# Tagged Questions

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

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### Does probability depend on knowledge?

There is at least $2/3$ probability that this question is rather silly, but being an almost absolute beginner in Probability, I will ask it anyway. Consider the following problem, proposed at AIME ...
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### How many possible $4$-digit integer $x$ are there if $y-x=3177$?

Given any $4$-digit positive integer $x$ not ending in '$0$', we can reverse the digits to obtain another integer $y$. How many possible $4$-digit integer $x$ are there if $y-x=3177$? Denote $x=abcd$...
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### Compute the sum $\sum_{k=1}^{10}{\dfrac{k}{2^k}}$ [duplicate]

Compute the sum $$\sum_{k=1}^{10}{\dfrac{k}{2^k}}$$ This question is taken from SMO junior (I can't remember which year it is). I have no idea how to start. Can anyone give some hint? By writing ...
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### Plane geometry problem, Suppose ABP,BCP,CAP have same area&perimeter…

I'm trying to solve following geometry question, but it is quite challenging.(at least for me!) Thanks for your help in advance. On plane, there is some triangle ABC. Also, there is a point P ...
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### Let $S$ be the smallest positive multiple of $15$, that comprises exactly $3k$ digits with $k$ $0$'s, $k$ $3$'s and $k$ $8$'s.

The following is taken from Singapore Mathematical Olympiad $2013$ Junior Round $1$. Let $S$ be the smallest positive multiple of $15$, that comprises exactly $3k$ digits with $k$ $0$'s, $k$ $3$'s ...
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### Variance of the random variable $|X \cup Y|$? [closed]

Let $X$ and $Y$ be random subsets of $\{1, 2, \dots, k-1, k\}$ picked uniformly at random from all $2^k$ subsets, independent of each other. What is the variance of the random variable $|X \cup Y|$?
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### How many sewings are there on a soccer ball?

A soccer ball is obtained by sewing $20$ hexagonal pieces of leather and $12$ pieces of leather of pentagonal shape. A sewing joins together the sides of two adjacent pieces. How many sewings are ...
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### Weight of watermelons after percentage of water is evaporated.

A stock of watermelons of the initial weight of $500 \space\text{kg}$ has been put in a store for a week. Initially the percentage of water in the watermelons makes up $99 \%$ of the weight,...
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### Arrangement of points in a circle

From the 2015 Moscow Mathematical Olympiad: The numbers $1$ to $1000$ are arranged on a circle such that each number divides the sum of its two neighbors. Suppose that the number $k$ has two odd ...
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### How do you find $∠XPC$ + $∠XPB$ such that $PB+PC$ is maximum where $P$ is a point on $f(x) = (x-1)(x-3)(x-5)$?

Problem: $f(x) = (x-1)(x-3)(x-5)$ intersects the x axis at $A(1,0)$, $B(3,0)$ and $C(5,0)$. A point $P(t,f(t))$ is selected on the curve such that $PB+PC$ is maximum and $t \in (3,5).$ Let $PX$ be ...
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### Bounding a strange function

Let $a>0$, show that for $x>0$, $1<f(x)<2$, where $$f(x)=\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1+a}}+\sqrt{\frac{ax}{ax+8}}$$ I could take the derivative, find the maximum of the function ...
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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, and ...
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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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### 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 this ...