Questions tagged [elementary-proofs]

For questions related to 'elementary' proofs in a technical sense, which has nothing to do with the difficulty of the argument or result. A typical example would be 'elementary' proofs of the Prime Number Theorem, which avoid complex analysis. The tag is however not limited to this particular notion of 'elementary.'

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3
votes
1answer
340 views

Two conjectural infinite series for $\pi$

I am looking for a proofs of the following two claims: Claim 1. $$\frac{2\pi}{\sqrt{3}}=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{\Omega_1(n)}}{n}$$ where $\Omega_1(n)$ is the number of prime ...
0
votes
0answers
44 views

Pseudo-Droz-Farny circles

I would like to present a construction of 2 circles. These 2 circles are somewhat similar in appearance to the well known Droz-Farny circles that can be drawn for every isogonal conjugate pairs of ...
8
votes
1answer
294 views

A conjectural infinite series for $\frac{\pi^2}{5\sqrt{5}}$

I am looking for a proof of the following claim: First define the function $\chi(n)$ as follows: $$\chi(n)=\begin{cases}1, & \text{if }n \equiv \pm 1 \pmod{10} \\ -1, & \text{if }n \equiv \pm ...
6
votes
2answers
380 views

An infinite series involving the mod-parity of Euler's totient function

Can you prove or disprove the following claim: First, define the function $\xi(n)$ as follows: $$\xi(n)=\begin{cases}-1, & \text{if }\varphi(n) \equiv 0 \pmod{4} \\ 1, & \text{if }\varphi(n) \...
8
votes
1answer
496 views

An infinite series involving harmonic numbers

I am looking for a proof of the following claim: Let $H_n$ be the nth harmonic number. Then, $$\frac{\pi^2}{12}=\ln^22+\displaystyle\sum_{n=1}^{\infty}\frac{H_n}{n(n+1) \cdot 2^n}$$ The SageMath ...
4
votes
1answer
184 views

Lemoine-Lozada circles

I made some rookie attempt to define the 4th Lemoine circle recently. The alternative name for this circle was suggested yesterday. Further investigation revealed a family of circles associated with ...
3
votes
0answers
141 views

Four infinite series involving Riemann zeta function

Can you provide a proof for at least one of the claims given below? It is known that $\pi=\displaystyle\sum_{n=1}^{\infty}\frac{3^n-1}{4^n} \cdot \zeta(n+1)$ where $\zeta$ denotes Riemann zeta ...
12
votes
1answer
3k views

The 4th vertex of a triangle?

I was immensely surprised and amused by the idea of the fourth side of a triangle that was introduced by B.F.Sherman in 1993. 'Sherman's Fourth Side of a Triangle' by Paul Yiu is available here. ...
5
votes
2answers
260 views

The 4th Lemoine circle

The first and second Lemoine circles are well-known to geometers. According to this article the third Lemoine circle has been first discovered by Jean-Pierre Ehrmann in 2002. It is worth noting that ...
5
votes
3answers
720 views

An infinite series that converges to $\frac{\sqrt{3}\pi}{24}$

Can you prove or disprove the following claim: Claim: $$\frac{\sqrt{3} \pi}{24}=\displaystyle\sum_{n=0}^{\infty}\frac{1}{(6n+1)(6n+5)}$$ The SageMath cell that demonstrates this claim can be found ...
12
votes
1answer
826 views

Nonstandard proofs of the fundamental theorem of arithmetic

Thirty or so years ago, someone showed me a clever proof of the Fundamental Theorem of Arithmetic that did not make use of the lemma "If $p\mid ab$ then $p\mid a$ or $p\mid b$". I'm unable ...
1
vote
1answer
47 views

Collinearity in tangential pentagon

I am looking for a proof of the following claim: Given tangential pentagon. Touching point of the incircle and the side of the pentagon,the vertex opposite to that side and the intersection point of ...
3
votes
1answer
114 views

The product of the lengths of two line segments that belong to Newton line

I am looking for the proof of the following claim: Consider a family of bicentric quadrilaterals with the same inradius length and the same distance between incenter and circumcenter. Denote by $P$ ...
5
votes
1answer
252 views

The square root of natural number expressed by an infinite series

Can you prove or disprove the following claim: Let $U(n,P,Q)$ be the nth generalized Lucas number of the first kind and let $m$ be a natural number. Then, $$\sqrt{m}=1+\displaystyle\sum_{n=1}^{\infty}...
4
votes
1answer
432 views

The constant $\pi$ expressed by an infinite series

I am looking for the proof of the following claim: First, define the function $\operatorname{sgn_1}(n)$ as follows: $$\operatorname{sgn_1}(n)=\begin{cases} -1 \quad \text{if } n \neq 3 \text{ and } n \...
4
votes
1answer
318 views

The constant $e$ represented by an infinite series

In this Wikipedia article the constant $\pi$ is represented by the following infinite series: $$\pi=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}-\frac{1}{5}+\frac{1}{6}+\frac{1}{7}+\frac{1}{8}+\frac{1}{9}-\...
4
votes
2answers
163 views

Six conelliptic points

Can you prove the following proposition: Proposition. Given an arbitrary triangle $\triangle ABC$. Let $D,E,F$ be the points on the sides $AB$,$BC$ and $AC$ respectively , such that $\frac{AB}{DA}=\...
3
votes
1answer
615 views

Three circles meet at a point

I am looking for the proof of the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with circumcenter $O$. Let $A',B',C'$ be a reflection points of the points $A,B,C$ ...
2
votes
0answers
60 views

Principal diagonals of octagon meet in a single point

Can you provide a proof for the following claim: Claim. Given octagon circumscribed about an ellipse. If the vertices of the octagon lie on another ellipse then its principal diagonals meet in a ...
1
vote
0answers
66 views

How to estimate the highest power of 2 in the partial sum of 2-adic $\log(-1)$ (i.e. $\sum_{i=1}^n\frac{2^i}{i}$)?

The estimate I wanna get is $$v_2(\sum_{i=1}^n\frac{2^i}{i})\geq\min_{t\geq n+1}\{t-v_2(t)\}\tag{*}$$ where $v_2$ is the 2-adic valuation, that is the highest power of 2 defined on $\mathbb{Q}$. Set $$...
1
vote
1answer
94 views

Generalizing Bottema's theorem

Can you provide another proof for the claim given below? Claim. In any triangle $\triangle ABC$ construct triangles $\triangle ACE$ and $\triangle BDC$ on sides $AC$ and $BC$ such that $\frac{AE}{AC}...
3
votes
1answer
87 views

Equal sums of line segments

I would like to see a proof of the following Claim. Let $A_1,A_2,A_3,A_4,A_5$ be vertices of bicentric pentagon. Let $B_1$ be the intersection point of $A_1A_3$ and $A_2A_5$, $B_2$ the intersection ...
6
votes
3answers
326 views

Necessary and sufficient condition for quadrilateral to be cyclic

Can you provide a proof for the following proposition: Proposition. Given any quadrilateral $ABCD$. Let $P,Q,R,S$ be nine-point centers of triangles $\triangle ABD$,$\triangle ABC$,$\triangle BCD$ ...
6
votes
2answers
793 views

Three circles intersecting at one point

Can you provide a proof for the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with nine-point center $N$ and circumcenter $O$. Let $A',B',C'$ be a reflection points ...
2
votes
1answer
60 views

Collinearity of three significant points of bicentric pentagon

Can you provide a proof for the following claim? Claim. Given bicentric pentagon. Consider the triangle whose sides are two diagonals drawn from the same vertex and side of pentagon opposite from ...
2
votes
0answers
84 views

Polyhedron - sphere intersection

newbie here. I'd like to ask you, if you know some brief, but somewhat solid proof of a convex polyhedron and a sphere centered at one of its vertices (with small enough radius, so it intersects only ...
11
votes
3answers
989 views

What is the limit of $a (n + 1) / a (n)$?

Let $a(n) = f(n,n)$ where $f(m,n) = 1$ if $m < 2 $ or $ n < 2$ and $f(m,n) = f(m-1,n-1) + f(m-1,n-2) + 2 f(m-2,n-1)$ otherwise. What is the limit of $a(n + 1) / a (n)$? $(2.71...)$
4
votes
1answer
147 views

Collinearity in bicentric polygons

Can you provide a proofs for the following two claims? Claim 1. The circumcenter, the incenter, and the intersection of the principal diagonals in a bicentric even-sided polygon are collinear. Claim ...
6
votes
1answer
153 views

Necessary and sufficient condition for tangential polygon to be cyclic

Can you prove or disprove the following claim? Claim. Let $A_1,A_2, \ldots ,A_n$ be the vertices of an $n$-sided tangential polygon and let $B_1,B_2, \ldots ,B_n$ be the contact points of the ...
2
votes
1answer
128 views

Stability estimates on quotients of the form $ \frac{\prod_{j=1}^n a_j}{\prod_{j=1}^n b_j} $

Suppose that $a_j,b_j \in \mathbb C$ are complex numbers, $j=1,\dots,n$, with the property that $|a_j|,|b_j| \geq c > d >0$ where $c,d$ are positive real numbers. I'm interested in the stability ...
4
votes
1answer
185 views

Point of concurrency [closed]

I am looking for the proof of the following claim: Claim: Let $\triangle ABC$ be an arbitrary triangle, $D$ its nine-point center and $E,F,G$ are the nine-point centers of the triangles $\triangle ...
9
votes
2answers
1k views

A tricky integral to evaluate

I came across this integral in some work. So, I would like to ask: QUESTION. Can you evaluate this integral with proofs? $$\int_0^1\frac{\log x\cdot\log(x+2)}{x+1}\,dx.$$
1
vote
1answer
114 views

A binomial convolution of Catalan numbers vs "utterly odd numbers"

An integer is called utterly odd if the terminal string of $1$’s in its binary representation has odd length. A number $2^{k+1}m+(2^k-1)$ where $m\geq0$ (every non-negative integer has this form) is ...
28
votes
1answer
894 views

Functional-analytic proof of the existence of non-symmetric random variables with vanishing odd moments

It is known that a random variable $X$ which is symmetric about $0$ (i.e $X$ and $-X$ have the same distribution) must have all its odd moments (when they exist!) equal to zero. The converse is a ...
1
vote
1answer
188 views

A generalization of Harcourt's theorem

This question is closely related to my previous question. Can you prove the claim given below? The following claim is a conjectured generalization of Harcourt's theorem. Claim. Let $A_1,A_2 \ldots ...
1
vote
1answer
222 views

A formula for the area of bicentric quadrilateral

Can you provide a proof for the claim given below? The following claim is inspired by Harcourt's theorem and can be seen as its generalization to quadrilaterals. Claim. Given bicentric quadrilateral $...
1
vote
1answer
243 views

A new perspective on Lehmer's totient problem

Lehmer's totient problem asks if there are any composite integers $n$ with $\phi(n) \ | \ n-1$. It is known that any such $n$ must be odd. It must also be a charmichael number. Assume $n=4m+3$ then $\...
2
votes
1answer
145 views

The centroid, the first and second Napoleon points and $X(930)$ lie on a circle

Can you provide an elementary proof for the claim given below? Preliminary definitions: $X(110)=$ focus of Kiepert parabola. $X(137)=X(110)$ of orthic triangle . $X(930)=$ anticomplement of $X(137)$ . ...
2
votes
1answer
128 views

Four concyclic triangle centers

Can you prove the claim given below? Inspired by Lester's theorem I have formulated the following claim: Claim. Given any scalene triangle $\triangle ABC$ . Let $D$ be the reflection of incenter in ...
51
votes
5answers
7k views

What is the simplest proof that the density of primes goes to zero?

By density of primes, I mean the proportion of integers between $1$ and $x$ which are prime. The prime number theorem says that this is asymptotically $1/\log(x)$. I want something much weaker, namely ...
1
vote
1answer
311 views

A generalization of Napoleon's theorem

Can you provide a proof for the following proposition? Proposition. Given an arbitrary $\triangle ABC$. The $\triangle AEB$, $\triangle BFC$ and $\triangle CDA$ are constructed on the sides of the $...
2
votes
2answers
166 views

Six concyclic points

Can you provide a proof for the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with excenters $J_A$,$J_B$ and $J_C$ . Let $G$ be the orthogonal projection of the $...
2
votes
2answers
158 views

Four concyclic points inside bicentric quadrilateral

Can you provide a proof for the following proposition: Proposition. Let quadrilateral $ABCD$ be inscribed into a circle with center $O$ and circumscribed around a circle with center $I$. Let $X$ be a ...
14
votes
2answers
606 views

Euclid-style proof of Dirichlet’s theorem on primes in certain arithmetic progression

The well-known theorem of Dirichlet on primes in arithmetic progression states that given coprime natural numbers $a\le q$, there are infinitely many prime numbers congruent to $a\pmod q$. The ...
14
votes
4answers
684 views

Six points on an ellipse

Can you prove the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with centroid $G$. Let $D,E,F$ be the points on the sides $AC$,$AB$ and $BC$ respectively , such ...
11
votes
1answer
724 views

Intersection point of three circles

Can you provide a proof for the following proposition: Proposition. Let $\triangle ABC$ be an arbitrary triangle with orthocenter $H$. Let $D,E,F$ be a midpoints of the $AB$,$BC$ and $AC$ , ...
13
votes
6answers
2k views

Alternative proofs sought after for a certain identity

Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so QUESTION. can you provide another verification for the problem below? Problem. Prove ...
-4
votes
2answers
192 views

An elementary-looking integral inequality

This might seem a bit easy but I still like to ask it for pedagogical reasons. QUESTION. Is this inequality true for non-negative integers $n$? $$\frac{\pi}2\int_0^1x^n\sin\left(\frac{\pi}2x\right)dx\...
2
votes
0answers
127 views

Asking for a combinatorial proof of a binomial-sum

QUESTION. Is there a combinatorial proof of the below identity? $$\sum_{k=0}^{n-1}\frac{2^{2k}}{2k+1}\frac{\binom{2n}n}{\binom{2k}k}=2^{2n}-\binom{2n}n.$$ REMARK. There are many other proofs (...
3
votes
2answers
232 views

Is there a combinatorial reason for variable-independence of this binomial-coefficient identity?

Consider the following identity $$\sum_{n=0}^{R-t}\binom{n+\ell}n\binom{R-\ell-n}{R-t-n}=\binom{R+1}{t+1}.\tag1$$ It is relatively easy to give an algebraic or mechanical proof of (1). But, I like to ...