Olympiad problems

2017 Saudi Arabia BMO TST, 2

Solve the following equation in positive integers $x, y$: $x^{2017} - 1 = (x - 1)(y^{2015}- 1)$

Russian TST 2018, P3

Tags: algebra , polynomial

Let $a < b$ be positive integers. Prove that there is a positive integer $n{}$ and a polynomial of the form \[\pm1\pm x\pm x^2\pm\cdots\pm x^n,\]divisible by the polynomial $1+x^a+x^b$.

JOM 2015 Shortlist, A2

Tags: inequality , inequalities

Let $ a, b, c $ be positive real numbers greater or equal to $ 3 $. Prove that $$ 3(abc+b+2c)\ge 2(ab+2ac+3bc) $$ and determine all equality cases.

1997 Austrian-Polish Competition, 7

Tags: inequalities , max , algebra

(a) Prove that $p^2 + q^2 + 1 > p(q + 1)$ for any real numbers $p, q$, . (b) Determine the largest real constant $b$ such that the inequality $p^2 + q^2 + 1 \ge bp(q + 1)$ holds for all real numbers $p, q$ (c) Determine the largest real constant c such that the inequality $p^2 + q^2 + 1 \ge cp(q + 1)$ holds for all integers $p, q$.

2010 China Team Selection Test, 2

Tags: algebra , polynomial , floor function , triangle inequality , inequalities

Given positive integer $n$, find the largest real number $\lambda=\lambda(n)$, such that for any degree $n$ polynomial with complex coefficients $f(x)=a_n x^n+a_{n-1} x^{n-1}+\cdots+a_0$, and any permutation $x_0,x_1,\cdots,x_n$ of $0,1,\cdots,n$, the following inequality holds $\sum_{k=0}^n|f(x_k)-f(x_{k+1})|\geq \lambda |a_n|$, where $x_{n+1}=x_0$.

1899 Eotvos Mathematical Competition, 1

Tags: algebra , geometry

The points $A_0, A_1, A_2, A_3, A_4$ divide a unit circle (circle of radius $1$) into five equal parts. Prove that the chords $A_0, A_1, A_0, A_2$ satisfy $$(A_0A_1 \cdot A_0A_2)^2= 5$$

2012 Princeton University Math Competition, A8

Tags: floor function , algebra

If $n$ is an integer such that $n \ge 2^k$ and $n < 2^{k+1}$, where $k = 1000$, compute the following: $$n - \left( \lfloor \frac{n -2^0}{2^1} \rfloor + \lfloor \frac{n -2^1}{2^2} \rfloor + ...+ \lfloor \frac{n -2^{k-1}}{2^k} \rfloor \right)$$

2021 Iranian Geometry Olympiad, 1

Tags: geometry , symmetry , reflection

With putting the four shapes drawn in the following figure together make a shape with at least two reflection symmetries. [img]https://cdn.artofproblemsolving.com/attachments/6/0/8ace983d3d9b5c7f93b03c505430e1d2d189fd.png[/img] [i]Proposed by Mahdi Etesamifard - Iran[/i]

1975 Kurschak Competition, 2

Tags: geometry , rhombus , inscribed , geometric inequalities

Prove or disprove: given any quadrilateral inscribed in a convex polygon, we can find a rhombus inscribed in the polygon with side not less than the shortest side of the quadrilateral.

2001 District Olympiad, 2

Tags: number theory

Consider the number $n=123456789101112\ldots 99100101$. a)Find the first three digits of the number $\sqrt{n}$. b)Compute the sum of the digits of $n$. c)Prove that $\sqrt{n}$ isn't rational. [i]Valer Pop[/i]

2006 Silk Road, 4

Tags: induction , geometry

A family $L$ of 2006 lines on the plane is given in such a way that it doesn't contain parallel lines and it doesn't contain three lines with a common point.We say that the line $l_1\in L$ is [i]bounding[/i] the line $l_2\in L$,if all intersection points of the line $l_2$ with other lines from $L$ lie on the one side of the line $l_1$. Prove that in the family $L$ there are two lines $l$ and $l'$ such that the following 2 conditions are satisfied simultaneously: [b]1)[/b] The line $l$ is bounding the line $l'$; [b]2)[/b] the line $l'$ is not bounding the line $l$.

2020 German National Olympiad, 1

Tags: maximal , geometry , angle , circles

Let $k$ be a circle with center $M$ and let $B$ be another point in the interior of $k$. Determine those points $V$ on $k$ for which $\measuredangle BVM$ becomes maximal.

2008 Miklós Schweitzer, 5

Tags: asymptotics , set theory , real analysis

Let $A$ be an infinite subset of the set of natural numbers, and denote by $\tau_A(n)$ the number of divisors of $n$ in $A$. Construct a set $A$ for which $$\sum_{n\le x}\tau_A(n)=x+O(\log\log x)$$ and show that there is no set for which the error term is $o(\log\log x)$ in the above formula. (translated by Miklós Maróti)

2008 Cuba MO, 4

Tags: algebra , functional

Determine all functions $f : R \to R$ such that $f(xy + f(x)) =xf(y) + f(x)$ for all real numbers $x, y$.

2015 VTRMC, Problem 5

Tags: calculus , integration

Evaluate $\int^\infty_0\frac{\operatorname{arctan}(\pi x)-\operatorname{arctan}(x)}xdx$ (where $0\le\operatorname{arctan}(x)<\frac\pi2$ for $0\le x<\infty$).

2013 Saudi Arabia Pre-TST, 4.4

Tags: perpendicular , isosceles , geometry

$\vartriangle ABC$ is a triangle, $M$ the midpoint of $BC, D$ the projection of $M$ on $AC$ and $E$ the midppoint of $MD$. Prove that the lines $AE,BD$ are orthogonal if and only if $AB = AC$.

2011 Tuymaada Olympiad, 1

Tags: combinatorics

Each real number greater than $1$ is coloured red or blue with both colours being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+b$ and $ab$ are of different colours.

2018 Purple Comet Problems, 12

Tags: geometry

Line segment $\overline{AB}$ has perpendicular bisector $\overline{CD}$, where $C$ is the midpoint of $\overline{AB}$. The segments have lengths $AB = 72$ and $CD = 60$. Let $R$ be the set of points $P$ that are midpoints of line segments $\overline{XY}$ , where $X$ lies on $\overline{AB}$ and $Y$ lies on $\overline{CD}$. Find the area of the region $R$.

2008 Harvard-MIT Mathematics Tournament, 7

Tags: geometry , 3d geometry , limit , function , calculus , derivative , calculus computations

([b]5[/b]) Find $ p$ so that $ \lim_{x\rightarrow\infty}x^p\left(\sqrt[3]{x\plus{}1}\plus{}\sqrt[3]{x\minus{}1}\minus{}2\sqrt[3]{x}\right)$ is some non-zero real number.

2022/2023 Tournament of Towns, P6

Tags: combinatorics

It is known that among several banknotes of pairwise distinct face values (which are positive integers) there are exactly $N{}$ fakes. In a single test, a detector determines the sum of the face values of all real banknotes in an arbitrary set we have selected. Prove that by using the detector $N{}$ times, all fake banknotes can be identified, if a) $N=2$ and b) $N=3$. [i]Proposed by S. Tokarev[/i]

2009 Princeton University Math Competition, 3

Tags: geometry , symmetry , rotation , geometric transformation , reflection

A polygon is called concave if it has at least one angle strictly greater than $180^{\circ}$. What is the maximum number of symmetries that an 11-sided concave polygon can have? (A [i]symmetry[/i] of a polygon is a way to rotate or reflect the plane that leaves the polygon unchanged.)

1993 Tournament Of Towns, (394) 2

Tags: number theory , digit

The decimal representation of all integers from $1$ to an arbitrary integer $n$ are written one after another as such: $$123... 91011... 99100... (n).$$ Does there exist $n$ such that each of the digits $0,1,2,...,9$ appears the same number of times in the given sequence? (A Andzans)

2017 NIMO Summer Contest, 1

Tags:

Let $x$ be the answer to this question. Find the value of $2017 - 2016x$. [i]Proposed by Michael Tang[/i]

2022 Cyprus JBMO TST, 4

Tags: combinatorics , counting

Consider the digits $1, 2, 3, 4, 5, 6, 7$. (a) Determine the number of seven-digit numbers with distinct digits that can be constructed using the digits above. (b) If we place all of these seven-digit numbers in increasing order, find the seven-digit number which appears in the $2022^{\text{th}}$ position.

1977 IMO Shortlist, 5

Tags: algorithm , combinatorics , combinatorics of words , graph theory

There are $2^n$ words of length $n$ over the alphabet $\{0, 1\}$. Prove that the following algorithm generates the sequence $w_0, w_1, \ldots, w_{2^n-1}$ of all these words such that any two consecutive words differ in exactly one digit. (1) $w_0 = 00 \ldots 0$ ($n$ zeros). (2) Suppose $w_{m-1} = a_1a_2 \ldots a_n,\quad a_i \in \{0, 1\}$. Let $e(m)$ be the exponent of $2$ in the representation of $n$ as a product of primes, and let $j = 1 + e(m)$. Replace the digit $a_j$ in the word $w_{m-1}$ by $1 - a_j$. The obtained word is $w_m$.