Olympiad problems

2005 Finnish National High School Mathematics Competition, 1

In the figure below, the centres of four squares have been connected by two line segments. Prove that these line segments are perpendicular.

VI Soros Olympiad 1999 - 2000 (Russia), 9.9

Tags: geometric inequality , right angle , geometry

The center of a circle, the radius of which is $r$, lies on the bisector of the right angle $A$ at a distance $a$ from its sides ($a > r$). A tangent to the circle intersects the sides of the angle at points $B$ and $C$. Find the smallest possible value of the area of triangle $ABC$.

1977 Miklós Schweitzer, 1

Tags: advanced fields , geometry

Consider the intersection of an ellipsoid with a plane $ \sigma$ passing through its center $ O$. On the line through the point $ O$ perpendicular to $ \sigma$, mark the two points at a distance from $ O$ equal to the area of the intersection. Determine the loci of the marked points as $ \sigma$ runs through all such planes. [i]L. Tamassy[/i]

Revenge EL(S)MO 2024, 2

Tags: ellipse , conic , geometry

Prove that for any convex quadrilateral there exist an inellipse and circumellipse which are homothetic. Proposed by [i]Benny Wang + Oron Wang[/i]

2005 Kazakhstan National Olympiad, 2

Tags: geometry

The line parallel to side $AC$ of a right triangle $ABC$ $(\angle C=90^\circ)$ intersects sides $AB$ and $BC$ at $M$ and $N$, respectively, so that the $CN / BN = AC / BC = 2$. Let $O$ be the intersection point of the segments $AN$ and $CM$ and $K$ be a point on the segment $ON$ such that $MO + OK = KN$. The perpendicular line to $AN$ at point $K$ and the bisector of triangle $ABC$ of $\angle B$ meet at point $T$. Find the angle $\angle MTB$.

2014 Romania Team Selection Test, 5

Tags: combinatorics

Let $n$ be an integer greater than $1$ and let $S$ be a finite set containing more than $n+1$ elements.Consider the collection of all sets $A$ of subsets of $S$ satisfying the following two conditions : [b](a)[/b] Each member of $A$ contains at least $n$ elements of $S$. [b](b)[/b] Each element of $S$ is contained in at least $n$ members of $A$. Determine $\max_A \min_B |B|$ , as $B$ runs through all subsets of $A$ whose members cover $S$ , and $A$ runs through the above collection.

1997 Tournament Of Towns, (553) 3

Tags: combinatorics

Initially there is a checker on every square of a $1\times n$ board. The first move consists of moving a checker to an adjacent square thus creating a stack of two checkers. Then each time when making a move, one can choose a stack and move it in either direction as many squares on the board as there are checkers in the stack. If after the move the stack lands on a non-empty square, it is placed on top of the stack which is already there. Prove that it is possible to stack all the checkers on one square in $n - 1$ moves. (A Shapovalov)

2011 Saudi Arabia Pre-TST, 3.3

Tags: geometry , geometric inequality

Let $P$ be a point in the interior of triangle $ABC$. Lines $AP$, $BP$, $CP$ intersect sides $BC$, $CA$, $AB$ at $L$, $M$, $N$, respectively. Prove that $$AP \cdot BP \cdot CP \ge 8PL \cdot PM \cdot PN.$$

2010 Today's Calculation Of Integral, 607

Tags: calculus , integration , analytic geometry , geometry , function , parameterization , logarithm

On the coordinate plane, Let $C$ be the graph of $y=(\ln x)^2\ (x>0)$ and for $\alpha >0$, denote $L(\alpha)$ be the tangent line of $C$ at the point $(\alpha ,\ (\ln \alpha)^2).$ (1) Draw the graph. (2) Let $n(\alpha)$ be the number of the intersection points of $C$ and $L(\alpha)$. Find $n(\alpha)$. (3) For $0<\alpha <1$, let $S(\alpha)$ be the area of the region bounded by $C,\ L(\alpha)$ and the $x$-axis. Find $S(\alpha)$. 2010 Tokyo Institute of Technology entrance exam, Second Exam.

1995 Taiwan National Olympiad, 4

Tags: polynomial , algebra

Let $m_{1},m_{2},...,m_{n}$ be mutually distinct integers. Prove that there exists a $f(x)\in\mathbb{Z}[x]$ of degree $n$ satisfying the following two conditions: a)$f(m_{i})=-1\forall i=1,2,...,n$. b)$f(x)$ is irreducible.

2014 Germany Team Selection Test, 2

Tags: number theory , algebra , functional equation

Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that \[ m^2 + f(n) \mid mf(m) +n \] for all positive integers $m$ and $n$.

2011 N.N. Mihăileanu Individual, 3

Tags: inequalities , real analysis , numerical analysis

Prove the inequalities $ 0<n\left( \sqrt[n]{2} -1 \right) -\left( \frac{1}{n+1} +\frac{1}{n+2} +\cdots +\frac{1}{n+n}\right) <\frac{1}{2n} , $ where $ n\ge 2. $ [i]Marius Cavachi[/i]

2025 239 Open Mathematical Olympiad, 4

Tags: algebra , inequality

Positive numbers $a$, $b$ and $c$ are such that $a^2+b^2+c^2+abc=4$. Prove that \[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]

2020 Thailand Mathematical Olympiad, 3

Tags: algebra , functional equation

Suppose that $f : \mathbb{R}^+\to\mathbb R$ satisfies the equation $$f(a+b+c+d) = f(a)+f(b)+f(c)+f(d)$$ for all $a,b,c,d$ that are the four sides of some tangential quadrilateral. Show that $f(x+y)=f(x)+f(y)$ for all $x,y\in\mathbb{R}^+$.

2013 Stanford Mathematics Tournament, 11

Tags:

What is the smalles positive integer with exactly $768$ divisors? Your answer may be written in its prime factorization.

2018 Sharygin Geometry Olympiad, 1

Tags: geometry

The incircle of a right-angled triangle $ABC$ ($\angle C = 90^\circ$) touches $BC$ at point $K$. Prove that the chord of the incircle cut by line $AK$ is twice as large as the distance from $C$ to that line.

1998 Portugal MO, 2

Tags: geometry , regular , octagon

The regular octagon of the following figure is inscribed in a circle of radius $1$ and $P$ is a arbitrary point of this circle. Calculate the value of $PA^2 + PB^2 +...+ PH^2$. [img]https://cdn.artofproblemsolving.com/attachments/4/c/85e8e48c45970556077ac09c843193959b0e5a.png[/img]

1984 Polish MO Finals, 5

Tags: hexagon , circles , disks , combinatorial geometry , geometry

A regular hexagon of side $1$ is covered by six unit disks. Prove that none of the vertices of the hexagon is covered by two (or more) discs.

2014 Iran Team Selection Test, 4

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R}^{+}\rightarrow \mathbb{R}^{+}$ such that $x,y\in \mathbb{R}^{+},$ \[ f\left(\frac{y}{f(x+1)}\right)+f\left(\frac{x+1}{xf(y)}\right)=f(y) \]

2015 BMT Spring, 3

Tags: combinatorics

How many ways are there to place the numbers $2, 3, . . . , 10$ in a $3 \times 3$ grid, such that any two numbers that share an edge are mutually prime?

CVM 2020, Problem 5

Tags: combinatorics

In a room with $9$ students, there are $n$ clubs with $4$ participants in each club. For any pairs of clubs no more than $2$ students belong to both clubs. Prove that $n \le 18$ [i]Proposed by Manuel Aguilera, Valle[/i]

2012 Indonesia TST, 1

Tags: function , polynomial , induction , algebra

Let $P$ be a polynomial with real coefficients. Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that there exists a real number $t$ such that \[f(x+t) - f(x) = P(x)\] for all $x \in \mathbb{R}$.

2016 Spain Mathematical Olympiad, 6

Tags: inequalities , jensen , rearrangement inequality , esp

Let $n\geq 2$ an integer. Find the least value of $\gamma$ such that for any positive real numbers $x_1,x_2,...,x_n$ with $x_1+x_2+...+x_n=1$ and any real $y_1+y_2+...+y_n=1$ and $0\leq y_1,y_2,...,y_n\leq \frac{1}{2}$ the following inequality holds: $$x_1x_2...x_n\leq \gamma \left(x_1y_1+x_2y_2+...+x_ny_n\right)$$

2015 ASDAN Math Tournament, 5

Tags: team test

Laurie loves multiplying numbers in her head. One day she decides to multiply two $2$-digit numbers $x$ and $y$ such that $x\leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering $51$ she remembers $15$). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?

2011 NIMO Summer Contest, 5

Tags: geometry , circumcircle , perimeter , perpendicular bisector

In equilateral triangle $ABC$, the midpoint of $\overline{BC}$ is $M$. If the circumcircle of triangle $MAB$ has area $36\pi$, then find the perimeter of the triangle. [i]Proposed by Isabella Grabski [/i]