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

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3
votes
3answers
107 views

Proving that $1^n+2^n+3^n+4^n$ $(n\in \Bbb N)$ is divisible by 10 when $n$ is not divisible by 4

I was solving some math problems to prepare for math contests and came across this one: Prove that $1^n+2^n+3^n+4^n$ $(n\in N)$ is divisible by 10 if and only if $n$ is not divisible by 4. So, from ...
1
vote
2answers
55 views

2015 AIME #3: Where did I go wrong?

This is a question conerning 2015 AIME #3. The problem goes as follows: There is a prime number $p$ such that $\displaystyle 16p+1$ is the cube of a positive integer. Find $p$. Here is my ...
2
votes
3answers
67 views

Do the lengths of all three segments necessarily have the same distribution?

Let $A$ and $B$ be independent $U(0, 1)$ random variables. Divide $(0, 1)$ into three line segments, where $A$ and $B$ are the dividing points. Do the lengths of all three segments necessarily have ...
1
vote
4answers
89 views

Among the following, which is closest to $\sqrt{0.016}$?

Among the following, which is closest in value to $\sqrt{0.016}$? A. $0.4$ B. $0.04$ C. $0.2$ D. $0.02$ E. $0.13$ My Approach: $(\frac{16}{1000})^\frac{1}{2} = (\frac{4}{250})^\frac{1}{2} = ...
3
votes
0answers
51 views

No eight digit perfect fourth powers with distinct digits and not containing 3.

Problem: Prove that there are no perfect fourth powers that have eight distinct digits in their base 10 representation and also don't contain 3 as a digit. My attempt: Since the problem is about ...
0
votes
3answers
38 views

Modular Arithmetic Related Question

I've been trying to solve this problem but I couldn't and I don't get the solutions either (I don't think I get how to use modular arithmetic to solve problems in general. And though I tried to ...
3
votes
1answer
114 views

$15$ integers $m_1 \ldots m_{15}$such that $ \sum _{k=1} ^{15} m_k \arctan {k} = \arctan 16$

Determine whether or not there exist $15$ integers $m_1 \ldots m_{15}$ such that $ \sum _{k=1} ^{15} m_k \cdot \arctan (k) = \arctan (16)$. This is a question from IMC 2015 Day 1 Problem. Here ...
1
vote
2answers
31 views

What is the probability that the two-seed makes the finals?

I can solve the following problem using brute-force combinatorics, but I'm looking for an elegant way to think about it, since there is a rather elegant answer. Suppose there is a tournament of ...
2
votes
3answers
82 views

Find the value of $\frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + \frac{1}{72} + \frac{1}{90}$

Find the value of $p+q$, where $p$ and $q$ are two positive integers such that $p$ and $q$ have no common factor larger than $1$ and $$\frac{1}{20} + \frac{1}{30} + \frac{1}{42} + \frac{1}{56} + ...
0
votes
2answers
39 views

Proving angles are supplementary in isosceles triangle

Let $ABC$ be a triangle with $AC=BC$, and let $P$ be a point inside $\triangle ABC$, satisfying $\angle PAB=\angle PBC$. If $M$ is the midpoint of $AB$, show that $\angle APM+\angle BPC=180^{\circ}$. ...
3
votes
2answers
54 views

how we can calculate $ \frac {\sqrt {x^2} + \sqrt {y^2} }{2 \sqrt {xyz}}$? [closed]

I teach math for Schools. How can Help me in the following past Olympiad question? If $y,z$ be two negative distinct number and $x$ and $y$ be negate of each other, how we can calculate $ ...
-1
votes
1answer
230 views

Set of all $n$; $n={d^2_1 + d^2_2 + d^2_3 +d^2_4}$

$A$ is the set of all $n$ numbers where $n={d^2_1 + d^2_2 + d^2_3 +d^2_4}$. Here $1=d_1<d_2<d_3<d_4$ where $d_1,d_2,d_3,d_4$ are the $4$ smallest divisors of $n$. As an example ...
6
votes
1answer
260 views

Game theory, olympiad question

I've seen the following question in the brazilian olympiad for university students, and I couldn't solve it. Thor and Loki play the game: Thor chooses an integer $n_1 \ge 1$ , Loki chooses $n_2 \gt ...
2
votes
2answers
59 views

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 ...
-1
votes
2answers
36 views

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 ...
1
vote
4answers
69 views

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 ...
1
vote
1answer
46 views

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 ...
1
vote
2answers
37 views

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 ...
3
votes
2answers
70 views

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 $|X \cup Y|$?
16
votes
3answers
840 views

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 ...
2
votes
2answers
230 views

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 ...
1
vote
2answers
55 views

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 ...
0
votes
0answers
27 views

$f\in \mathbb{Z}[x], f(x) = y^2, f(y) = z^2, f(z)=x^2 \implies x=y=z$?

Given that $f\in \mathbb{Z}[x], f(x) = y^2, f(y) = z^2, f(z)=x^2$ for some real numbers $(x,y,z)$, does it follow that $x=y=z$? It is well known that if $f\in \mathbb{Z}[x]$ and $f(x)=y, f(y)=z, ...
8
votes
4answers
140 views

Is there $n$ such that $n,n^2,n^3$ start with the same digit ($\neq 1)$

From the 2015 Moscow Mathematical Olympiad: Is there some $n>2$ such that $n,n^2$ and $n^3$ start with the same digit (this digit being different from $1$) Using a computer I found that $99$ ...
1
vote
1answer
67 views

Find minimum number of coins with Largest value coins?

There is a greedy algorithm for coin change problem : using most valuable coin as possible. How We can find a quick method to see which of following sets of coin values this algoithms cannot find ...
3
votes
1answer
40 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
74 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 ...
4
votes
1answer
33 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
83 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 ...
3
votes
3answers
61 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) ...
13
votes
3answers
567 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, ...
8
votes
3answers
187 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
55 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
60 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
1answer
66 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
46 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 ...
2
votes
1answer
43 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
70 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
31 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
69 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
38 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
63 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
40 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
83 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
38 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
49 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
3answers
107 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
38 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 ...