Olympiad problems

1962 Poland - Second Round, 2

Tags: algebra , polynomial

What conditions should real numbers $ a $, $ b $, $ c $, $ d $, $ e $, $ f $ meet in order for a polynomial of second degree $$ax^2 + 2bxy + cy^2 + 2dx + 2ey + f$$ was the product of two first degree polynomials with real coefficients ?

1998 Estonia National Olympiad, 4

Tags: system of equations , algebra

For real numbers $x, y$ and $z$ it is known that $$\begin{cases} x + y = 2 \\ xy = z^2 + 1\end {cases}$$ Find the value of the expression $x^2 + y^2+ z^2$.

2015 India Regional MathematicaI Olympiad, 1

Tags: right angle , circumcircle , geometry , cyclic quadrilateral , incenter

In a cyclic quadrilateral $ABCD$, let the diagonals $AC$ and $BD$ intersect at $X$. Let the circumcircles of triangles $AXD$ and $BXC$ intersect again at $Y$ . If $X$ is the incentre of triangle $ABY$ , show that $\angle CAD = 90^o$.

Kyiv City MO Seniors 2003+ geometry, 2008.10.4

Tags: cevian , midpoint , geometry

Given a triangle $ABC $, $A {{A} _ {1}} $, $B {{B} _ {1}} $, $C {{C} _ {1}}$ - its chevians intersecting at one point. ${{A} _ {0}}, {{C} _ {0}} $ - the midpoint of the sides $BC $ and $AB$ respectively. Lines ${{B} _ {1}} {{C} _ {1}} $, ${{B} _ {1}} {{A} _ {1}} $and ${ {B} _ {1}} B$ intersect the line ${{A} _ {0}} {{C} _ {0}} $ at points ${{C} _ {2}} $ , ${{A} _ {2}} $ and ${{B} _ {2}} $, respectively. Prove that the point ${{B} _ {2}} $ is the midpoint of the segment ${{A} _ {2}} {{C} _ {2}} $. (Eugene Bilokopitov)

2021-2022 OMMC, 3

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Evan has $10$ cards numbered $1$ through $10$. He chooses some of the cards and takes the product of the numbers on them. When the product is divided by $3$, the remainder is $1$. Find the maximum number of cards he could have chose. [i]Proposed by Evan Chang [/i]

2013 Paraguay Mathematical Olympiad, 3

Tags: number theory

We divide a natural number $N$, that has $k$ digits, by $19$ and get a residue of $10^{k-2} -Q$, where $Q$ is the quotient and $Q < 101$. Also, $10^{k-2}-Q$ is larger than $0$. How many possible values of $N$ are there?

ICMC 4, 3

Tags: sin , integration , calculus

Let $\displaystyle s_n=\int_0^1 \text{sin}^n(nx) \,dx$. (a) Prove that $s_n \leq \dfrac 2n$ for all odd $n$. (b) Find all the limit points of the sequence $s_1, s_2, s_3, \dots$. [i]Proposed by Cristi Calin[/i]

2005 Serbia Team Selection Test, 1

Tags: algebra

problem 1 :A sequence is defined by$ x_1 = 1, x_2 = 4$ and $ x_{n+2} = 4x_{n+1} -x_n$ for $n \geq 1$. Find all natural numbers $m$ such that the number $3x_n^2 + m$ is a perfect square for all natural numbers $n$

2014 Spain Mathematical Olympiad, 2

Tags: number theory

Given the rational numbers $r$, $q$, and $n$, such that $\displaystyle\frac1{r+qn}+\frac1{q+rn}=\frac1{r+q}$, prove that $\displaystyle\sqrt{\frac{n-3}{n+1}}$ is a rational number.

1992 National High School Mathematics League, 6

Tags: function

$f(x)$ is a function defined on $\mathbb{R}$, satisfying: $f(10+x)=f(10-x),f(20-x)=-f(20+x)$. Then, $f(x)$ is $\text{(A)}$even function, but not periodic function $\text{(B)}$even function, and periodic function $\text{(C)}$odd function, but not periodic function $\text{(D)}$odd function, and periodic function

1973 Bundeswettbewerb Mathematik, 1

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A positive integer has 1000 digits (decimal system), all but at most one of them being the digit $5$. Show that this number isn't a perfect square.

2022 Germany Team Selection Test, 3

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Consider a $100\times 100$ square unit lattice $\textbf{L}$ (hence $\textbf{L}$ has $10000$ points). Suppose $\mathcal{F}$ is a set of polygons such that all vertices of polygons in $\mathcal{F}$ lie in $\textbf{L}$ and every point in $\textbf{L}$ is the vertex of exactly one polygon in $\mathcal{F}.$ Find the maximum possible sum of the areas of the polygons in $\mathcal{F}.$ [i]Michael Ren and Ankan Bhattacharya, USA[/i]

2011 Junior Macedonian Mathematical Olympiad, 1

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Let $S(n)$ be the sum of digits of natural number $n{}$. Is there a natural number $n{}$ for which $n+S(n)+S(S(n))=2011?$

2018 Sharygin Geometry Olympiad, 7

Tags: geometry

Let $\omega_1,\omega_2$ be two circles centered at $O_1$ and $O_2$ and lying outside each other. Points $C_1$ and $C_2$ lie on these circles in the same semi plane with respect to $O_1O_2$. The ray $O_1C_1$ meets $\omega _2$ at $A_2,B_2$ and $O_2C_2$ meets $\omega_1$ at $A_1,B_1$. Prove that $\angle A_1O_1B_1=\angle A_2O_2B_2$ if and only if $C_1C_2||O_1O_2$.

2010 National Olympiad First Round, 4

Tags: factorial

How many positive integers less than $2010$ are there such that the sum of factorials of its digits is equal to itself? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

1978 IMO Longlists, 4

Tags: equilateral triangle , symmetry , geometry , similar triangles

Two identically oriented equilateral triangles, $ABC$ with center $S$ and $A'B'C$, are given in the plane. We also have $A' \neq S$ and $B' \neq S$. If $M$ is the midpoint of $A'B$ and $N$ the midpoint of $AB'$, prove that the triangles $SB'M$ and $SA'N$ are similar.

1978 Bundeswettbewerb Mathematik, 3

Tags: power of 2 , division , sum , number theory

For every positive integer $n$, define the remainder sum $r(n)$ as the sum of the remainders upon division of $n$ by each of the numbers $1$ through $n$. Prove that $r(2^{k}-1) =r(2^{k})$ for every $k\geq 1.$

2024 Junior Balkan Team Selection Tests - Moldova, 12

Tags: number theory

[b]Version 1.[/b] Find all primes $p$ satisfying the following conditions: (i) $\frac{p+1}{2}$ is a prime number. (ii) There are at least three distinct positive integers $n$ for which $\frac{p^2+n}{p+n^2}$ is an integer. [b]Version 2.[/b] Let $p \neq 5$ be a prime number such that $\frac{p+1}{2}$ is also a prime. Suppose there exist positive integers $a <b$ such that $\frac{p^2+a}{p+a^2}$ and $\frac{p^2+b}{p+b^2}$ are integers. Show that $b=(a-1)^2+1$.

2024 Azerbaijan IZhO TST, 4

Tags: number theory

Take a sequence $(a_n)_{n=1}^\infty$ such that $a_1=3$ $a_n=a_1a_2a_3...a_{n-1}-1$ [b]a)[/b] Prove that there exists infitely many primes that divides at least 1 term of the sequence. [b]b)[/b] Prove that there exists infitely many primes that doesn't divide any term of the sequence.

1994 Tournament Of Towns, (421) 2

Tags: point construction , fixed , geometric construction , geometry , ratio

Two circles, one inside the other, are given in the plane. Construct a point $O$, inside the inner circle, such that if a ray from $O$ cuts the circles at $A$ and $B$ respectively, then the ratio $OA/OB$ is constant. (Folklore)

1997 Brazil Team Selection Test, Problem 2

Tags: diophantine equation , number theory

We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.

2012 South East Mathematical Olympiad, 4

Tags: inequalities , trigonometry

Let $a, b, c, d$ be real numbers satisfying inequality $a\cos x+b\cos 2x+c\cos 3x+d\cos 4x\le 1$ holds for arbitrary real number $x$. Find the maximal value of $a+b-c+d$ and determine the values of $a,b,c,d$ when that maximum is attained.

2019 CCA Math Bonanza, I7

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How many permutations $\pi$ of $\left\{1,2,\ldots,7\right\}$ are there such that $\pi\left(k\right)\leq2k$ for $k=1,\ldots,7$? A permutation $\pi$ of a set $S$ is a function from $S$ to itself such that if $a\neq b$, then $\pi\left(a\right)\neq\pi\left(b\right)$. For example, $\pi\left(x\right)=x$ and $\pi\left(x\right)=8-x$ are permutations of $\left\{1,2,\ldots,7\right\}$ but $\pi\left(x\right)=1$ is not. [i]2019 CCA Math Bonanza Individual Round #7[/i]

MOAA Accuracy Rounds, 2023.9

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Let $\triangle{ABC}$ be a triangle with $AB = 10$ and $AC = 11$. Let $I$ be the center of the inscribed circle of $\triangle{ABC}$. If $M$ is the midpoint of $AI$ such that $BM = BC$ and $CM = 7$, then $BC$ can be expressed in the form $\frac{\sqrt{a}-b}{c}$ where $a$, $b$, and $c$ are positive integers. Find $a+b+c$. [color=#00f]Note that this problem is null because a diagram is impossible.[/color] [i]Proposed by Andy Xu[/i]

2022 Assam Mathematical Olympiad, 1

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For the positive integers $a, b$, $lcm(a, b) = \gcd(a, b) = p^2q^4$, where $p$ and $q$ are prime numbers. Find $lcm(ap, bq)$. Here lcm and gcd represent the least common multiple and the greatest common divisor respectively.