Olympiad problems

2005 Gheorghe Vranceanu, 3

Tags: function , calculus , integration , newton-leibniz , ftc , real analysis , periodic function

Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that: $$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$

2010 Tuymaada Olympiad, 2

Tags: geometry , circumcircle , geometric transformation , reflection , parallelogram , calculus , trigonometry

Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram. Show that $\angle BPC > \angle BAC$.

2016 CHMMC (Fall), 10

Tags: number theory , algebra

For a positive integer $n$, let $p(n)$ denote the number of prime divisors of $n$, counting multiplicity (i.e. $p(12)=3$). A sequence $a_n$ is defined such that $a_0 = 2$ and for $n > 0$, $a_n = 8^{p(a_{n-1})} + 2$. Compute $$\sum_{n=0}^{\infty} \frac{a_n}{2^n}$$

1999 Tuymaada Olympiad, 1

Tags: geometric transformation , combinatorics , geometry

50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table. The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine. [i]Proposed by A. Khrabrov, incorrect translation from Hungarian[/i]

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.