Olympiad problems

1992 Chile National Olympiad, 4

Tags: equilateral , parallel

Given three parallel lines, prove that there are three points, one on each line, which are the vertices of an equilateral triangle.

2008 IberoAmerican Olympiad For University Students, 2

Tags: polynomial , ratio , geometry , algebra

Prove that for each natural number $n$ there is a polynomial $f$ with real coefficients and degree $n$ such that $ p(x)=f(x^2-1)$ is divisible by $f(x)$ over the ring $\mathbb{R}[x]$.

2019 Iran Team Selection Test, 6

Tags: inequalities

$x,y$ and $z$ are real numbers such that $x+y+z=xy+yz+zx$. Prove that $$\frac{x}{\sqrt{x^4+x^2+1}}+\frac{y}{\sqrt{y^4+y^2+1}}+\frac{z}{\sqrt{z^4+z^2+1}}\geq \frac{-1}{\sqrt{3}}.$$ [i]Proposed by Navid Safaei[/i]

2012 IFYM, Sozopol, 5

Tags: number theory , sum of powers , sequence

We are given the following sequence: $a_1=8,a_2=20,a_{n+2}=a_{n+1}^2+12a_n a_{n+1}+11a_n$. Prove that none of the members of the sequence can be presented as a sum of three seventh powers of natural numbers.

2024 Alborz Mathematical Olympiad, P1

Tags: number theory , complete residue system , divisor

Find all positive integers $n$ such that if $S=\{d_1,d_2,\cdots,d_k\}$ is the set of positive integer divisors of $n$, then $S$ is a complete residue system modulo $k$. (In other words, for every pair of distinct indices $i$ and $j$, we have $d_i\not\equiv d_j \pmod{k}$). Proposed by Heidar Shushtari

2004 Putnam, A4

Tags: algebra , polynomial

Show that for any positive integer $n$ there is an integer $N$ such that the product $x_1x_2\cdots x_n$ can be expressed identically in the form \[x_1x_2\cdots x_n=\sum_{i=1}^Nc_i(a_{i1}x_1+a_{i2}x_2+\cdots +a_{in}x_n)^n\] where the $c_i$ are rational numbers and each $a_{ij}$ is one of the numbers, $-1,0,1.$

2023 Irish Math Olympiad, P8

Tags: algebra , inequalities

Suppose that $a, b, c$ are positive real numbers and $a + b + c = 3$. Prove that $$\frac{a+b}{c+2} + \frac{b+c}{a+2} + \frac{c+a}{b+2} \geq 2$$ and determine when equality holds.

1979 Yugoslav Team Selection Test, Problem 2

Tags: number theory

Find all integers $n$ with $1<n<1979$ having the following property: If $m$ is an integer coprime with $n$ and $1<m<n$, then $m$ is a prime number.

2009 Moldova National Olympiad, 7.3

Tags: point , geometry

On the lines $AB$ are located $2009$ different points that do not belong to the segment $[AB]$. Prove that the sum of the distances from point $A$ to these points is not equal to the sum of the distances from point $B$ to these points.

II Soros Olympiad 1995 - 96 (Russia), 11.1

Tags: calculus , algebra , integration , integral

Find some antiderivative of the function $y = 1/x^3$, the graph of which has exactly three common points with the graph of the function $y = |x|$.

PEN O Problems, 40

Tags: floor function , induction , logarithm

Let $X$ be a non-empty set of positive integers which satisfies the following: [list] [*] if $x \in X$, then $4x \in X$, [*] if $x \in X$, then $\lfloor \sqrt{x}\rfloor \in X$. [/list] Prove that $X=\mathbb{N}$.

2024 Azerbaijan IZhO TST, 3

Tags: geometry

In a triangle $ABC$, $I$ is the incenter. Line $CI$ intersects circumcircle of $ABC$ at $L$, and it is given that $CI=2IL$. $M;N$ are points chosen on $AB$ such that $\angle AIM=\angle BIN=90$. Prove that $AB=2MN$

2005 China Team Selection Test, 2

Tags: circumcircle , parallelogram , geometry

Let $\omega$ be the circumcircle of acute triangle $ABC$. Two tangents of $\omega$ from $B$ and $C$ intersect at $P$, $AP$ and $BC$ intersect at $D$. Point $E$, $F$ are on $AC$ and $AB$ such that $DE \parallel BA$ and $DF \parallel CA$. (1) Prove that $F,B,C,E$ are concyclic. (2) Denote $A_{1}$ the centre of the circle passing through $F,B,C,E$. $B_{1}$, $C_{1}$ are difined similarly. Prove that $AA_{1}$, $BB_{1}$, $CC_{1}$ are concurrent.

2013 Stanford Mathematics Tournament, 2

Tags: geometry , perimeter , rectangle

What is the perimeter of a rectangle of area $32$ inscribed in a circle of radius $4$?

2022 Romania National Olympiad, P1

Tags: algebra , logarithm

Let $a\neq 1$ be a positive real number. Find all real solutions to the equation $a^x=x^x+\log_a(\log_a(x)).$ [i]Mihai Opincariu[/i]

2012 Federal Competition For Advanced Students, Part 1, 3

Tags: induction , combinatorics

Consider a stripe of $n$ fieds, numbered from left to right with the integers $1$ to $n$ in ascending order. Each of the fields is colored with one of the colors $1$, $2$ or $3$. Even-numbered fields can be colored with any color. Odd-numbered fields are only allowed to be colored with the odd colors $1$ and $3$. How many such colorings are there such that any two neighboring fields have different colors?

2021 Azerbaijan Senior NMO, 1

Tags: factorial , prime factorization , number theory

At least how many numbers must be deleted from the product $1 \times 2 \times \dots \times 46 \times 47$ in order to make it a perfect square?

2022 Bulgarian Spring Math Competition, Problem 10.2

Tags: geometry , incenter , arc midpoint , ratio

Let $\triangle ABC$ have incenter $I$. The line $CI$ intersects the circumcircle of $\triangle ABC$ for the second time at $L$, and $CI=2IL$. Points $M$ and $N$ lie on the segment $AB$, such that $\angle AIM =\angle BIN = 90^{\circ}$. Prove that $AB=2MN$.

2006 Bulgaria Team Selection Test, 3

Tags: geometry

[b]Problem 3.[/b] Two points $M$ and $N$ are chosen inside a non-equilateral triangle $ABC$ such that $\angle BAM=\angle CAN$, $\angle ABM=\angle CBN$ and \[AM\cdot AN\cdot BC=BM\cdot BN\cdot CA=CM\cdot CN\cdot AB=k\] for some real $k$. Prove that: [b]a)[/b] We have $3k=AB\cdot BC\cdot CA$. [b]b)[/b] The midpoint of $MN$ is the medicenter of $\triangle ABC$. [i]Remark.[/i] The [b]medicenter[/b] of a triangle is the intersection point of the three medians: If $A_{1}$ is midpoint of $BC$, $B_{1}$ of $AC$ and $C_{1}$ of $AB$, then $AA_{1}\cap BB_{1}\cap CC_{1}= G$, and $G$ is called medicenter of triangle $ABC$. [i] Nikolai Nikolov[/i]

2007 Harvard-MIT Mathematics Tournament, 6

Tags: algebra , polynomial

Consider the polynomial $P(x)=x^3+x^2-x+2$. Determine all real numbers $r$ for which there exists a complex number $z$ not in the reals such that $P(z)=r$.

2015 AMC 10, 2

Tags:

A box contains a collection of triangular and square tiles. There are 25 tiles in the box, containing 84 edges total. How many square tiles are there in the box? $ \textbf{(A) }3 \qquad\textbf{(B) }5\qquad\textbf{(C) }7\qquad\textbf{(D) }9\qquad\textbf{(E) }\text{11} $

1999 German National Olympiad, 2

Tags: inequalities , algebra

Determine all real numbers $x$ for which $1+\frac{x}{2} -\frac{x^2}{8} \le \sqrt{1+x} \le 1+\frac{x}{2}$

VII Soros Olympiad 2000 - 01, 8.10

Tags: combinatorics

Place in the cells the boards measuring: a) $2 \times 2$, b) $4 \times 4$, c) $2n \times 2n$, numbers $0$, $1$ and $-1$ so that in each case all the sums of numbers in rows and columns are different.

2015 Taiwan TST Round 3, 3

Tags: prime divisor , number theory , sequence

Let $c \ge 1$ be an integer. Define a sequence of positive integers by $a_1 = c$ and \[a_{n+1}=a_n^3-4c\cdot a_n^2+5c^2\cdot a_n+c\] for all $n\ge 1$. Prove that for each integer $n \ge 2$ there exists a prime number $p$ dividing $a_n$ but none of the numbers $a_1 , \ldots , a_{n -1}$ . [i]Proposed by Austria[/i]

1963 Polish MO Finals, 5

Tags: algebra , polynomial

Prove that a fifth-degree polynomial $$ P(x) = x^5 - 3x^4 + 6x^3 - 3x^2 + 9x - 6$$ is not the product of two lower-degree polynomials with integer coefficients.