Olympiad problems

2022 Centroamerican and Caribbean Math Olympiad, 6

Tags: number theory

A positive integer $n$ is $inverosimil$ if there exists $n$ integers not necessarily distinct such that the sum and the product of this integers are equal to $n$. How many positive integers less than or equal to $2022$ are $inverosimils$?

2023 Chile National Olympiad, 4

Tags: combinatorics , combinatorial geometry , geometry

Inside a square with side $60$, $121$ points are drawn. Prove them are three points that are vertices of a triangle of area not exceeding $30$.

1992 Poland - Second Round, 3

Tags: geometry , isosceles

Through the center of gravity of the acute-angled triangle $ ABC $, lines are drawn perpendicular to the sides $ BC $, $ CA $, $ AB $, intersecting them at the points $ P $, $ Q $, $ R $, respectively. Prove that if $ |BP|\cdot |CQ| \cdot |AR| = |PC| \cdot |QA| \cdot |RB| $, then the triangle $ ABC $ is isosceles. Note: According to Ceva's theorem, the assumed equality of products is equivalent to the fact that the lines $ AP $, $ BQ $, $ CR $ have a common point.

2004 Purple Comet Problems, 21

Tags: number theory , relatively prime

Define $a_k = (k^2 + 1)k!$ and $b_k = a_1 + a_2 + a_3 + \cdots + a_k$. Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where $m$ and $n$ are relatively prime natural numbers. Find $n - m$.

1974 Spain Mathematical Olympiad, 4

Tags: geometry , equilateral , reflection

All three sides of an equilateral triangle are assumed to be reflective (except in the vertices), in such a way that they reflect the rays of light located in their plane, that fall on them and that come out of an interior point of the triangle. Determine the path of a ray of light that, starting from a vertex of the triangle reach another vertex of the same after reflecting successively on the three sides. Calculate the length of the path followed by the light assuming that the side of the triangle measures $1$ m.

2021 Israel TST, 3

Tags: geometry , incenter , circumcircle

In an inscribed quadrilateral $ABCD$, we have $BC=CD$ but $AB\neq AD$. Points $I$ and $J$ are the incenters of triangles $ABC$ and $ACD$ respectively. Point $K$ was chosen on segment $AC$ so that $IK=JK$. Points $M$ and $N$ are the incenters of triangles $AIK$ and $AJK$. Prove that the perpendicular to $CD$ at $D$ and the perpendicular to $KI$ at $I$ intersect on the circumcircle of $MAN$.

1986 IMO Longlists, 62

Tags: number theory

Determine all pairs of positive integers $(x, y)$ satisfying the equation $p^x - y^3 = 1$, where $p$ is a given prime number.

1997 Nordic, 3

Tags: ratio , distance , geometry , point set

Let $A, B, C$, and $D$ be four different points in the plane. Three of the line segments $AB, AC, AD, BC, BD$, and $CD$ have length $a$. The other three have length $b$, where $b > a$. Determine all possible values of the quotient $\frac{b}{a}$. .

2010 Contests, 3

Tags: geometry , graph theory , combinatorics

Given an integer $n\ge 2$, given $n+1$ distinct points $X_0,X_1,\ldots,X_n$ in the plane, and a positive real number $A$, show that the number of triangles $X_0X_iX_j$ of area $A$ does not exceed $4n\sqrt n$.

2024 MMATHS, 12

Tags:

Let $ABC$ be a triangle with $\angle{A}=60^\circ$ and orthocenter $H.$ Let $B'$ be the reflection of $B$ over $AC,$ $C'$ be the reflection of $C$ over $AB,$ and $A'$ be the intersection of $BC'$ and $B'C.$ Let $D$ be the intersection of $A'H$ and $BC.$ If $BC=5$ and $A'D=4,$ then the area of $\triangle{ABC}$ can be expressed as $a\sqrt{b}+\sqrt{c},$ where $a,b,$ and $c$ are positive integers, and $b$ and $c$ are not divisible by the square of any prime. Find $a+b+c.$

1995 May Olympiad, 4

Tags: geometry , pyramid , minimum , distance , 3d geometry

Consider a pyramid whose base is an equilateral triangle $BCD$ and whose other faces are triangles isosceles, right at the common vertex $A$. An ant leaves the vertex $B$ arrives at a point $P$ of the $CD$ edge, from there goes to a point $Q$ of the edge $AC$ and returns to point $B$. If the path you made is minimal, how much is the angle $PQA$ ?

2023 USAMTS Problems, 4

Tags: algebra , system of equations , vieta

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} $