Approx. Difficulty: 8/10 — No final answers provided.
Question 1
Prove that for any integer \(n\),
\[
\sin\bigl((2n+1)x\bigr) \;=\; \sin x \; U_{2n}\bigl(\cos x\bigr),
\]
where \(U_{2n}\) is the Chebyshev polynomial of the second kind.
Use the recursion \(\sin((m+1)x) = 2\cos x\,\sin(mx) - \sin((m-1)x)\) and match it with the definition of \(U_k\).
Question 2
In a triangle with angles \(A,B,C\), prove:
\[
\tan\left(\frac{A}{2}\right)\tan\left(\frac{B}{2}\right)
\;+\; \tan\left(\frac{B}{2}\right)\tan\left(\frac{C}{2}\right)
\;+\; \tan\left(\frac{C}{2}\right)\tan\left(\frac{A}{2}\right)
\;=\; 1.
\]
Use half-angle identities, e.g.
\(\tan\bigl(\frac{A}{2}\bigr)= \sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\),
where \(s=\frac{a+b+c}{2}\).
Use \(\tan(\alpha+\beta)=\frac{\tan\alpha+\tan\beta}{1-\tan\alpha\tan\beta}\). Set \(\alpha=2\theta, \beta=\theta\).
Question 8
Show that
\[
\prod_{k=1}^{n-1} \sin\!\Bigl(\tfrac{k\pi}{n}\Bigr)
\;=\; \frac{n}{2^{\,n-1}}.
\]
Classic product formula. Use roots of unity or the known identity for \(\sin(nx)\).
Question 9
If \(x,y,z\) are angles of a triangle, prove
\(\sin 2x + \sin 2y + \sin 2z \;=\; 4\sin x\sin y\sin z.\)
Set \(z=\pi-(x+y)\). Then \(\sin(2z)=\sin[2\pi-2(x+y)] = \sin(2(x+y))\). Use sum-to-product expansions.
Question 10
Show
\[
\sin^6 \theta + \cos^6 \theta
\;\ge\; \tfrac{3}{8}
\quad\text{for all real }\theta.
\]
Factor \(\sin^6\theta = (\sin^2\theta)^3\). Then use \(\sin^2\theta+\cos^2\theta=1\) and standard inequalities or expansions.
Question 11
In a triangle \(ABC\), prove
\(\cot A + \cot B + \cot C \;\le\; \frac{R}{r}\),
where \(R\) is the circumradius and \(r\) is the inradius.
Use \(\cot A = \frac{b^2 + c^2 - a^2}{4\Delta}\) and Euler's inequality \(R \ge 2r\). Carefully relate sides to \(R\) and \(r\).
Question 12
Evaluate in closed form:
\[
\sum_{k=1}^{n} (-1)^{k-1}\cos(2k\,x).
\]
Treat as the real part of \(\sum_{k=1}^n (-1)^{k-1} e^{i2kx}\). Factor the geometric series with ratio \(-\,e^{2ix}\).
Question 13
Let \(\alpha,\beta\in\bigl(0,\tfrac{\pi}{2}\bigr)\). Prove
\(\sin(\alpha+\beta) + \sin(\alpha-\beta) \;\ge\; 2\sin\alpha\cos\beta.\)
Expand \(\sin(\alpha+\beta)=\sin\alpha\cos\beta + \cos\alpha\sin\beta\) and \(\sin(\alpha-\beta)=\sin\alpha\cos\beta-\cos\alpha\sin\beta\).
Question 14
If \(0 < x < \tfrac{\pi}{2}\), prove
\(\cot x > 1 + \frac{x}{\pi-2x}\).
Compare \(\cot x\) with a rational function in \(x\). A direct approach involves bounding integrals or constructing a tangent-based inequality.
Question 15
Prove
\(\cos^3 A + \cos^3 B + \cos^3 C \;\le\; \tfrac{3}{4}\)
for acute angles \(A,B,C\) with \(A+B+C=\pi\).
Use \(\cos(A+B+C) = \cos\pi = -1\). Express one cosine in terms of the others, or apply Jensen’s inequality on \(\cos^3 x\) in a suitable domain.
Question 16
Show
\[
\sin\bigl(2x\bigr)\sin\bigl(4x\bigr)\sin\bigl(6x\bigr)
\;=\;\frac{\sin(6x)-3\,\sin(2x)+4\,\sin^3(2x)}{4},
\]
assuming \(\sin x \neq 0\).
Expand \(\sin(4x)=2\sin(2x)\cos(2x)\). Then multiply with \(\sin(6x)\) and simplify carefully.
Question 17
In a triangle \(ABC\), let sides be \(a,b,c\). Prove
\(\frac{a+b}{a+c} = \frac{\sin\bigl(\tfrac{A+B}{2}\bigr)}{\sin\bigl(\tfrac{A+C}{2}\bigr)}.\)
Use the sine rule and half-angle identities. Also note \(A+B=\pi-C\).
Question 18
Prove or disprove: If \(x+y+z=\pi\), then
\(\csc^2 x + \csc^2 y + \csc^2 z \;\ge\; 4.\)
Jensen’s inequality on \(f(t)=\csc^2 t\) for \(t\in(0,\pi)\), or transform \(\csc^2 t = 1+\cot^2 t\).
Question 19
Prove that
\[
\sin(20^\circ)\,\sin(40^\circ)\,\sin(60^\circ)\,\sin(80^\circ)
\;=\;
\frac{3\sqrt{3}}{16}.
\]
A useful approach is to note \(\sin 60^\circ = \tfrac{\sqrt{3}}{2}\).
Then pair angles cleverly: \(\sin(20^\circ)\,\sin(80^\circ)\) and
\(\sin(40^\circ)\,\sin(60^\circ)\).
Product-to-sum identities or known “special angle” transformations
can simplify each pair.
Question 20
Let \(f(\theta)=\sin^6\theta+\cos^6\theta\). Show
\(\min f(\theta)=\tfrac{3}{4}\) and \(\max f(\theta)=1.\)
Expand \(\sin^6\theta+\cos^6\theta\) in terms of \(\sin^2\theta+\cos^2\theta\) and \(\sin^2(2\theta)\). Find critical points.
Question 21
In \(\triangle ABC\) with sides \(a,b,c\), prove
\(\frac{a+b}{a+c} = \frac{\tan\bigl(\tfrac{B}{2}\bigr)}{\tan\bigl(\tfrac{C}{2}\bigr)}.\)
Similar to a known half-angle relation. Use \(\tan\bigl(\tfrac{B}{2}\bigr)=\sqrt{\frac{(s-b)(s-c)}{s(s-a)}}\).
Question 22
Show
\(\sin(3x)+\sin(5x)+\sin(7x) = \sin x\,[\,4\cos(3x)+4\cos(5x)\,].\)
Use sum-to-product. Try grouping \(\sin(3x)+\sin(7x)\) first, then factor out \(\sin x\).
\(1+\tan^2 x=\sec^2 x\). So the product is \(\prod \sec^2(\cdots)\). Look for known product identities of \(\sec\).
Question 25
Let \(\alpha,\beta,\gamma\) be angles in a triangle. Prove
\(\tan^2\alpha + \tan^2\beta + \tan^2\gamma \;\ge\; \tan\alpha\,\tan\beta + \tan\beta\,\tan\gamma + \tan\gamma\,\tan\alpha.\)
Since \(\alpha+\beta+\gamma=\pi\), we have \(\tan(\alpha+\beta)=-\tan\gamma\). Possibly rearrange or interpret as a symmetrical inequality in tangents.
Question 26
Show that the Chebyshev polynomial of the first kind \(T_n(x)=\cos(n\arccos x)\) satisfies
\[
T_n(x) = 2x\,T_{n-1}(x) - T_{n-2}(x),
\]
with \(T_0(x)=1,\, T_1(x)=x\).
Use the formula \(\cos(nx)=2\cos x\,\cos((n-1)x)-\cos((n-2)x)\).
Treat \(\cos\bigl(k-\frac12\bigr)x\) as the real part of \(e^{i(k-\frac12)x}\). Sum the geometric series.
Question 32
If \(|z|=1\), investigate whether
\(\bigl|1 + z + z^2 + \dots + z^n\bigr| \ge 1\)
always holds. Prove or provide a counterexample.
Geometric series sum. Check if certain roots of unity violate or satisfy the inequality. Possibly find \(z^m=-1\).
Question 33
Show that \(\sin x \;\ge\; \frac{2x}{\pi}\) for \(x\in\bigl[0,\frac{\pi}{2}\bigr]\) (no calculus approach).
Compare chord vs arc in the unit circle or consider a known geometric argument with sector areas.
Question 34
In \(\triangle ABC\), let \(I\) be the incenter with inradius \(r\). Prove
\(AI = \frac{r}{\sin(\tfrac{A}{2})}\).
\(I\) is where angle bisectors meet. Use right triangles formed by tangents from \(I\) and half-angle formulas.
Question 35
Show
\(\sin^4 x + \cos^4 x + \sin^4(2x) + \cos^4(2x) \;\ge\; 2.\)
Use \(\sin^4 t+\cos^4 t = 1 - \tfrac12\sin^2(2t)\). Summation for \(t=x\) and \(t=2x\). Look for minimal sum.
Question 36
Let \(\alpha+\beta+\gamma=\pi\). Prove
\(\sin 2\alpha+\sin 2\beta+\sin 2\gamma \;\le\; \tfrac{3\sqrt{3}}{2}.\)
Similar to bounding \(\sin A+\sin B+\sin C\). Try Jensen on \(\sin 2x\), or a known maximum at \(\alpha=\beta=\gamma=\tfrac{\pi}{3}\).
Question 37
If \(0 2.\)
Rearrangement or AM-GM on pairs \(\sin a,\sin b\) and \(\cos a,\cos b\). Also \(\sin a < \sin b\), \(\cos a > \cos b\).
Question 38
Prove
\(\displaystyle \prod_{k=1}^{n-1} \sin\Bigl(\frac{k\pi}{n}\Bigr) = \frac{n}{2^{\,n-1}}\)
for integer \(n>1\). (This is a classic result; re-verify.)
Use the zeroes of \(z^n = 1\) (roots of unity) or the known \(\sin\)-product approach. It's essentially the same as Q8, restated.
Question 39
In \(\triangle ABC\), let \(O\) be circumcenter, \(H\) be orthocenter, and let \(r\) be the circumradius. Prove
\[
OH^2 = 9r^2 - (a^2 + b^2 + c^2).
\]
Euler line relation \(OH^2=9OG^2\), or place in coordinates with \(O\) at origin. Use side-length expressions in terms of \(R\).
Question 40
Show
\(\sum_{k=1}^{n-1} \csc^2\Bigl(\frac{k\pi}{n}\Bigr) = \frac{n^2-1}{3}.\)
Known sum identity. Use \(\csc^2 x = 1 + \cot^2 x\). Summation over equally spaced angles in \((0,\pi)\).
Question 41
Let \(x,y,z\) be angles of a triangle. Prove
\(\cot x + \cot y + \cot z \;\ge\; \sqrt{\cot x\,\cot y + \cot y\,\cot z + \cot z\,\cot x}\).
Possibly apply Nesbitt-like inequality in trig form or transform via half-angle substitution.
Similar approach to sums of cosines at special angles. Use exponentials or known triple-angle expansions.
Question 43
Let \(a,b,c\) be sides of a triangle. Prove:
\[
\frac{a}{b-c} + \frac{b}{c-a} + \frac{c}{a-b}
= \frac{(a+b+c)(a^2 + b^2 + c^2 - bc - ca - ab)}{(b-c)(c-a)(a-b)}.
\]
Combine fractions over the common denominator \((b-c)(c-a)(a-b)\). Then simplify carefully.
Question 44
For \(n\) odd, prove
\(\displaystyle \prod_{k=1}^{\frac{n-1}{2}} \sin\Bigl(\tfrac{k\pi}{n}\Bigr) = \frac{\sqrt{n}}{2^{\frac{n-1}{2}}}.\)
Another standard result about half of the angles for odd \(n\).
Use \(\cos^3\alpha = \frac{1}{4}[\cos(3\alpha)+3\cos\alpha]\). Factor the sums.
Question 56
In a cyclic quadrilateral \(ABCD\), \(\angle A+\angle C=180^\circ\). Show
\[
AC\cdot BD = AB\cdot CD + BC\cdot AD
\]
if and only if \(ABCD\) is a right trapezoid.
From Ptolemy's theorem: \(AC\cdot BD = AB\cdot CD + BC\cdot AD\). The right trapezoid condition emerges from angle constraints.
Question 57
If \(\cos x + \cos 2x + \dots + \cos nx=0\) and
\(\sin x + \sin 2x + \dots + \sin nx=0\), prove
\(\cos(kx)+\cos((n-k)x)=0\) for \(k=1,\dots,n-1.\)
Real/imag parts of \(\sum_{m=1}^n e^{imx}=0\). Then partial sums must pair up to zero.
Question 58
In \(\triangle ABC\) with \(A=60^\circ\), prove
\(b^2 + c^2 - bc = a^2.\)
Law of Cosines: \(a^2=b^2+c^2-2bc\cos A\). Then \(\cos(60^\circ)=\tfrac12\).
Question 59
For \(|z|=1\), consider
\(\bigl|1 - z + z^2 - z^3 + \dots + z^{2n}\bigr|\). Show whether it is
always \(\ge 1\) or provide a counterexample.
Sum \(\sum_{k=0}^{2n} (-1)^k z^k\). This might fail for certain roots of unity. Check minimal absolute value.
Law of Cosines: \(a^2=b^2+c^2-2bc\cos A\). Solve for \(\cos A\).
Question 61
For \(|x|<\tfrac{\pi}{2}\), prove
\(\frac{\sin x}{x} \;\le\; \frac{\tan x}{x} \;\le\; 1+\frac{x^2}{2}\),
with no derivatives.
Use \(\sin x \le x \le \tan x\) in \((0,\tfrac{\pi}{2})\). Then bound \(\tan x\) from above by a parabola or geometric argument.
Question 62
For real \(\theta\), prove
\(\tan \theta + \sec \theta
= \csc\bigl(\tfrac{\pi}{2}-\theta\bigr) + \cot\bigl(\tfrac{\pi}{2}-\theta\bigr).\)
Use cofunction identities:
\(\csc(\tfrac{\pi}{2}-\theta)=\sec\theta\),
\(\cot(\tfrac{\pi}{2}-\theta)=\tan\theta\).
Question 63
Show
\[
\sum_{k=0}^{n} \binom{n}{k}\sin\!\bigl((n-2k)x\bigr)
= \frac{\sin\!\bigl((n+1)x\bigr)}{\sin x}.
\]
Use binomial expansions of \((e^{ix}+ e^{-ix})^n\). Then isolate imaginary parts.
Question 64
In an acute triangle \(ABC\), let \(H\) be the orthocenter. Show
\(\angle BHC = 180^\circ - A.\)
Use the fact that \(BH\) and \(CH\) are altars (altitudes). Also, \(AH\) is altitude, so \(\angle BHC\) subtends the same arc as \(\angle BAC\) in the circumcircle.
Question 65
Show
\(\displaystyle \sum_{k=1}^n (-1)^{k-1}\sin(2kx)
= \frac12 \Bigl[\cot x - \cot\bigl((2n+1)x\bigr)\Bigr].\)
Rewrite \(\sin(2kx)\) in exponential form. Factor out \((-1)^{k-1}\) and sum the geometric series. Then interpret in real/imag parts.
Question 66
Let \(z=\cos x + i\sin x\). Show
\(\frac{1}{\sin x}=\frac{z-z^{-1}}{2i},\;\frac{1}{\cos x}=\frac{z+z^{-1}}{2}\),
assuming \(\sin x,\cos x\neq 0\).
Recall \(\sin x=\frac{z-z^{-1}}{2i}\) and \(\cos x=\frac{z+z^{-1}}{2}\). Then invert them (provided neither is zero).
Question 67
Prove the reflection formula for cotangent:
\(\cot(\pi - x) = -\cot x.\)
Use \(\tan(\pi - x)=-\tan x\). Then invert to get \(\cot(\pi - x)=-\cot x\).
Question 68
In \(\triangle ABC\), if \(AB=2BC\) and \(\angle A=2\angle B\), prove \(AC=AB\).
Law of Sines: \(AB:BC:AC=\sin A:\sin B:\sin C\). Impose \(AB=2BC\) and \(A=2B\). Solve for \(\sin C\).
Question 69
Show
\(\displaystyle \sum_{k=1}^{n-1} \frac{1}{\sin^2\!\bigl(\tfrac{k\pi}{2n+1}\bigr)}
= n(2n+1).\)
Classic result. Use partial fraction expansions of \(\csc^2 x\) or consider the polynomial \(\sin((2n+1)x)\).
Question 70
Prove
\(\tan(75^\circ) + \tan(15^\circ) = 4.\)
\(\tan(75^\circ)=\tan(45^\circ+30^\circ)\), \(\tan(15^\circ)=\tan(45^\circ-30^\circ)\). Then sum.
Question 71
In \(\triangle ABC\) with sides \(a,b,c\), prove
\(\frac{a}{b+c-a} + \frac{b}{c+a-b} + \frac{c}{a+b-c} \;\ge\; 6.\)
Use Ravi substitution \(a=y+z,b=z+x,c=x+y\). Then each denominator simplifies, apply an inequality like AM-HM or Nesbitt-like approach.
Question 72
If \(|\sin x|\le|\cos x|\), prove
\(\sin(3x)\le \cos(3x)\).
Prove
\(\sin x + \sin 2x + \dots + \sin nx \;\le\; \frac{1}{\sin(\tfrac{x}{2})}\),
for \(0
Use the standard sum formula
\(\sin x + \dots + \sin nx = \frac{\sin(\frac{n x}{2})\sin(\frac{(n+1)x}{2})}{\sin(\frac{x}{2})}\). Then bound it.
Question 74
Let \(P(x)=\sin\bigl(n\arcsin x\bigr)\). Show \(P(x)\) is a polynomial of degree \(n\) in \(x\), and find the leading coefficient.
Expand via \(\sin(n\theta)= \Im\bigl(e^{i n\theta}\bigr)\). Then \(\theta=\arcsin x\). The leading term in \(\sin^n\theta\) is \(x^n\) times a constant.
Question 75
For angles \(\alpha,\beta\in(0,\pi)\), show
\(\sec\alpha + \sec\beta
\;\ge\; 4\sec\!\Bigl(\frac{\alpha + \beta}{2}\Bigr).\)
Check the convexity of \(\sec x\) on \((0,\pi/2)\cup(\pi/2,\pi)\). Alternatively, attempt direct expansions or transformations.
A famous triple-angle product. Often tackled via polynomial roots or product-to-sum repeatedly.
Question 77
In \(\triangle ABC\) with \(A=60^\circ\), prove
\(\frac{b+c}{a} + \frac{a}{b+c} = 2\cos B\,\cos C + 2.\)
Law of Cosines: \(a^2=b^2+c^2-bc\). Then express \(\cos B,\cos C\) in terms of sides and simplify.
Question 78
For integer \(n\ge 1\), prove
\(\displaystyle \prod_{k=1}^{n} \bigl(1+2\cos(\tfrac{2k\pi}{2n+1})\bigr) = 2n+1.\)
A known root-of-unity product. Factor \(x^{2n+1}=1\) and examine terms like \(x + 1 + x^{-1} = 2\cos\theta + 1\).
Question 79
In a cyclic quadrilateral \(ABCD\), Ptolemy's theorem holds. Extend it to show:
\[
AC \cdot BD + AB \cdot CD \;\ge\; BC\cdot AD + (AB + CD)^2,
\]
or argue if there’s an error.
Investigate standard Ptolemy: \(AC\cdot BD=AB\cdot CD+BC\cdot AD\). Then see how to incorporate \((AB+CD)^2\). Possibly a stronger or weaker statement.
Question 80
Show
\(\displaystyle \sum_{k=1}^{n-1}\cot^2\Bigl(\tfrac{k\pi}{2n}\Bigr)
= \frac{(n-1)(2n-1)}{3}.\)
Rewrite \(\cot^2 x = \csc^2 x -1\). Then apply known sum of \(\csc^2(\tfrac{k\pi}{2n})\).
Question 81
For real \(x\), show
\(\displaystyle
\frac{\sin^3 x}{\cos x - \sin x}
\;+\; \frac{\cos^3 x}{\sin x - \cos x}
\;=\;\sin x + \cos x.
\)
Combine fractions over \((\cos x - \sin x)(\sin x - \cos x)=-(\sin x + \cos x)^2\). Expand carefully.
Question 82
In \(\triangle ABC\), show
\(\tan(A+B)+\tan(B+C)+\tan(C+A) = \tan A\,\tan B\,\tan C.\)
\(A+B+C=\pi\implies \tan(A+B)=-\tan C\). Then sum up the tangents carefully.
Question 83
For \(x\in(0,\tfrac{\pi}{2})\), prove
\(x\cot x \;\ge\; 1 - \tfrac{2x^2}{\pi^2}\),
without derivatives.
Approximate \(\cot x\) or compare areas. Possibly use series expansions or a bounding approach akin to \(\sin x\le x\).
Question 84
Let \(m\) be an integer. Show
\(\sin(mx) = 2^{m-1}\displaystyle \prod_{k=0}^{m-1} \sin\Bigl(x + \tfrac{k\pi}{m}\Bigr)\)
if \(m\) is even (analogous expression if \(m\) is odd).
Factor \(\sin(mx)\) using \(\sin(mx)=\frac{1}{2i}(e^{imx}-e^{-imx})\). Then factor out each root.
Question 85
In a cyclic hexagon \(ABCDEF\), show
\[
AB\cdot CD\cdot EF + BC\cdot DE\cdot FA
\;\ge\; AC\cdot CE\cdot EA + BD\cdot DF\cdot FB.
\]
Possible extension of Ptolemy or a direct chord-length inequality. Decompose the hexagon or use complex-plane points on the unit circle.
Question 86
If \(|z|=1\) and
\(\sum_{k=1}^{n} z^k = 0\), prove
\(\sum_{k=1}^{n} k\,z^k=0.\)
Differentiate \(\frac{z^{n+1}-z}{z-1} =0\) w.r.t. \(z\) in a formal sense or manipulate discrete sums.
Use \(\tan 60^\circ=\sqrt{3}\). Pair \(\tan(60^\circ -x)\) and \(\tan(60^\circ +x)\). Or transform to sines/cosines.
Question 99
If \(A+B+C=\pi\) in \(\triangle ABC\), show
\[
\sum_{\text{cyc}} \frac{\sin A}{\sin B\,\sin C}
= \sum_{\text{cyc}} \frac{a}{bc}
= \frac{1}{r},
\]
where \(r\) is the inradius.
Use \(a=2R\sin A\), \(\Delta=\frac12 bc\sin A\), and \(r=\frac{\Delta}{s}\). Carefully manipulate the sums.
Question 100
Let \(x_1,x_2,\dots,x_{2n}\) be real angles with \(x_1+\dots+x_{2n}=\pi\). Prove
\(\displaystyle \prod_{k=1}^{2n}\sin x_k \;\le\; \frac{1}{2^{2n-1}}.\)
By AM-GM or Jensen, distributing angles equally maximizes product \(\sin(\tfrac{\pi}{2n})\).