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

learn more… | top users | synonyms (2)

2
votes
5answers
154 views

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 ...
-1
votes
3answers
71 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
42 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
71 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 have ...
0
votes
0answers
14 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
0answers
43 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 ...
0
votes
1answer
79 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 ...
4
votes
1answer
149 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.
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
22 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 BDF,\...
0
votes
2answers
20 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 ...
3
votes
1answer
64 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 $n>1$...
2
votes
2answers
59 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
60 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
38 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) $x+...
-2
votes
2answers
82 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 $∂y$?
4
votes
0answers
75 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.) ...
1
vote
2answers
46 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 ...
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 ...
1
vote
0answers
19 views

General Techniques - Number sets

There are many problems involving, proving numbers are irrational or not an integer and so forth (e.g roots of polynomials, size of an angle) What are some general techniques/tricks that I can use in ...
2
votes
3answers
105 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 ...
5
votes
2answers
85 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 ...
4
votes
2answers
74 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 ...
2
votes
1answer
47 views

Prove that $EF=\sqrt{3} AB$.

On a semicircle with diameter $AB$ and centre $S$, points $C$ and $D$ are given such that point $C$ belongs to arc $AD$. Suppose $\angle CSD=120^{\circ}$. Let $E$ be the point of intersection of the ...
1
vote
6answers
110 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 ...
4
votes
1answer
51 views

Characterize all continuous functions such that $\int_0^1 \left(f\left(\sqrt[i]{x}\right)\right)^{k - i}\,dx = {i\over k}$ [closed]

Let $k \ge 1$ be an odd integer. What are all continuous functions $f: [0, 1] \to \textbf{R}$ such that$$\int_0^1 \left(f\left(\sqrt[i]{x}\right)\right)^{k - i}\,dx = {i\over k}$$for every $i \in \{1, ...
5
votes
1answer
72 views

To prove a divisor to be a perfect square

Let $m$ and $n$ be positive integers, where $m$ is bigger than $n$ and $m+n$ is even. If $m^2-n^2+1\mid n^2-1$, prove that $m^2-n^2+1$ is a perfect square. I don't really have some valuable clues or ...
1
vote
1answer
18 views

Perturbation of tangent ball

As picture below, $A$ and $B$ are two balls, $\partial A\bigcap \partial B=\{k\}$, and $B$ contains $A$. How to show that $$ \forall h\in \partial B,\exists ~\varepsilon > 0 ~st~ A\subset B+\...
-2
votes
1answer
10 views

All the roots and all the coefficients of which are from S? [closed]

Does there exist a finite set S of non-zero real numbers such that for any positive integer n there exists a polynomial P(x) with degree at least n, all the roots and all the coefficients of which are ...
1
vote
2answers
114 views

Question about an inequality which seems right but not easy to prove

The origin problem is as follows: let $a,b,c,d$ are positive real numbers,and $a+b+c+d=4$ prove:$$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\geq 4+\frac{1}{4}[(a−b)^2+(b−c)^2+(c−d)^2+(d−...
9
votes
2answers
133 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, ...
1
vote
1answer
95 views

How many perfect squares exist? [closed]

Consider a set of $1985$ positive integers not necessarily distinct. Every number in set can be written in the form $p_1^{{\alpha _1}}p_2^{{\alpha _2}} \cdots p_9^{{\alpha _9}}$ where $p_1,p_2,\ldots,...
3
votes
1answer
27 views

Counting the set with restrictions

This is an interesting problem from a high school programming contest in China. Define a set $S=\{1,2,\ldots , n\}$ with $n$ elements. We need to choose some subset $A_{i,j}(A_{i,j}\subseteq S,1\...
3
votes
2answers
59 views

1987 AIME Problem #4

The Question: Find the area of the region enclosed by the graph of $|x-60|+|y|=|\frac x4|$. Answer: What I know: Because of all the absolute values I only need to find one side of the graph of ...
11
votes
2answers
320 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 are ...
2
votes
2answers
66 views

Combinatorics - Number of ways to fill a 3x3 grid with 0's and 1's such that there is at least one zero in each column and row

There seems to be a simple answer for this problem, but I just can't figure it out. I know there must be at least 3-9 zeros for a valid arrangement, and that there are $3!$ (6) possible combinations ...
1
vote
4answers
90 views

Find $N$ with $N^2=10^4M+N$

I'm a high school student in France and I participated in a math olympiad, and there was a question which I found impossible to solve. Maybe they are here some people who can help me: We take a ...
0
votes
2answers
43 views

Computing first k digits and last k digits of a large number using logarithm

How do we compute the first $k$ digits and last $k$ digits of a large number say $2^{N-1}$ for bigger values of $N$ using logarithms? An example for the algorithm will be greatly appreciated. I got ...
0
votes
0answers
55 views

Prove $(a-b)^2+(b-c)^2+(c-a)^2 \geq a^2+b^2+c^2-3\sqrt[3]{a^2b^2c^2}$ [duplicate]

For nonnegative real numbers $a,b$ and $c$, prove $$(a-b)^2+(b-c)^2+(c-a)^2 \geq a^2+b^2+c^2-3\sqrt[3]{a^2b^2c^2}$$ It is clear that the inequality is equivalent to $a^2+b^2+c^2 + 3\sqrt[3]{a^2b^2c^2}...
1
vote
2answers
80 views

Geometry - Tangent circles

Let chords AC and BD of a circle ω intersect at P. A smaller circle ω1 is tangent to ω at T and to segments AP and DP at E and F respectively. (a) Prove that ray T E bisects arc ABC of ω. (b) Let I ...
0
votes
1answer
29 views

Geometry - Angle chasing

An interior point $P$ is chosen in the rectangle ABCD such that $∠AP D + ∠BP C = 180◦$ . Find $∠DAP + ∠BCP$ Since also $\angle APB + \angle CPD = 180◦$ and by symmetry (you can swap $B$ and $D$) ...
6
votes
2answers
134 views

Show that $x=y+z$ for all $x \in S$

We are given a set $S$ as a subset of the rational numbers defined by: $0 \notin S$ If $s_1 , s_2 \in S$, then $\frac {s_1}{s_2} \in S$ There exists a nonzero rational number $q \notin S$ such ...
9
votes
1answer
541 views

2016 Spain Math Olympiad final stage, problem 2

Given a prime $p$. Prove that there exist $\alpha$ such that $p|\alpha(\alpha-1)+3$, if and only if there exist $\beta$ such that $p|\beta(\beta-1)+25$. My solution: Using quadratic residuu we have ...
3
votes
1answer
62 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) = ...
0
votes
1answer
54 views

Inequalities - AM-GM

Let $H_n = 1 + 1/2 + 1/3 + ... + 1/n$ Prove that; $H_n + n$ $\geq$ n$(n+1)^\frac{1}{n}$ for $n$ $\leq$ $2$ I have tried writing $H_n + n = 1/2 + 1/3 +...+ 1/n + (n+1)$ but am left with an $n!$ in ...
4
votes
1answer
82 views

Spanish Math Olympiad

In the circumscircle of a triangle $ABC$, let $A_1$ be the point diametrically opposed to the vertex $A$. Let $A'$ the intersection point of $AA'$ and $BC$. The perpendicular to the line $AA'$ from $A'...
0
votes
1answer
63 views

complex integrals springing from IMO question?

I was looking at problem $A2$ here: https://www.imo-official.org/problems/IMO2006SL.pdf The comment following $A2$ suggests that complex contours lead to a nice expression, but I don’t see the ...
11
votes
3answers
142 views

Doubt regarding divisibility of the expression: $1^{101}+2^{101} \cdot \cdot \cdot +2016^{101}$

In an interesting contest question I recently encountered, I chanced upon a question I couldn't solve. $$\sum^{2016}_{i=1}i^{101}$$ is divisible by: (a)2014 (b)2015 (c)2016 (d)2017 How would I ...
0
votes
0answers
16 views

Two polylines could form a convex quadrilateral

A close polyline $\Delta$ with length $n$ here means a sequence of segments $A_1A_2,\ldots, A_{n-1}A_n$ and $A_nA_1$ so that there are no two segments $A_iA_{i+1}$ and $A_jA_{j+1}$, with $1\leq i,j\...
0
votes
0answers
23 views

Projective Geometry - Pole/Polar

A circle is inscribed in quadrilateral $ABCD$ so that it touches sides $AB, BC, CD, DA$ at $E, F, G, H$ respectively. (a) Show that lines $AC, EF, GH$ are concurrent. In fact, they concur at ...