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

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Find all positive integer $n$ such that there exists $m$ with $2^n-1|m^2+17^2$.

Find all positive integer $n$ such that there exists $m$ with $2^n-1|m^2+17^2$. I have tried to mod $2^n-1$ and use the fact that $2^n \equiv 1 \pmod{2^n-1}$. I have also tried to factorize ...
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1answer
58 views

BMO1 2009/10 Question 5 Functional Equations Problem

Find all functions $f$, defined on the real numbers and taking real values, which satisfy the equation $f(x)f(y) = f(x + y) + xy$ for all real numbers $x$ and $y$. Thanks in advance for any ...
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1answer
64 views

Complete Solution (Icosahedron Proof Putnam)

I posted a similar question earlier, but then I noted an issue. Again the problem: A1: Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are ...
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1answer
24 views

How many integers can be made?

The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. How many integers $n$ can be made? Since there is: $$0, 1, 2, 3, 4, 5, 6, 7, ...
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1answer
61 views

BMO1 2009/10 Question 4 Geometry Problem

Two circles, of different radius, with centres at B and C, touch externally at A. A common tangent, not through A, touches the first circle at D and the second at E. The line through A which is ...
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2answers
142 views

2013 Putnam A1 Proof understanding (geometry)

Problem A1: Recall that a regular icosahedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icosahedron is ...
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2answers
28 views

Sum $\pmod{1000}$

Let $$N= \sum_{k=1}^{1000}k(\lceil \log_{\sqrt{2}}k\rceil-\lfloor \log_{\sqrt{2}}k \rfloor).$$ Find $N \pmod{1000}$. Let $\lceil x \rceil$ be represented by $(x)$ and $\lfloor x \rfloor$ be ...
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1answer
36 views

How many perfect squares exist (multiples of $24$)

How many positive perfect squares less than $10^6$ are multiples of 24? I quickly realized: $$24 = 2^{3}*3*5^0$$ $$10^6 = 2^6 * 5^6*3^0$$ We are finding numbers in the form $24(k^2)$. But I ...
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2answers
63 views

Probability Question (Colored Socks)

In a drawer Sandy has 5 pairs of socks, each pair a different color. On Monday Sandy selects two individual socks at random from the 10 socks in the drawer. On Tuesday Sandy selects 2 of the ...
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2answers
34 views

Probability using Combinations

I am confused on how this works. Normally, probability is: $$P = \frac{\text{Number of successes}}{\text{Number of total trials}}$$ For a problem like: If you flip a fair coin $8$ times, what is ...
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1answer
49 views

Difficult Probability mixed with combinatorics problem

Melinda has three empty boxes and $12$ textbooks, three of which are mathematics textbooks. One box will hold any three of her textbooks, one will hold any four of her textbooks, and one will hold ...
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77 views

INMO Problem with even function proof. [duplicate]

Let $n$ be a natural number. Show that $$\left[ \frac{n}{1} \right ] + \left[ \frac{n}{2} \right ] + \left[ \frac{n}{3} \right ] + \cdots + \left[ \frac{n}{n} \right ] + [\sqrt{n}]$$ is even. ...
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2answers
32 views

Interpretation of a Problem involving permutations

[USAMO 1999 submission, Titu Andreescu] Let $n$ be an odd integer greater than $1$. Find the number of permutations $p$ of the set $\{ 1, 2, …, n\}$ for which $$\def\x#1{\lvert p(#1)-#1\rvert} ...
2
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1answer
226 views

Sum of GCD and LCM

If $a,b \in \mathbb{N}$ and $ab > 2$ show that: $$\text{lcm}(a, b) + \gcd(a, b) \le ab + 1$$ Let the lcm be $l$ and let the gcd be $g$. We have to show: $$g + l \le ab + 1$$ I know that: ...
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1answer
134 views

Is this 5th root in the set of natural numbers?

Is $$\sqrt[5]{x(x+1)(x^4 + x^2 + 1)} \in \mathbb{N}$$ for some $x$? I am not asking for all $x$, but just for some natural number $x$? I don't believe so, but I may be wrong? Suppose $x=1$, ...
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1answer
43 views

AMC $12A$ Problem (Sequence lengths)

For each positive integer $n$, let $S(n)$ be the number of sequences of length $n$ consisting solely of the letters $A$ and $B$, with no more than three $A$s in a row and no more than three $B$s in ...
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1answer
35 views

Combinatorics Chess Spot Problem

Very tough problem, I must say. NOT CONSIDERING the squares both can go in from one of the black square not considering the squares both can go to. The horse can go to is: $$4 + 4 = 8 \space ...
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2answers
101 views

Product of repeated cosec.

$$P = \prod_{k=1}^{45} \csc^2(2k-1)^\circ=m^n$$ I realize that there must be some sort of trick in this. $$P = \csc^2(1)\csc^2(3).....\csc^2(89) = \frac{1}{\sin^2(1)\sin^2(3)....\sin^2(89)}$$ I ...
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2answers
141 views

Ball and urn method (counting problems)

How many ordered triples $(a, b, c)$ of positive integers exist with the property that $abc = 500$? Since, $500 = 2^2 5^3$ I believe this can be solved using Ball and Urn let $a = ...
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1answer
99 views

Putnam 2009 A1 Points in a plane

HINTS PLEASE! Let $f$ be a real-valued function on the plane such that for every square $ABCD$ in the plane, $f(A)+ f(B)+ f(C)+ f(D) = 0$. Does it follow that $f(P) = 0$ for all points $P$ in ...
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1answer
26 views

AMC12B Problem, probability

An unfair coin lands on heads with a probability of $\tfrac{1}{4}$. When tossed $n$ times, the probability of exactly two heads is the same as the probability of exactly three heads. What is the ...
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2answers
202 views

Putnam 2009 B1 (rational number as factorial)

Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}9=\frac{2!\cdot 5!}{3!\cdot 3!\cdot ...
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3answers
123 views

nonzero digits in decimal representation of $\sqrt{2}$

let $1,d_1d_2d_3\dots$ be a decimal representation of $\sqrt{2}$. Prove that at least one $d_i$ with $10^{1999}<i<10^{2000}$ is nonzero. I have no idea how to solve it. I think that the given ...
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62 views

Fraction of area covered by three circles

Take a square with edges of size $10$. Now take take three circles of radius $5$. Prove that you can't cover the square with these three circles. Find the maximum proportion of the area of the ...
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0answers
31 views

Determine points of tangency (Putnam 2007)

Find all values of $ \alpha$ for which the curves $ y=\alpha x^2+\alpha x+\frac1{24}$ and $ x=\alpha y^2+\alpha y+\frac1{24}$ are tangent to each other. This is an old Putnam Problem (2007 A1) ...
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3answers
72 views

All means integer

$a$ and $b$ are distinct positive integers such that $\frac{a+b}{2}$, $\sqrt{ab}$, and $\frac{2}{\frac{1}{a}+\frac{1}{b}}$ are integers. Find the smallest possible value of $|a-b|$. My work led me ...
1
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1answer
82 views

Find all nonnegative integers

Determine all nonnegative integers $x$ and $y$ so that $$3^x + 7^y$$ is a perfect square and $y$ is even. Without trial-and-error of course. $$3^x + 7^y = a^2$$ For some integer $a$. ...
3
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1answer
51 views

Geometry question posed in RMO 1999

Let $ ABCD $ be a square and $ M, N $ points on sides $AB, BC $ respectively, such that $\angle MDN = 45°$ . if $R$ is the midpoint of $MN$ show that $RP=RQ$ where $P,Q$ are the points of ...
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0answers
49 views

Putnam 2000 A1 Series square problem

Let $A$ be a positive real number. What are the possible values of $\displaystyle\sum_{j=0}^{\infty} x_j^2, $ given that $x_0, x_1, \cdots$ are positive numbers for which ...
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1answer
31 views

Find the solution to the system (not linear)

Find all $(x, y, z) \in \mathbb{R^3}$ satisfying: $$x^2 + 4y^2 = 4xz \tag1$$ $$y^2 + 4z^2 = 4xy \tag2$$ $$z^2 + 4x^2 = 4yz \tag3$$ This is a very difficult problem. I added $-4(1) + (3)$ to ...
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1answer
26 views

How to get real value after discount.

Suppose my item real value is 100. and i have given discount 10 % to my customer. Now the changed value of item id 90. If i set value 110 and give 10% discount then i got result 99. but i need result ...
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1answer
33 views

Would the powerset of $\mathbb{Z}$ also not denumerable?

Would the powerset of $\mathbb{Z}$ also be not denumerable?, Since Cantor's theorem says that the $\mathbb{N}$ is denumberable but the powerset of $\mathbb{N}$ is not denumberable because there does ...
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2answers
93 views

How would you prove that $2^{n-1} > n!$?

How do i prove that $2^{n-1} < n!$ for all $n \ge 1$ This is my proof: Base case: Let n=1 then $2^{1-1} =1$ is the same on the right side so it holds Inductive step: let $k \le 1$ we assume that ...
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0answers
94 views

An identity satisfying the divisors of a positive integer

I saw a hard competition problem with long and ugly proof in http://solmu.math.helsinki.fi/olympia/valmennus/2013/vt2013_12var.pdf ? The question is from Australian mathematical olympiad 1985. Is ...
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89 views

Posed in regional mathematics Olympiad 1995

Call a positive integer $n$ good if there are $n$ integers, positive or negative, and not necessarily distinct, such that their sum and product are both equal to $n$ For example, $8$ is good since ...
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2answers
67 views

2014 Putnam A1 Prime number factorial help

Question: Prove that every nonzero coefficient of the Taylor series of $(1-x+x^2)e^x$ about $x=0$ is a rational number whose numerator (in lowest terms) is either $1$ or a prime number. ...
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1answer
27 views

Taylor Series for $e^x(x^2 -x + 1)$

Find the Taylor Series for $e^x(x^2 -x + 1)$ about $x=0$. More importantly, find the COEFFICIENT (for nonzero terms) of the taylor series. The answer says: $$e^x(x^2 -x + 1) = 1 + ...
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1answer
33 views

How to tell if a function and a composite function is onto or one to one

For each of the following, f : A → B, g : B → C. Which one are true and which ones are false? So far i have, f is onto but g ◦ f is not onto. (False) f is 1-1 but g ◦ f is not 1-1. (False) g is onto ...
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2answers
34 views

How to prove if $A \times C \subseteq B \times D \implies A \subseteq B$

My proof Given $(x,y) \in A \times C \implies x \in A$ and $y \in C$ since $A \times C \subseteq B \times D$ then $(x,y) \in B \times D$ then $x \in B$ and $y \in D$ since $x \in A$ and $x \in B$ ...
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2answers
27 views

How to prove intersections and subsets of sets

Simple proofs for these are pretty straight forward such as proving if two sets are equal then they are subsets of each other or if you want to show one set is a subset of the other just show that ...
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0answers
26 views

How can you prove that a collection of union set is equal to the set of natural numbers?

For example $\bigcup_{n\in \mathbb{N}}A_n=\mathbb{N}$ My proof, to prove that two sets are equal i must show that they are subsets of each other. I understand how to show $\bigcup_{n\in ...
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1answer
90 views

Book recommendation to prepare for geometry in the International Mathematical Olympiad

What is the best book for preparation for "Geometry" for IMO? I've been searching one for past many weeks, got loads of names but couldn't finalize one, please help me.
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2answers
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Prove this inequality with $xyz\le 1$

if $x,y,z>0$ and $\color{red}{xyz\le 1}$, show that $$\color{blue}{\dfrac{x^2-x+1}{x^2+y^2+1}+\dfrac{y^2-y+1}{y^2+z^2+1} +\dfrac{z^2-z+1}{z^2+x^2+1}\ge 1}$$
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2answers
93 views

How to prove whether the equation set has a unique solution?

\begin{eqnarray} \begin{cases} \sin A \sin C-(\sin B)^2=0 \cr AC-B^2=0 \cr A+B+C-\pi=0 \cr A>0,B>0,C>0 \end{cases} \end{eqnarray} How to prove whether the equation set has a unique solution ...
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3answers
118 views

Putnam 2006 B1 Problem

Show that the curve $x^{3}+3xy+y^{3}=1$ contains only one set of three distinct points, $A,B,$ and $C,$ which are the vertices of an equilateral triangle, and find its area. Yikes. Without ...
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1answer
55 views

Length from tangent circles

A circle $Γ_1$ of radius $25$ is externally tangent to a circle $Γ_2$ of radius $16$ at $C$. Let $AB$ be a common direct tangent, so that $A$ lies on $Γ_1$ and $B$ lies on $Γ_2$. Draw the tangent to ...
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1answer
69 views

How to prove that there are no integers a,b such that $b^2=4a+2$

How to prove that there are no integers a,b such that $b^2=4a+2$ This seems like a very simple prof but when i tried to work through it i keep on hitting walls. I tried to prove this by ...
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1answer
21 views

Sum of numbers in a grouping question

A person grouped numbers in the following way: $$\left \{ 1 \right \},\left \{ 3,5 \right \},\left \{ 7,9,11 \right \},\left \{ 13,15,17,19 \right \},...$$ What is the sum of the numbers in the $9$th ...
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2answers
63 views

Volume and surface area of a drilled out cube (BM01 2010/11 Contest Question 2)

Let $s$ be an integer greater than $6$. A solid cube of side $s$ has a square hole of side $x < 6$ drilled directly through from one face to the opposite face (so the drill removes a cuboid). The ...
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2answers
621 views

1985 Putnam A1 Solution

I dont see what they mean by bijection of triples of subsets of $\{1, \ldots, 10\}$ and the $10\times3$ matrix with $0, 1$ entries? How is that created?