Olympiad problems

1989 IMO Longlists, 81

Tags: function , algebra

A real-valued function $ f$ on $ \mathbb{Q}$ satisfies the following conditions for arbitrary $ \alpha, \beta \in \mathbb{Q}:$ [b](i)[/b] $ f(0) \equal{} 0,$ [b](ii)[/b] $ f(\alpha) > 0 \text{ if } \alpha \neq 0,$ [b](iii)[/b] $ f(\alpha \cdot \beta) \equal{} f(\alpha)f(\beta),$ [b](iv)[/b] $ f(\alpha \plus{} \beta) \leq f(\alpha) \plus{} f(\beta),$ [b](v)[/b] $ f(m) \leq 1989$ $ \forall m \in \mathbb{Z}.$ Prove that \[ f(\alpha \plus{} \beta) \equal{} \max\{f(\alpha), f(\beta)\} \text{ if } f(\alpha) \neq f(\beta).\]

2005 Today's Calculation Of Integral, 38

Tags: calculus , integration , calculus computations , logarithm

Let $a$ be a constant number such that $0<a<1$ and $V(a)$ be the volume formed by the revolution of the figure which is enclosed by the curve $y=\ln (x-a)$, the $x$-axis and two lines $x=1,x=3$ about the $x$-axis. If $a$ varies in the range of $0<a<1$, find the minimum value of $V(a)$.

2010 AMC 8, 6

Tags: symmetry , geometry , rhombus , rectangle

Which of the following has the greatest number of line of symmetry? $ \textbf{(A)}\ \text{ Equilateral Triangle}$ $\textbf{(B)}\ \text{Non-square rhombus} $ $\textbf{(C)}\ \text{Non-square rectangle}$ $\textbf{(D)}\ \text{Isosceles Triangle}$ $\textbf{(E)}\ \text{Square} $

2021 BMT, T1

Tags: algebra

The arithmetic mean of $2, 6, 8$, and $x$ is $7$. The arithmetic mean of $2, 6, 8, x$, and $y$ is $9$. What is the value of $y - x$?

2022 Indonesia TST, N

Tags: integer , quadratic , greatest common divisor , number theory , diophantine

For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation: \[ x^2 - xy + y^2 = n. \] a) Determine $f(2022)$. b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.

2012 JBMO TST - Macedonia, 1

Tags: prime number , number theory

Find all prime numbers of the form $\tfrac{1}{11} \cdot \underbrace{11\ldots 1}_{2n \textrm{ ones}}$, where $n$ is a natural number.

2005 Sharygin Geometry Olympiad, 8

Tags: geometry , rectangle , inscribed

Around the convex quadrilateral $ABCD$, three rectangles are circumscribed . It is known that two of these rectangles are squares. Is it true that the third one is necessarily a square? (A rectangle is circumscribed around the quadrilateral $ABCD$ if there is one vertex $ABCD$ on each side of the rectangle).

1996 AMC 12/AHSME, 7

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A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $\$4.95$ for the father and $\$0.45$ for each year of a child's age, where age is defined as the age at the most recent birthday. If the bill is $\$9.45$, which of the following could be the age of the youngest child? $\textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 5$

1980 All Soviet Union Mathematical Olympiad, 301

Tags: floor function , algebra , equation

Prove that there is an infinite number of such numbers $B$ that the equation $\lfloor x^3/2\rfloor + \lfloor y^3/2 \rfloor = B$ has at least $1980$ integer solutions $(x,y)$. ($\lfloor z\rfloor$ denotes the greatest integer not exceeding $z$.)

2008 Romania National Olympiad, 2

Tags: inequalities , function , calculus , derivative , integration , geometry , real analysis

Let $ f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $ f'$ on $ [0,1]$. Prove that if $ f\left( \frac 12\right) \equal{} 0$, then \[ \int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.\]

2019 IMC, 1

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Evaluate the product $$\prod_{n=3}^{\infty} \frac{(n^3+3n)^2}{n^6-64}.$$ [i]Proposed by Orif Ibrogimov, ETH Zurich and National University of Uzbekistan and Karen Keryan, Yerevan State University and American University of Armenia, Yerevan[/i]

1998 All-Russian Olympiad, 7

Tags: combinatorics

A jeweller makes a chain consisting of $N>3$ numbered links. A querulous customer then asks him to change the order of the links, in such a way that the number of links the jeweller must open is maximized. What is the maximum number?

2014 National Olympiad First Round, 12

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If one can find a student with at least $k$ friends in any class which has $21$ students such that at least two of any three of these students are friends, what is the largest possible value of $k$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 12 $

2020 AMC 10, 7

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The $25$ integers from $-10$ to $14,$ inclusive, can be arranged to form a $5$-by-$5$ square in which the sum of the numbers in each row, the sum of the numbers in each column, and the sum of the numbers along each of the main diagonals are all the same. What is the value of this common sum? $\textbf{(A) }2 \qquad\textbf{(B) } 5\qquad\textbf{(C) } 10\qquad\textbf{(D) } 25\qquad\textbf{(E) } 50$

2003 IMO Shortlist, 5

Tags: function , algebra , functional equation

Let $\mathbb{R}^+$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ that satisfy the following conditions: - $f(xyz)+f(x)+f(y)+f(z)=f(\sqrt{xy})f(\sqrt{yz})f(\sqrt{zx})$ for all $x,y,z\in\mathbb{R}^+$; - $f(x)<f(y)$ for all $1\le x<y$. [i]Proposed by Hojoo Lee, Korea[/i]

2020-21 KVS IOQM India, 27

Tags: geometry , reflection , orthocenter , angle

Let $ABC$ be an acute-angled triangle and $P$ be a point in its interior. Let $P_A,P_B$ and $P_c$ be the images of $P$ under reflection in the sides $BC,CA$, and $AB$, respectively. If $P$ is the orthocentre of the triangle $P_AP_BP_C$ and if the largest angle of the triangle that can be formed by the line segments$ PA, PB$. and $PC$ is $x^o$, determine the value of $x$.

2024 ELMO Shortlist, N1

Tags: number theory

Find all pairs $(n,d)$ of positive integers such that $d\mid n^2$ and $(n-d)^2<2d$. [i]Linus Tang[/i]

2000 Slovenia National Olympiad, Problem 2

Tags: parameterization , absolute value

Find all real numbers $a$ for which the following equation has a unique real solution: $$|x-1|+|x-2|+\ldots+|x-99|=a.$$

2016 NIMO Problems, 5

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For positive integers $n,$ let $s(n)$ be the sum of the digits of $n.$ Over all four-digit positive integers $n,$ which value of $n$ maximizes the ratio $\frac{s(n)}{n}$? [i]Proposed by Michael Tang[/i]

2005 Alexandru Myller, 4

Tags: ratio , real analysis

Let $(a_n)_n$ be a sequence of positive irational numbers. a) Prove that for every $n\in\mathbb N^*$, the binomial development $(1+a_n)^n$ admits a unique maximum term and determine its rank $r_n\in\{1,2,\ldots,n+1\}$. b) We consider the sequences $x_n=a_n\sqrt n, n\in\mathbb N^*$ and $y_n=(1+a_n)^{r_n}, n\in\mathbb N^*$. Prove that $(x_n)_n$ is convergent if and only if the sequence $(y_n)_n$ is convergent. [i]Eugen Paltanea[/i]

2006 Team Selection Test For CSMO, 2

Tags: euler , geometry , circumcircle , parallelogram , projective geometry , perpendicular bisector

Let $AA_1$ and $BB_1$ be the altitudes of an acute-angled, non-isosceles triangle $ABC$. Also, let $A_0$ and $B_0$ be the midpoints of its sides $BC$ and $CA$, respectively. The line $A_1B_1$ intersects the line $A_0B_0$ at a point $C'$. Prove that the line $CC'$ is perpendicular to the Euler line of the triangle $ABC$ (this is the line that joins the orthocenter and the circumcenter of the triangle $ABC$).

1954 Putnam, A7

Tags: integer , number theory

Prove that there are no integers $x$ and $y$ for which $$x^2 +3xy-2y^2 =122.$$

1959 AMC 12/AHSME, 48

Tags: algebra , polynomial

Given the polynomial $a_0x^n+a_1x^{n-1}+\cdots+a_{n-1}x+a_n$, where $n$ is a positive integer or zero, and $a_0$ is a positive integer. The remaining $a$'s are integers or zero. Set $h=n+a_0+|a_1|+|a_2|+\cdots+|a_n|$. [See example 25 for the meaning of $|x|$.] The number of polynomials with $h=3$ is: $ \textbf{(A)}\ 3\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 9 $

2024 LMT Fall, 6

Tags: speed

Danyang is doing math. He starts to draw an isosceles triangle, but only manages to draws an angle of $70^{\circ}$ before he has to leave for recess. Find the sum of all possible values for the smallest angle in Danyang's triangle.

1996 Mexico National Olympiad, 5

Tags: combinatorics , square , perfect cube , sum

The numbers $1$ to $n^2$ are written in an n×n squared paper in the usual ordering. Any sequence of right and downwards steps from a square to an adjacent one (by side) starting at square $1$ and ending at square $n^2$ is called a path. Denote by $L(C)$ the sum of the numbers through which path $C$ goes. (a) For a fixed $n$, let $M$ and $m$ be the largest and smallest $L(C)$ possible. Prove that $M-m$ is a perfect cube. (b) Prove that for no $n$ can one find a path $C$ with $L(C ) = 1996$.