Questions tagged [contest-math]

For questions about mathematics competitions or the questions that typically appear in math competitions.

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2answers
33 views

Prove that three segments which intersect a circle pass through the same point

In a scalene triangle $\triangle ABC$ with $AB\ne AC$, I state we have $Y$ which is the point of intersection of the bisector of $\angle A$ with $BC$ and $D$ is the point where the perpendicular line ...
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1answer
35 views

What is the minimum number of broken queens required to cover an $n\times n$ board?

Consider an $n\times n$ board. Assume that the sides of the board are parallel to the north-south and the east-west directions. If a piece of "broken queen" is placed on this board, it &...
3
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2answers
86 views

Find all functions satisfying $ f \left( m ^ 2 + f ( n ) \right) = f ( m ) ^ 2 + n $, for all $ m , n \in \mathbb N $

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$f(m^{2}+f(n))=f(m)^{2}+n\text {, for all }m,n\in \mathbb N$$ I was initially unable to solve this problem, so I referred to a hint. The hint ...
1
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1answer
36 views

Symmedian and orthic triangle properties

I am trying to prove the following lemma that may be useful for junior-level international contests: Given $\Delta ABC$ an acute triangle and $(BE)$ and $(CF)$ its altitudes. Let's consider $(AM)$ ...
3
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0answers
46 views

How to “remember” the solution to previously solved problems?

Does anyone have tips on how to glean and remember, long-term, the key concepts from tricky brainteaser/Olympiad-style problems (or tough problems in general) that aren't just memorizing the specific ...
1
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1answer
35 views

Given a scalene triangle and its orthocenter, prove that a point on its Euler circle, has a constant line segment

Given a scalene triangle $\triangle ABC$ and $H$ the orthocenter of the triangle. $P$ is a point on the Euler circle of the triangle $\triangle ABC$. The segments $BH, CH$ intesect the opposite sides $...
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2answers
60 views

For which $n$ the first player has a strategy so that he wins no matter how the other player plays?

The Question From Junior O-level Tournament of the Towns paper, Fall 2020: There are $n$ stones in a heap. Two players play the game by alternatively taking either $1$ stone from the heap or a prime ...
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1answer
32 views

Given a scalene triangle $ABC$ with $H$ orthocenter, prove that two lines are parallel

Given a scalene triangle $\triangle ABC$ with $H$ the orthocenter of the triangle. The internal bisector of the angle $\angle BAC$ intersects the lines $BH$ and $CH$ at the points $Λ$ and $Θ$ ...
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0answers
70 views

The Board Football Problem (Probability)

A and B are playing " board football", a two player in which the objective is to score as many goals as possible. As the game does not have any terminating statement, an infinite number of ...
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0answers
37 views

Winning strategy of this Nim Game Variation?

In the Nim variation I'm looking at, two players alternate turns removing stones from N piles. The special condition is that each time a player removes k stones, k must be a multiple of the last k. ...
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1answer
79 views

Find the Relation between $a,b,c$

In the figure shown find the relation between $a,b,c$. My try: When two circles of radii $r_1,r_2$ touch externally, the length of their direct common tangent is $2\sqrt{r_1r_2}$ Let the radius of the ...
2
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1answer
52 views

Find all primes such that $\prod_{i=1}^{n} p_i=10\sum_{i=1}^{n}p_i$

Find all primes $p$ (it's not necessary to this primes to be different) such that $$\prod_{i=1}^{n} p_i=10\sum_{i=1}^{n}p_i$$ I’ve realized that $$10\mid \prod_{i=1}^{n} p_i\implies \prod_{i=1}^{n} ...
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1answer
61 views

Find all $n$ such that $2^n+1$ and $2^n-1$ are primes [duplicate]

Find all $n\in \mathbb N$ such that $2^n+1$ and $2^n-1$ are both primes . My Attempt: let $p=2^n+1$ and $q=2^n-1$ I will start with a claim. claim: $n=2\implies p=5, q=3$ is the only solution, ...
2
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1answer
55 views

If $a,b,c,d,e$ be real number other than $-2≤a≤b≤c≤d≤e≤2$, prove the following inequation $\frac{1}{b-a}+\frac{1}{c-b}+\frac{1}{d-c}+\frac{1}{e-d}≥4$

Question If $a,b,c,d,e$ be real numbers other than $-2≤a≤b≤c≤d≤e≤2$, prove the following inequation $$\frac{1}{b-a}+\frac{1}{c-b}+\frac{1}{d-c}+\frac{1}{e-d}≥4$$ I tried to use Cauchy-Schwarz like $$\...
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1answer
33 views

Subset permutations

Given that $B = (a_1, a_2 \cdots, a_{12})$ is a permutation of the set $(1, 2, \cdots, 12)$ such that $a_1>a_2>a_3>a_4>a_5>a_6$ and $a_6<a_7<a_8<a_9<a_{10}<a_{11}<a_{...
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1answer
63 views

proof check: $f$ maps positive integers to positive integers. $f(m)f(n)=f(mn)$ and $m+n$ divides $f(m)+f(n)$

Find all possible functions $f$ such that $f$ maps positive integers to positive integers, $f(m)f(n)=f(mn)$ and $m+n$ divides $f(m)+f(n)$. Can we safely say that $f(m)=m^k \cdot a$ (where $a$ is a ...
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0answers
23 views

A puzzle problem of Digital VLSI design?

My thoughts: Here price of chocolate is increasing in Geometric progression with common ratio 2 and first term 1, so on nth day price of chocolate will be $T_{n}=2^{n-1}$ irrespective of wheather ...
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1answer
41 views

Picking out unique solution of $x,y,z$ from two constraints

Let $x,y,z$ be positive real numbers in $\mathbb{R}$ satisfying the conditions:$x+y+z=12$ and $x^3 y^4z^5=(.1)600^3$. Then the value of $x^3 + y^3 +z^3=?$ This is a question from JEE mains, in it I ...
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0answers
155 views

$\sqrt{\frac{a^2+4bc}{b^2+c^2}}+\sqrt{\frac{b^2+4ac}{a^2+c^2}}+\sqrt{\frac{c^2+4ba}{b^2+a^2}}\ge 2+\sqrt{2}$

Prove that $\forall a,b,c\ge 0$ then $$\sqrt{\frac{a^2+4bc}{b^2+c^2}}+\sqrt{\frac{b^2+4ac}{a^2+c^2}}+\sqrt{\frac{c^2+4ba}{b^2+a^2}}\ge 2+\sqrt{2}$$ I have some ideas but they didn't lead to simpler ...
4
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1answer
56 views

Making sure $a$ and $b$ are relatively prime

I came across this interesting problem in the Olympiad Maths challenge practice problem, and it is really fascinating: Some $n$ numbers are selected randomly from the integers $1$ to $420$. $2$ ...
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1answer
27 views

A positive integer n is cool if the sum of the digits of n is equal to the sum of the digits of 11n. What is the sum of all cool three-digit integers? [closed]

A positive integer n is 'cool' if the sum of the digits of n is equal to the sum of the digits of 11n. What is the sum of all 'cool' three-digit integers? I can't figure out how to solve this one... ...
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1answer
17 views

KVPY Question : Series and Progression

There are $20$ urns such that the first urn contains $5$ balls, the second contains $10$ balls and in general the $k^{th}$ urn contains $2k+1$ balls more than that in the $(k-1)^{th}$ urn. Then the ...
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1answer
44 views

Question regarding a proof of a inequality

Given $a,b,c, \in (0,\infty)$, then the following inequality holds $$\sqrt{5a^2+12ab+7b^2}+\sqrt{5b^2+12bc+7c^2}+\sqrt{5c^2+12ca+7a^2} \leq 2 \sqrt6 (a+b+c)$$ What I've tried: First, I noticed that we ...
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1answer
37 views

z complex $p, q \in \mathbb{N}, p < q$ such that $|z ^ p + \frac {1}{z ^ p}| \ge |z ^ q + \frac {1}{z ^ q}|$, so $|z + \frac{1}{z}| < 2$

I have participated yesterday in a contest. One of the problems was this one: If $x \in \mathbb{C} \setminus \mathbb{R}$ and there exists $p, q \in \mathbb{N}, p < q$ such that $|z ^ p + \frac {1}{...
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1answer
82 views

Solve ${x}^{\lfloor x \rfloor} = a$

I have found this weird equation in a math book. Could you give me any hints? $${x}^{\lfloor x \rfloor} = a$$ for a given a. I have dealt with the trivial cases where $x$ is an integer but cannot ...
3
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1answer
83 views

Tilings of a 5×5 square with smaller squares.

There is a $5 × 5$ array of lights, such that at each step, we may toggle all the lights in any $2 × 2, 3 × 3, 4 × 4$ or $5 × 5$ sub-square. Initially all the lights are switched off. After a certain ...
3
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0answers
62 views

$\sum\sqrt{\frac{2a}{b+c}}\le\sqrt[3]{9\sum\frac{a}{b}}$

Let $a$, $b$ and $c$ be positive numbers. Prove that: $$\sqrt{\frac{2a}{b+c}}+\sqrt{\frac{2b}{c+a}}+\sqrt{\frac{2c}{a+b}}\le\sqrt[3]{9\left(\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\right)}$$ It is from ...
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0answers
50 views

Find all functions that satisfy $(f(x)+f(y))(f(z)+f(t))=f(xy-zt)+f(xt+yz)$ for all real $x, y, z, t$ and $f:\mathbb{R} \rightarrow \mathbb{R}$ [closed]

Find all functions that satisfy $(f(x)+f(y))(f(z)+f(t))=f(xy-zt)+f(xt+yz)$ for all real $x, y, z, t$ and $f:\mathbb{R} \rightarrow \mathbb{R}$ I got that $f(0)=0$ and $f(x)f(y)=f(xy)$.
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1answer
44 views

For what $k$ are the numbers $4n+1, kn+1$ coprime? [closed]

Already knowing the answer gives me sort of a hint ($k=\pm2^m+4 : m \in \mathbb{N} \cup \{0\}$), but I can't really find a way through it. Bonus question: how about a generalisation of this problem? ...
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1answer
22 views

There are 9 blocks whats the length and width of one block. the pic has the rest of the info for the question [closed]

The rest of the questions information is here I'm really confused. Please help me with this question.
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2answers
81 views

If positive reals $a$ and $b$ satisfy $a\sqrt{a}+b\sqrt{b}=183, a\sqrt{b}+b\sqrt{a}=182$, find $\frac{9}{5}(a+b)$.

Question from Math Olympiad: Suppose $a, b$ are positive real numbers such that $a\sqrt{a} + b\sqrt {b} = 183$ and $a\sqrt{b} + b\sqrt {a} = 182$. Find $\frac{9}{5}(a+b)$. My approach: $a\sqrt{a} + ...
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0answers
15 views

Maximum value of $a[i]$ $mod$ $a[j]$ such that $a[i]>=a[j]$ [closed]

Given an array $a[]$ of $N$ positive integers. Find the maximum possible value of $a[i]$ $mod$ $a[j]$ over all pairs of $i$ and $j$ such that $a[i]>=a[j]$. Example: ...
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1answer
53 views

31st IMO 1990 shortlist p1

Question : Is there a positive integer which can be written as the sum of 1990 consecutive positive integers and which can be written as a sum of two or more consecutive positive integers in just ...
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1answer
48 views

$\forall a,b,c,d\in \mathbb{R}$ prove that $(a-b)(c-d)+(a-c)(b-d)+(d-a)(b-c) \geq 0$ and $a + d = b + c$

$\forall a,b,c,d\in \mathbb{R}$ prove that $(a-b)(c-d)+(a-c)(b-d)+(d-a)(b-c)$ and $a + d = b + c$ Attempt consider $(a-b)(c-d)+(a-c)(b-d)+(d-a)(b-c)=2(a-b)(c-d)$ since $2 >0$ we also want show that ...
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0answers
18 views

Product is a perfect $k$th power iff each of $a,b,c$ is a perfect $k$th power.

Given that $k \geq 2$ is an integer. If $a,b,c \in N$ and they are pairwise coprime, then prove that $abc$ is a $k$th power if and only if each of $a,b,c$ is a $k$th power. My try: Let $$abc=p^k$$ for ...
2
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1answer
54 views

find all primes $p$ such that $3^p-(p+2)^2$ is a prime

find all primes $p$ such that $3^p-(p+2)^2$ is also a prime. My proof set $3^p-(p+2)^2=q$ case $1$:$p=2 \implies q = -7 \notin \mathbb N$. so there is no solution. case $2$: $p \geq 3$ notice that $p=...
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2answers
43 views

If $3a+4b=x^2$ and $4a+3b=y^2$ Show that $7\mid a,b$

$a,b\in \mathbb N$ And such that $3a+4b$ and $3b+4a$ are both perfect squares. Show that $7\mid a,b$. I don’t know how to solve this I’ve just made a system of equations: $$\cases{3a+4b=x^2 \\ 3b+4a=y^...
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1answer
69 views

Good algebra book for high school contest preparation. [closed]

So basically I need a book which I can learn Algebra from. I want to participate in a contest someday, so it would be good if that specific book would also have some really hard problems and good ...
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1answer
22 views

Prove that there exist integers $x, y$ so that $2x^2+3y^2-r$ is divisible by a given prime larger than 3. [closed]

Prove that there exist integers $x, y$ so that $2x^2+3y^2-r$ is divisible by some fixed prime larger than 3. $r$ is also an integer.
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1answer
81 views

How do i verify that $272727…2727$ ($100$ digits) can or cannot be written as a perfect square?? [duplicate]

I've been stuck on this question. I tried writing the number as as geometric progression plus $$2((10^{100}-1)/9)+5+5.10^2+5.10^4...$$ Got stuck in there.
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0answers
32 views

How to find number of “Multiplying Functions”?

Consider the set $A=\{1,2,3,4,5\}$. We call the function $f$ "Multiplying" if for each $x,y\in A$, at least one of these two conditions satisfied: $x\times y>5$ $f(x\times y)=f(x)\times ...
1
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0answers
36 views

Olympiad Combinatorics Advice. [duplicate]

I have been trying to improve my combinatorics for Math Olympiad. I am already familiar with like mid - AIME level, ie. essentially I only know short answer combo, and I wanted to improve to like USA(...
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1answer
61 views

a counting question (about even/odd number) [closed]

Suppose that there are $1994$ numbers on the blackboard:$1,2,\cdots,1994$. Each round, one erases any two numbers, $n_1$ and $n_2$, and then writes the sum ($n_1+n_2$) or the difference ($n_1-n_2$) of ...
5
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2answers
78 views

Difference between the maximum and minimum values ​of $a+b$ that satisfy $a+b+\frac{1}{a}+\frac{9}{b}=10, (a,b\in\mathbb{R}^+)$

Find the difference between the maximum and minimum values ​​of $a+b$ that satisfy $$a+b+\frac{1}{a}+\frac{9}{b}=10,\quad(a,b\in\mathbb{R}^+)$$ I'm trying to use Cauchy–Schwarz inequality, but I can'...
2
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1answer
56 views

Let $f(x)=\frac{9^x}{9^x+3}$. Evaluate $\sum_{i=1}^{1995}f(\frac{i}{1996})$.

Let $f(x)=\frac{9^x}{9^x+3}$. Evaluate $\sum_{i=1}^{1995}f(\frac{i}{1996})$. This problem seems extremely hard until you find out that $f(x)+f(1-x)=1$ and you can then evalute $f(\frac{1}{1996})+f(\...
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0answers
65 views

Solution to Putnam 1995 B6

I am stumped. I will give the problem and what I have so far. For a positive real number α, define $S(α)$ = {$\lfloor nα$$\rfloor$ : n = 1,2,3,...}. Prove that {1,2,3,...} cannot be expressed as the ...
0
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1answer
67 views

How to prove $a^2+b^2+c^2+d^2+ac+bd\ge\sqrt3$?

We have $ad-bc=1$ . prove $$a^2+b^2+c^2+d^2+ac+bd\ge\sqrt3$$ To solve it, I multiplied the inequality by $2$ and added $2(ad-bc)=2$ to it : $$2a^2+2b^2+2c^2+2d^2+2ac+2bd+2ad-2bc\ge2+2\sqrt3$$ $$(a+c)^...
2
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3answers
98 views

Midpoints of diagonals

Let $ABCD$ be convex quadrilateral such that $AB=CD$. And $E\neq F$ where $E, F$ is midpoint of $AC, BD$ respectively. Then prove that angle between$(AB, EF)$ and $(CD, EF)$ are equal. I'll prove $\...
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3answers
95 views

given $a+b+c=3$ prove that $abc(a^2+b^2+c^2) \leq 3$

I am preparing for inmo and I came accross this problem while solving a worksheet, but couldn't solve it, pl help me... Problem- Prove that if a,b,c are non negative real numbers such that a+b+c=3, ...
0
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2answers
68 views

Prove that there exists no bijective function $f: \Bbb{N} \to \Bbb{N}$ such that $f(mn)=f(m)+f(n)+3f(m)f(n)$ for $m,n \geqslant1.$

Prove that there exists no bijective function $f: \Bbb{N} \to \Bbb{N}$ such that $$f(mn)=f(m)+f(n)+3f(m)f(n)$$ for $m,n \geqslant1.$ This was a problem from a Putnam practice book and I couldn't seem ...

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