Olympiad problems

2022 MIG, 18

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If the six-digit number $\underline{2}\, \underline{0}\, \underline{2} \, \underline{1} \, \underline{a} \, \underline{b}$ is divisible by $9$, what is the greatest possible value of $a \cdot b$? $\textbf{(A) }18\qquad\textbf{(B) }20\qquad\textbf{(C) }36\qquad\textbf{(D) }40\qquad\textbf{(E) }42$

2019 JBMO Shortlist, N5

Tags: number theory

Find all positive integers $x, y, z$ such that $45^x-6^y=2019^z$ [i]Proposed by Dorlir Ahmeti, Albania[/i]

2000 All-Russian Olympiad, 5

Tags: induction , algebra

The sequence $a_1 = 1$, $a_2, a_3, \cdots$ is defined as follows: if $a_n - 2$ is a natural number not already occurring on the board, then $a_{n+1} = a_n-2$; otherwise, $a_{n+1} = a_n + 3$. Prove that every nonzero perfect square occurs in the sequence as the previous term increased by $3$.

2012 QEDMO 11th, 7

Tags: combinatorial geometry , combinatorics , rhombus , hexagon

In the following, a rhombus is one with edge length $1$ and interior angles $60^o$ and $120^o$ . Now let $n$ be a natural number and $H$ a regular hexagon with edge length $n$, which is covered with rhombuses without overlapping has been. The rhombuses then appear in three different orientations. Prove that whatever the overlap looks exactly, each of these three orientations can be viewed at the same time.

1995 China Team Selection Test, 3

Tags: vector , combinatorics

21 people take a test with 15 true or false questions. It is known that every 2 people have at least 1 correct answer in common. What is the minimum number of people that could have correctly answered the question which the most people were correct on?

2024 CCA Math Bonanza, L2.3

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Call an $8$-digit number [i]cute[/i] if its digits are a permutation of $1,2,\dots,8$. For example, $23615478$ is [i]cute[/i] but $31234587$ is not. Find the number of $8$-digit [i]cute[/i] numbers that are divisible by $11$. [i]Lightning 2.3[/i]

1969 Swedish Mathematical Competition, 1

Tags: number theory , diophantine equation , diophantine

Find all integers m, n such that $m^3 = n^3 + n$.

2014 PUMaC Geometry A, 4

Tags: geometric transformation , reflection , trigonometry , cyclic quadrilateral , perpendicular bisector , trig identities , geometry

Consider the cyclic quadrilateral with side lengths $1$, $4$, $8$, $7$ in that order. What is its circumdiameter? Let the answer be of the form $a\sqrt b+c$, for $b$ squarefree. Find $a+b+c$.

2022 Israel TST, 2

Tags: inequalities

The numbers $a$, $b$, and $c$ are real. Prove that $$(a^5+b^5+c^5+a^3c^2+b^3a^2+c^3b^2)^2\geq 4(a^2+b^2+c^2)(a^5b^3+b^5c^3+c^5a^3)$$

2007 Thailand Mathematical Olympiad, 12

Tags: combinatorics

An alien with four feet wants to wear four identical socks and four identical shoes, where on each foot a sock must be put on before a shoe. How many ways are there for the alien to wear socks and shoes?

2019 Kosovo National Mathematical Olympiad, 3

Tags: inequalities , algebra

Show that for any non-negative real numbers $a,b,c,d$ such that $a^2+b^2+c^2+d^2=1$ the following inequality hold: $$a+b+c+d-1\geq 16abcd$$ When does equality hold?

2014 Portugal MO, 2

Tags: geometry

Let $[ABCD]$ be a square, $M$ a point on the segment $[AD]$, and $N$ a point on the segment $[DC]$ such that $B\hat{M}A = N\hat{M}D = 60^{\circ}$. Calculate $M\hat{B}N$.

2007 National Olympiad First Round, 21

Tags: geometry

Let $ABCD$ be a quadrilateral such that $m(\widehat{A}) = m(\widehat{D}) = 90^\circ$. Let $M$ be the midpoint of $[DC]$. If $AC\perp BM$, $|DC|=12$, and $|AB|=9$, then what is $|AD|$? $ \textbf{(A)}\ 4 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ \text{None of the above} $

2015 JBMO Shortlist, 5

Tags: geometry , similar triangles , incircle

Let $ABC$ be an acute triangle with ${AB\neq AC}$. The incircle ${\omega}$ of the triangle touches the sides ${BC, CA}$ and ${AB}$ at ${D, E}$ and ${F}$, respectively. The perpendicular line erected at ${C}$ onto ${BC}$ meets ${EF}$ at ${M}$, and similarly the perpendicular line erected at ${B}$ onto ${BC}$ meets ${EF}$ at ${N}$. The line ${DM}$ meets ${\omega}$ again in ${P}$, and the line ${DN}$ meets ${\omega}$ again at ${Q}$. Prove that ${DP=DQ}$. Ruben Dario & Leo Giugiuc (Romania)

2008 District Olympiad, 2

Tags: algebra , polynomial , number theory

Determine $ x$ irrational so that $ x^2\plus{}2x$ and $ x^3\minus{}6x$ are both rational.

2023 MOAA, 17

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Call a polynomial with real roots [i]n-local[/i] if the greatest difference between any pair of its roots is $n$. Let $f(x)=x^2+ax+b$ be a 1-[i]local[/i] polynomial with distinct roots such that $a$ and $b$ are non-zero integers. If $f(f(x))$ is a 23-[i]local[/i] polynomial, find the sum of the roots of $f(x)$. [i]Proposed by Anthony Yang[/i]

1999 Dutch Mathematical Olympiad, 1

Tags: function

Let $f: \mathbb{Z} \rightarrow \{-1,1\}$ be a function such that \[ f(mn) =f(m)f(n),\ \forall m,n \in \mathbb{Z}. \] Show that there exists a positive integer $a$ such that $1 \leq a \leq 12$ and $f(a) = f(a + 1) = 1$.

2014 Iran Team Selection Test, 6

Tags: geometry , incenter , circumcircle , trigonometry , projective geometry , geometric transformation , homothety

$I$ is the incenter of triangle $ABC$. perpendicular from $I$ to $AI$ meet $AB$ and $AC$ at ${B}'$ and ${C}'$ respectively . Suppose that ${B}''$ and ${C}''$ are points on half-line $BC$ and $CB$ such that $B{B}''=BA$ and $C{C}''=CA$. Suppose that the second intersection of circumcircles of $A{B}'{B}''$ and $A{C}'{C}''$ is $T$. Prove that the circumcenter of $AIT$ is on the $BC$.

2020 ASDAN Math Tournament, 5

Tags: team test

Two quadratic polynomials $A(x)$ and $B(x)$ have a leading term of $x^2$. For some real numbers $a$ and $b$, the roots of $A(x)$ are $1$ and $a$, and the roots of $B(x)$ are $6$ and $b$. If the roots of $A(x) + B(x)$ are $a + 3$ and $b + \frac1 2$ , then compute $a^2 + b^2$.

2010 Stanford Mathematics Tournament, 17

Tags: geometry , perpendicular bisector

An equilateral triangle is inscribed inside of a circle of radius $R$. Find the side length of the triangle

2018 Bulgaria JBMO TST, 4

Tags: combinatorics , inequalities

Each cell of an infinite table (infinite in all directions) is colored with one of $n$ given colors. All six cells of any $2\times 3$ (or $3 \times 2$) rectangle have different colors. Find the smallest possible value of $n$.

2021 AMC 12/AHSME Spring, 4

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Ms. Blackwell gives an exam to two classes. The mean of the scores of the students in the morning class is $84$, and the afternoon class’s mean score is $70$. The ratio of the number of students in the morning class to the number of students in the afternoon class is $\frac{3}{4}$. What is the mean of the scores of all the students? $\textbf{(A) }74 \qquad \textbf{(B) }75 \qquad \textbf{(C) }76 \qquad \textbf{(D) }77 \qquad \textbf{(E) }78$

2007 Singapore Team Selection Test, 3

Tags: analytic geometry , number theory , greatest common divisor , modular arithmetic

Let $A,B,C$ be $3$ points on the plane with integral coordinates. Prove that there exists a point $P$ with integral coordinates distinct from $A,B$ and $C$ such that the interiors of the segments $PA,PB$ and $PC$ do not contain points with integral coordinates.

2022 Thailand TST, 3

Tags: geometry

Find all integers $n\geq 3$ for which every convex equilateral $n$-gon of side length $1$ contains an equilateral triangle of side length $1$. (Here, polygons contain their boundaries.)

2005 India Regional Mathematical Olympiad, 6

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Determine all triples of positive integers $(a,b,c)$ such that $a \leq b \leq c$ and $a +b + c + ab+ bc +ca = abc +1$.