Stack Exchange Network

Stack Exchange network consists of 175 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers.

Visit Stack Exchange

Questions tagged [contest-math]

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

4
votes
2answers
76 views

Find all ordered pairs $(a,b)$ such that $1/a + 1/b = 3/2018$ and $a,b$ are positive integers

I gave this problem my best attempt and am now trying to understand the solution for it. This is problem #1 to the 79th William Lowell Putnam Math Competition. This is the given solution by Kiran ...
0
votes
2answers
51 views

Number of possible ways in which $3$ divides $a_1+a_2+a_3+…+a_n$

I am solving this question from a contest on codeforces. How to solve this. Can we get any recurrence relation or generalized solution for this question? What I have understood is we have to find $...
2
votes
1answer
185 views

Question from the 2011 IMC key stage III paper, about determining which number has to be removed, so the multiplication is the square of an integer [on hold]

The product $$1!\cdot 2!\cdot \ldots\cdot 2011!\cdot 2012!$$ is written on the blackboard. Which factor in term of a factorial of an integer, should be erased so that the product of the remaining ...
0
votes
0answers
51 views

“New Year falls more often on Sundays than on Mondays”

This is a question from a hungarian math contest from the year 1948. It was Saturday on the 23rd October, 1948. Can one conclude that New Year falls more often on Sundays than on Mondays? Well, I ...
0
votes
3answers
144 views

Question from the 2011 IMC (international mathematics competition) key stage III paper, about the evaluation of $a+b+c$ [duplicate]

Let $a,b,c$ be positive integers such that $$ab+bc+ca+2(a+b+c)=8045\qquad abc-a-b-c=-2$$ Find the value of $a+b+c$. I originally saying that $a+b+c=abc+2$. However, it was to no avail, as I ...
1
vote
2answers
52 views

Question from the 2011 IMC (International Mathematics Competition) Key Stage III paper, about the evaluation of a quadratic equation

When $a=1, 2, 3, ..., 2010, 2011$, the roots of the equation $x^2-2x-a^2-a=0$ are $(a_1, b_1), (a_2, b_2), (a_3, b_3),\cdots, (a_{2010}, b_{2010}), (a_{2011}, b_{2011})$ respectively. Evaluate: $...
0
votes
0answers
42 views

continuous but non-differentiable function having countable points of non-differentiablity [on hold]

Consider function $f:\mathbb{R}\to\mathbb{R}$ satisfying $|f(x)-f(y)|\lt 4321|x-y|$ for all real numbers $x$ and $y$. Show that there exists a function $f(x)$ such that $f$ is continuous but is non-...
11
votes
1answer
139 views

Which positive integers can NOT be written as a sum of consecutive positive integers

I have been wondering about this question for a while (I am almost sure that I read it in some contest mathematics book a while ago) Determine which positive integers cannot be written as a sum of ...
2
votes
1answer
38 views

Find the limit $\lim_{t \rightarrow \infty} tx(t)$ where $x=x(t)$ is the least positive root of equation $x^2 + \frac12 = \cos(tx)$.

My attempts: First of all, if we draw graphs of left and right sides of the equation, will see that $x(t) \rightarrow 0$ as $t \rightarrow \infty$, because segment $[0, \frac{\pi}{2t}]$ shrinks with ...
2
votes
2answers
72 views

Showing that, if $n,10$ are coprime, then the last $3$ digits of $n^{101}$ will be the same as $n$'s. [duplicate]

Let $n$ be a positive integer where $n$ and $10$ are coprime number. Prove that then the last 3 digits of $n^\text{101}$ will be equal to the last 3 digits of $n$. SOURCE: Bangladesh Math Olympiad I ...
2
votes
1answer
91 views

Amidst $7$ prime numbers, difference of the largest and the smallest prime number is $d$. What is the highest possible value of $d$?

Let $a, b, c, b+c-a, c+a-b, a+b-c$ and $a+b+c$ be $7$ distinct prime numbers. Among $a+b, b+c$ and $c+a$, only one of the three numbers is equal to $800$. If the difference of the largest prime number ...
2
votes
1answer
76 views

Prove $\frac{m(1-m)}{n}+1\leq\frac{n^m-m^m}{x^m}\leq m(1-m)+n$

Let $f(x)=\frac{1}{m}x^m+\frac{1}{n}-x$,such that $\frac{1}{m}+\frac{1}{n}=1(n>m>1),x\in[m,n],f(x)\geq0 $ Prove that: $$\frac{m(1-m)}{n}+1\leq\frac{n^m-m^m}{x^m}\leq m(1-m)+n$$ This is a ...
-2
votes
0answers
31 views

Euclid Exam in Delhi [on hold]

I want to take the Euclid Mathematics exam in April 2019 but my school is not organising it. I am a resident of Delhi and I need a centre nearby where I can take it.
1
vote
2answers
40 views

Proving the divisibility of $4[(n-1)!+1]+n$ by $n(n+2)$ in the condition of $n,n+2 \in P$ where $P$ is the set of prime numbers [duplicate]

Let $n$ and ($n+2$) be two prime numbers. If any real value of $n$ satisfies that condition, then prove that $$\frac{4{[(n-1)!+1]}+n}{n(n+2)} = k$$ where $k$ is a positive integer. SOURCE: BANGLADESH ...
2
votes
2answers
47 views

If $a, b, c, d$ exists s.t. $p=a^2+kb^2$, $pn=c^2+kd^2$, proof that integer $x, y$ such that $n=x^2+ky^2$ exists.

Question. $p$ is a prime, $k$ is a given natural number. If $a, b, c, d$ exists s.t. $p=a^2+kb^2$, $pn=c^2+kd^2$, proof that integer $x, y$ such that $n=x^2+ky^2$ exists. My approach. Let $n=x^2+ky^2$...
1
vote
1answer
67 views

A question about INMO 2017, Question 3

Find all triples $(x,a,b)$ where $x$ is a real number, and $a,b$ are integers belonging to $\{1,2,\dots,9\}$ such that $$x^2-a\{x\}+b=0$$ Here $\{x\}$ denotes the fractional part of $x$. My ...
5
votes
0answers
38 views

Proving concurrence in a convex quadrilateral and circumcircles

Let $ABCD$ be a convex quadrilateral in which $AB = CD$ and $∠ABD + ∠ACD = 180^{\circ}$. Lines $AC$ and $BD$ intersect at $P$ and let $M$ be the midpoint of $AD$. Suppose that $MB$ and $MC$ intersect ...
3
votes
0answers
42 views

Find all the solutions $(x,y,z)$ $\in \Bbb Z^+$ such that $(x+1)^y-x^z=1$ [duplicate]

Find all the solutions $(x,y,z)$ $\in \Bbb Z^+$ such that $(x+1)^y-x^z=1$. This is an old problem from a math olympiad in Venezuela, in the year 2000. I don't know how to start solving this kind of ...
1
vote
2answers
42 views

Finding the ratio of a side of $\triangle ABC$ and its segment where one cevian line from the opposite vertex intersect the side in any point

In $\triangle ABC$, $L$ and $M$ are two points on $AB$ and $AC$ such that $AL = \frac{2AB}{5}$ and $AM = \frac{3AC}{4}$. $BM$ and $CL$ intersect at the point $P$ and the extension line of $AP$ and the ...
1
vote
1answer
42 views

What is the area of $OEAF$ in $ABC$ triangle in the following diagram?

The area of $ABC$ and $OBC$ triangle is $120$ and $24$ respectively. $BC=16$, $EF=8$. Find out the area of $OEAF$ quadrilateral. Source: Bangladesh Math Olympiad 2014 Junior Category. I can find ...
0
votes
2answers
35 views

value of $f(2008)$ in $4$ th degree polynomial

If $f(x)$ is a $4$ th degree polynomual such that $f(2003)=24, f(2004)=-6, f(2005)=4,f(2006)=-6,f(2007)=24$ Then value of $f(2008)$ is what i try assuming $f(x)=ax^4+bx^3+cx^2+dx+e\cdots \cdots ...
0
votes
2answers
28 views

What is the LCM of a and b if $(a+2 \sqrt{2} )/b$ is the ration of the area of larger circle and smaller circles?

Radius of all four smaller circles is $R$. If the ratio between the area of the larger circle and the sum of areas of the smallest circles is $(a+2 \sqrt{2} )/b$ then find the LCM of $a$ and $b$. ...
6
votes
0answers
116 views
+50

A more natural solution to finding the general terms of a recurrence relation in $2$ variables

A high school contest math problem in a problem book: Find the general terms of $$a_{1}=a,\quad b_{1}=b,\quad a_{ n + 1 }=\frac { 2 a _ { n } b _ { n } } { a _ { n } + b _ { n } },\quad b_{ n ...
0
votes
6answers
38 views

What is the area of $\triangle ABC$ if area of $ABEF$ and $BCFD$ is 15 and 23 in the following diagram?

$ABEF$ and $BCFD$ parallelograms have areas respectively $15$ and $23$. Find the area of $\triangle ABC$. Source: Bangladesh Math Olympiad 2014 Junior Category Is only areas of two parallelograms ...
2
votes
3answers
42 views

What is the area of $PXQY$ in the rectangle $ABCD$ in the following diagram?

In $ABCD$ rectangle $AB=6$, $AD=8$, $AE=ED$, $BF=FC$, $EP=PQ=QF$. Find the area of $PXQY$. Source: Bangladesh Math Olympiad 2014 Junior Category I cant prove that which type of quadrilateral it is....
5
votes
1answer
98 views

Solving $(x^2+4x+3)^x+(2x+4)^x=(x^2+4x+5)^x$ with $x\in(-1,\infty)$ [duplicate]

I've been struggling for a few hours on the below pre-calculus olympiad equation to which I still don't have an answer: $$(x^2+4x+3)^x+(2x+4)^x=(x^2+4x+5)^x$$ where $x \in (-1,\infty)$. Now, ...
4
votes
2answers
65 views

If $a$, $b$ and $c$ are sides of a triangle, then prove that $a^\text{2}(b+c-a) + b^\text{2}(c+a-b) + c^\text{2}(a+b-c)$ $\leqslant$ $3abc$

Let $a$, $b$ and $c$ be the sides of a triangle. Prove that $$a^\text{2}(b+c-a) + b^\text{2}(c+a-b) + c^\text{2}(a+b-c) \leqslant 3abc$$ SOURCE: BANGLADESH MATH OLYMPIAD (Preparatory Question.) I am ...
0
votes
1answer
31 views

Determinants Quadratic polynomials and iota in characteristic equation .

question: Given $A,B$ be two square matrices (with real entries )of order $2$ where $AB=BA$ $, \det(A)=\alpha>0$ , $\det(A+i\alpha B)=0$ then find value of L where , $L=\det(A^2-\alpha A B+...
3
votes
2answers
60 views

Understanding a proof from the APMO 1998 on inequalities.

I was having trouble with proving the following inequality.The question was from the book Secrets to Inequalities by Pham Kim Hung. $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} \geq \frac{x+y+z}{\sqrt[3]{...
3
votes
1answer
54 views

Compute: $\frac3{7\cdot2}+\frac3{7\cdot12}+\dots$

Compute: $$\frac3{7\cdot2}+\frac3{7\cdot12}+\frac3{17\cdot12}+\dots+\frac3{2017\cdot2012}$$ I couldn't really find the pattern in this one. I tried evaluating the first two terms which was $\frac14$, ...
1
vote
1answer
26 views

What is the probability if we throw dart towards a large square but it should hit only the inner part of small square $FEHG$ inscribed in it?

Let $ABCD$ be a square shaped board. 4 equal rectangles are drawn into it. The length of the sides of the rectangles are $x$ and $y$, where $\frac{x}{y}$ = $3$. A dart is thrown towards the square ...
3
votes
1answer
114 views

Prove that $(a+b) (a^2 + b^2) (a^4 + b^4)…(a^{32} + b^{32}) = a^{64} - b^{64}$ if $b = a-1$

Prove that if $b = a-1$, then $(a + b) (a^2 + b^2 ) (a^4 + b^4 ) ... (a^{32} + b^{32} ) = a^{64} - b^{64}$ . I saw this in a website and it wrote this: hint: Write down the equality $1 = a+b$ and ...
1
vote
1answer
78 views

Evaluate this question based on series and limits.

For $a \in \mathbb R,a≠-1$ $$\lim_{n\to\infty}\frac{1^a+2^a+\cdots +n^a}{(n+1)^{a-1}[(na+1)+(na+2)+(na+3)+\cdots+(na+n)]}=\frac{1}{60}$$ Then find the values of $a $. I tried to solve this ...
3
votes
1answer
108 views

Prove that inequality $\sqrt{2\sqrt{4\sqrt{8…\sqrt{2^n}}}} \leqslant n+1$

Let $n$ be the integer. Prove that $$\sqrt{2\sqrt{4\sqrt{8....\sqrt{2^n}}}} \leqslant n+1$$ SOURCE: BANGLADESH MATH OLYMPIAD I am a new beginner at the infinite radical and sequence. I don't know ...
-1
votes
1answer
24 views

Draw and study the discontinuous of $g(x)=\lfloor \sin(x) \rfloor$

Draw and study the discontinuous of $g(x)=\lfloor \sin(x) \rfloor$ $$g(x)=\begin{cases} -1 & x\in (-\pi,0) \\ 0 & x \in [0,\pi]\setminus \lbrace \frac{\pi}{2} \rbrace \\ 1 & x=\frac{\pi}{...
0
votes
1answer
53 views

What is the value of $a+b$ where the area of the square in the diagram is $\dfrac{a}{b}$ and both are co-primes?

The diagram shows two circles, each of radius $1$ and a square. The side length of the square can be written as $\dfrac{a}{b}$ ($a$ and $b$ are co-prime). Find $a+b$. Source: Bangladesh Math ...
2
votes
1answer
52 views

Find a 10 digit number that contains all digits from 0 to 9 once, starts with a 3, and is also divisible by all whole numbers from 2 to 18

Trial and error is always an option, but this question is from a timed math competition sheet, so it shouldn't take that long. Where do I start? Edit: As far as the competition aspect is involved, a ...
1
vote
0answers
103 views

Given that $[ABC]$ : Area of small circle = $\frac{3\sqrt3}{4}$ : $\pi$. How many parts of area of small circle is inscribed in large circle? [on hold]

In the common region of two circle, $\triangle ABC$ has been drawn with its maximum area such that the proportion of the maximum area of $\triangle ABC$ and the area of small circle is equal to $\frac{...
-4
votes
0answers
43 views

Inequality from iZhO 2008 [closed]

Solutions of iZhO before 2009 are not available, so I still can not prove this inequality. Let $a, b, c$ be positive real numbers such that abc = 1. Prove that: $\frac{1}{(a+b)b}+ \frac{1}{(b+c)c}+ \...
1
vote
3answers
48 views

What is the area of $ABCD$ parallelogram where $E$ is mid-point of BC and the area of $BEC$ is 126?

$ABCD$ is a parallelogram. Point $E$ divides $BC$ into two equal lengths. If the area of $BEF$ is 126, what is the area of $ABCD$? Source: Bangladesh Math Olympiad 2017 Junior Category. I can not ...
3
votes
3answers
82 views

In $\triangle ABC$, $AD$ $\perp$ $BC$ and $GE$ is the extended line of $DG$ where $G$ is centroid. Prove that $GD$ = $\frac{EG}{2}$

Let $ABC$ be a triangle and in $\triangle ABC$, $AD$ $\perp$ $BC$ and three median lines intersect at point $G$ where $G$ is the centroid of $\triangle ABC$. The extension of $DG$ intersects the ...
1
vote
4answers
99 views

a,b,c are three real numbers where $a + \frac{1}{b} = b + \frac{1}{c} = c + \frac{1}{a}$. Now $abc$ = ? Here will the answer be a number? [duplicate]

I want to know whether it is possible to get a real number (not an algebraic expression) as the product of $a$, $b$ and $c$. I tried for a long time and this is what I got. $$3abc = a^2 b + b^2 c + ...
1
vote
1answer
72 views

What is the value of $5 * 6$ in the following patttern?

Mr. Pascal built a computer for multiplying numbers and named it "Ramanujan". But Ramanujan multiplies $(3, 5), (2, 4), (3, 4)$ and $(4, 7)$and results are $17, 10, 14$ and $34$. If Ramanujan ...
0
votes
1answer
49 views

What is the area of the quadrilateral $ADEC$ in $ABC$ right triangle in the following diagram?

In the right angled triangle $ABC$, $\angle A = 90^\circ$, $AB=8$, $AC=6$, $BC = 10$. $D$ is a point on $AB$ in such way that if a perpendicular $DE$ is drawn on $BC$ from $D$ then $BE = 4$. What ...
2
votes
1answer
31 views

$A$ and $D$ are in circumference of a circle and $B$ and $C$ are its inner points such that $PA$= $12$, $\frac{AB}{CD}$ = $\frac{1}{2}$. Find $PC$

There is something misunderstanding with that question that I think it to have inadequte context or information (obviously for my little knowledge). So I couldn't solve the problem. SOURCE: ...
0
votes
2answers
85 views

Prove that $\frac{BC}{AH}×\frac{CA}{BH}×\frac{AB}{CH}$ = $\frac{BC}{AH}+\frac{CA}{BH}+\frac{AB}{CH}$, where $H$ is the orthocenter of $\triangle ABC$

It is an isolated problem which has a lack of context. I found that problem in a magazine which was stated such as below: In $\triangle ABC$, three altitude lines $AE$, $BF$ and $CD$ dropped from ...
0
votes
0answers
42 views

Show by non-graphical means that the infinite solutions to $\tan(x)=r$ lie within the $n\pi \le x_n \le(n+1/2)\pi$.

Show by non-graphical means that the infinite solutions to $$\tan(x) = r$$ lie within the $$n\pi \leq x_n \leq (n+1/2)\pi.$$ Please Explain.
1
vote
2answers
41 views

What is the value of $x+y$ if x and y are co-primes and $PR=\dfrac{x}{y}$ in the diagram?

$ST$ is the perpendicular bisector of $PR$ and $SP$ is the angle bisector of $\angle QPR$. If $QS=9cm$ and $SR=7cm$ then $PR=\dfrac{x}{y}$ where x, y are co-primes. $x+y$=? Source: Bangladesh ...
1
vote
1answer
35 views

Efficient method for computing the product of the first 8 terms of a recursive sequence

The problem I am trying to solve is the following: Let $x_1=97,$ and for $n>1,$ define $x_n=\frac{n}{x_{n-1}}.$ Calculate $x_1x_2 \cdots x_8.$ I tried the painstaking fail safe method for the ...
0
votes
1answer
44 views

Representing beauty contest with $n$ players in normal form game

I have a beauty contest question in which players must guess a number between $0$ and $5$. The closest score to ($p\times\text{average score}$) wins. Winners take 1 and losers take 0, whilst players ...