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

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2answers
51 views

Number theory recursion congruence problem.

this is a problem a friend of mine asked me: for any integer $n: a_1=n $ and for $a_k$ and $k$ an integer such that $k>1$ we have a_k the only integer such that $0\leq a_k<k$ and ...
4
votes
2answers
173 views

Exercise sum equal to 1 using only the digits 1,2, 3,…,9

Give a method to write the number one as the sum of three fractions, where each fraction the numerator is a one-digit number the denominator is a two-digit number and numbers that can be used are from ...
5
votes
1answer
197 views

Tricky Puzzle!! Please help.

I stumbled upon a puzzle I can't crack. It goes like this: In a certain Code language: 7321=6 5342=3 8645=15 Then 9312=? The Answer is 9. But I can't seem to find the logic behind it??
4
votes
1answer
124 views

Number Theory Contest Problem

Given that $x, y$ are positive integers with $x(x + 1)\mid y(y + 1)$, but neither $x$ nor $x + 1$ divides either of $y$ or $y + 1$, and $x^2+ y^2$ as small as possible, find $x^2+ y^2$. I have tried ...
3
votes
2answers
156 views

Number Theory Contest Math

Find the smallest positive integer $n$ such that $n^4 + (n + 1)^4$ is composite. Find the sum of the first $5$ positive integers $n$ such that $n^2 - 1$ is the product of 3 distinct primes. Answer to ...
7
votes
7answers
553 views

If $a^3 + b^3 +3ab = 1$, find $a+b$

Given that the real numbers $a,b$ satisfy $a^3 + b^3 +3ab = 1$, find $a+b$. I tried to factorize it but unable to do it.
4
votes
2answers
149 views

Minimum number of coins to ensure 10 coins of one type are selected

One coin is labelled with the number $1$, two different coins are labelled with the number $2$, three different coins are labelled with the number $3$, $\ldots$ , forty-nine different coins are ...
2
votes
2answers
104 views

Finding the index such that all partial sums are nonnegative

Given an array a[] of integers of arbitrary size N that sum to 0 (for example, a[] = {-1, 0, 5, 3, -9, 2}), does there always exists an index i ($0\le i\le N-1$) such that each partial sum $S_j = ...
2
votes
2answers
195 views

Solving two simultaneous equations

Suppose that $x$, $y$ and $z$ are three integers (positive,negative or zero) such that we get the following relationships simultaneously $x + y = 1 - z$ and $x^3 + y^3= 1 - z^2$ Find all such ...
0
votes
1answer
65 views

Partition of circumference into $3k$ arcs

The following problem is from 1982 Russian Mathematical Olympiad. If you go to this link, and scroll down to the section Russian Math Olympiad, then this is Problem 333 in that text-file. Let $k$ ...
4
votes
1answer
205 views

Online Math Open Contest 2 Problem 50

In tetrahedron $SABC$, the circumcircles of faces $SAB$, $SBC$, and $SCA$ each have radius $108$. The inscribed sphere of $SABC$, centered at $I$, has radius $35.$ Additionally, $SI = 125$. Let $R$ be ...
7
votes
2answers
138 views

Compute $\int_0^1\int_0^1…\int_0^1\lfloor{x_1+x_2+…+x_n}\rfloor dx_1dx_2…dx_n$

Compute $\int_0^1\int_0^1...\int_0^1\lfloor{x_1+x_2+...+x_n}\rfloor dx_1dx_2...dx_n$ where the integrand consists of the floor (or greatest integer less than or equal) function. The case $n=1,2,3$ ...
0
votes
1answer
80 views

Choosing a Set of r elements from a set having n elements.

Define a set $X$={$1$,$2$,$...$,$n$} . Determine the number of ways of selecting a subset of $X$ such that it contains no consecutive integers .
9
votes
4answers
448 views

“If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer”

If $1/a + 1/b = 1 /c$ where $a, b, c$ are positive integers with no common factor, $(a + b)$ is the square of an integer. I found this question in RMO 1992 paper ! Can anyone help me to prove ...
0
votes
1answer
56 views

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$.

Find all $(a,b,c)\in\mathbb{Z}^3$ such that $b^2-4ac=-20$, and either of the following is true: $-|a|<b\le|a|<|c|$, or $0\le b\le|a|=|c|$. We see that if $(a,b,c)$ is a solution, then so is ...
16
votes
1answer
177 views

$|3^a-2^b|\neq p$, from a contest

I recently came across an old contest problem: (I did not find the solution anywhere) Find the least prime number which cannot be written in the form $|3^a-2^b|$ where $a$ and $b$ are ...
1
vote
2answers
140 views

Prove that $\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0$.

Let $f$ be a continuous, nonnegative, real-valued function and $$\int_0^{\infty}f(x)dx<\infty.$$ Prove that $$\lim_{n\to\infty}\frac1{n}\int_0^{n}xf(x)dx=0.$$ A start: If ...
10
votes
2answers
259 views

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$, $f(x)f(yf(x))=f(x+y)$

Find all functions $f:\mathbb{R}^+\to \mathbb{R}^+$ such that for all $x,y\in\mathbb{R}^+$$$f(x)f(yf(x))=f(x+y)$$ A start: set y=0 to get $f(x)f(0)=f(x)$. So $f(0)=1$ unless $f$ is identically zero.
1
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2answers
79 views

Determine the value of $p>0$ for which $\sum_{n=1}^{\infty}(-1)^{\lfloor{\sqrt{n}}\rfloor}/n^p$ converges.

Determine the value of $p>0$ for which $$\sum_{n=1}^{\infty}\frac{(-1)^{\lfloor{\sqrt{n}}\rfloor}}{n^p}$$ converges. By considering $\lfloor{\sqrt{n}}\rfloor$, we see the series is $$\sum_{k\ge1} ...
34
votes
1answer
660 views

Integral $\int_0^1\frac{x^9\left(x^4+x^2-x-1-5\ln x\right)}{\left(x^{10}-1\right)\ln x}\mathrm dx$

A friend of mine sent me an integral that she had not been able to crack, and me neither. It comes with a result, but without a proof (I suppose it originated in some math contest). Could you please ...
1
vote
2answers
422 views

A truth teller and liar puzzle of Ramanujan mathematical olympiad 2013

On an island each person always tells the truth or each person always tells a lie. Three people say $A$ , $B$ and $C$ have a conversation. $A$ says that $B$ is lying , $B$ says that $C$ is lying and ...
1
vote
2answers
137 views

How find this maximum of $P_{1}+P_{n}$

Question $n$ students attend a test of $m$ problems where $m, n \ge 2$. The scoring rule for each problem is: If $x$ students answer a problem incorrectly, then a correct answer worth $x$ points ...
1
vote
2answers
74 views

Square root of decimal Places

I'm searching to find how to get the sqaure root of a number having decimal places. How to find the Square root of say $0.4$?
1
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1answer
89 views

Find functions such that under the Cartesian coordinate system $F(x, y) = f(x) g(y)$ but under the polar coordinate system $F(x, y) = h(r)$.

Find all non-constant function $F(x, y)\in C^2(\mathbb{R}^2)$ such that under the Cartesian coordinate system $F(x, y) = f(x)  g(y)$ but under the polar coordinate system $F(x, y) = h(r)$. My ...
10
votes
2answers
377 views

Another math contest problem: $\int_0^{\frac{\ln^22}4}\,\frac{\arccos\frac{\exp\sqrt x}{\sqrt2}}{1-\exp\sqrt{4\,x}}dx$

Prove: $$ {\Large\int_{0}^{\ln^{2}\left(2\right) \over4}}\, \frac{\arccos\left(\vphantom{\huge A} {\exp\left(\vphantom{\large A}\sqrt{x\,}\right) \over \sqrt{\vphantom{\large A}2\,}}\right)} ...
1
vote
1answer
87 views

Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge.

Let $\{a_n\}$ be any sequence of positive real numbers. Suppose for all $n$, $a_{n+1}\le a_n + \frac1{n^p}$. Find all positive $p$ such that we can guarantee $\{a_n\}$ always converge. For example, ...
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
346 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
248 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 ...
72
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
112 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
53 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
155 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
299 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
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2answers
109 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
61 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
405 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
84 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
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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
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1answer
93 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
167 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
132 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
213 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
60 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
98 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
163 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
81 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 ...