Olympiad problems

2022 BMT, 13

Real numbers $x$ and $y$ satisfy the system of equations $$x^3 + 3x^2 = -3y - 1$$ $$y^3 + 3y^2 = -3x - 1.$$ What is the greatest possible value of $x$?

2018 International Zhautykov Olympiad, 6

Tags: geometry , inequalities , geometric inequality

In a circle with a radius $R$ a convex hexagon is inscribed. The diagonals $AD$ and $BE$,$BE$ and $CF$,$CF$ and $AD$ of the hexagon intersect at the points $M$,$N$ and$K$, respectively. Let $r_1,r_2,r_3,r_4,r_5,r_6$ be the radii of circles inscribed in triangles $ ABM,BCN,CDK,DEM,EFN,AFK$ respectively. Prove that.$$r_1+r_2+r_3+r_4+r_5+r_6\leq R\sqrt{3}$$ .

2020 Princeton University Math Competition, 6

Tags: number theory

We say that a string of digits from $0$ to $9$ is valid if the following conditions hold: First, for $2 \le k \le 4$, no consecutive run of $k$ digits sums to a multiple of $10$. Second, between any two $0$s, there are at least $3$ other digits. Find the last four digits of the number of valid strings of length $2020$.

2000 National Olympiad First Round, 12

Tags:

$(a_n)$ is a sequence with $a_1=1$ and $|a_n| = |a_{n-1}+2|$ for every positive integer $n\geq 2$. What is the minimum possible value of $\sum_{i = 1}^{2000}a_{i}$? $ \textbf{(A)}\ -4000 \qquad\textbf{(B)}\ -3000 \qquad\textbf{(C)}\ -2000 \qquad\textbf{(D)}\ -1000 \qquad\textbf{(E)}\ \text{None} $

2019 AMC 10, 15

Tags: AMC , AMC 10 , AMC 12 , 2019 AMC 12A , 2019 AMC 10A , 2019 AMC , recursion

A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and $$a_n=\frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}$$for all $n \geq 3$ Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive inegers. What is $p+q ?$ $\textbf{(A) } 2020 \qquad\textbf{(B) } 4039 \qquad\textbf{(C) } 6057 \qquad\textbf{(D) } 6061 \qquad\textbf{(E) } 8078$

1988 IMO Longlists, 49

Tags: function , algebra , functional equation , IMO Shortlist

Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

2013 Hanoi Open Mathematics Competitions, 10

Tags: geometry , perimeter , maximum , area

Consider the set of all rectangles with a given area $S$. Find the largest value o $ M = \frac{16-p}{p^2+2p}$ where $p$ is the perimeter of the rectangle.

2023 Singapore Senior Math Olympiad, 1

Tags: geometry

Let $ABCD$ be a square, $E$ be a point on the side $DC$, $F$ and $G$ be the feet of the altitudes from $B$ to $AE$ and from $A$ to $BE$, respectively. Suppose $DF$ and $CG$ intersect at $H$. Prove that $\angle AHB=90^\circ$.

2015 Latvia Baltic Way TST, 7

Tags: perpendicular , geometry

Two circle $\Gamma_1$ and $\Gamma_2$ intersect at points $A$ and $B$, point $P$ is not on the line $AB$. Line $AP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $K$ and $L$ respectively, line $BP$ intersects again $\Gamma_1$ and $\Gamma_2$ at points $M$ and $N$ respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles $KMP$ and $LNP$ are $O_1$ and $O_2$ respectively. Prove that $O_1O_2$ is perpendicular to $AB$.

2012 Sharygin Geometry Olympiad, 4

Tags: Angle Chasing , isosceles , geometry

Let $ABC$ be an isosceles triangle with $\angle B = 120^o$ . Points $P$ and $Q$ are chosen on the prolongations of segments $AB$ and $CB$ beyond point $B$ so that the rays $AQ$ and $CP$ intersect and are perpendicular to each other. Prove that $\angle PQB = 2\angle PCQ$. (A.Akopyan, D.Shvetsov)

2009 Jozsef Wildt International Math Competition, W. 2

Tags: algebra

Find the area of the set $A = \{(x, y)\ |\ 1 \leq x \leq e,\ 0 \leq y \leq f (x)\}$, where \begin{tabular}{ c| c c c c |} &1 & 1& 1 & 1\\ $f(x)$=& $\ln x$ & 2$\ln x$ & 3$\ln x$ & 4$\ln x$ \\ &${(\ln x)}^2$ & $4{(\ln x)}^2 $& $9{(\ln x)}^2 $& $16{(\ln x)}^2$\\ &${(\ln x)}^3$ & $8{(\ln x)}^3$ &$ 27{(\ln x)}^3$ &$ 64{(\ln x)}^3$ \end{tabular}

2009 USA Team Selection Test, 7

Tags: algebra , polynomial , trigonometry , system of equations , algebra proposed

Find all triples $ (x,y,z)$ of real numbers that satisfy the system of equations \[ \begin{cases}x^3 \equal{} 3x\minus{}12y\plus{}50, \\ y^3 \equal{} 12y\plus{}3z\minus{}2, \\ z^3 \equal{} 27z \plus{} 27x. \end{cases}\] [i]Razvan Gelca.[/i]

2006 Stanford Mathematics Tournament, 7

Tags: analytic geometry

A lattice point in the plane is a point whose coordinates are both integers. Given a set of 100 distinct lattice points in the plane, find the smallest number of line segments $ \overline{AB}$ for which $ A$ and $ B$ are distinct lattice points in this set and the midpoint of $ \overline{AB}$ is also a lattice point (not necessarily in the set).

2010 Abels Math Contest (Norwegian MO) Final, 3

Tags: pigeonhole principle , combinatorics proposed , combinatorics

$ a)$ There are $ 25$ participants in a mathematics contest having four problems. Each problem is considered solved or not solved (that is, partial solutions are not possible). Show that either there are four contestants having solved the same problems (or not having solved any of them), or two contestants, one of which has solved exactly the problems that the other did not solve. $ b)$ There are $ k$ sport clubs for the students of a secondary school. The school has $ 100$ students, and for any selection of three of them, there exists a club having at least one of them, but not all, as a member. What is the least possible value of $ k$?

2011 239 Open Mathematical Olympiad, 3

Tags: inequalities

Positive reals $a,b,c,d$ satisfy $a+b+c+d=4$. Prove that $\sum_{cyc}\frac{a}{a^3 + 4} \le \frac{4}{5}$

2008 Iran MO (3rd Round), 2

Tags: function , complex analysis , complex analysis unsolved

Let $ g,f: \mathbb C\longrightarrow\mathbb C$ be two continuous functions such that for each $ z\neq 0$, $ g(z)\equal{}f(\frac1z)$. Prove that there is a $ z\in\mathbb C$ such that $ f(\frac1z)\equal{}f(\minus{}\bar z)$

2022 Germany Team Selection Test, 3

Tags: number theory , IMO Shortlist

Find all positive integers $n$ with the following property: the $k$ positive divisors of $n$ have a permutation $(d_1,d_2,\ldots,d_k)$ such that for $i=1,2,\ldots,k$, the number $d_1+d_2+\cdots+d_i$ is a perfect square.

2023 Bulgaria JBMO TST, 2

Tags: counting , combinatorics

On the coast of a circular island there are eight different cities. Initially there are no routes between the cities. We have to construct five straight two-way routes, which do not intersect, so that from each city there are one or two routes. In how many ways can this happen?

2006 Stanford Mathematics Tournament, 11

Tags: algebra , polynomial

Polynomial $P(x)=c_{2006}x^{2006}+c_{2005}x^{2005}+\ldots+c_1x+c_0$ has roots $r_1,r_2,\ldots,r_{2006}$. The coefficients satisfy $2i\tfrac{c_i}{c_{2006}-i}=2j\tfrac{c_j}{c_{2006}-j}$ for all pairs of integers $0\le i,j\le2006$. Given that $\sum_{i\ne j,i=1,j=1}^{2006} \tfrac{r_i}{r_j}=42$, determine $\sum_{i=1}^{2006} (r_1+r_2+\ldots+r_{2006})$.

2022 Stanford Mathematics Tournament, 7

Tags:

$\triangle ABC$ has side lengths $AB=20$, $BC=15$, and $CA=7$. Let the altitudes of $\triangle ABC$ be $AD$, $BE$, and $CF$. What is the distance between the orthocenter (intersection of the altitudes) of $\triangle ABC$ and the incenter of $\triangle DEF$?

2019 All-Russian Olympiad, 1

Tags: algebra , functional equation

Each point $A$ in the plane is assigned a real number $f(A).$ It is known that $f(M)=f(A)+f(B)+f(C),$ whenever $M$ is the centroid of $\triangle ABC.$ Prove that $f(A)=0$ for all points $A.$

1996 Canadian Open Math Challenge, 2

Tags:

The numbers $a,b,c$ are the digits of a three digit number which satisfy $49a+7b+c = 286$. What is the three digit number $(100a+10b+c)$?

2024 CMIMC Team, 6

Tags: team

Cyclic quadrilateral $ABCD$ has circumradius $3$. Additionally, $AC = 3\sqrt{2}$, $AB/CD = 2/3$, and $AD = BD$. Find $CD$. [i]Proposed by Justin Hsieh[/i]

2009 AMC 10, 14

Tags: AMC , AIME , search , AMC 8

On Monday, Millie puts a quart of seeds, $ 25\%$ of which are millet, into a bird feeder. On each successive day she adds another quart of the same mix of seeds without removing any seeds that are left. Each day the birds eat only $ 25\%$ of the millet in the feeder, but they eat all of the other seeds. On which day, just after Millie has placed the seeds, will the birds find that more than half the seeds in the feeder are millet? $ \textbf{(A)}\ \text{Tuesday}\qquad \textbf{(B)}\ \text{Wednesday}\qquad \textbf{(C)}\ \text{Thursday} \qquad \textbf{(D)}\ \text{Friday}\qquad \textbf{(E)}\ \text{Saturday}$

2007 Indonesia TST, 1

Tags: geometry , circumcircle , cyclic quadrilateral , angle bisector , geometry proposed

Let $ ABCD$ be a cyclic quadrilateral and $ O$ be the intersection of diagonal $ AC$ and $ BD$. The circumcircles of triangle $ ABO$ and the triangle $ CDO$ intersect at $ K$. Let $ L$ be a point such that the triangle $ BLC$ is similar to $ AKD$ (in that order). Prove that if $ BLCK$ is a convex quadrilateral, then it has an incircle.