Olympiad problems

2006 Junior Balkan Team Selection Tests - Romania, 4

Tags: combinatorics

The set of positive integers is partitionated in subsets with infinite elements each. The question (in each of the following cases) is if there exists a subset in the partition such that any positive integer has a multiple in this subset. a) Prove that if the number of subsets in the partition is finite the answer is yes. b) Prove that if the number of subsets in the partition is infinite, then the answer can be no (for a certain partition).

2017 IMC, 5

Tags: polynomial , complex analysis

Let $k$ and $n$ be positive integers with $n\geq k^2-3k+4$, and let $$f(z)=z^{n-1}+c_{n-2}z^{n-2}+\dots+c_0$$ be a polynomial with complex coefficients such that $$c_0c_{n-2}=c_1c_{n-3}=\dots=c_{n-2}c_0=0$$ Prove that $f(z)$ and $z^n-1$ have at most $n-k$ common roots.

2018 Tuymaada Olympiad, 7

Tags: inequalities , algebra

Prove the inequality $$(x^3+2y^2+3z)(4y^3+5z^2+6x)(7z^3+8x^2+9y)\geq720(xy+yz+xz)$$ for $x, y, z \geq 1$. [i]Proposed by K. Kokhas[/i]

1978 Chisinau City MO, 158

Tags: point , combinatorial geometry , combinatorics , geometry

Five points are selected on the plane so that no three of them lie on one straight line. Prove that some four of these five points are the vertices of a convex quadrilateral.

2014 Cezar Ivănescu, 2

Tags: antiderivative , function , real analysis , darboux , derivability , primitivability

[b]a)[/b] Give an example of function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ that admits a primitive $ F:\mathbb{R}\longrightarrow\mathbb{R}_{>0 } $ having the property that $ F^e $ is a primitive of $ f^e. $ [b]b)[/b] Prove that there is no derivable function $ g:\mathbb{R}\longrightarrow\mathbb{R} $ that has a primitive $ G:\mathbb{R}\longrightarrow\mathbb{R} $ such that $ e^G $ is a primitive of $ e^g. $

2021 Korea Winter Program Practice Test, 3

Tags: geometry

The acute triangle $ABC$ satisfies $\overline {AB}<\overline {BC}<\overline {CA}$. Let $H$ a orthocenter of $ABC$, $D$ a intersection point of $AH$ and $BC$, $E$ a intersection point of $BH$ and $AC$, and $M$ a midpoint of segment $BC$. A circle with center $E$ and radius $AE$ intersects the segment $AC$ at point $F$($\neq A$), and circumcircle of triangle $BFC$ intersects the segment $AM$ at point $S$. Let $P$($\neq D$), $Q$($\neq F$) a intersection point of circumcircle of triangle $ASD$ and $DF$, circumcircle of triangle $ASF$ and $DF$ respectively. Also, define $R$ as a intersection point of circumcircles of triangle $AHQ$ and $AEP$. Prove that $R$ lies on line $DF$.

2022 Iran Team Selection Test, 4

Tags: geometry

Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$. Let $M$ and $N$ be the midpoint of the sides $AD$ and $BC$, respectively. Suppose that $\omega_1$, $\omega_2$ and $\omega_3$ be the circumcircle of triangles $ADP$, $BCP$ and $OMN$, respectively. The intersection point of $\omega_1$ and $\omega_3$, which is not on the arc $APD$ of $\omega_1$, is $E$ and the intersection point of $\omega_2$ and $\omega_3$, which is not on the arc $BPC$ of $\omega_2$, is $F$. Prove that $OF=OE$. Proposed by Seyed Amirparsa Hosseini Nayeri

2010 IberoAmerican Olympiad For University Students, 2

Tags: trigonometry , limit , real analysis

Calculate the sum of the series $\sum_{-\infty}^{\infty}\frac{\sin^33^k}{3^k}$.

2009 Brazil National Olympiad, 2

Tags: geometry , circumcircle , perpendicular bisector

Let $ ABC$ be a triangle and $ O$ its circumcenter. Lines $ AB$ and $ AC$ meet the circumcircle of $ OBC$ again in $ B_1\neq B$ and $ C_1 \neq C$, respectively, lines $ BA$ and $ BC$ meet the circumcircle of $ OAC$ again in $ A_2\neq A$ and $ C_2\neq C$, respectively, and lines $ CA$ and $ CB$ meet the circumcircle of $ OAB$ in $ A_3\neq A$ and $ B_3\neq B$, respectively. Prove that lines $ A_2A_3$, $ B_1B_3$ and $ C_1C_2$ have a common point.

2005 Bosnia and Herzegovina Team Selection Test, 2

Tags: algebra , inequality , three variable inequality

If $a_1$, $a_2$ and $a_3$ are nonnegative real numbers for which $a_1+a_2+a_3=1$, then prove the inequality $a_1\sqrt{a_2}+a_2\sqrt{a_3}+a_3\sqrt{a_1}\leq \frac{1}{\sqrt{3}}$

2004 Tournament Of Towns, 5

Tags: induction , number theory

For which values of N is it possible to write numbers from 1 to N in some order so that for any group of two or more consecutive numbers, the arithmetic mean of these numbers is not whole?

1997 AMC 12/AHSME, 7

Tags:

The sum of seven integers is $ \minus{}1$. What is the maximum number of the seven integers that can be larger than $ 13$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

2008 Bulgarian Autumn Math Competition, Problem 8.4

Tags: geoemtry , combinatorics , obtuse triangle

Let $M$ be a set of $99$ different rays with a common end point in a plane. It's known that two of those rays form an obtuse angle, which has no other rays of $M$ inside in. What is the maximum number of obtuse angles formed by two rays in $M$?

2024-25 IOQM India, 27

Tags: geometry

In a triangle $ABC$, a point $P$ in the interior of $ABC$ is such that $$ \angle BPC - \angle BAC = \angle CPA - \angle CBA = \angle APB - \angle ACB.$$ Suppose $\angle BAC = 30^{\circ}$ and $AP = 12$. Let $D,E,F$ be the feet of perpendiculars from $P$ on to $BC,CA,AB$ respectively. If $m \sqrt{n}$ is the area of the triangle DEF where $m,n$ are integers with $n$ prime, then what is the value of the product $mn$?

2007 Stanford Mathematics Tournament, 1

Tags:

There are three bins: one with 30 apples, one with 30 oranges, and one with 15 of each. Each is labeled "apples," "oranges," or "mixed." Given that all three labels are wrong, how many pieces of fruit must you look at to determine the correct labels?

2018 Romanian Master of Mathematics, 2

Tags: algebra , polynomial

Determine whether there exist non-constant polynomials $P(x)$ and $Q(x)$ with real coefficients satisfying $$P(x)^{10}+P(x)^9 = Q(x)^{21}+Q(x)^{20}.$$

2016 CHKMO, 4

Tags: combinatorics

Given an integer $n\geq 2$. There are $N$ distinct circle on the plane such that any two circles have two distinct intersections and no three circles have a common intersection. Initially there is a coin on each of the intersection points of the circles. Starting from $X$, players $X$ and $Y$ alternatively take away a coin, with the restriction that one cannot take away a coin lying on the same circle as the last coin just taken away by the opponent in the previous step. The one who cannot do so will lost. In particular, one loses where there is no coin left. For what values of $n$ does $Y$ have a winning strategy?

2018 Czech-Polish-Slovak Junior Match, 3

Tags: trinomial , algebra , inequalities

Calculate all real numbers $r $ with the following properties: If real numbers $a, b, c$ satisfy the inequality$ | ax^2 + bx + c | \le 1$ for each $x \in [ - 1, 1]$, then they also satisfy the inequality $| cx^2 + bx + a | \le r$ for each $ x \in [- 1, 1]$.

1992 APMO, 3

Tags: combinatorics

Let $n$ be an integer such that $n > 3$. Suppose that we choose three numbers from the set $\{1, 2, \ldots, n\}$. Using each of these three numbers only once and using addition, multiplication, and parenthesis, let us form all possible combinations. (a) Show that if we choose all three numbers greater than $\frac{n}{2}$, then the values of these combinations are all distinct. (b) Let $p$ be a prime number such that $p \leq \sqrt{n}$. Show that the number of ways of choosing three numbers so that the smallest one is $p$ and the values of the combinations are not all distinct is precisely the number of positive divisors of $p - 1$.

1989 AIME Problems, 3

Tags: geometric series

Suppose $n$ is a positive integer and $d$ is a single digit in base 10. Find $n$ if \[ \frac{n}{810}=0.d25d25d25\ldots \]

2009 Hungary-Israel Binational, 1

Tags: trapezoid , circumcircle , parallelogram , power of a point , radical axis , complex number , geometry

Given is the convex quadrilateral $ ABCD$. Assume that there exists a point $ P$ inside the quadrilateral for which the triangles $ ABP$ and $ CDP$ are both isosceles right triangles with the right angle at the common vertex $ P$. Prove that there exists a point $ Q$ for which the triangles $ BCQ$ and $ ADQ$ are also isosceles right triangles with the right angle at the common vertex $ Q$.

2009 Indonesia TST, 3

Tags: power of a point , radical axis , geometry

Let $ ABC$ be an acute triangle with $ \angle BAC\equal{}60^{\circ}$. Let $ P$ be a point in triangle $ ABC$ with $ \angle APB\equal{}\angle BPC\equal{}\angle CPA\equal{}120^{\circ}$. The foots of perpendicular from $ P$ to $ BC,CA,AB$ are $ X,Y,Z$, respectively. Let $ M$ be the midpoint of $ YZ$. a) Prove that $ \angle YXZ\equal{}60^{\circ}$ b) Prove that $ X,P,M$ are collinear.

1948 Moscow Mathematical Olympiad, 148

Tags: diophantine , exponential , algebra , number theory

a) Find all positive integer solutions of the equation $x^y = y^x$ ($x \ne y$). b) Find all positive rational solutions of the equation $x^y = y^x$ ($x \ne y$).

2015 All-Russian Olympiad, 7

Tags: geometry , circumcircle

An acute-angled $ABC \ (AB<AC)$ is inscribed into a circle $\omega$. Let $M$ be the centroid of $ABC$, and let $AH$ be an altitude of this triangle. A ray $MH$ meets $\omega$ at $A'$. Prove that the circumcircle of the triangle $A'HB$ is tangent to $AB$. [i](A.I. Golovanov , A.Yakubov)[/i]

2019 China Second Round Olympiad, 4

Tags: combinatorics , coloring

Each side of a convex $2019$-gon polygon is dyed with red, yellow and blue, and there are exactly $673$ sides of each kind of color. Prove that there exists at least one way to draw $2016$ diagonals to divide the convex $2019$-gon polygon into $2017$ triangles, such that any two of the $2016$ diagonals don't have intersection inside the $2019$-gon polygon,and for any triangle in all the $2017$ triangles, the colors of the three sides of the triangle are all the same, either totally different.