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

learn more… | top users | synonyms (2)

4
votes
0answers
66 views

Smallest value that a certain variable can take in a system of equations.

Consider the solutions $(x,y,z,u)$ of the system of equations: $$\begin{cases} x+y=3(z+u)\\ x+z=4(y+u)\\ x+u=5(y+z)\\ \end{cases}$$ where $x,y,z \text{ and } u$ are positive integers. What ...
9
votes
4answers
2k views

Find coefficient of $x^8$ in $(1-2x+3x^2-4x^3+5x^4-6x^5+7x^6)^6$

Find coefficient of $x^8$ in $(1-2x+3x^2-4x^3+5x^4-6x^5+7x^6)^6$ how to do it? I think it should be $3^6$ since $(3x^2)^6=3^6x^8$. (this is false) Is this true?
1
vote
1answer
140 views

international mathematical competition for college students

I randomly came across with the following problems: Let $A,B \in M_n (\mathbb{C})$ such that $A^2B+B^2A=2ABA.$ Prove that $(AB-BA)^k=0$ for some positive integer $k$. The proof is as follows: Let ...
4
votes
2answers
106 views

Finding the ratio of areas produced by perpendiculars from the $3$ sides of an equilateral triangle.

A point O is inside an equilateral triangle $PQR$ and the perpendiculars $OL,OM,\text{and } ON$ are drawn to the sides $PQ,QR,\text{and } RP$ respectively. The ratios of lengths of the ...
2
votes
2answers
157 views

AIME number theory problem (unique factorization domains)

I'd greatly appreciate some help with the following problem, from a mock AIME I took. Compute the largest squarefree positive integer $n$ such that $\mathbb{Q}(\sqrt{-n})\cap \overline{\mathbb{Z}}$ ...
7
votes
1answer
180 views

A functional relation which is satisfied by $\cos x$ and $\sin x$

Assume that the functions $f,g : \mathbb R\to \mathbb R$ satisfy the relations \begin{align} \left\{ \begin{array}{ll} f(x+y) &=& f(x)f(y)-g(x)g(y), \\ g(x+y) &=& f(x)g(y)+f(y)g(x), ...
19
votes
2answers
704 views

Which functions satisfy the equation $\,\,f(x)\,f(y)-f(x+y)=\sin x\,\sin y\,$?

Find all continuous functions $f$ which satisfy the functional equation $$ f(x)\,f(y)-f(x+y)=\sin x\,\sin y, $$ for all $x,y\in\mathbb R$. I can prove that $f(n\pi)=\cos\left(n\pi\right)$ for all ...
2
votes
1answer
129 views

Characterization of arithmetic mean

Let $f_m$: $\mathbb{R}_{\geq 0}^m \to \mathbb{R}_{\geq 0}$ be a series of functions that satisfy symmetry (when permuting indices), strong monotonicity (in every entry), homogeneity of degree 1, ...
2
votes
1answer
138 views

How prove this convex quadrilateral $ABCD$ is rhombus.

In a convex quadrilateral $ABCD$ ,$AB\cap CD=O$, let $r_{1},r_{2},r_{3},r_{4}$ is the radius of inscribed circle in triangle $\Delta OAB,\Delta OAD,\Delta OBC,\Delta ODC$ respectively,such ...
11
votes
1answer
149 views

A Fantabulous integer is an integer which has another fantabulous integer smaller than it

BdMO 2013 problem-7: A positive integer is called “Fantabulous” if there is another fantabulous positive integer smaller than it. Find the number of fantabulous integers. I am bamboozled at ...
4
votes
0answers
68 views

Minimizing the distance between points in two sets

Given two sets $A, B\subset \mathbb{N}^2$, each with finite cardinality, what's the most efficient algorithm to compute $\min_{u\in A, v\in B}d(u, v)$ where $d(u,v)$ is the (Euclidean) distance ...
4
votes
1answer
83 views

Elegant proof of icosohedron property

This problem was question A1 on the 2013 Putnam contest. Is there a better way to solve this problem than just using pigeonhole principle? Specifically, is there a group theoretic way to interpret ...
4
votes
1answer
114 views

Question from Putnam '08: Given $F_n(x)$, find $\lim_{n\to\infty}\frac{n!F_n(1)}{\ln(n)}$

Problem Statement: Let $F_0(x) = \ln(x)$. For $n\ge0$ and $x\gt0$, let $F_{n+1}(x) = \int_0^xF_n(t)dt$. Evaluate $$\lim_{n\to\infty}\frac{n!F_n(1)}{\ln(n)}$$ Source: Putnam 2008, Problem B2. ...
0
votes
1answer
136 views

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$ for $|x-y|\le\alpha$.

Let $f(x)=\exp(-a|x|)$ and $a>0$. Show that there exists $C$ and $\alpha$ such that $$|f(x)-f(y)|\le\frac{C|x-y|}{1+x^2}$$ for $|x-y|\le\alpha$. From the mean value theorem, given any $x,y$ with ...
3
votes
1answer
95 views

Assuming on the AIME?

Is it OK to assume on the AIME competition? In geometry problems, could you assume that a trapezoid is isosceles or something like that? Could you give some examples, too? Thanks.
3
votes
1answer
286 views

Find a number leaving a particular remainder with 3 different numbers

I have the following question: Let $N$ be the greatest number that will divide $1305, 4665$ and $6905$, leaving the same remainder in each case. What is the sum of digits of $N$. My approach ...
0
votes
2answers
86 views

Inequality regarding areas of triangles

BdMO Nationals 2013: There is a point O inside ∆ABC. After joining A,O; B,O and C,O extend those line and they will intersect BC, AC and AB at points D, E and F respectively. ...
2
votes
1answer
68 views

Travelling to the point of origin without using the same road twice

BdMO 2013 Secondary: There are $n$ cities in a country. Between any two cities there is at most one road. Suppose that the total number of roads is $n$. Prove that there is a city such that ...
0
votes
4answers
70 views

How to prove this ineqality

prove that $1 \leq \frac{1}{1001} + \frac{1}{1002} + ......+\frac{1}{3001} \leq \frac{4}{3} $ it seems from some Olympiad. i tried using sum of series etc. but could not get it.
1
vote
1answer
49 views

Deducing a weight function from a set function.

I was working on this problem for which I think I have almost the solution, but if you could help me finish it, I would be so grateful. ${\bf{ Problem:}}$ So, $P$ is a nonempty collection of subsets ...
1
vote
1answer
72 views

Sum of an infinite geometric series?

BdMO Nationals 12: Each room of the Magic Castle has exactly one exit door.The rooms are designed such that when you can go from one room to the next one through a door, the second room's ...
1
vote
1answer
84 views

What kind of methods there are to solve a Diophantine equation from IMO longlist?

Namely, in IMO longlist 1987 were given the equation $3z^2=2x^3+385x^2+256x-58195$ and asked to find its integer points. How can I find those? I tried to substitute $z=12k,x=6t$ to get ...
3
votes
4answers
127 views

A Question regarding radius of circumcircle and sides of a triangle

NOTE: I am looking for a hint,not the whole solution. A question from BdMO Nationals 2012 Given triangle $ABC$, the square $PQRS$ is drawn such that $P$,$Q$ are on BC, $R$ is on $CA$ and $S$ on ...
7
votes
1answer
208 views

Korean Math Olympiad (Construct rectangle)

Prove that an $m$ × $n$ rectangle can be constructed using copies of the following shape if and only if $mn$ is a multiple of 8 where $m$ > 1 and $n$ > 1. My solution: starting from 2 × 4 and 3 × 8 ...
4
votes
2answers
71 views

finding the value of $f(\frac{1}{7})$

$f$ is a function mapping positive reals between $0$ and $1$ to reals. Let $f$ be given by, $f( \frac{x+y}{2} ) = (1-a)f(x)+af(y)$ where $y > x$ and $a$ being a constant. Also,$f(0) = 0$ and $f(1) ...
3
votes
2answers
166 views

How find this $a^3+b^3+c^3-20(a+3)(b+3)(c+3)=2013$ equation integer solution

if $a,b,c\in Z$,and $a\le b\le c$ and such $$\begin{cases} a+b+c=-3\\ a^3+b^3+c^3-20(a+3)(b+3)(c+3)=2013 \end{cases}$$ Find the value $3a+b+2c=?$ my try $$a+b+c=-3\Longrightarrow ...
4
votes
1answer
172 views

IMO Hong Kong TST 2014

Let $m,n$ be distinct positive integer not exceeding 2013 and $d$ be their gcd. Suppose $d^2|3(m-n)$. Find the greatest possible value of $d(m+n)$. I only know $m-n$ should be a perfect square, but ...
6
votes
2answers
103 views

Functional Equation f(x) = f(x/2)

Find all functions $f$ satisfying the property that $$ f(x) = f(x/2) $$ for all $x \in \mathbb{R}$ So far I've come up with the following assumptions: -$f$ is periodic, i.e of form $f(x) = A ...
1
vote
1answer
47 views

Checking whether the number is composite

Prove that $5^{125}-1$/ ($5^{25} - 1$) is composite I have written $5^{125}-1$ as $(5^{25}-1)(5^{100}+5^{75}+5^{50}+5^{25}+1)$ but what should I do after this? Sorry about earlier mistake in ...
2
votes
0answers
80 views

Is the number $n^7+7$ a perfect square ? [closed]

Prove that for any positive integer $n$, $n^7 + 7$ is never equal to a perfect square.
2
votes
1answer
66 views

sums of squares of integers

We have to prove that there exists infinitely many integers $a,b,c$ such that $a^2 + b^2 = c^2 + 3$ . This looked like a very straight-forward question . I did some algebraic manipulations but ...
12
votes
2answers
502 views

If 1 boy knows r girls and 1 girl knows r boys ,then number of boys=girls

Yet another question from BdMO 2013 Nationals: In a class,every boy knows $r$ number of girls and every girl knows $r$ number of boys.Show that there are equal number of boys and girls[Assume that ...
2
votes
1answer
81 views

Highest $n$ such that $2^n|a^{2012}+a^{2013}+a^{2014}+\cdots +a^{3012}$,$a=4k+2$

A question from BdMO 2013 Nationals: Let $a$ be an integer divisible by 2 but not divisible by 4. What is the largest positive integer n such that ...
2
votes
3answers
73 views

Proving $4(a^3 + b^3) \ge (a + b)^3$ and $9(a^3 + b^3 + c^3) \ge (a + b + c)^3$

Let $a$, $b$ and $c$ be positive real numbers. $(\mathrm{i})$ Prove that $4(a^3 + b^3) \ge (a + b)^3$. $(\mathrm{ii})$Prove that $9(a^3 + b^3 + c^3) \ge (a + b + c)^3.$ For the first one I ...
3
votes
2answers
105 views

2 is a primitive root mod $3^h$ for any positive integer $h$

It's easy to verify that 2 is a primitive root mod $3^2$. But then why does it follow that 2 is a primitive root mod $3^h$ for any positive integer $h$? This was used in the solution of 2009 Putnam ...
0
votes
2answers
78 views

How to prove $(F,+)$ and $(F\setminus \{0\},\cdot)$ aren't isomorphic, where $(F,+,\cdot)$ is an arbitrary field .

Assume $(F,+,\cdot)$ is an arbitrary field. How to prove $(F,+)$ and $(F\setminus \{0\},\cdot)$ aren't isomorphic? Thanks in advance.
2
votes
0answers
173 views

Korean Math Olympiad 2005 (trapezoid & tangent circles)

In a trapezoid $ABCD$ with $AD||BC$, $O_1$, $O_2$, $O_3$, $O_4$ denote the circles with diameters AB, BC, CD, DA, respectively. Show that there exists a circle with center inside the trapezoid which ...
0
votes
1answer
93 views

Figuring out an angle in an isosceles triangle

A problem from BdMO 2013: Let $ABC$ be an isoscles triangle with $AB=AC$.The bisector of $\angle B$ meets $AC$ at $D$.Given that $BC=BD+AD$,we need to figure out $\angle A$. If we consider ...
10
votes
2answers
124 views

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $\sum_{k=1}^n\frac{a_k-b_k}k$

Let $a_k=\frac1{\binom{n}k}$, $b_k=2^{k-n}$. Compute $$\sum_{k=1}^n\frac{a_k-b_k}k$$ By computing some partial sums, the answers are 0. It seems an inductive argument is possible.
1
vote
0answers
53 views

Find all rational solutions to $x^3 - y^2 = 2$. [duplicate]

Find all rational solutions to $x^3 - y^2 = 2$. The only integers solutions are $(3,\pm5)$: http://mathforum.org/library/drmath/view/51569.html
5
votes
2answers
87 views

Proof that b is not divisible by 6

$$b=\left \lfloor (\sqrt[3]{28}-3)^{-n} \right \rfloor$$ The brackets mean that the number is the largest integer smaller than $(\sqrt[3]{28}-3)^{-n} $ Proof that b is never divisible by 6. I have ...
16
votes
1answer
206 views

Closed form for $\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}}$

Let $$S=\sum_{n=0}^\infty\frac{\operatorname{Li}_{1/2}\left(-2^{-2^{-n}}\right)}{\sqrt{2^n}},\tag1$$ where $\operatorname{Li}_a(z)$ is the polylogarithm. For $a=1/2$ it can be represented as ...
0
votes
1answer
126 views

Why would the reflections of the orthocentre lie on the circumcircle?

Let ABC be a triangle which it not right-angled. Define a sequence of triangles $A_iB_iC_i$,with $i \geq 0$, as follows: $A_0B_0C_0$ is the triangle $ABC$; and, for $i \geq 0$, $A_{i+1}$, $B_{i+1}$, ...
2
votes
0answers
95 views

Rearranging numbered cards to reverse their order

I have been thinking about this question for a long time, but I can't solve it. Here is the question: We have $9$ cards, with numbers one to nine written on them (in the order $1, 2, \ldots , 9$). ...
0
votes
2answers
117 views

Coordinate Geometry Triangle

ABC is a triangle. BB$_1$ and CC$_1$ are angle bisectors of B and C respectively. E,F are feet of perpendiculars from A on BB$_1$ and CC$_1$ respectively. Suppose D is point at which incircle of ABC ...
1
vote
6answers
238 views

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root $a,b$ are real numbers. Roots may or may not be real. In this question, the aim is to find values of and b ...
1
vote
2answers
74 views

Prove $a = b = c$, given $P_1(x) = ax^2-bx-c$ , $P_2(x) = bx^2-cx-a$, $P_3(x)=cx^2-ax-b$ and $P_1(v)=P_2(v)=P_3(v)$

Prove $a = b = c$, given $P_1(x) = ax^2-bx-c$, $P_2(x) = bx^2-cx-a$, $P_3(x)=cx^2-ax-b$ and $P_1(v)=P_2(v)=P_3(v)$ where $v$ is a real number. $a,b,c$ are non zero real numbers.
8
votes
3answers
397 views

Proving $a^ab^b + a^bb^a \le 1$, given $a + b = 1$

Given $a + b = 1$, Prove that $a^ab^b + a^bb^a \le 1$; $a$ and $b$ are positive real numbers.
-1
votes
3answers
105 views

Show that $(A',B',C')$ form the vertices of an equilateral triangle.

Let $ABC$ be a triangle with $AB = AC $ and $angle BAC = 30.$ Let $(A')$ be the reflection of A in the line BC $(B')$ be the reflection of $B$ in the line CA $(C')$ be the reflection of C in the line ...
3
votes
1answer
129 views

Korean Math Olympiad 2000 (floor function, quadratic mod) [closed]

Let $p$ be a prime such that $p ≡ 1\ (\mathrm{mod}\ 4)$. Evaluate ...