Olympiad problems

2017 Iran MO (3rd round), 1

Tags: combinatorics

There's a tape with $n^2$ cells labeled by $1,2,\ldots,n^2$. Suppose that $x,y$ are two distinct positive integers less than or equal to $n$. We want to color the cells of the tape such that any two cells with label difference of $x$ or $y$ have different colors. Find the minimum number of colors needed to do so.

2007 Singapore Junior Math Olympiad, 2

Tags: geometry , equilateral , parallelogram , equilateral triangle

Equilateral triangles $ABE$ and $BCF$ are erected externally onthe sidess $AB$ and $BC$ of a parallelogram $ABCD$. Prove that $\vartriangle DEF$ is equilateral.

2009 Today's Calculation Of Integral, 473

Tags: calculus , integration , calculus computations

For nonzero real numbers $ r,\ l$ and the positive constant number $ c$, consider the curve on the $ xy$ plane : $ y \equal{} \left\{ \begin{array}{ll} x^2 & (0\leq x\leq r)\quad \\ r^2 & (r\leq x\leq l \plus{} r)\quad \\ (x \minus{} l \minus{} 2r)^2 & (l \plus{} r\leq x\leq l \plus{} 2r)\quad \end{array} \right.$ Denote $ V$ the volume of the solid by revolvering the curve about the $ x$ axis. Let $ r,\ l$ vary in such a way that $ r^2 \plus{} l \equal{} c$. Find the values of $ r,\ l$ which gives the maxmimum volume.

1969 IMO Shortlist, 68

Tags: point set , convex quadrilateral , combinatorial geometry , geometry

$(USS 5)$ Given $5$ points in the plane, no three of which are collinear, prove that we can choose $4$ points among them that form a convex quadrilateral.

2010 Puerto Rico Team Selection Test, 1

Tags: geometry , circles

The circles in the figure have their centers at $C$ and $D$ and intersect at $A$ and $B$. Let $\angle ACB =60$, $\angle ADB =90^o$ and $DA = 1$ . Find the length of $CA$. [img]https://cdn.artofproblemsolving.com/attachments/0/1/950a55984283091d15083fadcf35e8b95cb229.png[/img]

1980 AMC 12/AHSME, 23

Tags: trigonometry , pythagorean theorem , geometry , similar triangles

Line segments drawn from the vertex opposite the hypotenuse of a right triangle to the points trisecting the hypotenuse have lengths $\sin x$ and $\cos x$, where $x$ is a real number such that $0<x<\frac{\pi}2$. The length of the hypotenuse is $\text{(A)} \ \frac 43 \qquad \text{(B)} \ \frac 32 \qquad \text{(C)} \ \frac{3\sqrt{5}}{5} \qquad \text{(D)} \ \frac{2\sqrt{5}}{3} \qquad \text{(E)} \ \text{not uniquely determined}$

2021 Swedish Mathematical Competition, 4

Tags: algebra , functional , functional inequality

Give examples of a function $f : R \to R$ that satisfies $0 < f(x) < f(x + f(x)) <\sqrt2 x$, for all positive $x$, and show that there is no function $f : R \to R$ that satisfies $x < f(x + f(x)) <\sqrt2 f(x)$, for all positive $x$.

2021 AMC 12/AHSME Spring, 7

Tags:

Let $N=34 \cdot 34 \cdot 63 \cdot 270$. What is the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$? $\textbf{(A) }1:16 \qquad \textbf{(B) }1:15 \qquad \textbf{(C) }1:14 \qquad \textbf{(D) }1:8 \qquad \textbf{(E) }1:3$

1995 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 7

Tags: graph theory

In a class, the teacher discovers that every pupil has exactly three friends in the class, that two friends never have a common friend, and that every pair of two pupils who are not friends they have exactly one common friend. How many pupils are there in the class? A. 6 B. 9 C. 10 D. 12 E. 17

2009 Harvard-MIT Mathematics Tournament, 10

Tags:

Given a rearrangement of the numbers from $1$ to $n$, each pair of consecutive elements $a$ and $b$ of the sequence can be either increasing (if $a < b$) or decreasing (if $b < a$). How many rearrangements of the numbers from $1$ to $n$ have exactly two increasing pairs of consecutive elements? Express your answer in terms of $n$.

2017 Saudi Arabia IMO TST, 2

Tags: algebra , functional equation

Find all $f:\mathbb{R}\to\mathbb{R}$ satisfying: $$f(xf(y)-y)+f(xy-x)+f(x+y)=2xy,\quad\forall x,y\in\mathbb{R}.$$

2021 Canadian Mathematical Olympiad Qualification, 7

Tags: algebra , trigonometry

If $A, B$ and $C$ are real angles such that $$\cos (B-C)+\cos (C-A)+\cos (A-B)=-3/2,$$ find $$\cos (A)+\cos (B)+\cos (C)$$

2019 Ukraine Team Selection Test, 1

Tags: geometry , incenter , parallel , equal segments , collinear

In a triangle $ABC$, $\angle ABC= 60^o$, point $I$ is the incenter. Let the points $P$ and $T$ on the sides $AB$ and $BC$ respectively such that $PI \parallel BC$ and $TI \parallel AB$ , and points $P_1$ and $T_1$ on the sides $AB$ and $BC$ respectively such that $AP_1 = BP$ and $CT_1 = BT$. Prove that point $I$ lies on segment $P_1T_1$. (Anton Trygub)

Novosibirsk Oral Geo Oly IX, 2023.3

Tags: geometry , equal segments , isosceles , angle

An isosceles triangle $ABC$ with base $AC$ is given. On the rays $CA$, $AB$ and $BC$, the points $D, E$ and $F$ were marked, respectively, in such a way that $AD = AC$, $BE = BA$ and $CF = CB$. Find the sum of the angles $\angle ADB$, $\angle BEC$ and $\angle CFA$.

2023 Harvard-MIT Mathematics Tournament, 8

Tags: guts

Suppose $a,b,c$ are distinct positive integers such that $\sqrt{a\sqrt{b\sqrt{c}}}$ is an integer. Compute the least possible value of $a+b+c.$

1984 Iran MO (2nd round), 1

Tags: function , floor function , domain , limit , algebra

Let $f$ and $g$ be two functions such that \[f(x)=\frac{1}{\lfloor | x | \rfloor}, \quad g(x)=\frac{1}{|\lfloor x \rfloor |}.\] Find the domains of $f$ and $g$ and then prove that \[\lim_{x \to -1^+} f(x)= \lim_{x \to 1^- } g(x).\]

2015 Bundeswettbewerb Mathematik Germany, 1

Tags: geometry , rectangle , combinatorics

Let $a,b$ be positive even integers. A rectangle with side lengths $a$ and $b$ is split into $a \cdot b$ unit squares. Anja and Bernd take turns and in each turn they color a square that is made of those unit squares. The person that can't color anymore, loses. Anja starts. Find all pairs $(a,b)$, such that she can win for sure. [b]Extension:[/b] Solve the problem for positive integers $a,b$ that don't necessarily have to be even. [b]Note:[/b] The [i]extension[/i] actually was proposed at first. But since this is a homework competition that goes over three months and some cases were weird, the problem was changed to even integers.

2007 Mediterranean Mathematics Olympiad, 3

Tags: inradius , trigonometry , quadratic , geometry

In the triangle $ABC$, the angle $\alpha = \angle BAC$ and the side $a = BC$ are given. Assume that $a = \sqrt{rR}$, where $r$ is the inradius and $R$ the circumradius. Compute all possible lengths of sides $AB$ and $AC.$

1897 Eotvos Mathematical Competition, 1

Tags: geometry

Prove, for angles $\alpha$, $\beta$ and $\gamma$ of a right triangle, the following relation: $$\text{sin } \alpha \text{ sin } \beta \text{ sin } (\alpha-\beta) \text{ } + \text{ sin } \beta \text{ sin } \gamma \text{ sin } (\beta-\gamma) \text{ }+ \text{ sin } \gamma \text{ sin } \alpha \text{ sin } (\gamma-\alpha) \text{ }+ \text{ sin } (\alpha-\beta) \text{ sin } (\beta-\gamma) \text{ sin } (\gamma-\alpha) = 0.$$

2017 Iran Team Selection Test, 6

Tags: algebra , number theory

Let $k>1$ be an integer. The sequence $a_1,a_2, \cdots$ is defined as: $a_1=1, a_2=k$ and for all $n>1$ we have: $a_{n+1}-(k+1)a_n+a_{n-1}=0$ Find all positive integers $n$ such that $a_n$ is a power of $k$. [i]Proposed by Amirhossein Pooya[/i]

2022 Kosovo National Mathematical Olympiad, 4

Tags: number theory , prime number

Find all prime numbers $p$ and $q$ such that $pq-p-q+3$ is a perfect square.

2000 Belarus Team Selection Test, 4.2

Tags: geometry , circumcircle , perimeter , geometric inequality , triangle

Let ABC be a triangle and $M$ be an interior point. Prove that \[ \min\{MA,MB,MC\}+MA+MB+MC<AB+AC+BC.\]

2006 German National Olympiad, 5

Tags: root , absolute value , polynomial , inequalities

Let $x \neq 0$ be a real number satisfying $ax^2+bx+c=0$ with $a,b,c \in \mathbb{Z}$ obeying $|a|+|b|+|c| > 1$. Then prove \[ |x| \geq \frac{1}{|a|+|b|+|c|-1}. \]

2006 Taiwan National Olympiad, 3

Tags: induction , function , inequalities

$a_1, a_2, ..., a_{95}$ are positive reals. Show that $\displaystyle \sum_{k=1}^{95}{a_k} \le 94+ \prod_{k=1}^{95}{\max{\{1,a_k\}}}$

2022 Durer Math Competition Finals, 1

Tags: combinatorics

In duck language, only letters $q$, $a$, and $k$ are used. There is no word with two consonants after each other, because the ducks cannot pronounce them. However, all other four-letter words are meaningful in duck language. How many such words are there? In duck language, too, the letter $a$ is a vowel, while $q$ and $k$ are consonants.