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

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1
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3answers
43 views

Finding maximum $b$ in $x^5-20x^4+bx^3+cx^2+dx+e=0$

Let $b, c, d, e$ be real numbers such that the following equation $$x^5-20x^4+bx^3+cx^2+dx+e=0$$ has real roots only. Find the largest possibe value of $b$. What I have done is: Let $x_1, x_2, x_3, ...
0
votes
1answer
28 views

Maximum and minimum of a function.

Given a function $f(x) = C(x, 2) + C(N-x,2)$, where N is a constant and C(N, K) is the binomial coefficient choose K from N, we need to find minimum and maximum value. Also, $x > 0$. So, f(x) = $\...
3
votes
1answer
48 views

Proof of existing degree $n$ binomial

Let $P(x)$ be a polynomial with real coefficients such that $P(x) > 0$ for all $x \ge 0$. Prove that there exists a positive integer $n$ such that $(x + 1)^n P(x)$ is a polynomial with nonnegative ...
2
votes
1answer
70 views

Find polynomials $f(x), g(x)$, and $h(x)$

Find polynomials $f(x), g(x)$, and $h(x)$, if they exist, such that for all $x$, $$\mid f(x)\mid-\mid g(x) \mid+h(x)= \begin{cases} -1, & \text{if}~x<-1 \\ 3x+2, & \text{if}~-1\...
2
votes
1answer
37 views

Prove that $\frac{x_1 + x_2 + x_3 +x_4}{4}$ is independent of the line, and compute its value.

Consider the lines that meet the graph $y = 2x^4 + 7x^3 + 3x − 5$ in four distinct points $P_i=[x_i, y_i]$, $i = 1, 2, 3, 4$. Prove that $\frac{x_1 + x_2 + x_3 +x_4}{4}$ is independent of the line, ...
3
votes
1answer
52 views

How many solutions exist for a non-linear system

How many solutions exist to the following system: $$ \begin{eqnarray} xy+xz &=& 54+x^2 \\ yx+yz &=& 64+y^2 \\ xz+yz &=& 70+z^2 \end{eqnarray} $$ I have guessed that the ...
0
votes
1answer
24 views

Comparing absolute values

If $|i - (a + bi)| < 1$ does $|i - (a - bi)| < 1$ also? I would say yes, because the absolute value shouldn't differ by more than $1$? Where $i = \sqrt{-1}$
2
votes
1answer
49 views

find an invariant

I've been reading about the use of invariants in contest math. I saw the following problem (in my own words): There are $N = 2n$ numbers placed on a circle. Then we increase two any consecutive ...
1
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0answers
50 views

Prove that there is no integer $k$ with $P(k)=8$

Let $P(x)= x^n + a_{n-1}x^{n-1}+...+a_1x+a_0$be a polynomial with integral coefficients. Suppose that there exists four distinct integers $a$, $b$, $c$, $d$ with $P(a)=P(b)=P(c)=P(d)=5$. Prove ...
0
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1answer
41 views

Can I get three roots $a'$, $b'$ and $c'$ such that $P(x)=(x-a')(x-b')(x-c')$?

If I have $(x-a)(x-b)(x-c)=1$ ($a,b,c \in \mathbb{Z}$) for the polynomial $P(x)=(x-a)(x-b)(x-c)-1$, can I get three roots $a'$, $b'$ and $c'$ such that $P(x)=(x-a')(x-b')(x-c')$? This is only a ...
2
votes
1answer
120 views

Find a polynomial with integral coefficients whose zeros include $\sqrt{2} + \sqrt{5}$.

Find a polynomial with integral coefficients whose zeros include $\sqrt{2} + \sqrt{5}$. I think I can use $-3= (\sqrt{2} + \sqrt{5})(\sqrt{2} - \sqrt{5})$ and a certain telescopic factorisation. The ...
31
votes
8answers
3k views

Can a pre-calculus student prove this?

a and b are rational numbers satisfying the equation $a^3 + 4a^2b = 4a^2 + b^4$ Prove $\sqrt a - 1$ is a rational square So I saw this posted online somewhere, and I kind of understand what the ...
2
votes
0answers
216 views

How to prove this hard geometry

The incircle of triangle $ABC$ has center $I$ and touchs the sides $BC,CA,AB$ at the points $D,E,F$ respectively,and Let the centers of the excircles tangent to $BC,CA,AB$ be $I_{1},I_{2},I_{3}$ ...
1
vote
0answers
45 views

Prove intersection between side length and tangent to circumcircle at opposite vertex is collinear with points on perpendicular bisectors of sides

Let $ABC$ be a triangle with $AB\neq BC$. Point $E$ lies on the perpendicular bisector of $AB$ such that $BE\perp BC$. Point $F$ lies on the perpendicular bisector of $AC$ such that $CF\perp BC$. Let $...
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votes
1answer
42 views

Determine the symmetric sum of roots.

Please no complete solutions, ONLY HINTS REQUESTED! The complex numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the four distinct roots of the equation $x^4+2x^3+2=0$. Determine the ...
0
votes
2answers
49 views

Math Team Problem Involving Powers of Powers of 3

So I am in my high school math team and I was given the following expression $$3^{3^{3^{...}}}$$ Where there are multiple powers of 3 with a total of two thousand and fifteen 3's. The question ...
1
vote
2answers
77 views

Find all values of $x$

Determine all real values of $x$ such that: $$\log_{2}(2^{x-1} + 3^{x+1}) = 2x - \log_{2}(3^x) $$ Let $u = 2^x$ and let $y = 3^x$ For ease, let $\log_{2}$ be represented by just $\log$ so: Then, $\...
3
votes
2answers
53 views

Find $s^4-18s^2-8s$

Let $a,b,c$ be the roots of $x^3-9x^2+11x-1=0$, and let $s=\sqrt{a}+\sqrt{b}+\sqrt{c}$. Find $s^4-18s^2-8s$. $s^4 - 18s^2 - 8s = (s)(s + 4)(s - 2 + \sqrt{6})(s - 2 - \sqrt{6})$ $P(x) = (x - a)(x - ...
10
votes
3answers
849 views

sum of one hundred numbers

I saw this problem recently. It asks to prove that it is always possible to choose 100 numbers from 200 positive numbers such that their sum will be divisible by 100. Attempt to solve: my first step ...
2
votes
1answer
71 views

Show that there exist only $n$ solutions

Let $P(x)$ be a polynomial of degree $n>1$ with integer coefficients, and let $k$ be a positive integer. Consider the polynomial $Q(x) = P( P ( \ldots P(P(x)) \ldots ))$, where $P$ occurs $k$ times....
4
votes
1answer
132 views

Sum of powers of sine

Find $\displaystyle \sum_{n=1}^{89} \sin^6(n) = \frac{m}{n}$ Let $x = \sin(n)$ and let $y = \cos(n)$. Since $\cos(n) = \sin(90 - n)$ it follows that $= \sin^6(1) +\sin^6(1) + ... + \sin^6(45) + \...
4
votes
1answer
167 views

Leningrad Mathematical Olympiad $1991$

A finite sequence $a_1, a_2, ..., a_n$ is called $p$-balanced if any sum of the form $a_k+a_{k+p} + a_{k+2p}+...$ is the same for any $k = 1, 2, 3, ..., p$. For instance the sequence $a_1 = 1$, $a_2 ...
2
votes
1answer
57 views

How many sides from diagonals?

A polygon has $100$ diagonals, then it has at least: A-15, B-16, C-17, D-18 Sides? Using simple patterns, I noticed that all figures (even sides) have $\frac{n}{2}$ sides for $n$ diagonals; this ...
0
votes
3answers
76 views

Evaluate the nested square root

Evaluate: $x = \sqrt{11 - 2\sqrt{10}} - \sqrt{11 + 2\sqrt{10}}$ You may have seen my other Q/A here, but I am finding a different way, with perhaps perfect squares. If we seperate, $y = \sqrt{11 - ...
4
votes
3answers
37 views

Which of the constants A,B,C,D does T depend on?

Let $f(x)=cos(5x)+Acos(4x)+Bcos(3x)+Ccos(2x)+Dcos(x)+E$ and $T=f(0)-f(\pi/5)+f(2\pi/5)-f(3\pi/5)+..-f(9\pi/5)$.Then out of A,B,C,D which does T depend on? Hints please! P.S:KVPY 2011 question
5
votes
1answer
68 views

Olympiad problem about finding minimum value with $x^2y^2+y^2z^2+z^2x^2\ge x^2y^2z^2$

Let $x,y,z$ be positive real numbers such that $x^2y^2+y^2z^2+z^2x^2\ge x^2y^2z^2$. Find the minimum value of $$\frac{x^2y^2} {z^3(x^2+y^2)}+\frac {y^2z^2} {x^3(y^2+z^2)}+\frac {z^2x^2} {y^3(z^2+x^2)}...
16
votes
4answers
752 views

Why can't $p^p-(p-1)^{p-1}=n^2$ be a square?

Let $p$ be a prime number. Show that $p^p-(p-1)^{p-1}$ can't be a square. In other words, there is no $n\in\mathbb{N}^{+}$ such that $$p^p-(p-1)^{p-1}=n^2.$$
3
votes
2answers
81 views

What is the smallest possible value of $\lfloor (a+b+c)/d\rfloor+\lfloor (a+b+d)/c\rfloor+\lfloor (a+d+c)/b\rfloor+\lfloor (d+b+c)/a\rfloor$?

What is the smallest possible value of $$\left\lfloor\frac{a+b+c}{d}\right\rfloor+\left\lfloor\frac{a+b+d}{c}\right\rfloor+\left\lfloor\frac{a+d+c}{b}\right\rfloor+\left\lfloor\frac{d+b+c}{a}\right\...
1
vote
1answer
56 views

First Three Digits of Powers of 2 and 5

Suppose you know that there exists positive integer $n\in \mathbb{N}$ such that the first three digits of $2^n$ and $5^n$ are the same, and that $\forall n$ that do so, the first three digits are ...
0
votes
1answer
48 views

Prove existence of 5 non-attacking rooks

Problem: There are $41$ rooks on a $10\times10$ chessboard. Prove that there must exist $5$ rooks, none of which attack each other. I could only observe that at least one of rows and at least one ...
1
vote
2answers
170 views

The sum of two positive integers is 29 , find the minimum value of the sum of their squares.

If the sum of two positive integers a and b is 29 , find the minimum value of the sum of their squares. Of course I dont need a brute force answer. What is a quick way to find what a and b is?
1
vote
1answer
50 views

Find $\theta$ in the given triange

Given an isosceles triangle ABC, AB=AC, AD=BC, angle ACB = 80 degree. What is $\theta$? (I'm looking for ways to find this angle without using a calculator, sorry I added this late)
3
votes
1answer
82 views

General form of $\sqrt{a - b} - \sqrt{a + b}$?

General form of $\sqrt{a - b} - \sqrt{a + b}$? What I would do is: let $x = \sqrt{a - b} - \sqrt{a + b}$ $x^2 = 2a - 2\sqrt{a^2 - b^2}$ Then since $a + b > a - b$ $x = -\sqrt{2a - 2\sqrt{a^2 ...
1
vote
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
votes
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
votes
2answers
80 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
vote
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
votes
0answers
66 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
43 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 ...
55
votes
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 $k$ ...
1
vote
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 ...
1
vote
0answers
43 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
votes
0answers
168 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) ...
5
votes
3answers
172 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
60 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 ...
0
votes
1answer
44 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 ...
1
vote
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
vote
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
128 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 ...