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

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6
votes
3answers
121 views

Board game on a $m\times n$ board - winning strategy

Two friends, $A$ and $B$, play a game with one single game piece on a rectangular board with $m$ rows and $n$ columns. $A$ begins the game by moving the game piece from its starting point $(1, 1)$ to ...
3
votes
2answers
72 views

Finding an angle in a circle

In circle $O$, $PA\perp AO,AE\perp PO,\angle BCO=30^{\circ},\angle BFO=20^{\circ}$,find $\angle DAF$. It is obvious that $\angle EAD=\angle PAD=\frac{1}{2}\angle AOP$, but I can't get more ...
1
vote
1answer
62 views

Confused by a step in a solution to the problem

I'm pretty confused by the step $$ \prod_{n=1}^{45}\sin(2n^\circ)=\sum_{n=1}^{45}\frac{\omega^n-1}{2i\omega^{n/2}} $$ in the official solution of this problem from 2010 PUMaC Algebra A7: The ...
6
votes
2answers
316 views

How many solutions to the rational equation?

If $a$ and $b$ and $c$ are parameters, how many solutions for: $$\frac{(x-b)(x-c)}{(a-b)(a-c)} + \frac{(x-a)(x-c)}{(b-a)(b-c)} + \frac{(x-a)(x-b)}{(c-a)(c-b)} = 1$$ I would say $3 \implies ...
1
vote
2answers
88 views

How many numbers less than $1000$ with digit sum to $11$ and divisible by $11$

How many positive (integers) numbers less than $1000$ with digit sum to $11$ and divisible by $11$? There are $\lfloor 1000/11 \rfloor = 90$ numbers less than $1000$ divisible by $11$. $N = 100a + ...
0
votes
2answers
43 views

For how many integers $a$ does this equation have three solutions?

For how many integers $a$ does the equation $(x^2-a^2 ) \sqrt{(5-x)}=0$ have three different solutions? The options were: $10, 9, 8, $other. I say other. No matter what, $\sqrt{5-x} = 0$ ...
0
votes
1answer
34 views

Determine the maximum GCD

The sum of $10$ natural numbers is $2014$. Determine the greatest possible value of the GCD of these numbers. Is this a trial and error type of problem? $a_1 + a_2 + ... + a_{10} = 2014$. ...
0
votes
2answers
56 views

Which coefficient is greater?

If $b, c$ are integers, and $\sqrt{2} + \sqrt{3}$ is a root of the equation, $x^4 + bx^2 + c = 0$, which is greater, $b$ or $c$; where $b, c$ are both integers. Since $\sqrt{2} + \sqrt{3}$ is a ...
4
votes
0answers
57 views

Roots of a polynomial that is composed n times with itself

Let $f(x)=x(4x^2-3)(64x^6-96x^4+36x^2-3)$ and $f^{(n)}=f(f(f(\cdots f(x))\cdots)$ (composed with itself $n$ times). Prove that for all positive integers $n$, $f^{(n)}(x)=x$ has $9^n$ distinct ...
3
votes
3answers
76 views

Maximizing $\sin \beta \cos \beta + \sin \alpha \cos \alpha - \sin \alpha \sin \beta$

I need to maximize $$ \sin \beta \cos \beta + \sin \alpha \cos \alpha - \sin \alpha \sin \beta \tag{1}$$ where $\alpha, \beta \in [0, \frac{\pi}{2}]$. With numerical methods I have found that $$ ...
1
vote
0answers
28 views

LCM Challenge Range Query

I am trying to solve this question LCM Challenge. How can modulo be used when values of LCM goes quite high or what is the best approach for the question ?
0
votes
1answer
39 views

Which perfect squares can be written as the sum of two squares?

what perfect square number should be substracted from x so that resultant is perfect square number if solution doest not exist just tell not possible? note here x is also perfect square number ...
2
votes
2answers
60 views

$\sum_{k=0}^n {n \choose k} ^{2} = {2n \choose n}$ - Generating function $\sum_{k=0}^\infty \binom nk x^k = (1+x)^n$.

As part of a preparatory course in the contest PUTNAM, I have to show $\sum_{k=0}^n {n \choose k} ^{2} = {2n \choose n}$. I know that I can use the identity $\sum_{k=0}^n {n \choose k} {n \choose ...
0
votes
0answers
77 views

What is the volume of the largest cuboid with sides of integer length that can fit inside a sphere with a radius of 9?

I've come across the following problem: What is the volume of the largest cuboid with sides of integer length that can fit inside a sphere with a radius of 9? To attempt a solution, I first attempted ...
8
votes
3answers
336 views

Contest style inequality

Can anyone help me with this inequality? For $a,b,c>0:$ $$\sqrt{\frac{2}{3}}\left(\sqrt{\frac{a}{b+c}}+\sqrt{\frac{b}{c+a}}+\sqrt{\frac{c}{a+b}}\right)\leq ...
0
votes
1answer
28 views

Showing that an exponential-like function eventually catches up with a polynomial function

You are given a finite set $A$ of prime numbers. Let $A=\{a_1, a_2\cdots\}$. Let B be the set formed of numbers that are formed by multiplying powers of $a_i$ i.e. $B=\{t|t=a_1^{e_1}\cdots ...
1
vote
1answer
27 views

A property about harmonic quadrilateral

Point $A$ is the center of the circle. $BA\bot BE, FA\bot FE$. Prove $\displaystyle \frac{CG}{GD}=\frac{CE}{ED}$.
3
votes
0answers
88 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
180 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
76 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
82 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
51 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
43 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
100 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
43 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
116 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
102 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
56 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
37 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
106 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
43 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
23 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
101 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
33 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
60 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
42 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
45 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
45 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
46 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
74 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
139 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
122 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: $(f^{-1})'(2)=g(c)$ $1]$ $c=-1$ $2]$ $c=0$ ...