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

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1answer
50 views

Multiplying products of $p_1,p_2,\ldots,p_n$ gives a square.

Given $n+1$ ($n\ge 4$) arbitrary products of primes $p_1,p_2,\ldots, p_n$, prove multiplying some of the products gives a square. E.g., for $n=4$: $\{p_1,p_2,p_3,p_4,p_1p_3\}$ satisfies the ...
3
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1answer
68 views

A Diophantine Equation

Finding the number of $(a, b, c)$, where $a, b, c$ are positive integers, that $$ \frac{a^2+b^2-c^2}{ab}+\frac{c^2+b^2-a^2}{cb}+\frac{a^2+c^2-b^2}{ac}=2+\frac{15}{abc} $$ I factored it in ...
-1
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2answers
79 views

Find the minimum value of $a^2 + b^2 + 5 - ab - 2a - 2b$

Find the minimum value (integer) of $a^2 + b^2 + 5 - ab - 2a - 2b$. I believe the answer is $1$, I got this from trial-and-error. $= a^2 + b^2 - ab - 2(a + b) + 5 = (a + b)^2 - 2(a+ b) - 3ab + 5$ ...
1
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1answer
43 views

Find $k$ such that the area is an integer

For some positive integers k, the parabola with equation $y = \frac{x^2}{k} - 5$ intersects the circle with equation $x^2 + y^2 = 25$ at exactly three distinct points A, B and C. Determine all ...
2
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0answers
65 views

How many numbers can be written as a sum?

How many integers can be written as a sum of 4 consecutive integers such that the numbers are < $2015$. EDITED QUESTION: If $N = n + (n + 1) + (n+ 2) + (n + 3)$ then how many such $N$ exist ...
3
votes
1answer
42 views

Sum of smallest roots

If 2 and -3 are the roots of a biquadratic equation, then the sum of the two smallest roots of this equation is: $\{-1, -3, -5, \text{cannot be determined} \}$ those are the options. The answer ...
54
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7answers
1k views

Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers.

Let $k$ be a natural number . Then $3k+1$ , $4k+1$ and $6k+1$ cannot all be square numbers. I tried to prove this by supposing one of them is a square number and by substituting the corresponding $k$ ...
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3answers
57 views

How many participants required?

A test consisting of 20 problems is given at a math competition. Each correct answer to each problem gains 4 points; each wrong answer takes away 1 point, and each problem left without an answer gets ...
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0answers
42 views

puzzle-coloring problem-olmpiad

A $23\times23$ square is completely tiled by $1\times1, 2\times2$ and $3\times3$ tiles. What is the smallest number of $1\times1$ tiles needed? This is the solution If we color the rows of the $23\...
2
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0answers
157 views

Multivariable Factor Theorem

By my previous questions here and here I have been inspired to ask about the factor theorem; the multivariable case of it. So take $f(a, b, c) = (a-b)^3 + (b-c)^3 + (c-a)^3$ $f(a, a, c) = f(b, b, c) ...
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3answers
152 views

Olympiad inequality problem with $a+b+c+abc=4$

If $a,b,c \in \mathbb R_{> 0}$ and $a+b+c+abc=4$, prove that $$({a\over {\sqrt {b+c}}}+{b\over {\sqrt {c+a}}}+{c\over {\sqrt {a+b}}})^2(ab+bc+ca) \ge {\frac 12}(4-abc)^3$$ This can be solved by AM-...
2
votes
2answers
57 views

Factoring a polynomial (multivariable)

Factor $ (a - b)^3 + (b - c)^3 + (c-a)^3$ by SYMMETRY. Okay, this is the problem. Let $f(a) = (a - b)^3 + (b-c)^3 + (c-a)^3$ obviously, if you let $a = b$ then, $f(b) = 0$, thus $(a - b)$ is a ...
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1answer
41 views

Giving this formula in DNF and CNF propositional logic

The formula I am trying to turn into conjunctive normal form and disjunctive normal form is: $P \rightarrow (Q \land R)$ could anyone please help me give two answers, CNF and DNF? I have managed to ...
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3answers
52 views

Find the sum from the system of equations

If $x,y, z$ satisfy: $$x + y = z^2 + 1, y + z = x^2 + 1, x + z = y^2 + 1 $$ Find the value of $2x +3y + 4z$. This gives us (by getting $x + y + z$ that) $z^2 + z + 1 = x^2 + x + 1 = y^2 + y + 1 \...
4
votes
2answers
149 views

Prove every integer is obtained from functions

We are given the following operations: $$f(n)=10n, g(n)=10n+4, h(n)=\frac{n}{2}$$, where $n$ is a positive integer (n must be even for $h(n)$. Show that, beginning with $n=4$, every positive ...
1
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2answers
43 views

show this diophantine equation has at least is $3n+3\lfloor \frac{n+1}{3}\rfloor+1$ postive integer solution

For any postive integer $n\ge 4$, let $s(n)$ denote the number of ordered pairs $(x,y,z)$ of positive integers for which $$\color{red}{xy+yz+xz=n(x+y+z)}$$ show that $$s(n)\ge 3n+3\lfloor \dfrac{n+1}{...
6
votes
3answers
127 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
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2answers
77 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 ...
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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
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2answers
320 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 x = ...
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2answers
119 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 + ...
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votes
2answers
45 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, $\sqrt{5-x} = 0$ always ...
0
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1answer
35 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? $a_1 + a_2 + ... + a_{10} = 2014$. ...
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votes
2answers
59 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 root,...
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0answers
65 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 real ...
3
votes
3answers
77 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 $$ \...
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0answers
32 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 ?
1
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1answer
47 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
81 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 $\sum_{k=0}^n {n \choose k} {n \choose n-k}$...
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0answers
109 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
345 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 \sqrt{\frac{a}{b}+\frac{b}{c}+\frac{c}{a}...
0
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1answer
29 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 a_2^{e_2},...
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1answer
37 views

A property about harmonic quadrilateral

Point $A$ is the center of the circle. $BA\bot BE, FA\bot FE$. Prove $\displaystyle \frac{CG}{GD}=\frac{CE}{ED}$.
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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
199 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
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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 $$\...
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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 $\sum_{k\...
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1answer
29 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
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0answers
92 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 $y=b$...
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1answer
57 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 ...
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1answer
53 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
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1answer
106 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
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1answer
75 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 ...
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0answers
65 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 similar ...
1
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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
146 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
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1answer
28 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
124 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
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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 ...