Problems from or inspired by mathematics competitions. Questions regarding mathematics competitions.

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3
votes
0answers
89 views

Square of hockey stick identity: $\sum_{i=r}^n{i \choose r}^2$

Evaluate $\sum_{i=r}^n{i \choose r}^2$ where $n,r\in \mathbb{N},n>r$. This looks like the hockey stick identity but I can't find a way to evaluate it without a computer. Can someone help me out?
8
votes
1answer
193 views

An Inequality for sides and diagonal of convex quadrilateral from AMM

Let $\square ABCD$ be a convex quadrilateral. If the diagonals $AC$ and $BD$ have mid-points $E$ and $F$ respectively, show that: $$\overline{AB} + \overline{BC} +\overline{CD} + \overline{DA} \ge ...
3
votes
0answers
77 views

Prove that $\int^1_0 \frac{dx}{x^x} = 1+ \frac{1}{2^2} + \frac{1}{3^3}…$.

Prove that $\int^1_0 \frac{dx}{x^x} = 1+ \frac{1}{2^2} + \frac{1}{3^3}...$ Darboux theorem (integral) : Whatever the number $x(k,n) \in [a + \frac{k-1}{n}(b-a),a + \frac{k}{n}(b-a),]$, we have ...
0
votes
0answers
33 views

Find $\sum_{k\in S} \frac 1{2^k}$, where S is the set of numbers not divisible by 2 or 3.

Find $\sum_{k\in S} \frac 1{2^k}$, where S is the set of numbers not divisible by 2 or 3. This is a problem from CHMMC 2010. I was able to prove that this converges by comparing it to the sum ...
0
votes
1answer
28 views

Probability of scoring positive in a certain test .

In a math contest problem appeared which I have trouble solving . It goes as under - Consider an examination of $N$ questions - fully multiple choice questions . There are $c$ choices for each ...
3
votes
0answers
86 views

High School Problem on Differential Geometry (finding new curve's equation)

This is a question in a Differential Geometry test in the last year of high school (which I couldn't solve it!): Suppose there are two pieces of curves in the $x-y$ plane: one is $y=ax^2$ cut by ...
1
vote
1answer
56 views

Probability - A random point dividing a square into $4$ parts

A point P is chosen randomly in a square. Join P with the four vertices of the square so as to divide the square into four triangles. Find, correct to 2 decimal places, the probability that all ...
1
vote
1answer
50 views

If m postive integers such $\rm{lcm}[a_{i},a_{j}]\le 400,\forall i,j\in \{1,2,\cdots,m\}$,prove $m\le 40$

Let $a_{i}$ be postive integers,and such $1\le a_{1}\le a_{2}\le\cdots\le a_{m}\le 400$, and $$\operatorname{lcm}[a_{i},a_{j}]\le 400,\forall i,j\in \{1,2,\cdots,m\}$$ show that $m\le 40$ if we note ...
2
votes
1answer
104 views

Infinitely many primes of the form $pn+1$

Prove: Given a prime $p$, there are infinitely many $n\in \mathbb{Z}^+$, for which $pn+1$ is a prime. This is a simplified version of Dirichlet's theorem, so is there any elementary solution to ...
1
vote
1answer
73 views

Combinatorics olympiad problem

Twenty-five tennis players are numbered by the numbers $1,2,...,25$. The players are divided into five teams with five players on each team in such a way that the sum of the numbers of the players on ...
2
votes
0answers
64 views

Prove $\frac{5^{125}-1}{5^{25}-1}$ is not a prime [duplicate]

Prove $\displaystyle \frac{5^{125}-1}{5^{25}-1}$ is not a prime. Some obvious thoughts: $\displaystyle \frac{5^{125}-1}{5^{25}-1}={(5^{25})}^4+{(5^{25})}^3+{(5^{25})}^2+{5^{25}}+1$ UPD: A ...
1
vote
1answer
52 views

Solving functional equation $f:Q^+\to R^+$ where $f(xy)=f(x+y)(f(x)+f(y))$

Find all functions $f:\mathbb{Q}^+ \to \mathbb{R}^+$ with the property: $$f(xy)=f(x+y)(f(x)+f(y)),\qquad \forall x, y\in\mathbb{Q}^+ \tag{1}$$ This question is from the 2014 Bulgaria National ...
9
votes
2answers
135 views

Show that $2 \int f^2 \leq \int |f'| \cdot \int |f|$

Let $f(x)$ be a continuously differentiable function defined on closed interval $[0, 1]$ for which$$\int_0^1 f(x)\,dx = 0.$$How do I show that$$2 \int_0^1 f(x)^2\,dx \le \int_0^1 |f'(x)|\,dx \cdot ...
1
vote
1answer
27 views

Show for any permutation of $N$ there exist integers $\{a,a+d,a+2 d\}, (d>0)$ such that $f(a)<f(a+d)<f(a+2d)$

Show for any permutation there exist integers $\{a,a+d,a+2 d\}, (d>0)$ such that $f(a)<f(a+d)<f(a+2d)$
0
votes
1answer
119 views

Tiling a rectangle with L-tromino [duplicate]

Consider a $2^{1999} \times 2^{1999}$ square, with a single $1 \times 1$ square removed. Show that no matter where the small square is removed it is possible to tile this "giant square minus tiny ...
1
vote
1answer
59 views

Prove that L,M,N are collinear.

G, is the centroid of Triangle ABC; AG is produced to X such that GX = AG. If we draw parallels through X to CA,AB,BC meeting BC,CA,AB at L,M,N respectively, prove that L,M,N are collinear. I have an ...
0
votes
2answers
38 views

Show that the lines through the midpoints of BC,CA,AB respectively parallel to AD,BE,CF are concurrent

AD,BE,CF are concurrent lines in a triangle ABC. Show that the lines through the midpoints of BC,CA,AB respectively parallel to AD,BE,CF are concurrent. I am unable to proceed. Kindly comment on the ...
3
votes
1answer
111 views

Find the Product $abc$

if $a$,$b$,$c$ $\in$ $\mathbb{R}$ and if $$a+\frac{1}{b}=\frac{7}{3}$$ $$b+\frac{1}{c}=4$$ $$c+\frac{1}{a}=1$$ Then find the value of $abc$ I multiplied the three equations with $bc$, $ca$ and $ab$ ...
2
votes
1answer
44 views

$n$th degree polynomials $P(x) = Q(x)P''(x)$ with $Q$ quadratic, if $P$ has $\ge 2$ distinct roots then then $n$ distinct roots.

Let $P(x)$ be a polynomial of degree $n$ such that $P(x) = Q(x)P''(x)$ for some quadratic polynomial $Q$. Show that if $P$ has at least two distinct roots then it must have $n$ distinct roots.
0
votes
1answer
24 views

Average - Map - Infinite number of points

I have a problem to solve in the context of the preparation of the PUTNAM competition. I am asked to find the average of a certain map of $S \subset \mathbb{R^3}$ (domain $S$ is uncountable) into ...
1
vote
4answers
106 views

What is the sum of the cube of the roots of $ x^3 + x^2 - 2x + 1=0$?

I know there are roots, because if we assume the equation as a function and give -3 and 1 as $x$: $$ (-3)^3 + (-3)^2 - 2(-3) + 1 <0 $$ $$ 1^3 + 1^2 - 2(1) + 1 > 0 $$ It must have a root ...
0
votes
1answer
38 views

Prove that the perpendiculars from D,E,F to BC,CA,AB are concurrent

If two triangles ABC and DEF are such that the perpendicular from A,B,C to EF,FD,DE are congruent, prove that the perpendiculars from D,E,F to BC,CA,AB are concurrent. Source: Challenge and Thrills ...
-1
votes
1answer
65 views

Polynomial Problem from a Past Putnam Exam

Find polynomials $f(x)$, $g(x)$, and $h(x)$, if they exist, such that for all x, $|f(x)|$-$|g(x)|$+$|h(x)|$ = $ \begin{cases} -1 & x< -1 \\ \ 3x+2 & 1\leq x\leq 0 \\ ...
0
votes
2answers
46 views

Find all prime number solutions [duplicate]

Find all prime numbers $p$ and $q$ such that $p^{q+1} + q^{p+1}$ is a perfect square. Number theory problems like these are always difficult for me. So please insert the topics under which this ...
1
vote
1answer
52 views

Board game - winning strategy

Consider two friends, Alice and Bob, playing a game on a $1000 \times 1000$ board. Alice's game piece consists of a $2 \times 2$ square while Bob has to content himself with $3$ squares put together ...
0
votes
1answer
72 views

Differentiability of $ g(x)=f(2x)$ if $0 \leq x \leq \frac{1}{2}$, $g(x)= f(2x-1)$ if $\frac{1}{2}< x \leq 1$

This is from an MCQ contest. Let $f:[0,1] \longrightarrow \mathbb{R}$ be differentiable function. let $g:[0,1] \longrightarrow \mathbb{R}$ defined by: $$ g(x)=\begin{cases} f(2x) & ...
0
votes
3answers
46 views

Properties of the set of $x$ in $\mathbb{R}$ such that $\frac{|x^{2}-1|}{|x|+1}< \frac{1}{2}$

This is from an MCQ contest. Let $$\displaystyle A=\left\{ x\in\mathbb{R}\mid \dfrac{|x^{2}-1|}{|x|+1}< \dfrac{1}{2} \right\} $$ Then: $1]$ $A$ is an interval of $\mathbb{R}$ $2]$ ...
0
votes
2answers
73 views

Comparing $\int_{0}^{1}f(t)^2 dt$ to $\int_{0}^{1}f'(t)^{2} dt$ when $f(0)=0$

This is from an MCQ contest. Let $f: [0,1]\to \mathbb{R}$ be a function of class $C^{1}$ with $f(0)=0$. Which inequality is true? $1]$ ${\displaystyle \int_{0}^{1}\bigl[f(t)\bigr]^{2} ...
0
votes
1answer
47 views

Is this a Arithmetic or Geometric series?

$A_n = n^8+7$ Is the above equation an arithmetic or geometric progression? I would answer neither, but I'm not to sure.
1
vote
3answers
75 views

How to Compute $\lim _{x\to \:0}\frac{\ln \left(1+\sin \left(x^2\right)\right)-x^2}{\left(\arcsin \:x\right)^2-x^2}$

How to compute $$\lim _{x\to \:0}\frac{\ln \left(1+\sin \left(x^2\right)\right)-x^2}{\left(\arcsin \:x\right)^2-x^2}=-\dfrac{3}{2}$$ I'm interested in more ways of computing limit for ...
7
votes
3answers
141 views

How to compute$\int_{0}^{1}\dfrac{x\ln(x)}{(x^2+1)^2}dx$

How to compute $$\int_{0}^{1}\dfrac{x\ln(x)}{(x^2+1)^2}dx$$ I'm interested in more ways of computing this integral. My Thoughts \begin{align} ...
2
votes
6answers
133 views

Compute limit of $\lim_{n\to +\infty}n\left(\tan\left(\dfrac{\pi}{3}+\dfrac{1}{n} \right)-\sqrt{3}\right)$ without using L' Hôpital

Compute limit of $$\lim_{n\to +\infty}n\left(\tan\left(\dfrac{\pi}{3}+\dfrac{1}{n} \right)-\sqrt{3}\right)$$ without using L'Hospital's rule By using L'Hospital's rule and $$\tan'( \Diamond )=( ...
4
votes
2answers
40 views

Properties of $f(x)=\ln(1+x^2)+x+2$ vs $g(x)=\cosh x+\sinh x$

This is from an MCQ contest. Consider the two functions: $f(x)=\ln(1+x^2)+x+2$ et $g(x)=ch(x)+sh(x)$. The real number $c$ such that: $(f^{-1})'(2)=g(c)$ $1]$ $c=-1$ $2]$ $c=0$ ...
2
votes
3answers
122 views

Does $\sum\limits_{n\geq 1}\frac{1}{(3n-1)(3n+1)}$ converge or diverge?

How would you prove convergence/divergence of the following series? $$\sum\limits_{n\geq 1}\dfrac{1}{(3n-1)(3n+1)}$$ I'm interested in more ways of proving convergence/divergence for this ...
0
votes
1answer
103 views

If AB is a common tangent to two circles, prove that the circle in AB as a diameter cuts each of the circles orthogonally

If AB is a common tangent to two circles, prove that the circle on AB as a diameter cuts each of the circles orthogonally. Source: Challenge and Thrills in Pre College Mathematics.
0
votes
1answer
22 views

Two circles touch internally at X and a straight line cuts them at A, B, C, D in order. Prove that AB, CD subtend equal angles at X.

Two circles touch internally at X and a straight line cuts them at A, B, C, D in order. Prove that AB, CD subtend equal angles at X. Source: Challenge and Thrills in Pre College Mathematics.
2
votes
2answers
55 views

Convergence of $\sum\limits_n\frac{1}{a_n+1}$ and $\sum\limits_n\frac{a_n}{a_n+1}$ when $\sum\limits_na_n$ converges

This is from an MCQ contest. Let $\sum\limits_{n\geq 1}a_n$ be a convergent series of positive terms. Which of the following hold? $1]$ $\sum\limits_{n\geq 1}\dfrac{1}{a_n+1}$ and ...
5
votes
2answers
109 views

Discrete mathematics - Find all integer solutions of the equation $a^2+ b^2 + c^2=a^2 b^2$.

Find all integer solutions of the equation $a^2+ b^2 + c^2=a^2 b^2$. This is one of the questions we presented in one session to contest preparation PUTNAM. It turns out that I can't get from the ...
6
votes
4answers
155 views

Does $ \sum_{n\geq 2} \dfrac{\ln(1+n)}{\ln(n)}-1$ converge/diverge?

How would you prove convergence/divergence of the following series? $$ \sum_{n\geq 2}\left( \dfrac{\ln(1+n)}{\ln(n)}-1\right)$$ I'm interested in more ways of proving convergence/divergence for this ...
3
votes
1answer
65 views

Let $u_{n+2} = u_{n+1} + u_n$ where $u_1 = a, u_2 = b$. How many pairs exists $(a,b)$ such that $u_k = 21$ for some $k$?

A sequence of non negative integers $u_n$ is defined by $u_1 = a, u_2 = b$ and $u_{n+2} = u_{n+1} + u_n$. How many pairs of non-negative integers $(a,b)$ are there such that $21$ is a term of the ...
20
votes
5answers
5k views

Which week day(s) cannot be the first day of a century?

I think the question says everything. What I want is, a very short approach. What I did: Lets call the day which is not the part of a whole week, a free day. So in a normal year, there is $1$ free ...
1
vote
0answers
72 views

Properties of $\sum\limits_{n\ge2}\frac{n(n-1)x^n}{n!}$ for $x\in \mathbb{R}$

This is from an MCQ Contest. for all integer $n$ greater than or equal to $2$ Let $$\forall n\geq 2\qquad u_n=\dfrac{n(n-1)x^n}{n!}, \qquad x\in \mathbb{R}$$ for all $x$ in ...
1
vote
1answer
113 views

Circles and triangles

Put 4 identical circles inside an equilateral triangle of side length 2, such that a circle touches 2 others and only one side of the triangle. What are the radii? (ignore the vertical line)
10
votes
3answers
173 views

What value will take $f(100)$?

Let $f$ be a function from the positive integers into the positive integers and which satisfies $f(n + 1) > f(n)$ and $f(f(n)) = (f \circ f) (n) = 3n$ for all $n$. Find $f(100)$. This is one of ...
3
votes
0answers
37 views

Integrability of $f(x)=\left(1+\frac{1}{x} \right)^{1+\frac{1}{x}}-a-\frac{b}{x}$

This is from an MCQ contest. For all $x\geq 1$ let $$f(x)=\left(1+\dfrac{1}{x} \right)^{1+\dfrac{1}{x}}-a-\dfrac{b}{x}$$ note that ...
5
votes
1answer
65 views

Can we improve the constant $6$? $\inf_{0\le x\le 1}\sum_{j=1}^{n}\frac{1}{|x-p_{j}|}\le 6n\left(1+\frac{1}{3}+\cdots+\frac{1}{2n-1}\right)$

Some days ago, when I again read the William Lowell Putnam Mathematical Competition (1979), I found this nice problem: Let $p_{j}\in [0,1],j=1,2,\cdots,n$. Prove, that $$\inf_{0\le x\le ...
1
vote
1answer
87 views

On the solution of a olympics math problem.

The problem in question is The sequence $a_1,a_2,\ldots$ of integers satisfies the following conditions: $1\le a_j\le 2015$ for all $j\ge 1$; $k+a_k\ne\ell+a_\ell$ for all $1\le ...
1
vote
2answers
82 views

Behaviour of $\arctan(x) /x$ when $x\to0$

This is from an MCQ contest. let $f$ be the function defined on $\mathbb{R}$ by $$f(x) =\begin{cases} \dfrac{\arctan(x)}{x} & \text{ if } x \neq 0 \\ \\ 1 & x = 0 \end{cases} $$ ...
0
votes
0answers
65 views

About demonstration of sum of powers.

In the book "250 problems in Elementary Number Theory", the problem 9 asks to prove that $$ 1^3+2^3+\dots + n^3 \big\vert 3(1^5+2^5+\dots + n^5) $$ But in the demonstration (page 25 of Solutions) it ...
1
vote
1answer
83 views

Properties of $ f(x)=|x|^{1/(x-1)} $ (MCQ Contest)

Let $f$ be a function defined by: $$ f(x)=|x|^{\dfrac{1}{x-1}} $$ Then : Choose the correct option. more than one may be correct the domain of $f$ is ...