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

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3
votes
1answer
27 views

Probability: Finding the Number of Pears Given Two Scenarios

You have a bag containing 20 apples, 10 oranges, and an unknown number of pears. If the probability that you select 2 apples and 2 oranges is equal to the probability that you select 1 apple, 1 ...
1
vote
1answer
61 views

High-school group-theory problem(given in a contest)

Let $G$ be a finite group and let $ H \le G $ be a subgroup of $G$. Suppose there is some $ \emptyset \neq S \subset G$ such that for any $x\in S$ we have $x^2 \notin H$. Prove that there is $T ...
3
votes
1answer
26 views

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 ...
0
votes
2answers
71 views

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 ...
1
vote
3answers
48 views

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) ...
12
votes
3answers
487 views

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, ...
7
votes
3answers
156 views

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: $$|z^2 - 2iz+1|\le|z|^2+2|i||z|+1$$ $$\implies |z^2 - 2iz+1|\le16$$ ...
4
votes
2answers
39 views

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 ...
0
votes
1answer
37 views

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?
0
votes
2answers
54 views

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 $7$.
0
votes
0answers
36 views

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 ...
0
votes
1answer
26 views

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 ...
2
votes
1answer
65 views

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: $$\frac{26}{672-x} + \frac{24}{372-x} = \frac{50}{480-x}$$ ...
1
vote
1answer
29 views

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 ...
5
votes
1answer
52 views

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 ...
1
vote
1answer
33 views

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 ...
2
votes
1answer
54 views

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, ...
1
vote
1answer
21 views

$A(n) = f(m)$ 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 ...
1
vote
1answer
38 views

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 ...
3
votes
2answers
65 views

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 ...
0
votes
1answer
34 views

Prove that, for each $n$, $\int^{1}_{0} f_n(x)dx=\frac{1}{2}$

Problem: Define $f: [0, 1] \to [0, 1]$ by $f(x)= 2x$ for $0 \leq x \leq \frac{1}{2}$ and $f(x) = -2x+2$ for $\frac{1}{2} \leq x \leq 1$ Let, $f_1(x)=f(x)$ and $f_{n}(x) = f(f_{n-1}(x))$ for all $n ...
2
votes
2answers
92 views

Proving $x>\sin(x)$ without calculus for $x>0$

The starting problem was to prove $$\sin 26^{\circ}\sin 58^{\circ}\sin 74^{\circ}\sin 82^{\circ}\sin 86^{\circ}\sin 88^{\circ} \sin 89^{\circ}>\frac{45\sqrt{2}}{64\pi}\\\cos 1^{\circ}\cos ...
4
votes
2answers
48 views

Relation $S(2x)=2S(x)-9N(x)$.

Let $S(x)$ be the sum of digits of number $x$ and $N(x)$ be the number of digits of $x$ greater than $4$. Prove that $S(2x)=2S(x)-9N(x)$. For example, if $x=1992$ then $S(x)=1+9+9+2=21$ and ...
0
votes
2answers
55 views

2015 AMC 10B Problem 21

The problem and solutions I've attempted to solve another AMC 10 problem, and the problem is basically like this: Cozy the Cat and Dash the Dog are going up a staircase with a certain number of ...
1
vote
0answers
33 views

Prove $\frac{3}{64}(ab+bc+ca)^3\geq (de)^3+(ef)^3+(fd)^3$ where $a, b, c$ are three sides of and $d, e, f$ three angle bisectors of a triangle.

A triangle has sides $a, b,c$ and angle bisectors $d, e, f$ where each pair of $a$ and $d$, $b$ and $e$, $c$ and $f$ intersect. Prove that $\frac{3}{64}(ab+bc+ca)^3\geq(de)^3+(ef)^3+(fd)^3$. I was ...
2
votes
2answers
82 views

2015 AMC 10A Problems/Problem 14

The Clockblock Problem - problem and solutions I'm preparing myself for AMC 10 (which I'm sure a lot of other students would be doing too), but then I just don't know how to solve this problem (and ...
6
votes
0answers
42 views

What is the probability of a pen touching a bar given that the length of the pen is $10$ cm and the bars are regularly spaced at $15$ cm?

Problem: If a pen of length $10$ cm is thrown out of infinitely large window having vertical bars regularly spaced at $15$ cm, then find the probability that it will touch any of the bars. (Assume ...
5
votes
1answer
115 views

How do you find the probability of A winning if the probability of getting a favourable outcome in the $r^{th}$ turn is a function of $r$?

Problem: Two players A and B are playing snake and ladder. A is at 99 and he needs 1 to win in rolling of a dice. However, he is always allowed to re-throw the dice if 6 appears. What is the ...
0
votes
0answers
67 views

Integer solutions to $y^2 = \frac{x^5-1}{x-1}$

$$y^2 = \frac{x^5-1}{x-1}$$ has integer solutions. How many pairs $(x,y)$ are there? My Work If $\sqrt{x^4+x^3+x^2+x+1}$ is an integer then there is a solution. But what to do now. Note: This ...
-5
votes
0answers
70 views

Simple logical proof of Fermat's Last Theorem [closed]

My interest in the Fermat Conjecture (FC,) began as an interest in the Pythagorean theorem. I wasn't looking for integer solutions of n>2. I was more interested in the fact that odd integer values of ...
2
votes
1answer
20 views

Given that there is at least one error in the bit, what is the probability that it will be retransmitted?

A communication channel can increase the probability of successful transmission by using error-correcting codes. One of the simplest of these is called a "parity scheme". In such a scheme, the message ...
3
votes
1answer
33 views

Probability of having at least one error in block of three bits?

A communication channel can increase the probability of successful transmission by using error-correcting codes. One of the simplest of these is called a "parity scheme". In such a scheme, the message ...
2
votes
4answers
132 views

A Quadrilateral's area given four sides and a diagonal [closed]

Assume there exists a quadrilateral called ABCD and AB=5cm,BC=13cm,CD=16cm, DA=20cm and diagonal AC=12cm. The exercise now states that I should calculate the area of a quadrilateral. Thank you for ...
6
votes
4answers
105 views

# of partitions of $n$ into at most $r$ positive integers $=$ # of partitions of $n + r$ into exactly $r$ positive integers?

How do I see that the number of partitions of the integer $n$ into at most $r$ positive integers is equal to the number of partitions of $n + r$ into exactly $r$ positive integers?
4
votes
2answers
61 views

How many different strings can be made from letters in CHICAGOLAND, subject to constraints? [closed]

How many different strings can be made from the letters in CHICAGOLAND, using all letters, and such that no two vowels are adjacent to each other?
3
votes
1answer
60 views

Good set with $n$ elements must have element $\ge {2\over n}\binom{n}{n\over2}$?

Let $n$ be even. A set $\{a_1, \dots, a_n\}$ consisting of positive integer s is good if for every two different disjoint subsets $S$, $T \subseteq [n]$ of the same cardinality we have$$\sum_{i \in S} ...
3
votes
1answer
56 views

Putnam: Show that $a(n)=b(n+2)$

Let $a(n)$ be the number of representations of positive integer $n$ as a sum of 1's and 2's taking order into account. $$ \text{Example $n=4$: } (1+1+1+1), (1+2+1),(1+1+2),(2+1+1),(2+2)\implies ...
1
vote
1answer
53 views

Dual program is wrong. Authors claim is right.

In a well respected book, I found the following. The authors claim that it is correct. But I think it is wrong. This is the primal Linear Problem: $$ \begin{array}{cccc} ...
3
votes
1answer
94 views

1995 USAMO Problems/Problem 2

I tried to solve this problem: A calculator is broken so that the only keys that still work are the $\sin, \cos, \tan, \sin^{-1}, \cos^{-1}, \tan^{-1}$ buttons. The display initially shows $0$. ...
1
vote
1answer
76 views

math contest geometry problem

Consider a triangle $ABC$ with circumcircle $\omega$. Let $O$ be the center of $\omega$ and let $D, E, F$ be the midpoints of minor arcs $BC, CA, AB$ respectively. Let $DO$ intersect $\omega$ again at ...
6
votes
2answers
65 views

At least $P(m, n - 1) = {{m!}\over{(m - n+1)!}}$ surjective functions from $[m]$ to $[n]$?

How do I see that there are at least$$P(m, n - 1) = {{m!}\over{(m - n+1)!}}$$surjective functions from $[m]$ to $[n]$?
4
votes
1answer
93 views

How many integers between $1$ and $2016$ are divisible by a nontrivial cube $p^3$, $p > 1$? [closed]

How many integers between $1$ and $2016$ are divisible by a nontrivial cube $p^3$, $p > 1$?
2
votes
1answer
49 views

Find $N$ so that the sequence is the product of three consecutive numbers

Find the smallest natural number $N$ such that $13 \cdot 17 \cdot N$ is the product of three consecutive natural numbers. $x(x+1)(x+2) = 13 \cdot 17 \cdot N$. So let $x=N$, then, $N+1 = 13$ and ...
0
votes
2answers
34 views

How many cards need to be picked at least?

You have $50$ cards and you have the numbers from $1$ to $50$ written on them, and you randomly pick cards. How many cards do you need to pick out so you can ensure that at least $3$ cards with ...
3
votes
0answers
40 views

A number $n$ has $12$ divisors and $d_{d_4-1} = (d_1+d_2+d_4)d_8$.

Find a number $n$ which has - $1.$ $12$ divisors $(1 = d_1 < d_2 < \cdots <d_{12}=n )$ and $2.$ $d_{d_4-1}=(d_1+d_2+d_4)d_8$. Note: This is a problem from Russian Mathematical Olympiad ...
1
vote
2answers
57 views

How many three digit numbers exist such that the third digit is the geo mean

How many three digit numbers exist such that one of the digits is the geometric mean of the other two? A 12, B 18, C 24, D other So, $N = 100a + 10b + c$ let $c =\sqrt{ab}$. $ab$ must be a ...
0
votes
1answer
42 views

How to find the missing digit?

A student calculated the value of $1 \times 2\times 3\times \cdots \times 2015\times 2016=2016!$ Then he took the summation of all digits of that answer ! He got $24135$ , but later he realized ...
-1
votes
1answer
36 views

How to prove that there is no infinite arithmetic progression of perfect squares

How to prove that there is no infinite arithmetic progression of perfect squares This question from a school Olympiad paper ! How can I prove this directly or using contradiction ? For example : 1 ...
1
vote
1answer
38 views

The $n-th$ term of a sequence is the LCM of the integers from $1$ to $n$

The $n-th$ term of a sequence is the least common multiple (LCM) of the integers from $1$ to $n$. Which term of the sequence is the first one that is divisible by $100$? How I'll solve this? ...
6
votes
2answers
73 views

Compute $\lim_{n \to +\infty} n^{-\frac12 \left(1+\frac{1}{n}\right)} \left(1^1 \cdot 2^2 \cdot 3^3 \cdots n^n \right)^{\frac{1}{n^2}}$

How to compute $$\displaystyle \lim_{n \to +\infty} n^{-\dfrac12 \left(1+\dfrac{1}{n}\right)} \left(1^1\cdot 2^2 \cdot 3^3 \cdots n^n \right)^{\dfrac{1}{n^2}}$$ I'm interested in more ways of ...