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

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2
votes
1answer
27 views

Looking for a simpler solution about quadratic congruence

Here is the Problem: 1)Suppose $p$ is a prime. prove that for any integer $k$, there exist integers $x$ and $y$ such that $x^2+y^2 \equiv k\ \pmod p$. 2)Are there infinitely many composite ...
4
votes
1answer
81 views

IMO Shortlist 1995 G3 by inversion

The incircle of $\triangle ABC$ is tangent to sides $BC$, $CA$, and $AB$ at points $D$, $E$, and $F$, respectively. Point $X$ is chosen inside $\triangle ABC$ so that the incircle of $\triangle XBC$ ...
4
votes
1answer
197 views

An infinite series of a product of three logarithms

I was told this interesting question today, but I haven't managed to get very far: Evaluate $$\sum_{n=1}^\infty \log \left(1+\frac{1}{n}\right)\log \left(1+\frac{1}{2n}\right)\log ...
0
votes
2answers
21 views

Application of A.M. -G.M. inequality

Let x, y,z be positive numbers. The least value of $ \frac{x(1+y)+y(1+z)+z(1+x)}{(xyz)^{.5}}$ is a) $\frac{9}{2^{.5}}$ b) 6 c) $\frac{1}{6^{.5}}$ d.) None of the above I tried applying the A.M. ...
1
vote
1answer
22 views

Algebraic Manipulations [duplicate]

Let a, b and c be such that $ a+b+c = 0 $ and $ l^2 = \frac{a^2}{2a^2+bc} + \frac{b^2}{2b^2+ac} + \frac{c^2}{2c^2+ba} $ The what is the value of l My approach : I could just put in the adequate ...
2
votes
1answer
60 views

The number of integral solutions $(x,y)$ of $x^3+3x^2y+3xy^2+2y^3=50653$

This was a wonderful question given to me by professor in my last class test. He asked for the solution with the least number of steps. Find the number of integral solutions $(x,y)$ of the ...
2
votes
2answers
43 views

Find the minimum $k$

Find the minimum $k$, which $\exists a,b,c>0$, satisfies $$ \frac{kabc}{a+b+c}\geq (a+b)^2+(a+b+4c)^2$$ My Progress With the help of Mathematica, I found that when $k=100$, we can take ...
1
vote
5answers
127 views
+50

Find real parametar $a,b,c$ such that function $f$ become convex function $f(x) = \begin{cases}ax^2+bx+c,& x<0\\1 ,& x \ge 0\end{cases}$

Find real parametar $a,b,c$ such that function $f$ become convex function $$f(x) = \begin{cases}ax^2+bx+c,& x<0\\1 ,& x \ge 0\end{cases}$$ My work: If $f(x)$ is convex function that means ...
0
votes
0answers
61 views

What is the value of $k^2$

For all $f(x)$ and $g(x)$ functions that are differentiable in $\mathbb{R}$, and satisfy the following conditions: Condition A: $$f(1)=1,~f(3)=3.$$ Condition B: ...
4
votes
1answer
62 views

Any math competitions dedicated to calculations by hand (college level math)?

Most of the people consider hand calculations the thing of the past. However, I recently started thinking about it and there are many interesting ways to do basic arithmetics on large numbers, ...
0
votes
0answers
30 views

Combinatorial Nullstellenatz riddle

I've been unable to solve the last problem here: http://www.mit.edu/~evanchen/handouts/BMC_Combo_Null/BMC_Combo_Null.pdf Let $n ≥ 2$ be even and let $v_1, v_2, . . . , v_k ∈ \{±1\}^n$ be vectors of ...
1
vote
2answers
37 views

Power of a point proof

I found the question on page 13 of this link. Let $P$ be a point inside a circle such that there exist three chords through $P$ of equal length. Prove that $P$ is the center of the circle. I ...
0
votes
3answers
67 views

Number of real root of the equation $8x^3-6x+1$ lying between -1 and 1 is

Number of real root of the equation $8x^3-6x+1$ lying between -1 and 1 is: I am lagging in solving the inequality portion. Let the roots be $m_1,m_2,m_3$ then $m_1m_2m_3=-\frac{1}{8}$ which means ...
0
votes
1answer
80 views

Logarithm in the exponent

$$(2x)^{\log 2} = (3y)^{\log 3} \\ 3^{\log x} = 2^{\log y}$$ Solve for $x$ and $y$. My intuition for solving such problems is taking the logarithm on both sides but it does not work. I also ...
0
votes
1answer
31 views

Using Affine Transformation to prove Concurrency

Let $ABCDE$ be a convex pentagon with $F=BC\cap DE, G=CD\cap EA, H=DE\cap AB, I=EA\cap BC, J=AB\cap CD$, Suppose that the areas of $\triangle AHI, \triangle BIJ, \triangle CJF, \triangle DFG, ...
9
votes
2answers
121 views

A nice and hard colouring problem

This question is a generalization of a problem recently appeared in a Italian mathematical competition. $A$ and $B$ are two coprime integers, both greater than $2$. A non-constant colouring $$ ...
0
votes
0answers
28 views

mathematical formula to compute sum of all sub sequences of a number N

We have a number say N and we list down all its sub- sequences and sum them up.SAY for n=123 ,the sum is 177(123+12+23+13+1+2+3). I came across this mathematical formula which computes the sum taking ...
2
votes
1answer
133 views

A problem of olympiad. [closed]

This nice functional equation was proposed in the “VIII Olimpíada Iberoamericana de Matemáticas” held in Mexico (1993). Find all the functions $f:\mathbb N^* \to \mathbb N^*$ such that i) ...
-4
votes
0answers
50 views

Big list of mathematical facts [closed]

There is a lot of good books and articles dedicated to school competitive math problems solving. Sometimes they contain some list of methods and facts which can be used to solve problems. Those lists ...
1
vote
6answers
75 views

What are some good books on algebraic inequalities?

By algebraic inequalities I mean inequalities like Cauchy's inequality, the AM-GM inequality etc. I need it for the International Mathematics Olympiad (IMO), so I hope I can find some books that ...
30
votes
6answers
35k views

Expected Number of Coin Tosses to Get Five Consecutive Heads

A fair coin is tossed repeatedly until 5 consecutive heads occurs. What is the expected number of coin tosses?
2
votes
1answer
68 views

Olympiad Inequality AM-GM (easy)

Prove that $(1 + x + y)^2 + (1 + y + z)^2 + (1 + z + x)^2 ≤ 3(x + y + z)^2$, with equality if and only if $x = y = z = 1$ ($xyz \ge 1$) ($x,y,z$ positive reals) This simplifies to $x^2 + y^2 + z^2 ...
-1
votes
3answers
59 views

Find the middle number in the $29$th row in the Pascal's Triangle

This question is taken from the Singapore Mathematical Olmpiad training notes for Primary school. Find the middle number in the $29$th row of the Pascal's triangle. For example, the middle number ...
3
votes
0answers
37 views

$\alpha$ exists so that for any points $x_n$ there is a point at average distance $\alpha$ from the $x_n$.

Let $X$ be a connected and compact metric space. Prove a real number $\alpha$ exists so that for every finite set of points $x_1,x_2,\dots, x_n\in X$ (not necessarily distinct) there exists $x\in X$ ...
1
vote
2answers
64 views

If three cevians are concurrent at a point and form triangles of equal area, the point is the centroid

Let D,E,F be points on side BC,CA,AB of triangle ABC. The three cevians are concurrent at a point G. The areas of triangles BGD, CGE and AGF are equal. Prove that G is the centroid of ABC I ...
0
votes
0answers
11 views

Isogonal Conjugate of point outside of triangle

I was wondering about reflections of lines over the external bisectors instead of external bisectors in a triangle. Here is a problem that brought it up: Let $P$ be a given point inside quadrilateral ...
0
votes
1answer
75 views

Congruence - Number Theory

Prove that $2005^{2005}$ is not the sum of two perfect cubes. I have looked at some mods but none have given me anything useful as of yet. I looked at the usual mods such as $4, 5, 7, 11, 13$ but ...
0
votes
0answers
23 views

Prove that matrix $a_{ij}=|A_i\cap A_j|$ is positive semi-definite

Let $A_i, i=\overline{1,n}$ be finite sets. Define the elements of $n\times n$-matrix $A$ as $$ a_{ij}=\big|A_i\cap A_j\big|. $$ The problem is to prove that this matrix is positive semi-definite. I ...
6
votes
5answers
774 views

Do there exist several positive real numbers such that their sum is $1$ and sum of their squares is less than $0.01$

Do there exist several positive real numbers such that their sum is $1$ and sum of their squares is less than $0.01$? My Attempt: Let there are $n$ real numbers and we call them ...
1
vote
1answer
67 views

What is the depth of water above the prism?

I have been practising for a math competition and came across the following question: A fishtank with base $100\,\rm cm$by $200\,\rm cm$ and depth $100\,\rm cm$ contains water to a depth of ...
4
votes
1answer
114 views

Find the coefficient of $x^{19}$ in the expression $(x+1)(x+2)(x+3)\cdots (x+400)$

Find the coefficient of $x^{19}$ in the expression $(x+1)(x+2)(x+3)\cdots (x+400)$ I have no clue how to start. Any kind of help will be appreciated.
9
votes
2answers
132 views

What's the minimal $k$ satisfying these conditions? Graph theory problem.

I'm thinking following problem. There are five pairs of couples (So, ten people total) and $k$ clubs satisfying following three conditions. Let $A,B$ are arbitrary people among those 10, ...
0
votes
1answer
26 views

Find the value of $k$ in the equation [closed]

Find the value of $k$ for which the equation $$kx^2-2015x+(k-2015)=0$$ has one positive and one negative root.
0
votes
0answers
20 views

Prove Concurrency using Radical Axis of Circumcircles

Let the incircle of $\triangle ABC$ touch sides $BC,CA,AB$ at $D,E,F$, respectively. Let $\omega,\omega_1,\omega_2,\omega_3$ be the circumcircles of $\triangle ABCm,\triangle AEF,\triangle ...
2
votes
2answers
50 views

linear algebra (matrices) - challenging problem (determination of method/algorithm)

I wonder about the following method/algortithm about square matrices $A_{n \times n}$ $\in$ $M_{n\times n}(\mathbb{K})$, where $\mathbb{K} $ $\in$ {$ \mathbb{R}, \mathbb{C}$ }. Given certain value of ...
0
votes
2answers
15 views

Find the total number of Chair in the hall

In a School hall, $\frac{7}{31}$ of the chairs are arranged in rows of 5, and $\frac{11}{31}$ of the chairs are arranged in rows of 13. The rest of the chairs are stacked up. If there are less than ...
5
votes
4answers
1k views

Factorial question: number of trailing zeroes in 125! [duplicate]

How many zeros are after the last nonzero digit of 125! ? The answer is 31, but how do you solve it?
3
votes
1answer
62 views

Comparison of $ ( 1^a + 2^a+ … n^a)^n$ and $n^n(n!)^a $

For a given real number $a>0$ , define $ d_n =( 1^a + 2^a+ ... n^a)^n $ and $ b_n = n^n(n!)^a $ for $ n = 1,2,\ldots$ Then a) $ d_n< b_n $ for $ n> 1$, b) There exists an integer ...
5
votes
2answers
2k views

Maximizing the volume of a rectangular prism

A rectangular prism has a surface area of $300$ square inches. What whole number dimensions give the prism the greatest volume? This is a math olympiad problem. It involves the volume and surface ...
1
vote
2answers
44 views

Computing the coefficient of $x^n$ in the following expansion

The coefficient of $x^{-n}$ in the expansion of $\frac{2-3x}{1-3x+2x^2}$ is $a.)$ $(-3)^n - (2)^{\frac{1}{2}n -1} $ $b.)$ $2^n + 1 $ $c.)$ $ 3(2)^{\frac{1}{2}n - 1} - 2(3)^n $ $d.)$ None of the ...
0
votes
2answers
57 views

Inequality from IMO 2000 problem 4 question $\Pi_{cyc}\left(a-1+\frac{1}{b}\right)\leq 1$ $abc=1$

I know the problem is repeated but my question is somehow different. I want to know whether my proof is correct because I have troubles with the last part. Since $abc=1$ we can homogenize the ...
0
votes
1answer
36 views

A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola $x^2 -y^2 = 2000^2$

I found this answer here on AoPS. I agree with the answer till it multiplies $49$ by $2$. I think it should be multiplied by $4$ since there are $4$ possible cases: 1) $x+y, x-y$ is positive. 2) ...
-2
votes
2answers
79 views

can anybody please explain me the answer for Putnam Exam $2010 A-3$? [closed]

enter image description here How was $(x,y)$ transformed into $(au-bv,bu+av)$? and how did $∂g$ become $∂x$ and $∂y$?
3
votes
1answer
56 views

Functional Equation - Rational

Fing all functions $g: R \to R$ such that, $g(x+y) + g(x)g(y) = g(xy) + g(x) + g(y)$ I have shown that $g(x) = 0$ for all $x$ and $g(x) = 2$ for all $x$ are solutions. I have also show that $g(x) = ...
4
votes
0answers
74 views

What is the value of $ 1+ \frac{1}{3} + \frac{1\cdot3}{3\cdot6} + \frac{1\cdot3\cdot5}{3\cdot6\cdot9} +\dots $ [duplicate]

The infinite sum $$ 1+ \frac{1}{3} + \frac{1\cdot3}{3\cdot6} + \frac{1\cdot3\cdot5}{3\cdot6\cdot9} + \frac{1\cdot3\cdot5\cdot7}{3\cdot6\cdot9\cdot12} +\dots $$ is a.) $2^{1/2}$ b.) $3^{1/2}$ c.) ...
2
votes
1answer
45 views

A conditional inequality which itself implies a sharper version of it [duplicate]

Problem: Given that $m, n$ are positive integers such that $\sqrt{7} -\frac{m}{n} > 0$. Then show that $\sqrt{7}-\frac{m}{n} > \frac{1}{mn}$. I have failed to do this fascinating problem. My ...
10
votes
2answers
292 views

Russia (2000) contest:Prove the existence of a pair of rows and columns with intersections differently coloured

We have a $100\times100$ board divided into $10^4$ unit squares. These squares are coloured with four colours so that every row and every column has $25$ squares of each colour. Prove that there ...
2
votes
2answers
63 views

All-Russian Olympiad question (sum of symmetrical functions)

(All-Russian Olympiad, $1995$, $11^{th}$ Graders, Final Round) Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions each of which has a vertical ...
3
votes
2answers
74 views

All-Russian Olympiad question (composite of quadratics)

($1995$, All-Russian Olympiad, $9^{th}$ Graders, Final Round) Is it possible for the equation $f(g(h(x)))=0$, where $f, g$ and $h$ are quadratic functions, to have solutions $x=1,2,...,8$ ? I'm ...
2
votes
3answers
98 views

Number of pairs of rational numbers that satisfy the given relation

The number of pairs $(x,y)$ that satisfy : $2x^2 + y^2 + 2xy - 2y + 2 = 0$ is a.) $0$ b.) $1$ c.) $2$ d.) None of the foregoing numbers My attempt : I am not well versed in number theory , thus I ...