Olympiad problems

2005 AMC 12/AHSME, 12

Tags: quadratic , algebra

The quadratic equation $ x^2 \plus{} mx \plus{} n \equal{} 0$ has roots that are twice those of $ x^2 \plus{} px \plus{} m \equal{} 0$, and none of $ m,n,$ and $ p$ is zero. What is the value of $ n/p$? $ \textbf{(A)}\ 1\qquad \textbf{(B)}\ 2\qquad \textbf{(C)}\ 4\qquad \textbf{(D)}\ 8\qquad \textbf{(E)}\ 16$

2004 Czech and Slovak Olympiad III A, 3

Tags: combinatorics

Given a circle $S$ and its $121$ chords $P_i (i=1,2,\ldots,121)$, each with a point $A_i(i=1,2,\ldots,121)$ on it. Prove that there exists a point $X$ on the circumference of $S$ such that: there exist $29$ distinct indices $1\le k_1\le k_2\le\ldots\le k_{29}\le 121$, such that the angle formed by ${A_{k_j}}X$ and ${P_{k_j}}$ is smaller than $21$ degrees for every $j=1,2,\ldots,29$.

2011 LMT, 2

Tags:

Julia and Hansol are having a math-off. Currently, Julia has one more than twice as many points as Hansol. If Hansol scores $6$ more points in a row, he will tie Julia’s score. How many points does Julia have?

1966 AMC 12/AHSME, 30

Tags: vieta

If three of the roots of $x^4+ax^2+bx+c=0$ are $1$, $2$, and $3$, then the value of $a+c$ is: $\text{(A)}\ 35 \qquad \text{(B)}\ 24\qquad \text{(C)}\ -12\qquad \text{(D)}\ -61 \qquad \text{(E)}\ -63$

2013 China National Olympiad, 3

Tags: abstract algebra , modular arithmetic , number theory

Let $m,n$ be positive integers. Find the minimum positive integer $N$ which satisfies the following condition. If there exists a set $S$ of integers that contains a complete residue system module $m$ such that $| S | = N$, then there exists a nonempty set $A \subseteq S$ so that $n\mid {\sum\limits_{x \in A} x }$.

2005 Gheorghe Vranceanu, 4

Tags: real analysis , convergence , limit

Let be a sequence of real numbers $ \left( x_n \right)_{n\geqslant 0} $ with $ x_0\neq 0,1 $ and defined as $ x_{n+1}=x_n+x_n^{-1/x_0} . $ [b]a)[/b] Show that the sequence $ \left( x_n\cdot n^{-\frac{x_0}{1+x_0}} \right)_{n\geqslant 0} $ is convergent. [b]b)[/b] Prove that $ \inf_{x_0\neq 0,1} \lim_{n\to\infty } x_n\cdot n^{-\frac{x_0}{1+x_0}} =1. $

IV Soros Olympiad 1997 - 98 (Russia), 9.1

Tags: algebra , radical

Solve the equation $$2(x-6)=\dfrac{x^2}{(1+\sqrt{x+1})^2}$$

2019 Simurgh, 1

Tags: arithmetic sequence

Show that there exists a $10 \times 10$ table of distinct natural numbers such that if $R_i$ is equal to the multiplication of numbers of row $i$ and $S_i$ is equal to multiplication of numbers of column $i$, then numbers $R_1$, $R_2$, ... , $R_{10}$ make a nontrivial arithmetic sequence and numbers $S_1$, $S_2$, ... , $S_{10}$ also make a nontrivial arithmetic sequence. (A nontrivial arithmetic sequence is an arithmetic sequence with common difference between terms not equal to $0$).

2018 Mexico National Olympiad, 5

Tags: number theory , integer sequence

Let $n\geq 5$ an integer and consider a regular $n$-gon. Initially, Nacho is situated in one of the vertices of the $n$-gon, in which he puts a flag. He will start moving clockwise. First, he moves one position and puts another flag, then, two positions and puts another flag, etcetera, until he finally moves $n-1$ positions and puts a flag, in such a way that he puts $n$ flags in total. ¿For which values of $n$, Nacho will have put a flag in each of the $n$ vertices?

2003 Putnam, 1

Tags: inequalities , algorithm , floor function , partition

Let $n$ be a fixed positive integer. How many ways are there to write $n$ as a sum of positive integers, \[n = a_1 + a_2 + \cdots a_k\] with $k$ an arbitrary positive integer and $a_1 \le a_2 \le \cdots \le a_k \le a_1 + 1$? For example, with $n = 4$, there are four ways: $4$, $2 + 2$, $1 + 1 + 2$, $1 + 1 + 1 + 1$.

2019 Jozsef Wildt International Math Competition, W. 3

Tags: integration , trigonometry , calculus

Compute $$\int \limits_{-\frac{\pi}{4}}^{\frac{\pi}{4}}\frac{\cos x+1-x^2}{(1+x\sin x)\sqrt{1-x^2}}dx$$

1989 Greece Junior Math Olympiad, 1

Tags: number theory

Let $A$ be the sum of three consecutive integers and $B$ be the sum of the exact three consecutive integers. Is it possible to have $AB=33333$ ?

2014 BMT Spring, 19

Tags: calculus , integration

Evaluate the integral $\int_0^{\pi/2} \sqrt{\tan \theta} d\theta$.

2006 Princeton University Math Competition, 3

Tags: algebra

Find the minimum value of $x^2+2x+ \frac{24}{x}$ for $x > 0$.

2005 ITAMO, 3

Tags: circumcircle , incenter , greatest common divisor , angle bisector , geometry

Two circles $\gamma_1, \gamma_2$ in a plane, with centers $A$ and $B$ respectively, intersect at $C$ and $D$. Suppose that the circumcircle of $ABC$ intersects $\gamma_1$ in $E$ and $\gamma_2$ in $F$, where the arc $EF$ not containing $C$ lies outside $\gamma_1$ and $\gamma_2$. Prove that this arc $EF$ is bisected by the line $CD$.

2015 Balkan MO Shortlist, N1

Tags: number theory , sum of squares , combinatorics , square

Let $d$ be an even positive integer. John writes the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2 $ on the blackboard and then chooses three of them, let them be ${a_1}, {a_2}, {a_3}$, erases them and writes the number $1+ \displaystyle\sum_{1\le i<j\leq 3} |{a_i} -{a_j}|$ He continues until two numbers remain written on on the blackboard. Prove that the sum of squares of those two numbers is different than the numbers $1^2 ,3^2 ,\ldots,(2n-1)^2$. (Albania)

1998 Iran MO (3rd Round), 1

Tags: function , number theory

Find all functions $f: \mathbb N \to \mathbb N$ such that for all positive integers $m,n$, [b](i)[/b] $mf(f(m))=\left( f(m) \right)^2$, [b](ii)[/b] If $\gcd(m,n)=d$, then $f(mn) \cdot f(d)=d \cdot f(m) \cdot f(n)$, [b](iii)[/b] $f(m)=m$ if and only if $m=1$.

2015 China Second Round Olympiad, 2

Tags: geometry , incenter

In isoceles $\triangle ABC$, $AB=AC$, $I$ is its incenter, $D$ is a point inside $\triangle ABC$ such that $I,B,C,D$ are concyclic. The line through $C$ parallel to $BD$ meets $AD$ at $E$. Prove that $CD^2=BD\cdot CE$.

2013 Cuba MO, 5

Tags: algebra , polynomial

Let the real numbers be $a, b, c, d$ with $a \ge b$ and $c \ge d$. Prove that the equation $$(x + a) (x + d) + (x + b) (x + c) = 0$$ has real roots.

1996 Portugal MO, 1

Tags: geometry , square , right triangle

Consider a square on the hypotenuse of a right triangle $[ABC]$ (right at $B$). Prove that the line segment that joins vertex $B$ with the center of the square makes $45^o$ angles with legs of the triangle.

2007 Vietnam National Olympiad, 3

Tags: analytic geometry , hyperbola , geometry , conic

Let B,C be fixed points and A be roving point. Let H, G be orthecentre and centroid of triagle ABC. Known midpoint of HG lies on BC, find locus of A

2020 Taiwan TST Round 2, 4

Tags: algebra , polynomial

Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties: (a) for any integer $n$, $f(n)$ is an integer; (b) the degree of $f(x)$ is less than $187$. Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$, and Bob will tell Alice the value of $f(k)$. Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns. [i]Proposed by YaWNeeT[/i]

2013 Puerto Rico Team Selection Test, 2

Tags: combinatorics

How many 3-digit numbers have the property that the sum of their digits is even?

1999 AMC 12/AHSME, 12

Tags: polynomial , function , algebra

What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $ y \equal{} p(x)$ and $ y \equal{} q(x)$, each with leading coefficient $ 1$? $ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$

2006 Bulgaria Team Selection Test, 1

Tags: geometry , circumcircle

[b]Problem 4.[/b] Let $k$ be the circumcircle of $\triangle ABC$, and $D$ the point on the arc $\overarc{AB},$ which do not pass through $C$. $I_A$ and $I_B$ are the centers of incircles of $\triangle ADC$ and $\triangle BDC$, respectively. Proove that the circumcircle of $\triangle I_AI_BC$ touches $k$ iff \[ \frac{AD}{BD}=\frac{AC+CD}{BC+CD}. \] [i] Stoyan Atanasov[/i]