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.

1
vote
0answers
31 views

A problem about the field of rational functions over finite field

Let $p$ be a prime number, and let $F_{p}$ be the finite field with $p$ elements. Let $F=F_{p}(t)$ be the field of rational functions over $F_{p}$ . Consider all subfields of $F$ such that $F/C$ is a ...
0
votes
0answers
21 views

Finding the maximum number of moves of a game

I would like to know if my solution of the following problem is correct: I have $12$ coins with a black face and a white face, I put them in a row with faces of alternate colors(BWBW...). A move ...
2
votes
1answer
56 views

Suppose $f : [0, 1] \to [0, 1]$ is differentiable such that $|f'(x)|\ne1$ for all $x\in[0, 1]$. Prove $\exists !a,b\in[0,1]:f(a)=a,f(b)=1-b$

Suppose $f : [0, 1] \to [0, 1]$ is differentiable such that $|f'(x)|\ne1$ for all $x\in[0, 1]$. Prove there exist unique $a,b\in[0,1]$ such that $f(a)=a$ and $f(b)=1-b$. Not sure how I'm ...
0
votes
0answers
53 views

A question from a past BMO exam, on modulo [duplicate]

If the last three digits of the number $7^{400}$ are 001 find the last three digits of the number $7^{9999}$. Guys, I tried working out this question today, however to no success. I tried using ...
0
votes
3answers
55 views

If $a^3-b^3\equiv 0\bmod3$ prove that $a^3-b^3\equiv 0\bmod 9$ [on hold]

If $a^3-b^3\equiv 0\bmod3$ prove that $a^3-b^3\equiv 0\bmod 9$. Guys, this question appeared in a preparation leaflet for a national exam, which I recently did. However, I was incapable of doing it. ...
1
vote
1answer
61 views
+50

Find the numbers of ordered array $(a,b,c,d)$ such $a^2+b^2\equiv c^3+d^3\pmod p$

Let $p$ be prime number,and such $p\equiv 1\pmod {12}$,Find the numbers of ordered array $(a,b,c,d)$ that satisfies the following conditions: (1):$a,b,c,d\in \{0,1,2,\cdots,p-1\}$ (2):$a^2+b^2\equiv ...
-1
votes
0answers
26 views

An object distribution problem

There is A and B boxes. There is papers that written on it from 1 to 27. A box cannot contain any paper that written multiply by any A box paper and B box paper. B box cannot contain any paper that ...
4
votes
3answers
124 views

Why is this proof of a congruence relation valid?

The following question comes from the 2012 Singapore Mathematical Olympiad (Open Section), Round 2. Let $p$ be an odd prime. Show that $$1^{p-2}+2^{p-2}+3^{p-2}+\dots+\left(\frac{p-1}2\right)^{p-2}=...
0
votes
2answers
36 views

the possible permutations of a number, following a set of rules

We can change a natural number $n$ in the following ways A) If the number $n$ has at least 2 digits, we can delete the last digit and subtract it from the number remaining (for example, if we have ...
4
votes
2answers
34 views

Minimizing lengths of cevians in an isosceles right triangle

Consider isosceles right triangle $ABC$ with $BC$ as the hypotenuse and $AB=AC=6$. $E$ is on $BC$ and $F$ is on $AB$ such that $AE+EF+FC$ is minimized. Compute $EF$. My thought process: I reflected ...
-3
votes
0answers
30 views

finding all possible combinations of a, b, c, such that a number, is a power of 2016 [closed]

Find all the possible trios of integers (a, b, c), such that the number $N=\frac{(a-b)(b-c)(c-a)}{2}+2$, is a power of 2016 The question above is from a past JBMO exam. I attempted to solve it, with ...
-2
votes
2answers
62 views

A question, from a past JBMO exam, involving, finding all possible solutions [on hold]

Find all the solutions to the equation $x^2-3xy+p^2y^2=12p$, where x and y are integers and p is a prime number. Guys, I tried solving the question above, however, to no avail. The question is from a ...
1
vote
2answers
24 views

A quest involving modulo and coins [closed]

A piggy-bank contains exactly 1000 coins (of 2, 5, 10, 20 and 50 cents), of total value $100. Prove that the piggy-bank contains at least one 10 cent coin. I was attempting this question, in ...
0
votes
1answer
26 views

A question invoving combinatorics and many limitations [on hold]

Three kids, Andrew, Basil and George, were listening to four different songs. None of the four songs was appreciated, by all three kids. For each of the three possible pairs of kids (Andrew and Basil ...
1
vote
1answer
36 views

proving that x+y+z=0 and that $x^2-yz=y^2-zx=z^2-xy$ [on hold]

For the real numbers x, y, z which are different between themselves and the real numbers k, the following equalities are true: $x^2+y^2+kxy=y^2+z^2+kyz=z^2+x^2+kzx$, prove that $x+y+z=0$ $x^2-yz=y^...
-2
votes
2answers
67 views

A question, from the 2014 IMC selection exam

Andrew is getting prepared for exams in four different subjects, Greek, Math, physics and computers. He is preparing a weekly programme ($7$ days) so that he studies only one lesson every day and ...
-2
votes
4answers
77 views

proving that $a^{2009}+b^{2009}=c^{2009}+d^{2009}$ [closed]

Knowing that $a+b=c+d$ and $a^3+b^3=c^3+d^3$, prove that $a^{2009}+b^{2009}=c^{2009}+d^{2009}$. Can you guys please help me complete this proof, as I was attempting it yesterday, without getting far. ...
1
vote
2answers
40 views

Proving that $a=b=c=d$

If $a,b,c,d>0$ and $a^4+b^4+c^4+d^4=4abcd$, prove that a=b=c=d I've been trying this question all day, but I am not able, to thing of a solution. Can you guys please help me? Thanking you in ...
0
votes
1answer
22 views

Minimum value of an equation

For the real numbers a, b, c, find the smallest value which the following expression can take. $3*a^2+27b^2+5c^2-18*a*b-30*c+327$ I am having difficulties working out the problem above. The only ...
0
votes
1answer
40 views

Proving the independence of an equation, from certain variables

The different and unequal to zero real numbers x, y, z satisfy the equation $$x^3+y^3+m(x+y)=y^3+z^3+m(y+z)=z^3+x^3+m(z+x).$$ Prove that $$K=\left(\frac{x-y}{z}+\frac{y-z}{x}+\frac{z-x}{y}\right)\...
3
votes
2answers
110 views

A constraint that implies convexity

Let $f:\mathbb{R} \to \mathbb{R} $ be a function such that $\forall x<y, \exists z\in(x, y) $ with $(y-x) f(z) \le (y-z) f(x) +(z-x) f(y) $. a) Give an example of a non-convex function $f$ ...
0
votes
0answers
30 views

A point such that Areas of triangle formed is equal to area of given quadrilateral

'ABCD is a quadrilateral and X is a given Point in AD. Find a point Y in AB such that the area of triangle AXY is equal to that of ABCD' Since it is a quadrilateral, I haven't reached anywhere using ...
2
votes
1answer
67 views

$4^x + 5^x = 6^x$

In a recent MathCounts Contest, the following question was asked If $x$ satisfies $4^x + 5^x= 6^x$ then find the greatest integer not greater than $x$ I tried to take logarithms of both sides, but ...
1
vote
1answer
46 views

evaluation of a fraction, a preperation question for the JBMO

Given that: $$\frac{x}{x^2+3x+1}=a$$ Evaluate $$\frac{x^2}{x^4+3x^2+1}$$ I tried solving the question above by factorizing it, however, I didn't get far.
0
votes
2answers
32 views

Question for preparation for the internationals

If the numbers $a$ and $b$ are such that $a^2+b^2+8a-14b+65=0$, find the value of $a^2+ab+b^2$. I tried all every method I know for the above question, but I just aren't able to work it out. There ...
0
votes
1answer
27 views

Question for preparation for the IMC internationals

Work out the addition of: (1/(1+2))+(1/(1+2+3))+...+(1/(1+2+3+...+51)) Guys, I'm having difficulties working this one out. Using a calculator, it would have been easy, however in the internationals ...
6
votes
1answer
114 views

Minimal number of questions to identify a subset

This is a curiosity question. Recently I stumbled across the following problem : Given three integers $k,m, n$ such that $m+k\leq n$. A friend chooses a subset $S\subseteq\lbrace1,\ldots,N\rbrace$...
6
votes
1answer
77 views

Finding the maximum relative misalignment of numbered rings on a combination lock?

I've been trying to figure out a general formula to calculate the maximum relative misalignment of m identical rings with n symbols each on a combination lock like the one shown below. By "maximum ...
1
vote
1answer
65 views

For positive $a$, $b$, $c$, $d$ with $a+b+c+d=4$, show that $\sum_{cyc} \left(\frac{1}{a^2}-a^2\right)\ge0$

From MOP 2012: If $a,b,c,d>0$, $a+b+c+d=4$, show $$\sum_{cyc} \left(\frac{1}{a^2}-a^2\right)\ge0$$ My attempt: Using the derivative and tangent line, I obtain $$(x-1)(-x^3+7x^2-x-1)=0$$ ...
2
votes
1answer
59 views

Find all real functions such that $(x + 1)f(xf(y)) = xf(y(x + 1))$

Find all real functions of real variable such that $$(x + 1)f(xf(y)) = xf(y(x + 1))$$ Let $a=f(0)$. For $y=0$ we get $(x+1)f(ax) = ax$, so if $a\ne 0$ we get $$f(x) = {ax\over x+a}$$ which is actual ...
3
votes
1answer
89 views

Number of equal triangles in a chessboard

$1\times 1$ is cut and taken out from every corner of a $8\times 8$ chess board. At least, how many equal triangles (equal triangles means congruent triangles, and color is not important) can be drawn ...
4
votes
2answers
106 views

Find the positive root of $100^{2}=x^{2}+ \left( \frac{100x}{100+x} \right)^{2}$

I was struggling with this problem: $$100^{2}=x^{2}+ \left( \frac{100x}{100+x} \right)^{2}$$ It came up when i was developing a solution to a geometry problem. I've already checked in Mathematica ...
2
votes
1answer
93 views

prove this inequality with $x^3+y^3+z^3$

Let $x,y,z>0$. Show that $$x^3+y^3+z^3+2(xy^2+yz^2+zx^2)\ge 3(x^2y+y^2z+z^2x).$$ I have a solution by using $y=x+a,z=x+a+b$; my question is: can this be solved using simple methods (such as AM-...
3
votes
2answers
140 views

Find the minimum of $(1+a^2)(1+b^2)(1+c^2)$ where $a,b,c\geq 0$

Find the minimum of: $$(1+a^2)(1+b^2)(1+c^2) \ \ \ a,b,c\geq 0$$ Knowing that $$ab+bc+ac=27$$ I tried my best using QM-AM-GM inequalities, Cauchy-Schwarz, etc. I tried also to do it with ...
0
votes
1answer
56 views

Help in solving the geometry question

I having the following question with me which is a part of the 2013 IMO shortlist "Let $ABC$ be an acute-angled triangle with orthocenter $H$, and let $W$ be a point on side $BC$. Denote by $M$ and $...
16
votes
6answers
354 views

Proving surjectivity of some map from a power set to a subset of integers.

We assign every element $i$ from $N=\{1,2,...,n\}$ a positive integer $a_i$. Suppose $$a_1+a_2+...+a_n = 2n-2$$ then prove that map $T: \mathcal{P}(N) \to \{1,2,...,2n-2\}$ defined with $$T(X) = \sum ...
0
votes
0answers
23 views

Find all differentiable functions $f:(0, \infty) \to (0, \infty)$ for which $f'(\frac{a}{x})=\frac{x}{f(x)}$ [duplicate]

Question: Find all differentiable functions $f:(0, \infty) \to (0, \infty)$ for which there is a positive real number $a$ such that $f'(\frac{a}{x})=\frac{x}{f(x)}$ for all $x>0$ [Putnam 2005] ...
3
votes
0answers
79 views

For positive $a$, $b$, $c$ with $abc=1$, show $\sum_{cyc} \left(\frac{a}{a^7+1}\right)^7\leq \sum_{cyc}\left(\frac{a}{a^{11}+1}\right)^7$

I'm interested by the following problem : Let $a,b,c$ be real positive numbers such that $abc=1$ and $a \leq 1$ and $b\leq 1$ then we have : $$\left(\frac{a}{a^7+1}\right)^7+\left(\frac{b}{b^7+1}\...
0
votes
1answer
53 views

question which was used for preperation, for the JBMO internationals

Find positive integers $a, b$, such that $(a, b)=1$ and the number $\frac {9a^2+14b^2}{9ab}$ is an integer. Guys, I tried solving the question above, with the use of modular mathematics. However, it ...
12
votes
2answers
227 views

tiling a square with rectangles

consider the set of all the rectangles with dimensions $2^a\times 2^b\,a,b\in \mathbb{Z}^{\ge 0}$.we want to tile a $n\times n$ square by rectangles from this set (you can use a rectangle several ...
2
votes
2answers
219 views

show this inequality to $\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $

Let $a,b$ and $c$ be positive real numbers. Prove that $$\sum_{cyc} \frac {a^3b}{(3a+2b)^3} \ge \sum_{cyc} \frac {a^2bc}{(2a+2b+c)^3} $$ This problem is from Iran 3rd round-2017-Algebra final exam-P3,...
0
votes
0answers
24 views

How to prove that the required number of guesses suffice?

The following question is there in the IMO Shortlist 2006: "The leader of an IMO team chooses positive integers $n$ and $k$ with $n > k$, and announces them to the deputy leader and a contestant. ...
0
votes
1answer
35 views

How to think about this type of game problem?

I am practising some olympiad problems. Not getting how to solve type of problems where we have a game and we need to answer who will win. Here is one such problem: Amy and Bob play the game. At ...
3
votes
1answer
132 views

Geometry problem - areas of triangles (contest math)

This problem is from 2019 Math Kangaroo competition for 9th-10th graders that took place last week, problem #29. I was able to solve it using coordinate geometry, both triangles have the same area. ...
3
votes
2answers
78 views

Functional equation: $(x+y)f(x,y)=yf(x,x+y)$

Find functions $f$ on pairs of positive natural numbers satisfying: $f(x,x)=x$ $f(x,y)=f(y,x)$ $(x+y)f(x,y)=yf(x,x+y)$ It is quite easy to find that $f(1,k)=k$ for all $k$ by induction: if $f(...
0
votes
1answer
31 views

Why is that particular kind of sequence relevant to the question not repeat

The following is a question from the IMO 2012 shortlist: Several positive integers are written in a row. Iteratively, Alice chooses two adjacent numbers $x$ and $y$ such that $x > y$ and $x$ ...
3
votes
2answers
143 views

exist infinitely many positive integers $n$ such $\omega (n)+\omega (n+1)\equiv 0\pmod 3$

For integer $n>1$, $\omega (n)$ denotes number of distinct prime factors of $n$, and $\omega (1)=0$. Prove that:there exist infinitely many positive integers $n$ satisfying $$\omega (n)+\omega (n+...
1
vote
0answers
81 views

Ring Theoretical Method of Solving a Math Olympiad Problem

These paragraphs are from Steve Olson's book Count Down: Six Kids Vie for Glory at the World's Toughest Math Competition. On page 170, the author said: The sixth and last problem on the Forty-...
2
votes
1answer
70 views

Find the angle in an isosceles triangle

Let triangle $\Delta ABC$ have $AB=AC$. Then we draw the angle bisector from $B$ to $AC$ intersecting at $D$. Find the angle $\angle BAC$ if $BC=AD+BD$. My attempts: I know that the answer is 100° ...
1
vote
1answer
54 views

Help in solving this geometric inequality

I have the following geometry problem with me: "The altitudes through the vertices A,B,C of an acute angles triangle meet the opposite sides at D,E and F respectively, and AB > AC. The line EF meets ...