begin{aligned} &A=\left\\{(x, y): x^{2}-y^{2}=\frac{x}{x^{2}+y^{2}}\right\\} \\ &B=\left\\{(x, y): 2 x y+\frac{y}{x^{2}+y^{2}}=3\right\\} \\ &C=\left\\{(x, y): x^{3}-3 x y^{2}+3 y=1\right\\} \\ &D=\left\\{(x, y): 3 x^{2} y-3 x-y^{3}=0\right\\} \end{aligned} $$ Prove that \(A \cap B=C \cap D\).

$$ \text { Calculate the sum } S_{n}=\sum_{k=0}^{n}\left(\begin{array}{l} n \\ k \end{array}\right) \cos k \alpha, \text { where } \alpha \in[0, \pi] . $$

Let \(A_{1} A_{2} \cdots A_{n}\) be a polygon and let \(a_{1}, a_{2}, \ldots, a_{n}\) be the coordinates of the vertices \(A_{1}, A_{2}, \ldots, A_{n} .\) If \(\left|a_{1}\right|=\left|a_{2}\right|=\cdots=\left|a_{n}\right|=R\), prove that $$ \sum_{1 \leq i

$$ \text { Consider the cube root of unity } $$ $$ \varepsilon=\cos \frac{2 \pi}{3}+i \sin \frac{2 \pi}{3} $$ Compute $$ (1+\varepsilon)\left(1+\varepsilon^{2}\right) \cdots\left(1+\varepsilon^{1987}\right) $$

$$ \text { Compute the product } P=\cos 20^{\circ} \cdot \cos 40^{\circ} \cdot \cos 80^{\circ} $$

Let \(P\) be an arbitrary point in the plane of a triangle \(A B C .\) Then $$ \alpha \cdot P B \cdot P C+\beta \cdot P C \cdot P A+\gamma \cdot P A \cdot P B \geq \alpha \beta \gamma $$ where \(\alpha, \beta, \gamma\) are the sides of \(A B C\).

$$ \text { Let } a, b, c, z \text { be complex numbers such that }|a|=|b|=|c|>0 \text { and } $$ $$ \begin{aligned} &a z^{2}+b z+c=0 . \text { Prove that } \\ &\qquad \frac{\sqrt{5}-1}{2} \leq|z| \leq \frac{\sqrt{5}+1}{2} . \end{aligned} $$

Given a point on the circumcircle of a cyclic quadrilateral, prove that the products of the distances from the point to any pair of opposite sides or to the diagonals are equal.

Let \(z_{1}, z_{2}\) be complex numbers such that $$ \left|z_{1}\right|=\left|z_{2}\right|=r>0 $$ Prove that $$ \left(\frac{z_{1}+z_{2}}{r^{2}+z_{1} z_{2}}\right)^{2}+\left(\frac{z_{1}-z_{2}}{r^{2}-z_{1} z_{2}}\right)^{2} \geq \frac{1}{r^{2}} $$

Prove the identity $$ \left(\left(\begin{array}{l} n \\ 0 \end{array}\right)-\left(\begin{array}{l} n \\ 2 \end{array}\right)+\left(\begin{array}{l} n \\ 4 \end{array}\right)-\cdots\right)^{2}+\left(\left(\begin{array}{l} n \\ 1 \end{array}\right)-\left(\begin{array}{l} n \\ 3 \end{array}\right)+\left(\begin{array}{l} n \\ 5 \end{array}\right)-\cdots\right)^{2}=2^{n} $$

Let \(z_{1}, z_{2}, \ldots, z_{n}\) be the coordinates of the vertices of a regular polygon with the circumcenter at the origin of the complex plane. Prove that there are \(i, j, k \in\) \(\\{1,2, \ldots, n\\}\) such that \(z_{i}+z_{j}=z_{k}\) if and only if 6 divides \(n .\)

Let \(\varepsilon \neq 1\) be a cube root of unity. Compute $$ \left(1-\varepsilon+\varepsilon^{2}\right)\left(1-\varepsilon^{2}+\varepsilon^{4}\right) \cdots\left(1-\varepsilon^{n}+\varepsilon^{2 n}\right) $$

$$ \begin{aligned} &\text { Let } x, y, z \text { be real numbers such that }\\\ &\sin x+\sin y+\sin z=0 \quad \text { and } \quad \cos x+\cos y+\cos z=0 \end{aligned} $$ Prove that $$ \sin 2 x+\sin 2 y+\sin 2 z=0 \quad \text { and } \quad \cos 2 x+\cos 2 y+\cos 2 z=0 $$

Three equal circles \(\mathcal{C}_{1}\left(O_{1} ; r\right), \mathcal{C}_{2}\left(O_{2} ; r\right)\) and \(\mathcal{C}_{3}\left(O_{3} ; r\right)\) have a common point O. Circles \(\mathcal{C}_{1}\) and \(\mathcal{C}_{2}, \mathcal{C}_{2}\) and \(\mathcal{C}_{3}, \mathcal{C}_{3}\) and \(\mathcal{C}_{1}\), meet again at points \(A, B, C\) respectively. Prove that the circumradius of triangle \(A B C\) is equal to \(r\).

$$ \text { Let } p, q \text { be complex numbers such that }|p|+|q|<1 \text { . Prove that the moduli } $$ $$ \text { of the roots of the equation } z^{2}+p z+q=0 \text { are less than } 1 . $$

Let \(z_{1}, z_{2}, z_{3}\) be complex numbers such that $$ \left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1 $$ and $$ \frac{z_{1}^{2}}{z_{2} z_{3}}+\frac{z_{2}^{2}}{z_{1} z_{3}}+\frac{z_{3}^{2}}{z_{1} z_{2}}+1=0 $$ Prove that $$ \left|z_{1}+z_{2}+z_{3}\right| \in\\{1,2\\} . $$

A tourist takes a trip through a city in stages. Each stage consists of three segments of length 100 meters separated by right turns of \(60^{\circ} .\) Between the last segment of one stage and the first segment of the next stage, the tourist makes a left turn of \(60^{\circ}\). At what distance will the tourist be from his initial position after 1997 stages?

If \(m\) and \(p\) are positive integers and \(m>p\), then $$ \begin{gathered} \left(\begin{array}{c} m \\ 0 \end{array}\right)+\left(\begin{array}{c} m \\ p \end{array}\right)+\left(\begin{array}{c} m \\ 2 p \end{array}\right)+\left(\begin{array}{c} m \\ 3 p \end{array}\right)+\cdots \\ =\frac{2^{m}}{p}\left(1+\sum_{k=1}^{\left[\frac{p-1}{2}\right]}\left(\cos \frac{k \pi}{p}\right)^{m} \cos \frac{m k \pi}{p}\right) . \end{gathered} $$

Let \(z_{1}, z_{2}, \ldots, z_{n}\) be the coordinates of the vertices of a regular polygon. Prove that $$ z_{1}^{2}+z_{2}^{2}+\cdots+z_{n}^{2}=z_{1} z_{2}+z_{2} z_{3}+\cdots+z_{n} z_{1} $$

$$ \text { Prove that the complex number } $$ $$ z=\frac{2+i}{2-i} $$ has modulus equal to 1 , but \(z\) is not an \(n^{\text {th }}-\) root of unity for any positive integer \(n .\)

Prove that $$ \cos ^{2} 10^{\circ}+\cos ^{2} 50^{\circ}+\cos ^{2} 70^{\circ}=\frac{3}{2} $$

On the sides \(A B\) and \(B C\) of triangle \(A B C\) draw squares with centers \(D\) and E such that points \(C\) and \(D\) lie on the same side of line \(A B\) and points \(A\) and \(E\) lie opposite sides of line \(B C .\) Prove that the angle between lines \(A C\) and \(D E\) is equal to \(45^{\circ}\).

Let \(G\) be the centroid of triangle \(A B C\) and let \(R_{1}, R_{2}, R_{3}\) be the circumradii of triangles \(G B C, G C A, G A B\), respectively. Then $$ R_{1}+R_{2}+R_{3} \geq 3 R $$ where \(R\) is the circumradius of triangle \(A B C .\)

$$ \text { Let } f=x^{2}+a x+b \text { be a quadratic polynomial with complex coefficients } $$ $$ \text { with both roots having modulus 1. Prove that } f=x^{2}+|a| x+|b| \text { has the same property. } $$

Let \(z_{1}, z_{2}\) be complex numbers with \(\left|z_{1}\right|=\left|z_{2}\right|=1 .\) Prove that $$ \left|z_{1}+1\right|+\left|z_{2}+1\right|+\left|z_{1} z_{2}+1\right| \geq 2 $$

Let \(A, B, C\), be fixed points in the plane. A man starts from a certain point \(P_{0}\) and walks directly to A. At A he turns by \(60^{\circ}\) to the left and walks to \(P_{1}\) such that \(P_{0} A=A P_{1} .\) After he performs the same action 1986 times successively around points \(A, B, C, A, B, C, \ldots\), he returns to the starting point. Prove that \(A B C\) is an equilateral triangle, and that the vertices \(A, B, C\), are arranged counterclockwise. \(\left(27^{\text {th }} \mathrm{IMO}\right)\)

The following identity holds: $$ \left(\begin{array}{c} n \\ m \end{array}\right)+\left(\begin{array}{c} n \\ m+p \end{array}\right)+\left(\begin{array}{c} n \\ m+2 p \end{array}\right)+\cdots=\frac{2^{n}}{p} \sum_{k=0}^{p-1}\left(\cos \frac{k \pi}{p}\right)^{n} \cos \frac{(n-2 m) k \pi}{p} $$

Let \(n \geq 4\) and let \(a_{1}, a_{2}, \ldots, a_{n}\) be the coordinates of the vertices of a regular polygon. Prove that $$ a_{1} a_{2}+a_{2} a_{3}+\cdots+a_{n} a_{1}=a_{1} a_{3}+a_{2} a_{4}+\cdots+a_{n} a_{2} $$

$$ \text { Let } U_{n} \text { be the set of } n^{\text {th }} \text { -roots of unity. Prove that the following statements } $$ are equivalent: a) there is \(\alpha \in U_{n}\) such that \(1+\alpha \in U_{n}\) b) there is \(\beta \in U_{n}\) such that \(1-\beta \in U_{n} .\)

Solve the equation $$ \cos x+\cos 2 x-\cos 3 x=1 $$

On the sides \(A B\) and \(B C\) of triangles \(A B C\) equilateral triangles \(A B N\) and \(A C M\) are drawn external to the figure. If \(P, Q, R\) are the midpoints of segments \(B C, A M, A N\), respectively, prove that triangle \(P Q R\) is equilateral.

Let \(A B C\) be a triangle and let \(P\) be a point in its interior. Let \(R_{1}, R_{2}, R_{3}\) be the radii of the circumcircles of triangles \(P B C, P C A, P A B\), respectively. Lines \(P A, P B, P C\) intersect sides \(B C, C A, A B\) at \(A_{1}, B_{1}, C_{1}\), respectively. Let $$ k_{1}=\frac{P A_{1}}{A A_{1}}, \quad k_{2}=\frac{P B_{1}}{B B_{1}}, \quad k_{3}=\frac{P C_{1}}{C C_{1}} . $$ Prove that \(k_{1} R_{1}+k_{2} R_{2}+k_{3} R_{3} \geq R\), where \(R\) is the circumradius of triangle \(A B C .\)

$$ \text { Let } a, b \text { be nonzero complex numbers. Prove that the equation } $$$$ a z^{3}+b z^{2}+\bar{b} z+\bar{a}=0 $$ has at least one root with absolute value equal to \(1 .\)

$$ \text { Let } n>0 \text { be an integer and let } z \text { be a complex number such that }|z|=1 \text { . } $$ Prove that $$ n|1+z|+\left|1+z^{2}\right|+\left|1+z^{3}\right|+\cdots+\left|1+z^{2 n}\right|+\left|1+z^{2 n+1}\right| \geq 2 n $$

Let \(a, n\) be integers and let \(p\) be prime such that \(p>|a|+1 .\) Prove that the polynomial \(f(x)=x^{n}+a x+p\) cannot be represented as a product of two nonconstant polynomials with integer coefficients.

Consider the integers \(a_{n}, b_{n}, c_{n}\), where $$ \begin{aligned} &a_{n}=\left(\begin{array}{l} n \\ 0 \end{array}\right)+\left(\begin{array}{l} n \\ 3 \end{array}\right)+\left(\begin{array}{l} n \\ 6 \end{array}\right)+\cdots, \\ &b_{n}=\left(\begin{array}{l} n \\ 1 \end{array}\right)+\left(\begin{array}{l} n \\ 4 \end{array}\right)+\left(\begin{array}{l} n \\ 7 \end{array}\right)+\cdots, \\ &c_{n}=\left(\begin{array}{l} n \\ 2 \end{array}\right)+\left(\begin{array}{l} n \\ 5 \end{array}\right)+\left(\begin{array}{l} n \\ 8 \end{array}\right)+\cdots . \end{aligned} $$ Show that: 1) \(a_{n}^{3}+b_{n}^{3}+c_{n}^{3}-3 a_{n} b_{n} c_{n}=2^{n}\). 2) \(a_{n}^{2}+b_{n}^{2}+c_{n}^{2}-a_{n} b_{n}-b_{n} c_{n}-c_{n} a_{n}=1\) 3) Two of integers \(a_{n}, b_{n}, c_{n}\) are equal and the third differs by one.

Let \(z_{1}, z_{2}, \ldots, z_{n}\) be distinct complex numbers such that $$ \left|z_{1}\right|=\left|z_{2}\right|=\cdots=\left|z_{n}\right|=1 $$ Consider the statements: a) \(z_{1}, z_{2}, \ldots, z_{n}\) are the coordinates of the vertices of a regular polygon. b) \(z_{1}^{n}+z_{2}^{n}+\cdots+z_{n}^{n}=n(-1)^{n+1} z_{1} z_{2} \cdots z_{n}\) Decide with proof if the implications \(a) \Rightarrow b\) ) and \(b) \Rightarrow a\) ) are true.

$$ \text { Let } n>3 \text { be a positive integer and let } \varepsilon \neq 1 \text { be an } n^{\text {th }} \text { root of unity. } $$ 1) Show that \(|1-\varepsilon|>\frac{2}{n-1}\). 2) If \(k\) is a positive integer such that \(n\) does not divides \(k\), then $$ \left|\sin \frac{k \pi}{n}\right|>\frac{1}{n-1} $$

Compute the sums $$ S=\sum_{k=1}^{n} q^{k} \cdot \cos k x \quad \text { and } \quad T=\sum_{k=1}^{n} q^{k} \cdot \sin k x . $$

Let \(A A^{\prime} B B^{\prime} C C^{\prime}\) be a hexagon inscribed in the circle \(\mathcal{C}(O ; R)\) such that $$ A A^{\prime}=B B^{\prime}=C C^{\prime}=R $$

For any point \(M\) in the plane of triangle \(A B C\) the following inequality holds: $$ A M^{3} \sin A+B M^{3} \sin B+C M^{3} \sin C \geq 6 \cdot M G \cdot \operatorname{area}[A B C] $$ where \(G\) is the centroid of triangle \(A B C .\)