Olympiad problems

2021-IMOC, G9

Tags: geometry , incenter , excenter , reflection , tangent circle

Let the incenter and the $A$-excenter of $\triangle ABC$ be $I$ and $I_A$, respectively. Let $BI$ intersect $AC$ at $E$ and $CI$ intersect $AB$ at $F$. Suppose that the reflections of $I$ with respect to $EF$, $FI_A$, $EI_A$ are $X$, $Y$, $Z$, respectively. Show that $\odot(XYZ)$ and $\odot(ABC)$ are tangent to each other.

2023 Belarusian National Olympiad, 8.6

Tags: geometry

On the side $BC$ of a triangle $ABC$ the midpoint $M$ and arbitrary point $K$ is marked. Lines that pass through $K$ parallel to the sides of the triangle intersect the line $AM$ at $L$ and $N$. Prove that $ML=MN$.

2006 Vietnam Team Selection Test, 3

Tags: induction , combinatorics

In the space are given $2006$ distinct points, such that no $4$ of them are coplanar. One draws a segment between each pair of points. A natural number $m$ is called [i]good[/i] if one can put on each of these segments a positive integer not larger than $m$, so that every triangle whose three vertices are among the given points has the property that two of this triangle's sides have equal numbers put on, while the third has a larger number put on. Find the minimum value of a [i]good[/i] number $m$.

1972 All Soviet Union Mathematical Olympiad, 171

Tags: combinatorics , square table

Is it possible to put the numbers $0,1$ or $2$ in the unit squares of the cross-lined paper $100\times 100$ in such a way, that every rectangle $3\times 4$ (and $4\times 3$) would contain three zeros, four ones and five twos?

1988 Iran MO (2nd round), 1

Tags: polynomial , induction , factorial , algebra

[b](a)[/b] Prove that for all positive integers $m,n$ we have \[\sum_{k=1}^n k(k+1)(k+2)\cdots (k+m-1)=\frac{n(n+1)(n+2) \cdots (n+m)}{m+1}\] [b](b)[/b] Let $P(x)$ be a polynomial with rational coefficients and degree $m.$ If $n$ tends to infinity, then prove that \[\frac{\sum_{k=1}^n P(k)}{n^{m+1}}\] Has a limit.

PEN E Problems, 38

Tags: floor function

Prove that if $c > \dfrac{8}{3}$, then there exists a real number $\theta$ such that $\lfloor\theta^{c^n}\rfloor$ is prime for every positive integer $n$.

2009 IMO Shortlist, 5

Tags: combinatorics , game , invariant

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

2017 China Western Mathematical Olympiad, 3

Tags: geometry , circumcenter

D is the a point on BC，I1 is the heart of a triangle ABD, I2 is the heart of a triangle ACD，O1 is the Circumcenter of triangle AI1D, O2 is the Circumcenter of the triangle AI2D，P is the intersection point of O1I2 and O2I1，Prove: PD is perpendicular to BC.

2009 May Olympiad, 5

Tags: combinatorics

An ant walks along the lines of a grid made up of $55$ horizontal lines and $45$ vertical lines. You want to paint some sections of lines so that the ant can go from any intersection to any other intersection, walking exclusively along painted sections. If the distance between consecutive lines is $10$ cm, what is the least possible number of centimeters that should be painted? What is the higher value?

2021 Novosibirsk Oral Olympiad in Geometry, 2

Tags: concyclic , geometry

The robot crawls the meter in a straight line, puts a flag on and turns by an angle $a <180^o$ clockwise. After that, everything is repeated. Prove that all flags are on the same circle.

1992 Baltic Way, 17

Tags: angle bisector , geometry

Quadrangle $ ABCD$ is inscribed in a circle with radius 1 in such a way that the diagonal $ AC$ is a diameter of the circle, while the other diagonal $ BD$ is as long as $ AB$. The diagonals intersect at $ P$. It is known that the length of $ PC$ is $ 2/5$. How long is the side $ CD$?

1997 Federal Competition For Advanced Students, P2, 6

Tags: polynomial , algebra

For every natural number $ n$, find all polynomials $ x^2\plus{}ax\plus{}b$, where $ a^2 \ge 4b$, that divide $ x^{2n}\plus{}ax^n\plus{}b$.

2010 Puerto Rico Team Selection Test, 2

Tags: algebra

There is the sequence of numbers $1, a_2, a_3, ...$ such that satisfies $1 \cdot a_2 \cdot a_3 \cdot ... \cdot a_n = n^2$, for every integer $n> 2$. Determine the value of $a_3 + a_5$.

2018 Romania Team Selection Tests, 4

Tags: pell equation , sophie germain identity , factorial , number theory

Given an non-negative integer $k$, show that there are infinitely many positive integers $n$ such that the product of any $n$ consecutive integers is divisible by $(n+k)^2+1$.

CNCM Online Round 3, 1

Tags: v4913 orz

Harry, who is incredibly intellectual, needs to eat carrots $C_1, C_2, C_3$ and solve [i]Daily Challenge[/i] problems $D_1, D_2, D_3$. However, he insists that carrot $C_i$ must be eaten only after solving [i]Daily Challenge[/i] problem $D_i$. In how many satisfactory orders can he complete all six actions? [i]Proposed by Albert Wang (awang2004)[/i]

1973 AMC 12/AHSME, 4

Tags: geometry , trigonometry , trig identities , law of sines , law of cosines

Two congruent $ 30^{\circ}$-$ 60^{\circ}$-$ 90^{\circ}$ are placed so that they overlap partly and their hypotenuses coincide. If the hypotenuse of each triangle is 12, the area common to both triangles is $ \textbf{(A)}\ 6\sqrt3 \qquad \textbf{(B)}\ 8\sqrt3 \qquad \textbf{(C)}\ 9\sqrt3 \qquad \textbf{(D)}\ 12\sqrt3 \qquad \textbf{(E)}\ 24$

2021 Harvard-MIT Mathematics Tournament., 8

Tags: combinatorics , floor function , number theory

For each positive real number $\alpha$, define $$\lfloor \alpha \mathbb{N}\rfloor :=\{\lfloor \alpha m \rfloor\; |\; m\in \mathbb{N}\}.$$ Let $n$ be a positive integer. A set $S\subseteq \{1,2,\ldots,n\}$ has the property that: for each real $\beta >0$, $$ \text{if}\; S\subseteq \lfloor \beta \mathbb{N} \rfloor, \text{then}\; \{1,2,\ldots,n\} \subseteq \lfloor \beta\mathbb{N}\rfloor.$$ Determine, with proof, the smallest positive size of $S$.

2009 India IMO Training Camp, 9

Tags: polynomial , algebra

Let $ f(x)\equal{}\sum_{k\equal{}1}^n a_k x^k$ and $ g(x)\equal{}\sum_{k\equal{}1}^n \frac{a_k x^k}{2^k \minus{}1}$ be two polynomials with real coefficients. Let g(x) have $ 0,2^{n\plus{}1}$ as two of its roots. Prove That $ f(x)$ has a positive root less than $ 2^n$.

1996 India Regional Mathematical Olympiad, 6

Tags:

Given any positive integer $n$ , show that there are two positive rational numbers $a$ and $b$ , $a \not= b$, which are not integers and which are such that $a - b, a^2 - b^2 , \ldots a^n - b^n$ are all integers.

2025 Harvard-MIT Mathematics Tournament, 30

Tags: guts

Let $a,b,$ and $c$ be real numbers satisfying the system of equations \begin{align*} a\sqrt{1+b^2}+b\sqrt{1+a^2}&=\tfrac{3}{4},\\ b\sqrt{1+c^2}+c\sqrt{1+b^2}&=\tfrac{5}{12}, \ \text{and} \\ c\sqrt{1+a^2}+a\sqrt{1+c^2}&=\tfrac{21}{20}. \end{align*} Compute $a.$