Olympiad problems

2011 Iran MO (3rd Round), 2

Tags: polynomial , algebra

[b]a)[/b] Prove that for every natural numbers $n$ and $k$, we have monic polynomials of degree $n$, with integer coefficients like $A=\{P_1(x),.....,P_k(x)\}$ such that no two of them have a common factor and for every subset of $A$, the sum of elements of $A$ has all its roots real. [b]b)[/b] Are there infinitely many monic polynomial of degree $n$ with integer coefficients like $P_1(x),P_2(x),....$ such that no two of them have a common factor and the sum of a finite number of them has all it's roots real? [i]proposed by Mohammad Mansouri[/i]

2005 Alexandru Myller, 2

Tags: geometry , circumcircle

Let $ ABC $ be a triangle with $ \angle BAC <90^{\circ } . $ In the exterior of $ ABC, $ choose the points $ D,E $ such that $ DA=DB,EA=EC $ and $ \angle ADB =\angle AEC =2\angle BAC . $ Show that the symmetric of $ A $ with respect to the midpoint of the segment $ DE $ is the circumcircle of $ ABC. $

2018 Ukraine Team Selection Test, 6

Tags: number theory , vieta jumping

Find the smallest positive integer $n$ or show no such $n$ exists, with the following property: there are infinitely many distinct $n$-tuples of positive rational numbers $(a_1, a_2, \ldots, a_n)$ such that both $$a_1+a_2+\dots +a_n \quad \text{and} \quad \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n}$$ are integers.

2002 SNSB Admission, 2

Tags: differential equation , function , algebra , polynomial

Provided that the roots of the polynom $ X^n+a_1X^{n-1} +a_2X^{n-2} +\cdots +a_{n-1}X +a_n:\in\mathbb{R}[X] , $ of degree $ n\ge 2, $ are all real and pairwise distinct, prove that there exists is a neighbourhood $ \mathcal{V} $ of $ \left( a_1,a_2,\ldots ,a_n \right) $ in $ \mathbb{R}^n $ and $ n $ functions $ x_1,x_2,\ldots ,x_n\in\mathcal{C}^{\infty } \left( \mathcal{V} \right) $ whose values at $ \left( a_1,a_2,\ldots ,a_n \right) $ are roots of the mentioned polynom.

2013 VTRMC, Problem 7

Tags: calculus

Evaluate $\sum_{n=1}^\infty \frac{n}{(2^n-2^{-n})^2}+\frac{(-1)^nn}{(2^n-2^{-n})^2}$

2020 Malaysia IMONST 2, 5

Tags: algebra , quadratic equation

Let $p$ and $q$ be real numbers such that the quadratic equation $x^2 + px + q = 0$ has two distinct real solutions $x_1$ and $x_2$. Suppose $|x_1-x_2|=1$, $|p-q|=1$. Prove that $p, q, x_1, x_2$ are all integers.

2001 China National Olympiad, 1

Tags: number theory

Let $a,b,c$ be positive integers such that $a,b,c,a+b-c,a+c-b,b+c-a,a+b+c$ are $7$ distinct primes. The sum of two of $a,b,c$ is $800$. If $d$ be the difference of the largest prime and the least prime among those $7$ primes, find the maximum value of $d$.

1999 Gauss, 17

Tags: gauss

In a “Fibonacci” sequence of numbers, each term beginning with the third, is the sum of the previous two terms. The first number in such a sequence is 2 and the third is 9. What is the eighth term in the sequence? $\textbf{(A)}\ 34 \qquad \textbf{(B)}\ 36 \qquad \textbf{(C)}\ 107 \qquad \textbf{(D)}\ 152 \qquad \textbf{(E)}\ 245$

2013 Princeton University Math Competition, 2

Tags:

(Following question 1) Now instead consider an infinite strip of squares, labeled with the integers $0, 1, 2, \ldots$ in that order. You start at the square labeled $0$. You want to end up at the square labeled $3$. In how many ways can this be done in exactly $15$ moves?

2021 IMO Shortlist, A8

Tags: algebra , functional equation , function

Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ that satisfy $$(f(a)-f(b))(f(b)-f(c))(f(c)-f(a)) = f(ab^2+bc^2+ca^2) - f(a^2b+b^2c+c^2a)$$for all real numbers $a$, $b$, $c$. [i]Proposed by Ankan Bhattacharya, USA[/i]

2012 Peru IMO TST, 2

Tags: geometry

Let $a, b, c$ be the lengths of the sides of a triangle, and $h_a, h_b, h_c$ the lengths of the heights corresponding to the sides $a, b, c,$ respectively. If $t \geq \frac{1} {2}$ is a real number, show that there is a triangle with sidelengths $$ t\cdot a + h_a, \ t\cdot b + h_b , \ t\cdot c + h_c.$$

2005 Taiwan TST Round 3, 2

Tags: incenter , circumcircle , geometry

Given a triangle $ABC$, we construct a circle $\Gamma$ through $B,C$ with center $O$. $\Gamma$ intersects $AC, AB$ at points $D$, $E$, respectively($D$, $E$ are distinct from $B$ and $C$). Let the intersection of $BD$ and $CE$ be $F$. Extend $OF$ so that it intersects the circumcircle of $\triangle ABC$ at $P$. Show that the incenters of triangles $PBD$ and $PCE$ coincide.

2013 Stanford Mathematics Tournament, 3

Tags: calculus , limit , trigonometry , analytic geometry

Suppose $a$ and $b$ are real numbers such that \[\lim_{x\to 0}\frac{\sin^2 x}{e^{ax}-bx-1}=\frac{1}{2}.\] Determine all possible ordered pairs $(a, b)$.

1997 All-Russian Olympiad Regional Round, 8.1

Tags: number theory , perfect square , combinatorics

Prove that the numbers from $1$ to $16$ can be written in a line, but cannot be written in a circle so that the sum of any two adjacent numbers is square of a natural number.

2004 India National Olympiad, 1

Tags: parallelogram , geometry

$ABCD$ is a convex quadrilateral. $K$, $L$, $M$, $N$ are the midpoints of the sides $AB$, $BC$, $CD$, $DA$. $BD$ bisects $KM$ at $Q$. $QA = QB = QC = QD$ , and$\frac{LK}{LM} = \frac{CD}{CB}$. Prove that $ABCD$ is a square

2017 ASDAN Math Tournament, 2

Tags: geometry test

An equilateral triangle $ABC$ shares a side with a square $BCDE$. If the resulting pentagon has a perimeter of $20$, what is the area of the pentagon? (The triangle and square do not overlap).

1950 Moscow Mathematical Olympiad, 183

Tags: geometry , perimeter , square , circles

A circle is inscribed in a triangle and a square is circumscribed around this circle so that no side of the square is parallel to any side of the triangle. Prove that less than half of the square’s perimeter lies outside the triangle.

1968 Miklós Schweitzer, 7

Tags: function , limit , advanced fields

For every natural number $ r$, the set of $ r$-tuples of natural numbers is partitioned into finitely many classes. Show that if $ f(r)$ is a function such that $ f(r)\geq 1$ and $ \lim _{r\rightarrow \infty} f(r)\equal{}\plus{}\infty$, then there exists an infinite set of natural numbers that, for all $ r$, contains $ r$-triples from at most $ f(r)$ classes. Show that if $ f(r) \not \rightarrow \plus{}\infty$, then there is a family of partitions such that no such infinite set exists. [i]P. Erdos, A. Hajnal[/i]

2001 Balkan MO, 4

Tags: 3d geometry , analytic geometry , combinatorics , geometry

A cube side 3 is divided into 27 unit cubes. The unit cubes are arbitrarily labeled 1 to 27 (each cube is given a different number). A move consists of swapping the cube labeled 27 with one of its 6 neighbours. Is it possible to find a finite sequence of moves at the end of which cube 27 is in its original position, but cube $n$ has moved to the position originally occupied by $27-n$ (for each $n = 1, 2, \ldots , 26$)?

2007 IMAR Test, 2

Tags: geometry , 3d geometry , sphere , combinatorics

Denote by $ \mathcal{C}$ the family of all configurations $ C$ of $ N > 1$ distinct points on the sphere $ S^2,$ and by $ \mathcal{H}$ the family of all closed hemispheres $ H$ of $ S^2.$ Compute: $ \displaystyle\max_{H\in\mathcal{H}}\displaystyle\min_{C\in\mathcal{C}}|H\cap C|$, $ \displaystyle\min_{H\in\mathcal{H}}\displaystyle\max_{C\in\mathcal{C}}|H\cap C|$ $ \displaystyle\max_{C\in\mathcal{C}}\displaystyle\min_{H\in\mathcal{H}}|H\cap C|$ and $ \displaystyle\min_{C\in\mathcal{C}}\displaystyle\max_{H\in\mathcal{H}}|H\cap C|.$

2023 Tuymaada Olympiad, 6

Tags: number theory , euclidean algorithm

An $\textit{Euclidean step}$ transforms a pair $(a, b)$ of positive integers, $a > b$, to the pair $(b, r)$, where $r$ is the remainder when a is divided by $b$. Let us call the $\textit{complexity}$ of a pair $(a, b)$ the number of Euclidean steps needed to transform it to a pair of the form $(s, 0)$. Prove that if $ad - bc = 1$, then the complexities of $(a, b)$ and $(c, d)$ differ at most by $2$.

2000 Tournament Of Towns, 6

Tags: tournament , combinatorics

In the spring round of the Tournament of Towns this year, $6$ problems were posed in the Senior A-Level paper. In a certain country, each problem was solved by exactly $1000$ participants, but no two participants solved all $6$ problems between them. What is the smallest possible number of participants from this country in the spring round Senior A-Level paper? (R Zhenodarov)

2005 Estonia Team Selection Test, 5

Tags: coloring , combinatorics

On a horizontal line, $2005$ points are marked, each of which is either white or black. For every point, one finds the sum of the number of white points on the right of it and the number of black points on the left of it. Among the $2005$ sums, exactly one number occurs an odd number of times. Find all possible values of this number.

2020 Canadian Mathematical Olympiad Qualification, 8

Tags: radical , rational , equation , algebra

Find all pairs $(a, b)$ of positive rational numbers such that $\sqrt[b]{a}= ab$

2013 Today's Calculation Of Integral, 897

Tags: calculus , integration , geometry , geometric transformation , rotation , calculus computations , logarithm

Find the volume $V$ of the solid formed by a rotation of the region enclosed by the curve $y=2^{x}-1$ and two lines $x=0,\ y=1$ around the $y$ axis.