Olympiad problems

2001 Hungary-Israel Binational, 5

Tags: modular arithmetic , graph theory , algebra , linear equation , combinatorics unsolved , combinatorics

Here $G_{n}$ denotes a simple undirected graph with $n$ vertices, $K_{n}$ denotes the complete graph with $n$ vertices, $K_{n,m}$ the complete bipartite graph whose components have $m$ and $n$ vertices, and $C_{n}$ a circuit with $n$ vertices. The number of edges in the graph $G_{n}$ is denoted $e(G_{n})$. (a) Let $p$ be a prime. Consider the graph whose vertices are the ordered pairs $(x, y)$ with $x, y \in\{0, 1, . . . , p-1\}$ and whose edges join vertices $(x, y)$ and $(x' , y')$ if and only if $xx'+yy'\equiv 1 \pmod{p}$ . Prove that this graph does not contain $C_{4}$ . (b) Prove that for inﬁnitely many values $n$ there is a graph $G_{n}$ with $e(G_{n}) \geq \frac{n\sqrt{n}}{2}-n$ that does not contain $C_{4}$.

2020 USA TSTST, 3

Tags: TST , Niven theorem

We say a nondegenerate triangle whose angles have measures $\theta_1$, $\theta_2$, $\theta_3$ is [i]quirky[/i] if there exists integers $r_1,r_2,r_3$, not all zero, such that \[r_1\theta_1+r_2\theta_2+r_3\theta_3=0.\] Find all integers $n\ge 3$ for which a triangle with side lengths $n-1,n,n+1$ is quirky. [i]Evan Chen and Danielle Wang[/i]

2022 Greece JBMO TST, 3

Tags: algebra , inequalities

The real numbers $x,y,z$ are such that $x+y+z=4$ and $0 \le x,y,z \le 2$. Find the minimun value of the expression $$A=\sqrt{2+x}+\sqrt{2+y}+\sqrt{2+z}+\sqrt{x+y}+\sqrt{y+z}+\sqrt{z+x}$$.

2017 Latvia Baltic Way TST, 12

Tags: geometry , circumcircle , collinear

A diameter $AK$ is drawn for the circumscribed circle $\omega$ of an acute-angled triangle $ABC$, an arbitrary point $M$ is chosen on the segment $BC$, the straight line $AM$ intersects $\omega$ at point $Q$. The foot of the perpendicular drawn from $M$ on $AK$ is $D$, the tangent drawn to the circle $\omega$ through the point $Q$, intersects the straight line $MD$ at $P$. A point $L$ (different from $Q$) is chosen on $\omega$ such that $PL$ is tangent to $\omega$. Prove that points $L$, $M$ and $K$ lie on the same line.

2008 Harvard-MIT Mathematics Tournament, 7

Tags: geometry , 3D geometry , limit , function , calculus , derivative , calculus computations

([b]5[/b]) Find $ p$ so that $ \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right)$ is some non-zero real number.

1996 Bundeswettbewerb Mathematik, 2

Tags: modular arithmetic , induction , number theory unsolved , number theory

Define the sequence $(x_n)$ by $x_0 = 0$ and for all $n \in \mathbb N,$ \[x_n=\begin{cases} x_{n-1} + (3^r - 1)/2,&\mbox{ if } n = 3^{r-1}(3k + 1);\\ x_{n-1} - (3^r + 1)/2, & \mbox{ if } n = 3^{r-1}(3k + 2).\end{cases}\] where $k \in \mathbb N_0, r \in \mathbb N$. Prove that every integer occurs in this sequence exactly once.

2007 Thailand Mathematical Olympiad, 6

Tags: geometry , Geometric Inequalities

A triangle has perimeter $2s$, inradius $r$, and incenter $I$. If $s_a, s_b$ and $s_c$ are the distances from $I$ to the three vertices, then show that $$\frac34 +\frac{r}{s_a}+\frac{r}{s_b}+\frac{r}{s_c} \le \frac{s^2}{12r^2}$$

2018 Puerto Rico Team Selection Test, 1

Tags: Arithmetic Progression , algebra

Omar made a list of all the arithmetic progressions of positive integer numbers such that the difference is equal to $2$ and the sum of its terms is $200$. How many progressions does Omar's list have?

2011 Paraguay Mathematical Olympiad, 1

Tags: algebra proposed , algebra

Find the value of the following expression: $\frac{1}{2} + (\frac{1}{3} + \frac{2}{3}) + (\frac{1}{4} + \frac{2}{4} + \frac{3}{4}) + \ldots + (\frac{1}{1000} + \frac{2}{1000} + \ldots + \frac{999}{1000})$

2023 4th Memorial "Aleksandar Blazhevski-Cane", P1

Tags: number theory , polynomial , floor function , Perfect Square

Let $a, b, c, d$ be integers. Prove that for any positive integer $n$, there are at least $\left \lfloor{\frac{n}{4}}\right \rfloor $ positive integers $m \leq n$ such that $m^5 + dm^4 + cm^3 + bm^2 + 2023m + a$ is not a perfect square. [i]Proposed by Ilir Snopce[/i]

2013 Dutch BxMO/EGMO TST, 2

Tags: Triple , positive integers , combinatorics , number theory , Sum

Consider a triple $(a, b, c)$ of pairwise distinct positive integers satisfying $a + b + c = 2013$. A step consists of replacing the triple $(x, y, z)$ by the triple $(y + z - x,z + x - y,x + y - z)$. Prove that, starting from the given triple $(a, b,c)$, after $10$ steps we obtain a triple containing at least one negative number.

2023 Greece JBMO TST, 2

Tags: geometry , cyclic quadrilateral

Consider a cyclic quadrilateral $ABCD$ in which $BC = CD$ and $AB < AD$. Let $E$ be a point on the side $AD$ and $F$ a point on the line $BC$ such that $AE = AB = AF$. Prove that $EF \parallel BD$.

2007 Greece National Olympiad, 4

Tags: combinatorics unsolved , combinatorics

Given a $2007\times 2007$ array of numbers $1$ and $-1$, let $A_{i}$ denote the product of the entries in the $i$th row, and $B_{j}$ denote the product of the entries in the $j$th column. Show that \[A_{1}+A_{2}+\cdots +A_{2007}+B_{1}+B_{2}+\cdots +B_{2007}\neq 0.\]

2023 Malaysian IMO Team Selection Test, 4

Tags: number theory

Do there exist infinitely many triples of positive integers $(a, b, c)$ such that $a$, $b$, $c$ are pairwise coprime, and $a! + b! + c!$ is divisible by $a^2 + b^2 + c^2$? [i]Proposed by Anzo Teh Zhao Yang[/i]

2015 Lusophon Mathematical Olympiad, 5

Tags: geometry , tangent circles , isosceles

Two circles of radius $R$ and $r$, with $R>r$, are tangent to each other externally. The sides adjacent to the base of an isosceles triangle are common tangents to these circles. The base of the triangle is tangent to the circle of the greater radius. Determine the length of the base of the triangle.

2006 Macedonia National Olympiad, 3

Tags: inequalities , inequalities proposed

Let $a,b,c$ be real numbers distinct from $0$ and $1$, with $a+b+c=1$. Prove that \[8\left(\frac{1}{2}-ab-bc-ca\right)\left(\frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} \right)\ge 9 \]

2022 South East Mathematical Olympiad, 5

Tags: inequalities , Number sequence , GCD

Positive sequences $\{a_n\},\{b_n\}$ satisfy:$a_1=b_1=1,b_n=a_nb_{n-1}-\frac{1}{4}(n\geq 2)$. Find the minimum value of $4\sqrt{b_1b_2\cdots b_m}+\sum_{k=1}^m\frac{1}{a_1a_2\cdots a_k}$,where $m$ is a given positive integer.

1996 Bulgaria National Olympiad, 1

Tags: number theory , prime numbers

Find all prime numbers $p,q$ for which $pq$ divides $(5^p-2^p)(5^q-2^q)$.

2018 Thailand TST, 1

Tags: geometry , parallel , incircle

Let $E$ and $F$ be points on side $BC$ of a triangle $\vartriangle ABC$. Points $K$ and $L$ are chosen on segments $AB$ and $AC$, respectively, so that $EK \parallel AC$ and $FL \parallel AB$. The incircles of $\vartriangle BEK$ and $\vartriangle CFL$ touches segments $AB$ and $AC$ at $X$ and $Y$ , respectively. Lines $AC$ and $EX$ intersect at $M$, and lines $AB$ and $FY$ intersect at $N$. Given that $AX = AY$, prove that $MN \parallel BC$.

2022 European Mathematical Cup, 4

Tags: combinatorics , Sets , intersections

A collection $F$ of distinct (not necessarily non-empty) subsets of $X = \{1,2,\ldots,300\}$ is [i]lovely[/i] if for any three (not necessarily distinct) sets $A$, $B$ and $C$ in $F$ at most three out of the following eight sets are non-empty \begin{align*}A \cap B \cap C, \ \ \ \overline{A} \cap B \cap C, \ \ \ A \cap \overline{B} \cap C, \ \ \ A \cap B \cap \overline{C}, \\ \overline{A} \cap \overline{B} \cap C, \ \ \ \overline{A} \cap B \cap \overline {C}, \ \ \ A \cap \overline{B} \cap \overline{C}, \ \ \ \overline{A} \cap \overline{B} \cap \overline{C} \end{align*} where $\overline{S}$ denotes the set of all elements of $X$ which are not in $S$. What is the greatest possible number of sets in a lovely collection?

2004 Purple Comet Problems, 1

Tags:

How many different positive integers divide $10!$ ?

1975 Spain Mathematical Olympiad, 4

Tags: algebra , inequalities , Sum , Product

Prove that if the product of $n$ real and positive numbers is equal to $1$, its sum is greater than or equal to $n$.

1949 Kurschak Competition, 3

Tags: number theory , consecutive

Which positive integers cannot be represented as a sum of (two or more) consecutive integers?

2014 Singapore Senior Math Olympiad, 11

Tags: logarithms

Suppose that $x$ is real number such that $\frac{27\times 9^x}{4^x}=\frac{3^x}{8^x}$. Find the value of $2^{-(1+\log_23)x}$

2000 Regional Competition For Advanced Students, 4

Tags: rational , number theory , algebra , Sequence , recurrence relation

We consider the sequence $\{u_n\}$ defined by recursion $u_{n+1} =\frac{u_n(u_n + 1)}{n}$ for $n \ge 1$. (a) Determine the terms of the sequence for $u_1 = 1$. (b) Show that if a member of the sequence is rational, then all subsequent members are also rational numbers. (c) Show that for every natural number $K$ there is a $u_1 > 1$ such that the first $K$ terms of the sequence are natural numbers.