Questions tagged [contest-math]

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

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Australian Maths Competition Area Question

PQRS is a square. T and U are midpoints of the sides PS and PQ respectively. TQ, SU and PR intersect at V. This is a question from the 2009 Intermediate Division AMC paper. I was given it in a ...
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1answer
78 views

My Solution for IMO 1988 Problem 3

The problem states: $\text{ 3. A function f is defined on the positive integers}$$ $$\text{ (and taking positive integer values) is given by }$ $$f(1) = 1, f(3) = 3,$$ $$f(2n) = f(n),$$ $$f(...
0
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1answer
62 views

If $a^{2} + b^{2} = c^{2}$ then show that $ 3 | ab $ [duplicate]

If $a^{2} + b^{2} = c^{2}$ then show that $ 3 | ab $ Attempt I'll try by contradiction. If $3$ does not divide $ab$ then $ab = 3m + n$. So we have $$ (a+b)^{2} = a^{2} + b^{2} + 2ab = a^{2} + b^{...
-1
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3answers
97 views

$n$ and $m$ are integers and $m >1$. Given: $2^n −2=m(m+1)$, Prove that $n$ can’t be an even number . [on hold]

$n$ and $m$ are integers and $m >1$ we have : $2^n −2=m(m+1).$ Prove that $n$ can’t be an even number .
2
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1answer
77 views

Finding number of cycles in permutation corresponding to translation in a group [University Olympiad]

here is a question from our university olympiad practice manual which is intended to help us prepare for upcoming inter-university math Olympiad. I think I can see, solution would be similar to Proof ...
0
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2answers
54 views

Prove that $A, P, Q$ are collinear.

Two circles $\omega_1$, $\omega_2$ intersect at $A, B$. An arbitrary line through $B$ meets $\omega_1$, $\omega_2$ at $C, D$ respectively. The points $E, F$ are chosen on $\omega_1$, $\omega_2$ ...
4
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3answers
84 views

Providing divisibility condition given fraction identity

If $x,y,z$ are positive integers satisfying $$\frac{1}{x^2}+\frac{1}{y^2}=\frac{1}{z^2}$$ prove that $20{\,\mid\,}xy$. My work: Expanding, we find $$(xz)^2+(yz)^2=(xy)^2$$ I know the Pythagorean ...
5
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3answers
94 views

Minimising expression with absolute values

Let $x,y,z$ be distinct reals and consider the expression $$L=\frac{(|x|+|y|+|z|)^3}{|(x-y)(x-z)(y-z)|}$$ Find the minimum possible value of $L$ over all $(x,y,z)$. My work Using a calculator, the ...
3
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1answer
84 views

Using induction, how to find all positive number $n$ such that $5^{n} > n!$?

Using induction, how to find all positive number $n$ such that $$ 5^{n} > n!$$ Attempt: I found this problem in the Induction topic in an education material. I know starting that $n=12$ then $5^...
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1answer
27 views

Showing that the number of queues is uniquely expresssed as the product of $2010$ positive integers.

This problem came from a friend in preparation for a contest. There are $2011$ people in a queue lining up for a conference, and no two people have the same height. Bob is the $27$th tallest person ...
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3answers
56 views

Let $m$ and $n$ be positive integers such that $m(n-m)=-11n+8$ . Find the sum of all possible values of $m-n$.

Let $m$ and $n$ be positive integers such that $m(n-m)=-11n+8$ . Find the sum of all possible values of $m-n$. after manipulation you get the quadratic $0=m^2-mn+(8-11n)$ from that you get $m=\frac{n ...
2
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1answer
69 views

Tips for discrete mathematics in Contests and a example problem.

In the last time I did lot's of preparations for future math contests and I discovered a problem where I don't know how to start with. Here is the problem: In a spa there are 100 showers. In ...
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0answers
176 views

Inequality $a^{\tan(b)}+b^{\tan(c)}+c^{\tan(a)}\leq 3(\frac{1}{3})^{\tan(\frac{1}{3})}$ [on hold]

Hi I have a new problem this is the following : Let $a,b,c>0$ such that $a+b+c=1$ then we have : $$a^{\tan(b)}+b^{\tan(c)}+c^{\tan(a)}\leq 3(\frac{1}{3})^{\tan(\frac{1}{3})}$$ I know that $f(...
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0answers
75 views

Inequality $\frac{x}{x^{10}+1}+\frac{y}{y^{10}+1}+\frac{z}{z^{10}+1}\leq \frac{3}{2}$

I try to prove this : Let $x,y,z>0$ such that $xyz=1$ then we have : $$\frac{x}{x^{10}+1}+\frac{y}{y^{10}+1}+\frac{z}{z^{10}+1}\leq \frac{3}{2}$$ You can follow my proof here.It works too but ...
4
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1answer
129 views

Find the five digit number $N$ by the clues!

I was preparing for a competition, and I encountered the following problem. In a classroom, the teacher said to five students, Alan, Bob, Carl, Dick and Eason, ‘I have written down a five-digit ...
3
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3answers
162 views

Find all additive real valued functions such that $f(x^{2019})=f(x)^{2019}$

The following is the final problem from this page: Find all the functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $$f(x+y)=f(x)+f(y) \; \; \; \forall \,x,y\in \mathbb{R}$$ and also (this is ...
1
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1answer
76 views

Find all the integer pairs $(x,y)$ that satisfy the equation $7x^2-40xy+7y^2=(|x-y|+2)^3$ [duplicate]

Find all the integer pairs $(x,y)$ that satisfy the equation $7x^2-40xy+7y^2=(|x-y|+2)^3$ it is clear the equation is symmetric therefore you can assume w.l.o.g that $x\ge y$ which makes the new ...
1
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1answer
62 views

$ \cos(2 \pi x)$, $ \cos(4 \pi x)$, $ \cos(8 \pi x)$, $ \cos(16 \pi x)$, and $ \cos(32 \pi x)$ are all nonpositive

$ \cos(2 \pi x)$, $ \cos(4 \pi x)$, $ \cos(8 \pi x)$, $ \cos(16 \pi x)$, and $ \cos(32 \pi x)$ are all nonpositive. What is the smallest positive $ x$? I tried ranges for each of cos(?)..that did not ...
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2answers
34 views

points $R$ and $T$ lie on the side $CD$ of the parallelogram $ABCD$ such that $DR= RT= TC$ what is the area, in $cm^2$ , of the shaded region?

points $R$ and $T$ lie on the side $CD$ of the parallelogram $ABCD$ such that $DR= RT= TC$ . Lines $AR$ and $AT$ intersect the extension of $BC$ at points $M$ and $L$ respectively, and the lines $BT$ ...
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1answer
103 views

Brazilian Math Olympiad question about batteries and a torch [closed]

This question comes from the 27th Brazilian Mathematical Olympiad (2005). We have four charged batteries, four uncharged batteries, and a radio which needs two charged batteries to work. We do not ...
6
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1answer
71 views

Constructions | Olympiad Geometry

Let $ABCD$ be a parallelogram with no angle equal to $60^\circ$. Find all pairs of points $(E,F)$ such that $AE = BE$, $BF = CF$, and triangle $DEF$ is equilateral. So far after making a couple of ...
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0answers
72 views

Problem in downvote system

Problem For my game, I'm building a system where players have power/weight, and they can downvote each other, players with 66% of downvote weight are banned. The weight of the votes is calculated ...
10
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1answer
109 views

Understanding a complex number proof that there is a convex equiangular $1990$-gon with sides $1^2$, $2^2$, $\ldots$, $1990^2$

IMO 1990, Problem B3 reads: Prove that there exists a convex $1990$-gon such that all its angles are equal and the lengths of the sides are the numbers $1^2$, $2^2$, $\ldots$, $1990^2$ in some ...
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2answers
86 views

For positive reals $a_i$ and $b_i$, if $\sum a_i \geq \sum a_i b_i$, then $\sum a_i \leq \sum\frac{a_i}{b_i}$

Given $a_1$, $a_2$, $a_3$, $\ldots$, $a_n$ and $b_1$, $b_2$, $b_3$, $\ldots$, $b_n$ positive real numbers such that $$a_1 + a_2 + a_3 +\cdots+ a_n \geq a_1b_1+a_2b_2+a_3b_3...a_nb_n$$ show that: $$...
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2answers
82 views

there are some integer pairs $(m,n)$ that satisfy $\frac{m^2+mn+n^2}{m+2n}=\frac{13}{3}$ Find the value of $m + 2n$

there are some integer pairs $(m,n)$ that satisfy $$\frac{m^2+mn+n^2}{m+2n}=\frac{13}{3}.$$ Find the value of $m + 2n.$ I tried expressing the numerator in terms of $m+2n$ but it resulted in nothing ...
4
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1answer
168 views

For positive $a$, $b$, $c$ with $abc=1$, show that $\sum_{cyc}\sqrt{a^2-a+1}\geq a+b+c$

Let $a,b,c$ are positive number such that $abc=1$. Prove that: $$\sqrt{a^2-a+1}+\sqrt{b^2-b+1}+\sqrt{c^2-c+1}\;\geq\; a+b+c$$ This problem froms my Math teacher. I have attempted to let $$(a,b,c)=(\...
2
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0answers
40 views

Redux: An interesting four sum inequality (feat. probability)

This post asked for help in proving the inequality \begin{align*} \left(\sum \limits_{k=1}^n (2k-1)\frac{k+1}{k}\right) \left( \sum \limits_{k=1}^n (2k-1)\frac{k}{k+1}\right) \le n^2 \left(\sum \...
2
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1answer
37 views

Prove $ (x^{3} + x^{4} +y^{3} + y^{4} + z^{3} + z^{4})^{3} \le (x^{9} + y^{9} + z^{9})[ (1+x)^{3/2} + (1+y)^{3/2} + (1+z)^{3/2} ]^{2} $

Prove $$ (x^{3} + x^{4} +y^{3} + y^{4} + z^{3} + z^{4})^{3} \le (x^{9} + y^{9} + z^{9})[ (1+x)^{3/2} + (1+y)^{3/2} + (1+z)^{3/2} ]^{2} $$ if $x,y,z \ge 0$ I made this problem using Holders ...
1
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2answers
55 views

Prove $(x^{3} + x^{4} +y^{3} + y^{4} + z^{3} + z^{4})^{2} \le (x^{6} + y^{6} + z^{6})[ (1+x)^{2} + (1+y)^{2} + (1+z)^{2} ] $

Prove $$ (x^{3} + x^{4} +y^{3} + y^{4} + z^{3} + z^{4})^{2} \le (x^{6} + y^{6} + z^{6})[ (1+x)^{2} + (1+y)^{2} + (1+z)^{2} ] $$ if $x,y,z \ge 0$ I made this problem using Holders inequality, notice ...
5
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3answers
99 views

Given $x, y$ that $xy-\frac{x}{y^2}-\frac{y}{x^2}=3$, work out $xy-x-y$.

Today I had a competition in Xiamen, China. I know how to do the questions except this strange equation. Given $x, y$ that $xy-\dfrac{x}{y^2}-\dfrac{y}{x^2}=3$, work out $xy-x-y$. Such a strange ...
1
vote
1answer
55 views

Determine all positive integer solution sets $(k, n, l, m)$ to the equation $(1 + n^k)^l=1+n^m$ where $l>1$

Determine all positive integer solution sets $(k, n, l, m)$ to the equation $(1 + n^k)^l=1+n^m$ where $l>1$ I worked out that (1,2,2,3) is a solution and is the only solution where $n=l=2$ because ...
0
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2answers
62 views

How many solution are there to the equation $n_1^4+n_2^4+…+n_{16}^4=65536$ with non-negative integers

How many solution are there to the equation $n_1^4+n_2^4+...+n_{16}^4=65536$ with non-negative integers ($n_1,n_2,...,n_{16}$), of which at least two are consecutive? I know $65536=2^{16}=16^4$ but ...
0
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1answer
63 views

Integration of $\frac{1}{1-\cos(\alpha)\cos x }$

Integration of $\dfrac{1}{1-\cos(\alpha)\cos x }$ w.r.t $x$ How to do this problem? I was trying to reduce it $\dfrac{1}{1-\cos(\alpha)\cos (x) }=\dfrac{\sec (\alpha)}{\sec (\alpha)-\cos (x) }$. But ...
0
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1answer
36 views

An inequality with sides of a triangle

Let $a,b$ and $c$ be the sides of a triangle. Prove that $3a^2b + 3b^2c + 3c^2a - 3abc -2b^2a - 2c^2b - 2a^2c \ge0$ I tried to use $a = x+y, b = x + z$ and $c = y+z$ substitutions but it didn't ...
1
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2answers
60 views

$f_1(k)=$ square of sum of digits of $k$; $f_n(k)=f_1(f_{n-1}(k))$; find $f_{1988}(11)$

For any positive integer $k$, let $f_1(k)$ denote the square of the sum of the digits of $k$. For $n>1$, let $f_n(k)=f_1(f_{n-1}(k))$. Find $f_{1988}(11)$ I got the answer as 256 can anybody ...
0
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1answer
61 views

Inequalities: BMO 2009/10 Round 2 [duplicate]

BMO 2009/10 Round 2 Q.4 asks Prove that for all positive reals $x,y, z$: $$4(x+y+z)^3 >27(x^2y + y^2z + z^2x)$$ My try:- Using AM-GM, LHS is greater than $108xyz$. Using AM-GM, RHS is greater ...
0
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1answer
650 views

No.of pairs of parallel diagonals

Today (11/08/19) was PRMO 2019. Q.15 asks In how many ways can a pair of parallel diagonals of a regular polygon of 10 sides be selected? I could not visualise the polygon during the test, so I ...
5
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1answer
63 views

4 distinct integers with prime sum for each triple

Here is a nice high school olympiad math problem: Can you choose 4 distinct positive integers so that the sum of each 3 of them is prime? How about 5? It looks that just by looking at reminders mod ...
1
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1answer
28 views

Find the number of possible partitions.

Find the number of possible partitions of word ‘MATHEMATICS’ such that each contains atleast one vowel.For example MA-THEMATICS ,MATHEMATICS ,MATH-EMA-TICS ,MATHEMATICS are some of the possible ...
2
votes
2answers
126 views

Inequality $\frac{\ln(7a+b)}{7a+b}+\frac{\ln(7b+c)}{7b+c}+\frac{\ln(7c+a)}{7c+a}\leq \frac{3\ln(8\sqrt{3})}{8\sqrt{3}}$

I'm interested by the following problem : Let $a,b,c>0$ such that $abc=a+b+c$ then we have : $$\frac{\ln(7a+b)}{7a+b}+\frac{\ln(7b+c)}{7b+c}+\frac{\ln(7c+a)}{7c+a}\leq \frac{3\ln(8\sqrt{3})}{8\...
1
vote
1answer
56 views

Find four distinct positive integers whose product is divisible by the sum of every pair of them.

The teacher gave us British Mathematical olympiad $1992$ Round $1$ Problem $3$. Find four distinct positive integers whose product is divisible by the sum of every pair of them. Can you find a ...
9
votes
2answers
88 views

Are functions that sum to zero over vertices of similar polygons identically zero?

This is a generalization of problem A1 from the 2009 Putnam competition. The original problem asks for proof that any function $f:\mathbb{R}^2\rightarrow \mathbb{R}$ that sums to zero over the ...
0
votes
4answers
129 views

Given $x+y+z=3, x,y,z>0 $ how to prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x} >= x^2+y^2+z^2$

Given $x+y+z=3, x,y,z>0 $ how to prove that $\frac{x}{y}+\frac{y}{z}+\frac{z}{x} >= x^2+y^2+z^2$ ? I tried something basic like $(x+y+z)^2 = x^2+y^2+z^2 + 2xy+2zx+2yz$, so we just need to ...
1
vote
2answers
112 views

From $D$ on any side of $\triangle ABC$, drop perpendiculars $DP$, $DQ$ to other sides. Find minimum value of $|PQ|$.

Israel Olympiad: Lengths of the sides of $\triangle ABC$ are $4$, $5$, and $6$. At any point $D$ on any side, drop perpendiculars $DP$ and $DQ$ to the other sides. Determine the minimum value of $|...
1
vote
1answer
115 views

Max of $(a-x^2)(b-y^2)(c-z^2)$ when $x+y+z=a+b+c=1$, $x,y,z,a,b,c \geq 0$

What's the maximum value of $$(a-x^2)(b-y^2)(c-z^2)$$ given $x+y+z=a+b+c=1$, $x,y,z,a,b,c \geq 0$ The tricky part, as you could see in one of the attempted answer, is how to handle the case when ...
3
votes
0answers
96 views

Constant function given an inequality [closed]

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ that satisfies $f(x + y)\leq yf(x) + f(f(x))$ for all $x,y\in\mathbb{R}$. Show that $f(x)=0$ for all $x\leq0$. I tried setting y = 0 in an attempt to prove ...
3
votes
2answers
54 views

Find every $n$: $n^2 + 340 = m^2$

Let $n$, $m \in N$. The problem asks to find every natural number $ n $ such that: $ n^2 + 340 = m^2 $ I tried to solve the equation like this: $ n^2 - m^2 = 340 $ $ (n + m)(n - m) = 2^2 * 5 * 17 ...
2
votes
3answers
184 views

Solve $a!b!=a!+b!+c!$ where $a$, $b$ and $c$ are nonnegative integers.

My teacher in Math Team gave the following question to us. Solve $$a!b!=a!+b!+c!$$ where $a$, $b$ and $c$ are nonnegative integers. I found only one solution by trial and error and it is $(a,b,c)=(...
2
votes
1answer
58 views

Find the number of polynomials $P(x)$ with coefficients $0,1,2,3$ such that $P(2) = n$.

Find the number of polynomials $P(x)$ with coefficients $0,1,2,3$ such that $P(2) = n$. Only what I currently know is that I should consider polynomials with the smallest degree $k$, where $k$ is ...
2
votes
0answers
73 views

Perfect squares of the form $ab^n+c$ and a Diophantine equation

The motivation for this question comes from the following problem from an international Team selection test of 2007 from Chile: Problem: Let $p$ be a prime number. Find all pairs of positive ...