# Tagged Questions

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

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### Cauchy like inequality $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$

Problem: Prove that for real $x, y, \alpha, \beta$, $(5\alpha x+\alpha y+\beta x + 3\beta y)^2 \leq (5\alpha^2 + 2\alpha \beta +3\beta ^2)(5x^2+2xy+3y^2)$. I am looking for an elegant (non-bashy) ...
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### Expected value problem with cars on a highway

There is a very long, straight highway with $N$ cars placed somewhere along it, randomly. The highway is only one lane, so the cars can’t pass each other. Each car is going in the same direction, ...
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### How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality?

Problem: How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality? My attempt: $$|z^2 - 2iz+1|\le|z|^2+2|i||z|+1$$ $$\implies |z^2 - 2iz+1|\le16$$ ...
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### Pair of friends and a pair of “enemies” in each group of three students

The problem: There is a class. In each group of three students in the class there is a pair of friends and a pair of "enemies". Find the maximum number of students in the class. I tried to play with ...
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### How many non-congruent triangles with perimeter 11 have integer side lengths? [closed]

How many non-congruent triangles with perimeter 11 have integer side lengths? I failed to solve it. Can anyone help?
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### A number theory contest problem

I have come across a problem I can't solve. Can anyone help? Here is the problem Find least integer $N$ such that sum of the digits of both $N$ and $N+1$ is divisible by $7$.
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### Books or website about solving IMO problems

Hey I want to solve IMO problems like the problem in the image below, but I cannot solve the problem or any of the problems in the IMO, so do you guys have some good website or books that teach how to ...
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### Four Spheres Intersect Along Circles: Prove That Circles' Planes Are Either $\parallel$ to The Same Line, Or Have a Common Point

Problem: Let $\,A,\,B,\,C,\,D\,$ be four distinct spheres in a space. Suppose the spheres $A$ and $B$ intersect along a circle which belongs to some plane $P$, the spheres $B$ and $C$ intersect ...
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### Math Contests: How to Solve Equation with $x$ in the Denominator

Okay, I realize this seems like a really stupid question, but on a math contest (without calculators) I got down to this equation: $$\frac{26}{672-x} + \frac{24}{372-x} = \frac{50}{480-x}$$ ...
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### Show impossibility of a perfect covering

Problem: Show that a $8 \times 8$ chessboard cannot be perfectly covered by $1$ square tetramino, and 15 other tetraminoes chosen from straight tetraminoes and Z-tetraminoes. My attempt: I tried to ...
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### Number of $n$-digit permutations with exactly $n-2$ digits smaller than the next

How many permutations of $1,2,\cdots, n$ contain exactly $n-2$ digits that are smaller than the digit immediately to their right? My solution proceeded with recursion. It has some chance of being ...
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### Sum of Reciprocals

I wonder if someone help me with this: I have $\pi_1+\pi_2+ \pi_3 +\pi_4=A$ and $\pi_1\pi_2\pi_3\pi_4=B$ where $\pi_i \;\forall i=1,2,3,4$ are unknown but $A,B$ are known numbers. Can I find for ...
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### Joining the Midpoints of the Sides of a Quadrilateral

$ABCD$ is a quadrilateral. $P$, $Q$ and $R$ are the midpoints of $AB$, $BC$ and $CD$ respectively. If $PQ = 3$, $QR = 4$ and $PR = 5$; find the area of $ABCD$. Since, $5^2 = 3^2+4^2$, So, ...
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### $A(n) = f(m)$ numbers of $f(m)$ followed by $f(m)$ numbers of $0$. $f(m)$ is the remainder when $m$ is divided by $9$.

A series is formed in the following manner: $A(1) = 1;$ $A(n) = f(m)$ numbers of $f(m)$ followed by $f(m)$ numbers of $0$; $m$ is the number of digits in $A(n-1).$ Find $A(30)$. Here ...
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### Greatest common divisor of $(2^{21}-1,2^{27}-1)$ [duplicate]

Find $\text{gcd}(2^{21}-1,2^{27}-1).$ My proof: We know that $2^{21}-1=(2^3)^7-1=8^7-1=(8-1)(8^6+\dots+8+1)=7(8^6+\dots+8+1)=7N_1$ and ...
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### Largest number of consecutive positive integers whose sum is exactly $2014$.

$97+98+ ...........+114+115 = 2014$. Here sum of $19$ consecutive numbers is $2014$. Find the largest number of consecutive positive integers whose sum is exactly 2014 and justify why you think ...
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### Mr.Smith commute word problem. Solved through logic, where is the argument unsound?

Mr. Smith commutes to the city regularly and invariably takes the same train home which arrives at the his home station at 5 pm. At this time, his chauffeur always just arrives, promptly picks him up ...