Olympiad problems

Estonia Open Senior - geometry, 2005.1.2

Two circles $c_1$ and $c_2$ with centres $O_1$ and $O_2$, respectively, are touching externally at $P$. On their common tangent at $P$, point $A$ is chosen, rays drawn from which touch the circles $c_1$ and $c_2$ at points $P_1$ and $P_2$ both different from $P$. It is known that $\angle P_1AP_2 = 120^o$ and angles $P_1AP$ and $P_2AP$ are both acute. Rays $AP_1$ and $AP_2$ intersect line $O_1O_2$ at points $G_1$ and $G_2$, respectively. The second intersection between ray $AO_1$ and $c_1$ is $H_1$, the second intersection between ray $AO_2$ and $c_2$ is $H_2$. Lines $G_1H_1$ and $AP$ intersect at $K$. Prove that if $G_1K$ is a tangent to circle $c_1$, then line $G_2A$ is tangent to circle $c_2$ with tangency point $H_2$.

2017 Simon Marais Mathematical Competition, A4

Tags: vector

Let $A_1,A_2,\ldots,A_{2017}$ be the vertices of a regular polygon with $2017$ sides.Prove that there exists a point $P$ in the plane of the polygon such that the vector $$\sum_{k=1}^{2017}k\frac{\overrightarrow{PA}_k}{\left\lVert\overrightarrow{PA}_k\right\rVert^5}$$ is the zero vector. (The notation $\left\lVert\overrightarrow{XY}\right\rVert$ represents the length of the vector $\overrightarrow{XY}$.)

2021 Cyprus JBMO TST, 4

Tags: geometry

Let $\triangle AB\varGamma$ be an acute-angled triangle with $AB < A\varGamma$, and let $O$ be the center of the circumcircle of the triangle. On the sides $AB$ and $A \varGamma$ we select points $T$ and $P$ respectively such that $OT=OP$. Let $M,K$ and $\varLambda$ be the midpoints of $PT,PB$ and $\varGamma T$ respectively. Prove that $\angle TMK = \angle M\varLambda K$.

1980 IMO, 2

Tags: algebra , polynomial , algebra proposed

Let $p: \mathbb C \to \mathbb C$ be a polynomial with degree $n$ and complex coefficients which satisfies \[x \in \mathbb R \iff p(x) \in \mathbb R.\] Show that $n=1$

2015 Latvia Baltic Way TST, 8

Tags: algebra , rational , number theory

Given a fixed rational number $q$. Let's call a number $x$ [i]charismatic [/i] if we can find a natural number $n$ and integers $a_1, a_2,.., a_n$ such that $$x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} \cdot ... \cdot(q + n)^{a_n} .$$ i) Prove that one can find a $q$ such that all positive rational numbers are charismatic. ii) Is it true that for all $q$, if the number $x$ is charismatic, then $x + 1$ is also charismatic?

2001 District Olympiad, 1

Tags: induction , arithmetic sequence , algebra proposed , algebra

Let $(a_n)_{n\ge 1}$ be a sequence of real numbers such that \[a_1\binom{n}{1}+a_2\binom{n}{2}+\ldots+a_n\binom{n}{n}=2^{n-1}a_n,\ (\forall)n\in \mathbb{N}^*\] Prove that $(a_n)_{n\ge 1}$ is an arithmetical progression. [i]Lucian Dragomir[/i]

2014 Harvard-MIT Mathematics Tournament, 1

Tags: probability

There are $100$ students who want to sign up for the class Introduction to Acting. There are three class sections for Introduction to Acting, each of which will fit exactly $20$ students. The $100$ students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is the probability that Alex and Zhu end up getting into the same section for the class?

2016 CMIMC, 1

Tags: 2016 , CMIMC , geometry , Tiebreaker

Point $A$ lies on the circumference of a circle $\Omega$ with radius $78$. Point $B$ is placed such that $AB$ is tangent to the circle and $AB=65$, while point $C$ is located on $\Omega$ such that $BC=25$. Compute the length of $\overline{AC}$.

2010 Canada National Olympiad, 5

Tags: algebra , polynomial

Let $P(x)$ and $Q(x)$ be polynomials with integer coefficients. Let $a_n = n! +n$. Show that if $\frac{P(a_n)}{Q(a_n)}$ is an integer for every $n$, then $\frac{P(n)}{Q(n)}$ is an integer for every integer $n$ such that $Q(n)\neq 0$.

2023 UMD Math Competition Part II, 2

Tags: combinatorics , induction , UMD

Let $n \ge 2$ be an integer. There are $n$ houses in a town. All distances between pairs of houses are different. Every house sends a visitor to the house closest to it. Find all possible values of $n$ (with full justification) for which we can design a town with $n$ houses where every house is visited.

1991 Arnold's Trivium, 82

Tags:

For what values of the velocity $c$ does the equation $u_t = u -u^2 + u_{xx}$ have a solution in the form of a traveling wave $u = \varphi(x-ct)$, $\varphi(-\infty) = 1$, $\varphi(\infty) = 0$, $0 \le u \le 1$?

2010 Contests, 2

Tags: induction , number theory proposed , number theory

Determine if there are positive integers $a, b$ such that all terms of the sequence defined by \[ x_{1}= 2010,x_{2}= 2011\\ x_{n+2}= x_{n}+ x_{n+1}+a\sqrt{x_{n}x_{n+1}+b}\quad (n\ge 1) \] are integers.

2017 Princeton University Math Competition, B1

Tags: number theory

Let $a_n$ be the least positive integer the sum of whose digits is $n$. Find $a_1 + a_2 + a_3 + \dots + a_{20}$.

2014 Saint Petersburg Mathematical Olympiad, 5

Tags: combinatorics , geometry

On a cellular plane with a cell side equal to $1$, arbitrarily $100 \times 100$ napkin is thrown. It covers some nodes (the node lying on the border of a napkin, is also considered covered). What is the smallest number of lines (going not necessarily along grid lines) you can certainly cover all these nodes?

MOAA Gunga Bowls, 2021.18

Tags: MOAA 2021 , Gunga

Find the largest positive integer $n$ such that the number $(2n)!$ ends with $10$ more zeroes than the number $n!$. [i]Proposed by Andy Xu[/i]

1979 IMO Longlists, 15

Tags: graph theory , number theory , maximization , Extremal Graph Theory , Extremal combinatorics , IMO Shortlist , IMO Longlist

Let $n \geq 2$ be an integer. Find the maximal cardinality of a set $M$ of pairs $(j, k)$ of integers, $1 \leq j < k \leq n$, with the following property: If $(j, k) \in M$, then $(k,m) \not \in M$ for any $m.$

1967 Spain Mathematical Olympiad, 7

Tags: algebra

On a road a caravan of cars circulates, all at the same speed, maintaining the minimum separation between one and the other indicated by the Code of Circulation. This separation is, in meters, $\frac{u^2}{100}$, where $u$ is the speed expressed in km/h. Assuming that the length of each car is $2.89$ m, calculate the speed at which they must circulate so that the capacity of traffic is maximum, that is, so that in a fixed time the maximum number pass of vehicles at a point on the road.

2004 Cono Sur Olympiad, 4

Tags: number theory , cono sur

Arnaldo selects a nonnegative integer $a$ and Bernaldo selects a nonnegative integer $b$. Both of them secretly tell their number to Cernaldo, who writes the numbers $5$, $8$, and $15$ on the board, one of them being the sum $a+b$. Cernaldo rings a bell and Arnaldo and Bernaldo, individually, write on different slips of paper whether they know or not which of the numbers on the board is the sum $a+b$ and they turn them in to Cernaldo. If both of the papers say NO, Cernaldo rings the bell again and the process is repeated. It is known that both Arnaldo and Bernaldo are honest and intelligent. What is the maximum number of times that the bell can be rung until one of them knows the sum? Personal note: They really phoned it in with the names there…

2003 Singapore MO Open, 3

Tags: diophantine , Diophantine equation , number theory

For any given prime $p$, determine whether the equation $x^2 + y^2 + p^z = 2003$ always has integer solutions in $x, y, z$. Justify your answer

2017 Greece National Olympiad, 2

Tags: combinatorics

Let $A$ be a point in the plane and $3$ lines which pass through this point divide the plane in $6$ regions. In each region there are $5$ points. We know that no three of the $30$ points existing in these regions are collinear. Prove that there exist at least $1000$ triangles whose vertices are points of those regions such that $A$ lies either in the interior or on the side of the triangle.

2017 Sharygin Geometry Olympiad, P18

Tags: geometry , Lemoine point , angles

Let $L$ be the common point of the symmedians of triangle $ABC$, and $BH$ be its altitude. It is known that $\angle ALH = 180^o -2\angle A$. Prove that $\angle CLH = 180^o - 2\angle C$.

2005 Georgia Team Selection Test, 8

Tags: geometry , circumcircle

In a convex quadrilateral $ ABCD$ the points $ P$ and $ Q$ are chosen on the sides $ BC$ and $ CD$ respectively so that $ \angle{BAP}\equal{}\angle{DAQ}$. Prove that the line, passing through the orthocenters of triangles $ ABP$ and $ ADQ$, is perpendicular to $ AC$ if and only if the triangles $ ABP$ and $ ADQ$ have the same areas.

2017 F = ma, 17

Tags: kinematics

17) An object is thrown directly downward from the top of a 180-meter-tall building. It takes 1.0 seconds for the object to fall the last 60 meters. With what initial downward speed was the object thrown from the roof? A) 15 m/s B) 25 m/s C) 35 m/s D) 55 m/s E) insufficient information

2001 Estonia Team Selection Test, 6

Tags: incircle , circumcircle , geometry

Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.

2015 CHMMC (Fall), 3

Tags: algebra

A trio of lousy salespeople charge increasing prices on tomatoes as you buy more. The first charges you $x^1_1$ dollars for the $x_1$[i]th [/i]tomato you buy from him, the second charges $x^2_2$ dollars for the $x_2$[i]th[/i] tomato, and the third charges $x^3_3$ dollars for the $x_3$[i]th [/i]tomato. If you want to buy $100$ tomatoes for as cheap as possible, how many should you buy from the first salesperson?