Olympiad problems

2001 All-Russian Olympiad Regional Round, 10.8

There are a thousand non-intersecting arcs on a circle, and on each of them contains two natural numbers. Sum of numbers of each arc is divided by the product of the numbers of the arc following it clockwise arrow. What is the largest possible value of the largest number written?

2016 Abels Math Contest (Norwegian MO) Final, 4

Tags: functional equation , algebra

Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ f(x) f(y) = |x - y| \cdot f \left( \frac{xy + 1}{x - y} \right) \] Holds for all $x \not= y \in \mathbb{R}$

2001 Cono Sur Olympiad, 3

Tags: rectangle , combinatorics , geometry

Three acute triangles are inscribed in the same circle with their vertices being nine distinct points. Show that one can choose a vertex from each triangle so that the three chosen points determine a triangle each of whose angles is at most $90^\circ$.

2017 May Olympiad, 3

Tags: geometry , parallelogram

Let $ABCD$ be a quadrilateral such that $\angle ABC = \angle ADC = 90º$ and $\angle BCD$ > $90º$. Let $P$ be a point inside of the $ABCD$ such that $BCDP$ is parallelogram, the line $AP$ intersects $BC$ in $M$. If $BM = 2, MC = 5, CD = 3$. Find the length of $AM$.

2015 Brazil Team Selection Test, 2

Tags: floor function , number theory , sequence , parity

Let $n > 1$ be a given integer. Prove that infinitely many terms of the sequence $(a_k )_{k\ge 1}$, defined by \[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\] are odd. (For a real number $x$, $\lfloor x\rfloor$ denotes the largest integer not exceeding $x$.) [i]Proposed by Hong Kong[/i]

2017 All-Russian Olympiad, 3

Tags: combinatorics

There are 3 heaps with $100,101,102$ stones. Ilya and Kostya play next game. Every step they take one stone from some heap, but not from same, that was on previous step. They make his steps in turn, Ilya make first step. Player loses if can not make step. Who has winning strategy?

2002 Tuymaada Olympiad, 3

Tags: circumcircle , incenter , geometry

A circle having common centre with the circumcircle of triangle $ABC$ meets the sides of the triangle at six points forming convex hexagon $A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}$ ($A_{1}$ and $A_{2}$ lie on $BC$, $B_{1}$ and $B_{2}$ lie on $AC$, $C_{1}$ and $C_{2}$ lie on $AB$). If $A_{1}B_{1}$ is parallel to the bisector of angle $B$, prove that $A_{2}C_{2}$ is parallel to the bisector of angle $C$. [i]Proposed by S. Berlov[/i]

1976 IMO Longlists, 17

Tags: 3d geometry , sphere , ratio , geometry

Show that there exists a convex polyhedron with all its vertices on the surface of a sphere and with all its faces congruent isosceles triangles whose ratio of sides are $\sqrt{3} :\sqrt{3} :2$.

2003 Junior Tuymaada Olympiad, 5

Tags: algebra , radical inequality , inequality

Prove that for any real $ x $ and $ y $ the inequality $x^2 \sqrt {1+2y^2} + y^2 \sqrt {1+2x^2} \geq xy (x+y+\sqrt{2})$ .

2012 China Girls Math Olympiad, 1

Tags: induction , marvio , inequalities

Let $ a_1, a_2,\ldots, a_n$ be non-negative real numbers. Prove that $\frac{1}{1+ a_1}+\frac{ a_1}{(1+ a_1)(1+ a_2)}+\frac{ a_1 a_2}{(1+ a_1)(1+ a_2)(1+ a_3)}+$ $\cdots+\frac{ a_1 a_2\cdots a_{n-1}}{(1+ a_1)(1+ a_2)\cdots (1+ a_n)} \le 1.$

2023 IRN-SGP-TWN Friendly Math Competition, 2

Tags: functional equation , algebra , geometry

Let $f: \mathbb{R}^{2} \to \mathbb{R}^{+}$such that for every rectangle $A B C D$ one has $$ f(A)+f(C)=f(B)+f(D). $$ Let $K L M N$ be a quadrangle in the plane such that $f(K)+f(M)=f(L)+f(N)$, for each such function. Prove that $K L M N$ is a rectangle. [i]Proposed by Navid.[/i]

2008 Romania Team Selection Test, 1

Tags: function , inequalities

Let $ n \geq 3$ be an odd integer. Determine the maximum value of \[ \sqrt{|x_{1}\minus{}x_{2}|}\plus{}\sqrt{|x_{2}\minus{}x_{3}|}\plus{}\ldots\plus{}\sqrt{|x_{n\minus{}1}\minus{}x_{n}|}\plus{}\sqrt{|x_{n}\minus{}x_{1}|},\] where $ x_{i}$ are positive real numbers from the interval $ [0,1]$.

2024 239 Open Mathematical Olympiad, 1

Tags: combinatorics , set

We will say that two sets of distinct numbers are $\textit{linked}$ to each other if between any two numbers of each set lies at least one number of the other set. Is it possible to fill the cells of a $100 \times 200$ rectangle with distinct numbers so that any two rows of the rectangle are linked to one another, and any two columns of the rectangle are linked to one another?

2023 Princeton University Math Competition, 12

Tags: combinatorics

12. What is the sum of all possible $\left(\begin{array}{l}i \\ j\end{array}\right)$ subject to the restrictions that $i \geq 10, j \geq 0$, and $i+j \leq 20$ ? Count different $i, j$ that yield the same value separately - for example, count both $\left(\begin{array}{c}10 \\ 1\end{array}\right)$ and $\left(\begin{array}{c}10 \\ 9\end{array}\right)$.

2009 India IMO Training Camp, 2

Tags: induction , number theory

Let us consider a simle graph with vertex set $ V$. All ordered pair $ (a,b)$ of integers with $ gcd(a,b) \equal{} 1$, are elements of V. $ (a,b)$ is connected to $ (a,b \plus{} kab)$ by an edge and to $ (a \plus{} kab,b)$ by another edge for all integer k. Prove that for all $ (a,b)\in V$, there exists a path fromm $ (1,1)$ to $ (a,b)$.

II Soros Olympiad 1995 - 96 (Russia), 10.3

Tags: triangle inequality , geometry

Each side of an acute triangle is multiplied by the cosine of the opposite angle. a) Prove that a triangle can be formed from the resulting segments. 6) Find the radius of the circle circumscribed around the resulting triangle if the radius of the circle circumscribed around the original triangle is equal to $R$.

2012 India PRMO, 19

Tags: diophantine equation , number theory

How many integer pairs $(x,y)$ satisfy $x^2+4y^2-2xy-2x-4y-8=0$?

2022 BMT, 10

Tags: algebra

Let $p, q,$ and $r$ be the roots of the polynomial $f(t) = t^3 - 2022t^2 + 2022t - 337.$ Given $$x = (q-1)\left ( \frac{2022 - q}{r-1} + \frac{2022 - r}{p-1} \right )$$ $$y = (r-1)\left ( \frac{2022 - r}{p-1} + \frac{2022 - p}{q-1} \right )$$ $$z = (p-1)\left ( \frac{2022 - p}{q-1} + \frac{2022 - q}{r-1} \right )$$ compute $xyz - qrx - rpy - pqz.$

2016 Turkey EGMO TST, 3

Tags: geometry

Let $X$ be a variable point on the side $BC$ of a triangle $ABC$. Let $B'$ and $C'$ be points on the rays $[XB$ and $[XC$, respectively, satisfying $B'X=BC=C'X$. The line passing through $X$ and parallel to $AB'$ cuts the line $AC$ at $Y$ and the line passing through $X$ and parallel to $AC'$ cuts the line $AB$ at $Z$. Prove that all lines $YZ$ pass through a fixed point as $X$ varies on the line segment $BC$.

2018 Thailand TST, 3

Tags: combinatorics

Let $n > 1$ be a given integer. An $n \times n \times n$ cube is composed of $n^3$ unit cubes. Each unit cube is painted with one colour. For each $n \times n \times 1$ box consisting of $n^2$ unit cubes (in any of the three possible orientations), we consider the set of colours present in that box (each colour is listed only once). This way, we get $3n$ sets of colours, split into three groups according to the orientation. It happens that for every set in any group, the same set appears in both of the other groups. Determine, in terms of $n$, the maximal possible number of colours that are present.

2009 Purple Comet Problems, 25

Tags: algebra , polynomial , function , number theory , relatively prime , rational function

The polynomial $P(x)=a_0+a_1x+a_2x^2+...+a_8x^8+2009x^9$ has the property that $P(\tfrac{1}{k})=\tfrac{1}{k}$ for $k=1,2,3,4,5,6,7,8,9$. There are relatively prime positive integers $m$ and $n$ such that $P(\tfrac{1}{10})=\tfrac{m}{n}$. Find $n-10m$.

1963 Dutch Mathematical Olympiad, 2

Tags: geometry , locus , geometric inequality , midpoint

The straight lines $k$ and $\ell$ intersect at right angles. A line intersects $k$ in $A$ and $\ell$ in $B$. Consider all straight line segments $PQ$ ($P$ on $k$ and $Q$ on $\ell$), which makes an angle of $45^o$ with $AB$. (a) Determine the locus of the midpoints of the line segments $PQ$, (b) If the perpendicular bisector of such a line segment $PQ$ intersects the line $k$ at $K$ and the line $\ell$ at $L$, then prove that $KL \ge PQ$. [hide=original wording of second sentence]De loodrechte snijlijn van k en l snijdt k in A en t in B[/hide]

2015 ITAMO, 3

Tags: angle chasing , angle bisector , geometry

Let ABC a triangle, let K be the foot of the bisector relative to BC and J be the foot of the trisectrix relative to BC closer to the side AC (3* m(JAC)=m(CAB) ). Let C' and B' be two point on the line AJ on the side of J with respect to A, such that AC'=AC and AB=AB'. Prove that ABB'C is cyclic if and only if lines C'K and BB' are parallel.

1987 AMC 8, 25

Tags: probability

Ten balls numbered $1$ to $10$ are in a jar. Jack reaches into the jar and randomly removes one of the balls. Then Jill reaches into the jar and randomly removes a different ball. The probability that the sum of the two numbers on the balls removed is even is $\text{(A)}\ \frac{4}{9} \qquad \text{(B)}\ \frac{9}{19} \qquad \text{(C)}\ \frac{1}{2} \qquad \text{(D)}\ \frac{10}{19} \qquad \text{(E)}\ \frac{5}{9}$

1981 Putnam, A3

Tags: limit , double integral

Find $$ \lim_{t\to \infty} e^{-t} \int_{0}^{t} \int_{0}^{t} \frac{e^x -e^y }{x-y} \,dx\,dy,$$ or show that the limit does not exist.