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

learn more… | top users | synonyms (2)

4
votes
2answers
365 views

Counting ordered triples of non-negative integers not greater than 100

Can we find the number of ordered triples $(x,y,z)$ of non-negative integers satisfying (i) $x \leq y \leq z$ (ii) $x + y + z \leq 100$? Source:Regional Mathematics Olympiad India (2003) Thank you.I ...
2
votes
1answer
156 views

Question about infinite descent method in this proof

How do we conclude that:"We have therefore constructed another pair $(a_1, b_1)$ in $S(c, k)$ with $a_1 + b_1 < a + b$. However, $S(c, k)$ is contain in $Z^+ × Z^+$, so using the argument of ...
-3
votes
3answers
189 views

How to write a complete solution to this putnam problem (1985 , B3)?

$a_{ij}$ is a positive integer for $i,j=1,2,3,\ldots$, and for each positive integer we can find exactly eight $a_{ij}$ equal to it. Prove that $a_{ij}\gt ij$ for some $i,j$. Proposed solution. ...
3
votes
0answers
104 views

If $0\leq \cdots \lt s'''\lt s''\lt s'\lt s$ and $s''=((s')^2-k)/s$, $s'''=((s'')^2-k)/s',\ldots$, then $k$ is a perfect square

This is an IMO problem from 1988, problem 6. The book does not provide a proof of this part and it is eluding me. Let $$\cdots \lt s''' \lt s'' \lt s' \lt s$$ all be ...
13
votes
2answers
411 views

What's the smallest area a square can have that a cube can still be wrapped with it?

My task is to wrap a unit cube with the smallest square sheet of paper possible. The paper is assumed to be infinitely thin of course and no cutting or stretching is permitted. I must be able to ...
1
vote
2answers
101 views

Interscholastic Mathematics League Senior B #12

Compute the product of the nonreal roots of the equation $x^4+4x^3+6x^2+1004x+1001=0$. So here is what I have done so far. I got two of the roots to be zero and 4 since ...
-1
votes
1answer
71 views

Interscholastic Mathematic League Senior B Division [closed]

The number 2011 has the property that one of its digits is the sum of its other digits, i.e., 0+1+1=2. Compute the sum of the two largest integers less than 2011 with this property.
1
vote
2answers
118 views

Interscholastic Mathematic League Senior B Division #10

In traingle ABC, Angle A=45 degrees, Angle B is 60 degrees, and AC= radical 15. D is also a point on AB so that AB is perpendicular to CD. The circle with diameter AB intersects CD at point E. Compute ...
1
vote
1answer
76 views

Interscholastic Mathematic League Senior B Division #11

The roots of the equation 3x^3-38x^2+cx-192=0 form a geometric progression. Compute c.
0
votes
1answer
102 views

Interscholastic Mathematics League Senior B Division #2

Points P,Q,R, and S are chosen on the sides of parallelogram ABCD, so that P is on line AB, Q is on line BC, R is on line CD, S is on line DA, and AP=BQ=CR=DS=1/3 AB. Compute the ratio of the area of ...
0
votes
1answer
85 views

Interscholastic Mathematic League Senior B Division #1

Let n be a positive integer less than 1000. If n^3 has 10 factors, compute the largest value of n.
0
votes
1answer
31 views

Terminology concerning a certain solution to a certain system of equations

Say you have a solution $\textbf{x}=(x_1,x_2,\ldots,x_n)$ to a system of equations. It turns out that $-\textbf{x}$ is also a solution. Is there accepted terminology for such a pair of solutions? (I ...
2
votes
1answer
358 views

Concentric circles(IMO 1988/Problem 1)

Let us consider 2 concentric circle radii R and r (R > r) with centre O.We fix P on the small circle and consider the variable chord PA of the small circle. Points B and C lie on the large circle; ...
2
votes
3answers
201 views

Edited:what is a way to solve this problem other than use algebra way?

I am getting tired of using expansion to solve this problem. I wonder if there is any non-algebra ways to solving it. Problem: Suppose that the number $x$ satisfies the equation $x+x^{-1}=3$. ...
6
votes
2answers
2k views

How to improve mathematics for Programming Contests?

You might close this question or vote "-1" for this but I just can't stop myself from asking this question from the experts of mathematics which solves thousands of problem related to mathematics and ...
1
vote
1answer
281 views

Another Trigonometry problem, sum of products of sine function over partitions of N

I don't know how to write the summation symbol so I'm providing you the original link to problem http://www.codechef.com/OCT11/problems/PARSIN .My approach to solve this problem is first reduce the ...
2
votes
1answer
529 views

Average and minimum Values of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$, $\forall x \in \mathbb{R}$

A problem was asked at Putnam Competition in 2003 (Problem 3), about finding the minimum Value of $|\sin x+ \cos x + \tan x + \cot x +\sec x +\csc x|$ where $x$ is Real. the question paper and ...
17
votes
3answers
805 views

Examples of math contest problems that can be solved in a 'cheap' way?

What are some examples of math contest problems that can be solved by using a nonrigorous, 'cheap' shortcut? For instance, a problem on the 2011 AMC went: A raft and a motorboat left dock A and ...
8
votes
3answers
971 views

Olympiad Inequality Problem

Consider three positive reals $x,y,z$ such that $xyz=1$. How would one go about proving: $$\frac{x^5y^5}{x^2+y^2}+\frac{y^5z^5}{y^2+z^2}+\frac{x^5z^5}{x^2+z^2}\ge \frac{3}{2}$$ I really dont know ...
5
votes
1answer
288 views

Is it possible for the number created by ordering $1$ to $n$ where $n > 1$ be a palindrome?

Is it possible for the number created by the consecutive numbers $1$ to $n$ where $n > 1$ be a palindrome eg. $1234567\ldots n$? I believe this is a contest problem, but how would one solve ...
5
votes
2answers
1k views

Let a; b; c and d be non-negative numbers such that a+b+c+d = 4. Prove that 4/(abcd) ≥ a/b + b/c + c/d + d/a

How would I approach this using only the AM - GM inequality? Are there any other methods that does not involve the AM-GM inequality?
6
votes
2answers
352 views

What are the different international mathematics competitions that are held?

Apart from the IMO which is for the younger students, I am looking for competitions which are maybe for undergraduates or graduate students.
2
votes
5answers
294 views

Finding the smallest positive integer a

Can we find the smallest positive integer $a$ such that $1971|50^n+a.23^n$ where n is odd? Source:Problem Solving Strategies by Arthur Engel
12
votes
2answers
2k views

IMO 2011 problem 6 Geometry

The is year's IMO problem 6 was a geometry problem that only 6 participants managed to solve completely. The problem is formulated like this: Let $ABC$ be an acute triangle with circumcircle ...
4
votes
1answer
562 views

IMO 2011 Problem 5 - Show that $f(m) \mid f(n)$ if $f(m) \leq f(n)$

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for ...
6
votes
1answer
239 views

Putnam A-6: 1978: Upper bound on number of unit distances

Let n distinct points in the plane be given.prove that fewer than $2n^\frac{3}{2}$ pairs of them are at unit distance apart
1
vote
1answer
278 views

Just wondering what this imo problem is asking and how to solve

Just wondering what this imo problem is asking(it looks simple but i don't understand what's important in the question) and how to solve: Suppose that $s_1,s_2,s_3,\ldots$ is a strictly increasing ...
15
votes
3answers
2k views

The easy(?) part of IMO 2011 Problem 3

Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies $$f(x + y) \leq yf(x) + f(f(x))$$ for all real numbers $x$ and $y$. How can I prove that ...
1
vote
1answer
301 views

Find number of interesting numbers (China TST 2011)

A positive integer $n$ is known as an interesting number if $n$ satisfies $$ \left\{\frac{n}{10^k}\right\} > \frac{n}{10^{10}} $$ for all $k=1, 2, \ldots, 9$, where $\{x\}=x - \lfloor x \rfloor$. ...
4
votes
2answers
192 views

Trouble understanding proof that the unit interval cannot be partitioned in a certain way

From the book "Putnam and Beyond." The problem: Show that the interval [0, 1] cannot be partitioned into two disjoint sets A and B such that B = A + a for some real number a. Proof: Assume ...
9
votes
2answers
482 views

Group theory intricate problem

This is Miklos Schweitzer 2009 Problem 6. It's a group theory problem hidden in a complicated language. A set system $(S,L)$ is called a Steiner triple system if $L \neq \emptyset$, any pair $x,y ...
8
votes
5answers
774 views

Favorite Math Competition Problems

I'm running a weekly math contest for a summer camp and would like to compile a list of interesting problems. The problems may presuppose mathematical knowledge up to but not including Calculus. For ...
12
votes
4answers
733 views

Solving for the implicit function $f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$ and $f(1)=1$

How can I find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(1)=1$ and $$f\left(f(x)y+\frac{x}{y}\right)=xyf\left(x^2+y^2\right)$$ for all real numbers $x$ and $y$ with $y\neq0$? PS. This is ...
1
vote
2answers
406 views

Functional equation $f(x^2+y)=f(x)+f(y^2)$ from Olympiad

How do i deal with stuff like that? tried to just write many equalities but it just doesnt help... The equation is $f(x^2+y)=f(x)+f(y^2)$. EDIT: the question is to find all functions such that this ...
8
votes
2answers
822 views

Help with olympiad question: Solve $x^3 + y^3 + z^3 = 2011 \text{ for }x,y,z \in \mathbb{Z}$

Solve the equation: $x^3 + y^3 + z^3 = 2011$ in integer numbers. I'm trying to solve problems I couldnt on the competition itself but I'm totally stuck.
11
votes
2answers
984 views

Preparation for Putnam?

If all the training I have right now is Calc 1-3, Linear Algebra and some introduction to set theory/discrete math, what would you recommend focusing on over summer in preparation to Putnam? Real ...
0
votes
1answer
255 views

How can I find all increasing sequences $\{a_i\}_{i=1}^{\infty}$ such that $d(x_1+x_2+\cdots+x_k)=d(a_{x_{1}}+a_{x_{2}}+\cdots + a_{x_{k}})$?

How can one find all increasing sequences $\{a_i\}_{i=1}^{\infty}$ such that $$d(x_1+x_2+\cdots+x_k)=d(a_{x_{1}}+a_{x_{2}}+\cdots + a_{x_{k}}),$$ holds for all $k$-tuples $(x_1,x_2,\cdots,x_k)$ of ...
17
votes
2answers
1k views

Olympiad calculus problem

This problem is from a qualifying round in a Colombian math Olympiad, I thought some time about it but didn't make any progress. It is as follows. Given a continuous function $f : [0,1] \to ...
15
votes
2answers
611 views

Find a way from 2011 to 2 in four steps using a special movement

USAMTS 6/2/22 states: The roving rational robot rolls along the rational number line. On each turn, if the robot is at $\frac{p}{q}$, he selects a positive integer $n$ and rolls to ...
12
votes
1answer
528 views

Contest problems with connections to deeper mathematics

We all know that problems from for example the IMO and the Putnam competition can sometimes have lovely connections to "deeper parts of mathematics". I would want to see such problems here which you ...
4
votes
1answer
235 views

CMO 2011 Sum of integers

The following problem was asked in the CMO 2011 and I'd be interested in finding various solutions for it. Here's the problem: Fix a positive integer $d$, then for any integer $k$ there exists a ...
1
vote
2answers
105 views

Non-Standard Deviation

Given a list of integers, how to find the sum of the differences of all possible pairs of numbers ? For example if the number are $3,1,2$ then, answer should be $$\lvert 3-1 \rvert + \lvert 3 -2 ...
2
votes
1answer
243 views

Counting digits in an arithmetic sequence

Given $a, d, n, x$. Suggest me a suitable algorithm to compute the number of times the digit $x$ appearing in the arithmetic sequence $a, a + d, a + 2 \times d, \cdots, a + n \times d$. For ...
2
votes
3answers
445 views

Solutions to interesting problems with elegant and unintuitive methods

I am looking to build a repertoire of olympiad type problems which have non-intuitive elegant solutions, If possible instead of a resource, I think problems would be the best. (i.e. select the best ...
1
vote
1answer
2k views

Solving systems of linear congruences (modular equations)

This is a task from an old programming contest the task is as follows, A list of system of linear equations is given in the inputs.If the system is solvable we have to output the solution in the form ...
1
vote
2answers
718 views

Direct formula for a variation of Josephus problem:

What is the direct formula for the following variation of Josephus problem? There are $n$ persons, numbered $1$ to $n$, around a circle.. Starting from $kth$ person, every second person is ...
2
votes
2answers
291 views

Finding GCD of all permutations of a decimal number [duplicate]

Possible Duplicate: Computing GCD of all permutations (of the digits) of a given number. How can we find the greatest common divisor (GCD) of all numbers that can be obtained by permuting ...
1
vote
3answers
158 views

Minimum and maximum

There are five real numbers $a,b,c,d,e$ such that $$a + b + c + d + e = 7$$$$a^2+b^2+c^2+d^2+e^2 = 10$$ How can we find the maximum and minimum possible values of any one of the numbers ? Source
5
votes
3answers
955 views

Olympiad - sequence of sum of complex numbers

This is the other problem I couldn't solve in the olympiad test I took today. Let $c_1, ... , c_n$ be complex numbers with unitary norm, and $S_k=\sum_{i=1}^n c_i^k$, $k\in \mathbb{N}$. Suppose $S_1, ...
6
votes
4answers
751 views

Fast method for Nth Squarefree number (using mathematica)

I am trying to compute Nth Squarefree numbers using Mathematica. What I am trying to utilize is the SquareFreeQ[i] function. Here is my solution : ...