Olympiad problems

2003 Gheorghe Vranceanu, 1

Let $ M $ be a set of nonzero real numbers and $ f:M\longrightarrow M $ be a function having the property that the identity function is $ f+f^{-1} . $ [b]1)[/b] Prove that $ m\in M\iff -m\in M. $ [b]2)[/b] Show that $ f $ is odd. [b]3)[/b] Determine the cardinal of $ M. $

2018 CMIMC CS, 4

Tags: computer science

Consider the grid of numbers shown below. 20 01 96 56 16 37 48 38 64 60 96 97 42 20 98 35 64 96 40 71 50 58 90 16 89 Among all paths that start on the top row, move only left, right, and down, and end on the bottom row, what is the minimum sum of their entries?

2005 Purple Comet Problems, 9

Tags:

Find the number of nonnegative integers $n$ for which $(n^2 - 3n + 1)^2 + 1$ is a prime number

2000 Harvard-MIT Mathematics Tournament, 5

Tags: geometry , 3d geometry

Find all $3$-digit numbers which are the sums of the cubes of their digits.

OMMC POTM, 2024 3

Tags: geometry

Define acute triangle $ABC$ with $AB = AC$ and circumcenter $O$. Define point $D$ inside $ABC$ on the circumcircle of $BOC$. Prove that the distance from $A$ to line $DO$ is half $BD+DC$..

2024 LMT Fall, 25

Tags: speed

Let $a_n$ be a sequence such that $a_1=1$, $a_2=1$, and $a_{n+2}=\tfrac{a_{n+1}a_n}{a_{n+1}+a_n}$. Find the value of \[\sum_{n=1}^\infty \frac{1}{a_n3^n}.\]

1983 IMO Longlists, 63

Tags: trigonometric equation , trigonometry , number theory , equation , trigonometric identities

Let $n$ be a positive integer having at least two different prime factors. Show that there exists a permutation $a_1, a_2, \dots , a_n$ of the integers $1, 2, \dots , n$ such that \[\sum_{k=1}^{n} k \cdot \cos \frac{2 \pi a_k}{n}=0.\]

2006 MOP Homework, 1

Tags: circumcircle , geometry

In isosceles triangle $ABC$, $AB=AC$. Extend segment $BC$ through $C$ to $P$. Points $X$ and $Y$ lie on lines $AB$ and $AC$, respectively, such that $PX \parallel AC$ and $PY \parallel AB$. Point $T$ lies on the circumcircle of triangle $ABC$ such that $PT \perp XY$. Prove that $\angle BAT = \angle CAT$.

1965 Dutch Mathematical Olympiad, 3

Tags: geometry , isosceles , combinatorial geometry

Given are the points $A$ and $B$ in the plane. If $x$ is a straight line is in that plane, and $x$ does not coincide with the perpendicular bisectror of $AB$, then denote the number of points $C$ located at $x$ such that $\vartriangle ABC$ is isosceles, as the "weight of the line $x$”. Prove that the weight of any line $x$ is at most $5$ and determine the set of points $P$ which has a line with weight $1$, but none with weight $0$.

2024 Brazil Team Selection Test, 2

Tags: algebra , functional inequality

Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.

2007 Stanford Mathematics Tournament, 4

Tags: floor function

How many positive integers $n$, with $n\le 2007$, yield a solution for $x$ (where $x$ is real) in the equation $\lfloor x \rfloor+\lfloor 2x\rfloor+\lfloor 3x\rfloor=n$?

1989 Greece National Olympiad, 4

Tags: geometry , trapezoid , tangential , geometric inequality , geometric inequalities

A trapezoid with bases $a,b$ and altitude $h$ is circumscribed around a circl.. Prove that $h^2\le ab$.

2010 Tuymaada Olympiad, 2

Tags: inequalities , circumcircle , geometry

In acute triangle $ABC$, let $H$ denote its orthocenter and let $D$ be a point on side $BC$. Let $P$ be the point so that $ADPH$ is a parallelogram. Prove that $\angle DCP<\angle BHP$.

2024 LMT Fall, 11

Tags: team

Let $\phi=\tfrac{1+\sqrt 5}{2}$. Find \[\left(4+\phi^{\frac12}\right)\left(4-\phi^{\frac12}\right)\left(4+i\phi^{-\frac12}\right)\left(4-i\phi^{-\frac12}\right).\]

1985 Federal Competition For Advanced Students, P2, 5

Tags: number theory

A sequence $ (a_n)$ of positive integers satisfies: $ a_n\equal{}\sqrt{\frac{a_{n\minus{}1}^2\plus{}a_{n\plus{}1}^2}{2}}$ for all $ n \ge 1$. Prove that this sequence is constant.

2006 China Team Selection Test, 3

Tags: geometry

$\triangle{ABC}$ can cover a convex polygon $M$.Prove that there exsit a triangle which is congruent to $\triangle{ABC}$ such that it can also cover $M$ and has one side line paralel to or superpose one side line of $M$.

2010 Contests, 3

Tags: trigonometry , angle bisector , geometry

Let $ABC$ be an isosceles triangle with apex at $C.$ Let $D$ and $E$ be two points on the sides $AC$ and $BC$ such that the angle bisectors $\angle DEB$ and $\angle ADE$ meet at $F,$ which lies on segment $AB.$ Prove that $F$ is the midpoint of $AB.$

2021 AMC 10 Spring, 15

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The real number $x$ satisfies the equation $x+\frac{1}{x}=\sqrt{5}$. What is the value of $x^{11}-7x^7+x^3$? $\textbf{(A)}\ -1 \qquad\textbf{(B)}\ 0 \qquad\textbf{(C)}\ 1 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 8$

2006 Cono Sur Olympiad, 3

Tags: inequalities , number theory

Let $n$ be a natural number. The finite sequence $\alpha$ of positive integer terms, there are $n$ different numbers ($\alpha$ can have repeated terms). Moreover, if from one from its terms any we subtract 1, we obtain a sequence which has, between its terms, at least $n$ different positive numbers. What's the minimum value of the sum of all the terms of $\alpha$?

PEN M Problems, 9

Tags: floor function , recursive sequences

An integer sequence $\{a_{n}\}_{n \ge 1}$ is defined by \[a_{1}=2, \; a_{n+1}=\left\lfloor \frac{3}{2}a_{n}\right\rfloor.\] Show that it has infinitely many even and infinitely many odd integers.

2021 MMATHS, 5

Tags:

Suppose that $a_1 = 1$, and that for all $n \ge 2$, $a_n = a_{n-1} + 2a_{n-2} + 3a_{n-3} + \ldots + (n-1)a_1.$ Suppose furthermore that $b_n = a_1 + a_2 + \ldots + a_n$ for all $n$. If $b_1 + b_2 + b_3 + \ldots + b_{2021} = a_k$ for some $k$, find $k$. [i]Proposed by Andrew Wu[/i]

2005 Serbia Team Selection Test, 5

Tags: inequalities

Let $a,b,c$ be positive reals such that $abc=1$ .Prove the inequality $\frac{a}{a^2+2}+\frac{b}{b^2+2}+\frac{c}{c^2+2}\leq 1$

1971 IMO Longlists, 9

Tags: 3d geometry , prism , trigonometry , geometry

The base of an inclined prism is a triangle $ABC$. The perpendicular projection of $B_1$, one of the top vertices, is the midpoint of $BC$. The dihedral angle between the lateral faces through $BC$ and $AB$ is $\alpha$, and the lateral edges of the prism make an angle $\beta$ with the base. If $r_1, r_2, r_3$ are exradii of a perpendicular section of the prism, assuming that in $ABC, \cos^2 A + \cos^2 B + \cos^2 C = 1, \angle A < \angle B < \angle C,$ and $BC = a$, calculate $r_1r_2 + r_1r_3 + r_2r_3.$

2022 Grosman Mathematical Olympiad, P4

Tags: combinatorics , inequality

Along a circle-shaped path are $100$ boys and $100$ girls. The distance between two points on the path is defined as the length of the smaller arc through which it is possible to get from one point to the other. Prove that the sum of distances between pairs of the same gender is always less than or equal to the sum of distances between all pairs of a boy and a girl.

2012 India IMO Training Camp, 3

Tags: function , algebra

Let $f:\mathbb{R}\longrightarrow \mathbb{R}$ be a function such that $f(x+y+xy)=f(x)+f(y)+f(xy)$ for all $x, y\in\mathbb{R}$. Prove that $f$ satisfies $f(x+y)=f(x)+f(y)$ for all $x, y\in\mathbb{R}$.