Olympiad problems

2023 USAMTS Problems, 4

Prove that for any real numbers $1 \leq \sqrt{x} \leq y \leq x^2$, the following system of equations has a real solution $(a, b, c)$: \[a+b+c = \frac{x+x^2+x^4+y+y^2+y^4}{2}\] \[ab+ac+bc = \frac{x^3 + x^5 + x^6 + y^3 + y^5 + y^6}{2}\] \[abc=\frac{x^7+y^7}{2}\]

2016 HMNT, 2

Tags: geometry

What is the smallest possible perimeter of a triangle whose side lengths are all squares of distinct positive integers?

2023 JBMO Shortlist, G1

Tags: geometry

Let $ABC$ be a triangle with circumcentre $O$ and circumcircle $\Omega$. $\Gamma$ is the circle passing through $O,B$ and tangent to $AB$ at $B$. Let $\Gamma$ intersect $\Omega$ a second time at $P \neq B$. The circle passing through $P,C$ and tangent to $AC$ at $C$ intersects with $\Gamma$ at $M$. Prove that $|MP|=|MC|$.

2001 Belarusian National Olympiad, 7

Tags: geometry

The convex quadrilateral $ABCD$ is inscribed in the circle $S_1$. Let $O$ be the intersection of $AC$ and $BD$. Circle $S_2$ passes through $D$ and $ O$, intersecting $AD$ and $CD$ at $ M$ and $ N$, respectively. Lines $OM$ and $AB$ intersect at $R$, lines $ON$ and $BC$ intersect at $T$, and $R$ and $T$ lie on the same side of line $BD$ as $ A$. Prove that $O$, $R$,$T$, and $B$ are concyclic.

2023 Belarusian National Olympiad, 9.5

Tags: algebra , polynomial

The polynomial $P(x)=a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\ldots+a_1x+a_0$ ($a_{2n} \neq 0$) doesn't have any real roots. Prove that the polynomial $Q(x)=a_{2n}x^{2n}+a_{2n-2}x^{2n-2}+\ldots+a_2x^2+a_0$ also doesn't have any real roots.

2003 USA Team Selection Test, 5

Tags: trigonometry , function , inequalities

Let $A, B, C$ be real numbers in the interval $\left(0,\frac{\pi}{2}\right)$. Let \begin{align*} X &= \frac{\sin A\sin (A-B)\sin (A-C)}{\sin (B+C)} \\ Y &= \frac{\sin B\sin(B-C)\sin (B-A)}{\sin (C+A)} \\ Z &= \frac{\sin C\sin (C-A)\sin (C-B)}{\sin (A+B)} . \end{align*} Prove that $X+Y+Z \geq 0$.

2015 Postal Coaching, Problem 2

Tags: algebra , functional equation , function

Find all functions $f: \mathbb{Q} \to \mathbb{R}$ such that $f(xy)=f(x)f(y)+f(x+y)-1$ for all rationals $x,y$

2002 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory , integer , divisor

Let $p, q$ be two distinct primes. Prove that there are positive integers $a, b$ such that the arithmetic mean of all positive divisors of the number $n = p^aq^b$ is an integer.

2009 Tournament Of Towns, 5

Tags: rhombus , circumcenter , geometry

In rhombus $ABCD$, angle $A$ equals $120^o$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ so that angle $NAM$ equals $30^o$. Prove that the circumcenter of triangle $NAM$ lies on a diagonal of of the rhombus.

1996 Czech and Slovak Match, 2

Tags: binary operation , algebra

Let ⋆ be a binary operation on a nonempty set $M$. That is, every pair $(a,b) \in M$ is assigned an element $a$ ⋆$ b$ in $M$. Suppose that ⋆ has the additional property that $(a $ ⋆ $b) $ ⋆$ b= a$ and $a$ ⋆ $(a$ ⋆$ b)= b$ for all $a,b \in M$. (a) Show that $a$ ⋆ $b = b$ ⋆ $a$ for all $a,b \in M$. (b) On which finite sets $M$ does such a binary operation exist?

1967 Spain Mathematical Olympiad, 7

Tags: algebra

On a road a caravan of cars circulates, all at the same speed, maintaining the minimum separation between one and the other indicated by the Code of Circulation. This separation is, in meters, $\frac{u^2}{100}$, where $u$ is the speed expressed in km/h. Assuming that the length of each car is $2.89$ m, calculate the speed at which they must circulate so that the capacity of traffic is maximum, that is, so that in a fixed time the maximum number pass of vehicles at a point on the road.

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

Tags:

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

Tags:

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

Tags:

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

Tags:

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? $\textbf{(A) }2 \qquad\textbf{(B) } 5\qquad\textbf{(C) } 10\qquad\textbf{(D) } 25\qquad\textbf{(E) } 50$