Olympiad problems

2004 Brazil National Olympiad, 6

Tags: algebra

Let $a$ and $b$ be real numbers. Define $f_{a,b}\colon R^2\to R^2$ by $f_{a,b}(x;y)=(a-by-x^2;x)$. If $P=(x;y)\in R^2$, define $f^0_{a,b}(P) = P$ and $f^{k+1}_{a,b}(P)=f_{a,b}(f_{a,b}^k(P))$ for all nonnegative integers $k$. The set $per(a;b)$ of the [i]periodic points[/i] of $f_{a,b}$ is the set of points $P\in R^2$ such that $f_{a,b}^n(P) = P$ for some positive integer $n$. Fix $b$. Prove that the set $A_b=\{a\in R \mid per(a;b)\neq \emptyset\}$ admits a minimum. Find this minimum.

2022/2023 Tournament of Towns, P1

Tags: combinatorics

There are $N{}$ mess-loving clerks in the office. Each of them has some rubbish on the desk. The mess-loving clerks leave the office for lunch one at a time (after return of the preceding one). At that moment all those remaining put half of rubbish from their desks on the desk of the one who left. Can it so happen that after all of them have had lunch the amount of rubbish at the desk of each one will be the same as before lunch if a) $N = 2{}$ and b) $N = 10$? [i]Alexey Zaslavsky[/i]

2016 Junior Balkan Team Selection Tests - Romania, 1

Tags: geometry

Let $ABC$ be a acute triangle where $\angle BAC =60$. Prove that if the Euler's line of $ABC$ intersects $AB,AC$ in $D,E$, then $ADE$ is equilateral.

2008 AIME Problems, 14

Tags: trigonometry , calculus , derivative , geometry , circumcircle , function , conic

Let $ a$ and $ b$ be positive real numbers with $ a\ge b$. Let $ \rho$ be the maximum possible value of $ \frac{a}{b}$ for which the system of equations \[ a^2\plus{}y^2\equal{}b^2\plus{}x^2\equal{}(a\minus{}x)^2\plus{}(b\minus{}y)^2\]has a solution in $ (x,y)$ satisfying $ 0\le x<a$ and $ 0\le y<b$. Then $ \rho^2$ can be expressed as a fraction $ \frac{m}{n}$, where $ m$ and $ n$ are relatively prime positive integers. Find $ m\plus{}n$.

1990 Flanders Math Olympiad, 1

Tags: geometry

On the standard unit circle, draw 4 unit circles with centers [0,1],[1,0],[0,-1],[-1,0]. You get a figure as below, find the area of the colored part. [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=277[/img]

1998 Chile National Olympiad, 3

Tags: radical , nested radical , algebra

Evaluate $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+...}}}}$.

1998 National High School Mathematics League, 10

Tags: arithmetic sequence

Arithmetic sequence with all items real, and the common difference is $4$. If the sum of the square of the first item and all items else is not more than $100$, then there are________items at most.

2020 Spain Mathematical Olympiad, 4

Tags: combinatorics , game

Ana and Benito play a game which consists of $2020$ turns. Initially, there are $2020$ cards on the table, numbered from $1$ to $2020$, and Ana possesses an extra card with number $0$. In the $k$-th turn, the player that doesn't possess card $k-1$ chooses whether to take the card with number $k$ or to give it to the other player. The number in each card indicates its value in points. At the end of the game whoever has most points wins. Determine whether one player has a winning strategy or whether both players can force a tie, and describe the strategy.

Geometry Mathley 2011-12, 15.1

Tags: geometry , circumcircle , concurrent circles , incircle , equal segments , concurrency

Let $ABC$ be a non-isosceles triangle. The incircle $(I)$ of the triangle touches sides $BC,CA,AB$ at $A_0,B_0$, and $C_0$. Points $A_1,B_1$, and $C_1$ are on $BC,CA,AB$ such that $BA1 = CA_0, CB_1 = AB_0, AC_1 = BC_0$. Prove that the circumcircles $(IAA1), (IBB_1), (ICC_1)$ pass all through a common point, distinct from $I$. Nguyễn Minh Hà

2015 India IMO Training Camp, 3

Tags: floor function , number theory , sequence , parity

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2018 Thailand TST, 1

Tags: geometry , pentagon , inscribed circle

Let $ABCDE$ be a convex pentagon such that $AB=BC=CD$, $\angle{EAB}=\angle{BCD}$, and $\angle{EDC}=\angle{CBA}$. Prove that the perpendicular line from $E$ to $BC$ and the line segments $AC$ and $BD$ are concurrent.

Kyiv City MO Seniors Round2 2010+ geometry, 2018.11.2

Tags: circumcircle , geometry , right angle

In the quadrilateral $ABCD $, $AB = BC $, the point $K $ is the midpoint of the side $CD $, the rays $BK $ and $AD $ intersect at the point $M $ , the circumscribed circle $ \Delta ABM $ intersects the line $AC $ for the second time at the point $P $. Prove that $\angle BKP = 90 {} ^ \circ $. (Anton Trygub)

2007 Iran Team Selection Test, 1

Tags: algebra , polynomial , number theory

Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$, such that for each $i\neq j, (a_{i},a_{j})=1$, and for each $n$, the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$? [i]By Omid Hatami[/i]

2000 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , modular arithmetic , algebra

Let be a natural power of two. Find the number of numbers equivalent with $ 1 $ modulo $ 3 $ that divide it. [i]Dan Brânzei[/i]

2014 AIME Problems, 12

Tags: function , probability , algebra , domain , number theory

Let $A=\{1,2,3,4\}$, and $f$ and $g$ be randomly chosen (not necessarily distinct) functions from $A$ to $A$. The probability that the range of $f$ and the range of $g$ are disjoint is $\tfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.

2013 IFYM, Sozopol, 7

Tags: number theory , lucas theorem

Let $n\in \mathbb{N}$. Prove that $lcm [1,2,..,n]=lcm [\binom{n}{1},\binom{n}{2},...,\binom{n}{n}]$ if and only if $n+1$ is a prime number.

1995 South africa National Olympiad, 3

Tags: circumcircle , geometry

The circumcircle of $\triangle ABC$ has radius $1$ and centre $O$ and $P$ is a point inside the triangle such that $OP=x$. Prove that \[AP\cdot BP\cdot CP\le(1+x)^2(1-x),\] with equality if and only if $P=O$.

2020 Online Math Open Problems, 30

Tags:

Suppose that $F$ is a field with exactly $5^{14}$ elements. We say that a function $f:F \rightarrow F$ is [i]happy[/i], if, for all $x,y \in F$, $$\left(f(x+y)+f(x)\right)\left(f(x-y)+f(x)\right)=f(y^2)-f(x^2).$$ Compute the number of elements $z$ of $F$ such that there exist distinct happy functions $h_1$ and $h_2$ such that $h_1(z)=h_2(z).$ [i]Proposed by Luke Robitaille[/i]

2009 Harvard-MIT Mathematics Tournament, 8

Tags: calculus , integration , logarithm

Compute \[\int_1^{\sqrt{3}} x^{2x^2+1}+\ln\left(x^{2x^{2x^2+1}}\right)dx.\]

2007 ISI B.Stat Entrance Exam, 3

Tags: calculus , integration , function , calculus computations

Let $f(u)$ be a continuous function and, for any real number $u$, let $[u]$ denote the greatest integer less than or equal to $u$. Show that for any $x>1$, \[\int_{1}^{x} [u]([u]+1)f(u)du = 2\sum_{i=1}^{[x]} i \int_{i}^{x} f(u)du\]

2019 AMC 12/AHSME, 22

Tags: real analysis

Define a sequence recursively by $x_0=5$ and \[x_{n+1}=\frac{x_n^2+5x_n+4}{x_n+6}\] for all nonnegative integers $n.$ Let $m$ be the least positive integer such that \[x_m\leq 4+\frac{1}{2^{20}}.\] In which of the following intervals does $m$ lie? $\textbf{(A) } [9,26] \qquad\textbf{(B) } [27,80] \qquad\textbf{(C) } [81,242]\qquad\textbf{(D) } [243,728] \qquad\textbf{(E) } [729,\infty]$

2007 Bulgarian Autumn Math Competition, Problem 12.1

Tags: trigonometric equation , parameter , parameterization , algebra

Determine the values of the real parameter $a$, such that the equation \[\sin 2x\sin 4x-\sin x\sin 3x=a\] has a unique solution in the interval $[0,\pi)$.

2003 Tournament Of Towns, 7

Tags: combinatorics

Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After $1000$ turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play?

2019 CCA Math Bonanza, L5.1

Tags:

Let $F_1=F_2=1$ and $F_n=F_{n-1}+F_{n-2}$ for any integer $n\geq3$. For some integer $k>1$, Johnny converts $F_k$ kilometers to miles, then rounds to the nearest integer. Assume that $1$ mile is exactly $1.609344$ kilometers. Estimate the smallest value of $k$ such that Johnny [i]does not[/i] get that this is $F_{k-1}$ miles. An estimate of $E$ earns $2^{1-\left|A-E\right|}$ points, where $A$ is the actual answer. [i]2019 CCA Math Bonanza Lightning Round #5.1[/i]

2024 Indonesia TST, N

Tags: number theory

A natural number $n$ is called "good" if there exists natural numbers $a$ and $b$ such that $a+b=n$ and $ab \mid n^2+n+1$. Show that there are infinitely many "good" numbers