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

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2
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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
143 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
123 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
61 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
39 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
113 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
109 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
66 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
53 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
49 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
113 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
23 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
56 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
111 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
156 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
114 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
66 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
89 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
67 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 ...
0
votes
3answers
64 views

An infinite sum in the product of sines

This is an undergrad or lower level question I need help with. Evaluate $$\quad \sum_{n=1}^{\infty} \sin{\left(\frac{a}{3^n}\right)}\sin{\left(\frac{2a}{3^n}\right)}$$ where a is just some real ...
3
votes
1answer
94 views

Algorithm to uniquely determine a number using two adjacent digits

(Russia) Arutyun and Amayak perform a magic trick as follows. A spectator writes down on a board a sequence of $N$ (decimal) digits. Amayak covers two adjacent digits by a black disc. Then ...
1
vote
0answers
28 views

Finding the speed of the faster of two machines.

This is a question from a maths contest and is as follows: Two machines move at constant speeds around a circle of circumference $600$ cm, starting together from the same point. If they travel in ...
0
votes
1answer
63 views

Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by $\binom{n-r+1}{r}$

Let $X=\{1,2,3,\dots,n\}$ where $n\in N$. Show that the number or $r$ combinations of $X$ which contain no consecutive integers is given by $\binom{n-r+1}{r}$ where $0\le r\le n-r+1$. I am unable ...
-5
votes
2answers
388 views

How many zeros are obtained if we multiply all the natural numbers from 1 to 100? [duplicate]

Options are: 20 21 22 23 24 I recently came across the above question in a competitive exam, where we get about 30 seconds to 1 minute for solving each problem. I want to if there are quick and ...
3
votes
0answers
63 views

Find all pairs of prime numbers $p , q$ for which: $p^2 | q^3 + 1$ and $q^2 | p^6 − 1$.

Find all pairs of prime numbers $p , q$ for which: $$p^2 \mid q^3 + 1 \tag{A}$$ and $$q^2 \mid p^6 − 1 \tag{B}$$ The question is from the Bulgaria National Olympiad 2014. I'm looking for ...
4
votes
5answers
147 views

Solve for $x^{11} + \frac{1}{x^{11}}$ if $x^3 + \frac{1}{x^3} = 18.$

Solve for $x^{11} + \frac{1}{x^{11}}$ if $x^3 + \frac{1}{x^3} = 18.$ I'm familiar with variants of this problem, especially those where the exponent in the given is a factor of the exponent in what ...
6
votes
1answer
124 views

Proving that $\pi(2n) < \pi(n)+\frac{2n}{\log_2(n)}$

Given that $\pi(x)$ is the prime-counting function, prove that, for $n\geq 3$, 1: $\pi(2n) < \pi(n)+\frac{2n}{\log_2(n)}$ 2:$ \pi(2^n) < \frac{2^{n+1}\log_2(n-1)}{n}$ For $x\geq8$ a real ...
0
votes
1answer
67 views

Ratio of Area of Quadrilaterals

This is a question related to the last one I asked yesterday: Ratio of Areas of Quadrilateral Again I tried to make use of the same theorem (*) but have failed. I believe this time the question is ...
6
votes
0answers
100 views

Find the least possible value of $n$ such that there exist $P(x), Q(x) \in \mathbb{Z}[x]$

Find the least possible value of $n, n \geq 2015$ such that there exists polynomial $P(x)$ with degree $n$, integer coefficients, the coefficient of the term $x^n$ is positive and polynomial $Q(x)$ ...