Olympiad problems

1954 Moscow Mathematical Olympiad, 263

Define the maximal value of the ratio of a three-digit number to the sum of its digits.

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 $x, y\in\mathbb{R}$.

1998 Nordic, 4

Tags: combinatorics , modular arithmetic , binomial coefficient

Let $n$ be a positive integer. Count the number of numbers $k \in \{0, 1, 2, . . . , n\}$ such that $\binom{n}{k}$ is odd. Show that this number is a power of two, i.e. of the form $2^p$ for some nonnegative integer $p$.

1977 Miklós Schweitzer, 9

Tags: vector , function , integration , real analysis

Suppose that the components of he vector $ \textbf{u}=(u_0,\ldots,u_n)$ are real functions defined on the closed interval $ [a,b]$ with the property that every nontrivial linear combination of them has at most $ n$ zeros in $ [a,b]$. Prove that if $ \sigma$ is an increasing function on $ [a,b]$ and the rank of the operator \[ A(f)= \int_{a}^b \textbf{u}(x)f(x)d\sigma(x), \;f \in C[a,b]\ ,\] is $ r \leq n$, then $ \sigma$ has exactly $ r$ points of increase. [i]E. Gesztelyi[/i]

2013 Harvard-MIT Mathematics Tournament, 9

Tags: hmmt

Let $z$ be a non-real complex number with $z^{23}=1$. Compute \[\sum_{k=0}^{22}\dfrac{1}{1+z^k+z^{2k}}.\]

2015 Estonia Team Selection Test, 3

Tags: number theory , rational

Let $q$ be a fixed positive rational number. Call number $x$ [i]charismatic [/i] if there exist a positive integer $n$ and integers $a_1, a_2, . . . , a_n$ such that $x = (q + 1)^{a_1} \cdot (q + 2)^{a_2} ...(q + n)^{a_n}$. a) Prove that $q$ can be chosen in such a way that every positive rational number turns out to be charismatic. b) Is it true for every $q$ that, for every charismatic number $x$, the number $x + 1$ is charismatic, too?

1998 Iran MO (3rd Round), 2

Tags: reflection , trigonometry , circumcircle , isogonal conjugate , geometry

Let $ M$ and $ N$ be two points inside triangle $ ABC$ such that \[ \angle MAB \equal{} \angle NAC\quad \mbox{and}\quad \angle MBA \equal{} \angle NBC. \] Prove that \[ \frac {AM \cdot AN}{AB \cdot AC} \plus{} \frac {BM \cdot BN}{BA \cdot BC} \plus{} \frac {CM \cdot CN}{CA \cdot CB} \equal{} 1. \]

2012 Saint Petersburg Mathematical Olympiad, 1

Tags: algebra , number theory

Find all integer $b$ such that $[x^2]-2012x+b=0$ has odd number of roots.

2007 AMC 12/AHSME, 2

Tags: geometry

An aquarium has a rectangular base that measures $ 100$ cm by $ 40$ cm and has a height of $ 50$ cm. It is filled with water to a height of $ 40$ cm. A brick with a rectangular base that measures $ 40$ cm by $ 20$ cm and a height of $ 10$ cm is placed in the aquarium. By how many centimeters does the water rise? $ \textbf{(A)}\ 0.5 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 1.5 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 2.5$

2009 Canadian Mathematical Olympiad Qualification Repechage, 2

Tags: pythagorean theorem , trigonometry , geometry

Triangle $ABC$ is right-angled at $C$ with $AC = b$ and $BC = a$. If $d$ is the length of the altitude from $C$ to $AB$, prove that $\dfrac{1}{a^2}+\dfrac{1}{b^2}=\dfrac{1}{d^2}$

Cono Sur Shortlist - geometry, 2012.G3

Tags: circumcircle , power of a point , radical axis , geometry

Let $ABC$ be a triangle, and $M$, $N$, and $P$ be the midpoints of $AB$, $BC$, and $CA$ respectively, such that $MBNP$ is a parallelogram. Let $R$ and $S$ be the points in which the line $MN$ intersects the circumcircle of $ABC$. Prove that $AC$ is tangent to the circumcircle of triangle $RPS$.

2006 Mediterranean Mathematics Olympiad, 1

Tags: combinatorics

Every point of a plane is colored red or blue, not all with the same color. Can this be done in such a way that, on every circumference of radius 1, (a) there is exactly one blue point; (b) there are exactly two blue points?

2003 Romania Team Selection Test, 11

Tags: trigonometry , pigeonhole principle , area of a triangle , geometry

In a square of side 6 the points $A,B,C,D$ are given such that the distance between any two of the four points is at least 5. Prove that $A,B,C,D$ form a convex quadrilateral and its area is greater than 21. [i]Laurentiu Panaitopol[/i]

2005 Today's Calculation Of Integral, 33

Tags: calculus , integration , trigonometry , function , derivative , calculus computations , logarithm

Evaluate \[\int_{-\ln 2}^0\ \frac{dx}{\cos ^2 h x \cdot \sqrt{1-2a\tanh x +a^2}}\ (a>0)\]

2018 Saint Petersburg Mathematical Olympiad, 4

Tags: algebra , polynomial , abstract algebra

$f(x)$ is polynomial with integer coefficients, with module not exceeded $5*10^6$. $f(x)=nx$ has integer root for $n=1,2,...,20$. Prove that $f(0)=0$

2024 India IMOTC, 24

Tags: combinatorics , geometry , combinatorial geometry

There are $n > 1$ distinct points marked in the plane. Prove that there exists a set of circles $\mathcal C$ such that [color=#FFFFFF]___[/color]$\bullet$ Each circle in $\mathcal C$ has unit radius. [color=#FFFFFF]___[/color]$\bullet$ Every marked point lies in the (strict) interior of some circle in $\mathcal C$. [color=#FFFFFF]___[/color]$\bullet$ There are less than $0.3n$ pairs of circles in $\mathcal C$ that intersect in exactly $2$ points. [i]Note: Weaker results with $\it{0.3n}$ replaced by $\it{cn}$ may be awarded points depending on the value of the constant $\it{c > 0.3}$.[/i] [i]Proposed by Siddharth Choppara, Archit Manas, Ananda Bhaduri, Manu Param[/i]

2006 Junior Balkan Team Selection Tests - Romania, 2

Tags: geometry , combinatorial geometry , combinatorics , triangle area , area , trapezoid

In a plane $5$ points are given such that all triangles having vertices at these points are of area not greater than $1$. Show that there exists a trapezoid which contains all point in the interior (or on the sides) and having the area not exceeding $3$.

2019 Saudi Arabia Pre-TST + Training Tests, 3.1

Tags: circumcircle , perpendicular , geometry

In triangle $ABC, \angle B = 60^o$, $O$ is the circumcenter, and $L$ is the foot of an angle bisector of angle $B$.The circumcirle of triangle $BOL$ meets the circumcircle of $ABC$ at point $D \ne B$. Prove that $BD \perp AC$.

2007 Mathematics for Its Sake, 2

Tags: limit , arithmetic sequence , real analysis , arithmetic progression , sequence

Let $ \left( a_n \right)_{n\ge 1} $ be an arithmetic progression of positive real numbers, and $ m $ be a natural number. Calculate: [b]a)[/b] $ \lim_{n\to\infty } \frac{1}{n^{2m+2}} \sum_{1\le i<j\le n} a_i^ma_j^m $ [b]b)[/b] $ \lim_{n\to\infty } \frac{1}{a_n^{2m+2}} \sum_{1\le i<j\le n} a_i^ma_j^m $ [i]Dumitru Acu[/i]

2013 Romania Team Selection Test, 2

Tags: geometry

Let $K$ be a convex quadrangle and let $l$ be a line through the point of intersection of the diagonals of $K$. Show that the length of the segment of intersection $l\cap K$ does not exceed the length of (at least) one of the diagonals of $K$.

2008 Danube Mathematical Competition, 1

Tags: inequalities

$x,y,z,t \in \mathbb R_+^*$: \[ (xy)^{1/2}+(yz)^{1/2}+(zt)^{1/2}+(tx)^{1/2}+(xz)^{1/2}+(yt)^{1/2} \ge 3(xyz+xyt+xzt+yzt)^{\frac{1}{3}} \]

1957 Poland - Second Round, 6

Tags: tangential , geometry

Prove that if a convex quadrilateral has the property that there exists a circle tangent to its sides (i.e. an inscribed circle), and also a circle tangent to the extensions of its sides (an excircle), then the diagonals of the quadrilateral are perpendicular to each other.

2009 District Olympiad, 2

Tags: geometry , geometric inequality

Hiven an acute triangle $ABC$, consider the midpoints $M$ and $N$ of the sides $AB$ and $AC$, respectively. If point $S$ is variable on side $BC$, prove that $$(MB - MS)(NC - NS) \le 0$$

2012 Bulgaria National Olympiad, 3

Tags: polynomial , algebra

We are given a real number $a$, not equal to $0$ or $1$. Sacho and Deni play the following game. First is Sasho and then Deni and so on (they take turns). On each turn, a player changes one of the “*” symbols in the equation: \[*x^4+*x^3+*x^2+*x^1+*=0\] with a number of the type $a^n$, where $n$ is a whole number. Sasho wins if at the end the equation has no real roots, Deni wins otherwise. Determine (in term of $a$) who has a winning strategy

1982 Canada National Olympiad, 2

Tags:

If $a$, $b$ and $c$ are the roots of the equation $x^3 - x^2 - x - 1 = 0$, (i) show that $a$, $b$ and $c$ are distinct: (ii) show that \[\frac{a^{1982} - b^{1982}}{a - b} + \frac{b^{1982} - c^{1982}}{b - c} + \frac{c^{1982} - a^{1982}}{c - a}\] is an integer.