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

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4
votes
1answer
83 views

Minimum difference of roots of a polynomial and its derivative

Let $P(x) = (x-x_1)(x-x_2)...(x-x_n)$ where all the n roots are real and distinct. Let $y_1,y_2,...,y_{n-1}$ be the roots of $P'$. Show that $\min_{i\neq j}|x_i-x_j|<\min_{i\neq j}|y_i-y_j|$. My ...
3
votes
1answer
343 views

How find this value of $x,y$

let $x,y\in R$, such $$\begin{cases} \sqrt{1+(x+y)^2}=-y^6+2x^2y^3+4x^4\\ \sqrt{2x^2y^2-x^4y^4}\ge 4x^2y^3+5x^3 \end{cases}$$ find the value of $x,y$. My try: since ...
1
vote
1answer
214 views

Newton's problem of cows and fields

I encountered this problem about Newton's problem of cows and fields: In a field, 17 cows can finish the whole grass in the field for 30 days. 19 cows can finish in 24 days. If a group of cows eat ...
67
votes
3answers
2k views

A math contest problem $\int_0^1\ln\left(1+\frac{\ln^2x}{4\,\pi^2}\right)\frac{\ln(1-x)}x \ \mathrm dx$

A friend of mine sent me a math contest problem that I am not able to solve (he does not know a solution either). So, I thought I might ask you for help. Prove: ...
0
votes
2answers
108 views

KVPY Scholarship Exam Problem on finding the area of a rectangle

In a rectangle $ABCD$, the coordinates of $A$ and $B$ are $(1,2)$ and $(3,6)$ respectively and some diameter of the circumscribing circle of $ABCD$ has equation $2x-y+4=0$. Then the area of the ...
0
votes
0answers
51 views

placing integers on circle with no repeated differences.

Is it possible to place 2008 numbers from 1 to 2009 on a circle such that the absolute values of the differences between numbers and their inmediate neighbors are all different? I think this is from ...
0
votes
1answer
52 views

The intersections of three polygons in a square with area $=6$

Let three convex polygons with areas equal to $3$, in a square with area equals to $6$. We need to prove that there are two of them which has their intersection with area is at least $1$. I have no ...
3
votes
3answers
152 views

Question from Spring 2012 AMATYC Student Mathematics League

How would you go about solving a problem like this: Let a, b, and c be positive integers which satisfy $a^3+b^3+c^2=2012$. Find $a+b+c.$ It doesn't appear that there's enough information to ...
10
votes
2answers
294 views

How prove this equation have infinite solution?

Let $x,y,z\in Z$, such that $\gcd(x,y)=\gcd(y,z)=\gcd(x,z)=1$. Show that the number of solutions to $$2013x^2+y^3=z^4$$ is infinite. This problem is from the China Mathematical Olympiad ...
1
vote
2answers
108 views

Determine the least natural number $k$ such that $a(k)>1$

Let $a(n)$ be a sequence with $a(0)=1/2$ and $a(n+1)=a(n)+(a(n)^2)/2013$, $n$ natural number. Determine the least natural number $k$ such that $a(k)>1$. This problem is from Poland proposed to ...
1
vote
1answer
60 views

Let $f,g$ be two distinct functions from $[0,1]$ to $(0, +\infty)$ such that $\int_{0}^{1} g = \int_{0}^{1} f $.

Let $f,g$ be two continuous, distinct functions from $[0,1]$ to $(0, +\infty)$ such that $\int_{0}^{1} g = \int_{0}^{1} f $. Given $n\in \mathbb{N},$ let $y_n = \int_{0}^{1} \frac{f^{(n+1)}}{g^{(n)}} ...
3
votes
5answers
400 views

What is the value of $f(0)+f(8)$?

Suppose $f$ is a polynomial of degree $7$ which satisfies $f(1) =2$, $f(2)=5$, $f(3)=10$, $f(4)=17$, $f(5)=26$, $f(6)=37$ and $f(7)=50$. What is the value of $f(0)+f(8)$?
0
votes
1answer
81 views

Basis and dimensions for quadratic polynomials

How do I find the basis and dimension for the set of all quadratic polynomials p(x)=ax^2+bx+c that satisfy p(1)=0.
0
votes
0answers
40 views

Why is $A^{m} - 1 = (A^{m'} - 1)(A^{m'(a-1)} + A^{m'(a-2)} + … + A^{m'} + 1).$

Show that if $m$ is a multiple of $a^n$, then $(a + 1)^m -1$ is a multiple of $a^{n+1}$. Here is a solution, but I don't understand it: We use induction on $n$. For $n = 0$ we have to show that ...
1
vote
3answers
83 views

Finding the possible location of points

The numbers 1,2,....6 are to be placed in some order at the points A,B,.....F in the figure below. How many ways can the numbers be placed so that each sum of consecutive pairs of points is odd?
1
vote
1answer
92 views

Bernie's Breakfast

Moderator Note: This is a current contest question on Brilliant.org. Bernie's Breakfast Buffet offers omelettes as part of their buffet on Saturday and Sunday. They offer 6 different toppings ...
4
votes
2answers
161 views

An Olympiad Problem (tiling a rectangle with the L-tetromino)

An L block that is 3 unit blocks high and 2 unit blocks wide . It is true that if an n by m rectangle can be covered by such L blocks with out overlap that we would require an even amount of L blocks, ...
2
votes
1answer
127 views

Lines $ MF, DE, QR$ in a triangle intersect at one point

In a triangle ABC, a circle is inscribed with center in $I$. The inscribed circle touches sides $BC,CA,AB$ in $D,E,F$ respectively. Join the point $C$ and $F$, $B$ and $E$. Let $Q$ and $R$ be the ...
3
votes
1answer
210 views

Minimum period of function such that $f\left(x+\frac{13}{42}\right)+f(x)=f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right) $

Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1,f(x)\neq constant $ and ...
2
votes
1answer
59 views

Find the result of a weird looking sum

How do I find the value of such ridiculous-looking sum? $$\sum^{100}_{i=1}\lfloor \sqrt{i}\rfloor$$
0
votes
1answer
97 views

solving a hardcore limit with product

Can this expression be simplified? $$\lim_{x\to0}\left(\prod^{\frac{1}{x}-1}_{i=1}\frac{1}{\sec\frac{xi\pi}{2}}\right)^x$$
8
votes
3answers
160 views

How can we find the gcd for elements (binomial coefficient)?

$\gcd\left(\binom{2n}1,\binom{2n}3,\binom{2n}5,\ldots,\binom{2n}{2n-1}\right)$ i want to know what is specialty of such a series.I am not able to generalize the problem solution.Is there any rule for ...
-1
votes
1answer
24 views

Problem about composite and divisibliity.

If $n$ is an integer that is composite, then $n$ is divisible by a prime number $p:p<\sqrt{n}$. Can someone help me prove this question or give me some hints? Appreciated!
2
votes
3answers
79 views

Related Theorem of Binomial Theorem

Proving that for any whole number $n$, the following identity holds: $$\sum^{n}_{i=1}{n\choose{i}}i=n\times2^{n-1}$$ So, I memorized this formula for preparing for math contests, but I think it's ...
2
votes
2answers
106 views

What is the largest value of $n$ for which $2n + 1$ is a factor of $122 + n^{2}$?

Given that $n$ is a natural number, what is its largest value such that $2n + 1$ is a factor of $122 + n^{2}$?
1
vote
1answer
65 views

Prove $1 + \sum_{i=0}^n(\frac1{x_i}\prod_{j\neq i}(1+\frac1{x_j-x_i}))=\prod_{i=0}^n(1+\frac1{x_i})$

Prove the identity $$1 + \sum_{i=0}^n \left(\frac1{x_i}\prod_{j\neq i} \left(1+\frac1{x_j-x_i} \right) \right)=\prod_{i=0}^n \left(1+\frac1{x_i} \right)$$ and hence deduce the inequality in Problem ...
9
votes
2answers
291 views

1965 Putnam, B2

The problem statement: Suppose $n$ players engage in a tournament in which each player plays every other player in exactly one game, to a win or a loss. Let $w_i$ and $l_i$ denote the wins and ...
7
votes
1answer
98 views

Find the sum of the first ten terms

How do I find the sum below? $$\sum_{i=1}^{10}\frac{2i+1}{i^2(i+1)^2}$$ I think there should be a simpler way instead of just adding the ten terms up using brute force, since it's impossible to do ...
2
votes
3answers
108 views

Factor $(x+y)^7-(x^7+y^7)$

So I was doing some practice problems to prepare upcoming math contests. This is one of the problems: Factor $(x+y)^7-(x^7+y^7)$ I got zero for $(x+y)^7-(x^7+y^7)$, however, the solutions ...
3
votes
1answer
153 views

Solve the simultaneous equations $x + \frac{3x-y}{x^2+y^2} = 3 $, $ y – \frac{x+3y}{x^2+y^2} = 0$

Find all solution in $\mathbb{R}$ for the following system of equations: \begin{cases} x + \frac{3x-y}{x^2+y^2} = 3 \\ y – \frac{x+3y}{x^2+y^2} = 0 \end{cases} I've tried few method, but none ...
1
vote
1answer
126 views

Find conditions on a,b,c so that p(x) and q(x) have exactly 2 roots in common. Also solve the equation p(x)=0

Let $p(x) = x^4 +ax^3 +bx^2+cx +1$ and $q(x) = x^4 +cx^3 +bx^2+ax+1$ with a,b,c real numbers.Find conditions on a,b,c so that p(x) and q(x) have exactly 2 roots in common. Also solve the equation ...
2
votes
2answers
69 views

Solving $y^2(x^2+1) +x^2(y^2+16) =448$

$y^2(x^2+1) +x^2(y^2+16) =448$ The task is to find all solutions in integers $(x,y)$. This is the fourth question of rmo 1st stage.The solution here is not complete. I have tried to solve unable to. ...
-4
votes
3answers
198 views

In how many ways can $16 be divided among 4 people? [closed]

In how many ways can $16 be divided among 4 people, assuming that each person has to get something and there are 5 cent coins and up
1
vote
2answers
220 views

Mathematical olympiad combinatorics question

In a problem set about various topics on combinatorics, geometry and algebra, I found this one There is a $6\times6$ grid, each square filled with a grasshopper. After the bell rings, each ...
3
votes
2answers
135 views

Contest math integer doublet equation

Can anyone help me with this? Find all ordered pairs $(x, y)$ of positive integers $x$, $y$ such that $$x^2 + 4y^2 = (2xy − 7)^2$$
6
votes
4answers
199 views

Contest math problem algebra proof

Let $r, s$ be integers and let $$a = (2011)^2 + (2011)r + s$$ and $$b = (2012)^2 + (2012)r + s$$ Show that there exists an integer $c$ with $c^2 + rc + s = ab$. Can anyone help me with this?
2
votes
1answer
129 views

Where can I find Putnam competition questions and solutions online?

Math people: Until recently, at least, there existed at least one Web page containing complete Putnam competition problems and solutions from the past twenty years or so. In retrospect, I see that I ...
7
votes
2answers
211 views

Compositeness of $n^4+4^n$ [duplicate]

My coach said that for all positive integers $n$, $n^4+4^n$ is never a prime number. So we memorized this for future use in math competition. But I don't understand why is it?
0
votes
1answer
97 views

Is it appropriate to use conjectures in contest?

Is it appropriate to use conjectures in math contest? I've been to fair amount of math contest and as far as I know the judges want a solid proof for every step take to make to solve the problem. But ...
4
votes
1answer
119 views

Let ABC be an acute angled triangle

Let ABC be an acute angled triangle; AD be the bisector of ∠BAC with D on BC; BE be the altitude from B on AC. Show that ∠CED > 45°
0
votes
0answers
61 views

How would you call this? (Language-Question)

if you are required to proof something in mathematical problem solving, like Olympiade style to-proof problems. For example, "Proof that you can divide every triangle can be divided into 5 ...
0
votes
3answers
106 views

Geometry proof contest math 2b

Could someone help me with this? Let A, B, C, D, L, M, N be distinct points in the plane such that A, B, C, D are the vertices of a square with sides AB, BC, CD, DA and L, M, N lie on the sides AB, ...
4
votes
2answers
124 views

sum of 14 4th powers and sum of 14 cubes

Prove that $4(x_1^4 + x_2^4 + x_3^4 + \dots + x_{14}^4) = 7(x_1^3+ x_2^3 + x_3^3 + \dots + x_{14}^3)$ has no solution in positive integers. Hint : suppose on the contrary $\sum_{k=1}^{14} {(x_k^4 - ...
3
votes
0answers
47 views

Math contest geometry proof problem 2 [duplicate]

Could someone help me with this? Let A, B, C, D, L, M, N be distinct points in the plane such that A, B, C, D are the vertices of a square with sides AB, BC, CD, DA and L, M, N lie on the sides AB, ...
6
votes
2answers
553 views

Olympiad number theory problem

I found this problem in previous problems of the olympiads of my country If $t^2+n^2=r^2$, where $t$ has $3$ positive divisors, $n$ has $30$ positive divisors and $t,n,r$ are natural numbers, ...
11
votes
4answers
366 views

Find all functions $f(x+y)=f(x^{2}+y^{2})$ for positive $x,y$

Find all functions $f:\mathbb{R}^{+}\to \mathbb{R}$ such that for any $x,y\in \mathbb{R}^{+}$ the following holds: $$f(x+y)=f(x^{2}+y^{2}).$$
8
votes
1answer
155 views

An exponential “rearrangement” inequality: $x^x+y^y>x^y+y^x$

Let $x,y$ be distinct real numbers greater than $0$. Prove $$x^x+y^y>x^y+y^x .$$ Source: I think it comes from a Russian test given in 1991, but I haven't been able to verify this.
11
votes
5answers
517 views

$g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$.

Find all functions $g:\mathbb{R}\to\mathbb{R}$ with $g(x+y)+g(x)g(y)=g(xy)+g(x)+g(y)$ for all $x,y$. I think the solutions are $0, 2, x$. If $g(x)$ is not identically $2$, then $g(0)=0$. I'm trying ...
6
votes
2answers
125 views

Arguing about a homework problem correctness

I've recently completely a homework in a problem solving class, I think my reasoning is correct but my teacher insisted that my answer is incorrect. I'm not sure if I'm correct or not. Question: ...
3
votes
1answer
131 views

Math contest proof equation problem

Could someone help me with this? If $m$ and $n$ are positive integers, then show that $$\frac{m}{ \sqrt n}+ \frac{m}{\sqrt[4]{n}} \neq 1$$.