Olympiad problems

2001 China Team Selection Test, 2

$a$ and $b$ are natural numbers such that $b > a > 1$, and $a$ does not divide $b$. The sequence of natural numbers $\{b_n\}_{n=1}^\infty$ satisfies $b_{n + 1} \geq 2b_n \forall n \in \mathbb{N}$. Does there exist a sequence $\{a_n\}_{n=1}^\infty$ of natural numbers such that for all $n \in \mathbb{N}$, $a_{n + 1} - a_n \in \{a, b\}$, and for all $m, l \in \mathbb{N}$ ($m$ may be equal to $l$), $a_m + a_l \not\in \{b_n\}_{n=1}^\infty$?

1981 Romania Team Selection Tests, 6.

Tags: geometry

In the plane of traingle $ABC$ we consider a variable point $M$ which moves on line $MA$ towards $A$. Halfway there, it stops and starts moving in a straight line line towards $B$. Halfway there, it stops and starts moving in a straight line towards $C$, and halfway there it stops and starts moving in a straight line towards $A$, and so on. Show that $M$ will get as close as we want to the vertices of a fixed triangle with area $\text{area}(ABC)/7$.

2010 Contests, 4

Tags: number theory , integer

(a) Determine all pairs $(x, y)$ of (real) numbers with $0 < x < 1$ and $0 <y < 1$ for which $x + 3y$ and $3x + y$ are both integer. An example is $(x,y) =( \frac{8}{3}, \frac{7}{8}) $, because $ x+3y =\frac38 +\frac{21}{8} =\frac{24}{8} = 3$ and $ 3x+y = \frac98 + \frac78 =\frac{16}{8} = 2$. (b) Determine the integer $m > 2$ for which there are exactly $119$ pairs $(x,y)$ with $0 < x < 1$ and $0 < y < 1$ such that $x + my$ and $mx + y$ are integers. Remark: if $u \ne v,$ the pairs $(u, v)$ and $(v, u)$ are different.

2023 India National Olympiad, 3

Tags: function , number theory

Let $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying: [list] [*] for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$; [*] for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$. [/list] [i]Proposed by Sutanay Bhattacharya[/i]

Putnam 1939, A7

Tags:

Do either $(1)$ or $(2)$: $(1)$ Let $C_a$ be the curve $(y - a^2)^2 = x^2(a^2 - x^2).$ Find the curve which touches all $C_a$ for $a > 0.$ Sketch the solution and at least two of the $C_a.$ $(2)$ Given that $(1 - hx)^{-1}(1 - kx)^{-1} = \sum_{i\geq0}a_i x^i,$ prove that $(1 + hkx)(1 - hkx)^{-1}(1 - h^2x)^{-1}(1 - k^2x)^{-1} = \sum_{i\geq0} a_i^2 x^i.$

2020 Estonia Team Selection Test, 1

Tags: arithmetic sequence , algebra , arithmetic progression

Let $a_1, a_2,...$ a sequence of real numbers. For each positive integer $n$, we denote $m_n =\frac{a_1 + a_2 +... + a_n}{n}$. It is known that there exists a real number $c$ such that for any different positive integers $i, j, k$: $(i - j) m_k + (j - k) m_i + (k - i) m_j = c$. Prove that the sequence $a_1, a_2,..$ is arithmetic

2015 Romania Team Selection Tests, 3

Tags: least common multiple , function , number theory , rip

If $k$ and $n$ are positive integers , and $k \leq n$ , let $M(n,k)$ denote the least common multiple of the numbers $n , n-1 , \ldots , n-k+1$.Let $f(n)$ be the largest positive integer $ k \leq n$ such that $M(n,1)<M(n,2)<\ldots <M(n,k)$ . Prove that : [b](a)[/b] $f(n)<3\sqrt{n}$ for all positive integers $n$ . [b](b)[/b] If $N$ is a positive integer , then $f(n) > N$ for all but finitely many positive integers $n$.

2019 Bangladesh Mathematical Olympiad, 4

Tags: algebra , inequalities

$A$ is a positive real number.$n$ is positive integer number.Find the set of possible values of the infinite sum $x_0^n+x_1^n+x_2^n+...$ where $x_0,x_1,x_2...$ are all positive real numbers so that the infinite series $x_0+x_1+x_2+...$ has sum $A$.

2006 Estonia National Olympiad, 3

Tags: induction , number theory

The sequence $ (F_n)$ of Fibonacci numbers satisfies $ F_1 \equal{} 1, F_2 \equal{} 1$ and $ F_n \equal{} F_{n\minus{}1} \plus{}F_{n\minus{}2}$ for all $ n \ge 3$. Find all pairs of positive integers $ (m, n)$, such that $ F_m . F_n \equal{} mn$.

2010 Thailand Mathematical Olympiad, 3

Tags: angle bisector , projection , geometry

Let $\vartriangle ABC$ be a scalene triangle with $AB < BC < CA$. Let $D$ be the projection of $A$ onto the angle bisector of $\angle ABC$, and let $E$ be the projection of $A$ onto the angle bisector of $\angle ACB$. The line $DE$ cuts sides $AB$ and AC at $M$ and $N$, respectively. Prove that $$\frac{AB+AC}{BC} =\frac{DE}{MN} + 1$$

2014 Iran Team Selection Test, 6

Tags: circumcircle , trigonometry , ratio , symmetry , geometry

The incircle of a non-isosceles triangle $ABC$ with the center $I$ touches the sides $BC$ at $D$. let $X$ is a point on arc $BC$ from circumcircle of triangle $ABC$ such that if $E,F$ are feet of perpendicular from $X$ on $BI,CI$ and $M$ is midpoint of $EF$ we have $MB=MC$. prove that $\widehat{BAD}=\widehat{CAX}$

2023 Paraguay Mathematical Olympiad, 1

Tags: algebra

In the following sequence of numbers, each term, starting with the third, is obtained by adding three times the previous term plus twice the previous term to the previous one: $$a_1, a_2, 78, a_4, a_5, 3438, a_7, a_8,…$$ As seen in the sequence, the third term is $78$ and the sixth term is $3438$. What is the value of the term $a_7$?

2014 Romania Team Selection Test, 1

Tags: geometry , circumcircle , projective geometry , trigonometry

Let $ABC$ a triangle and $O$ his circumcentre.The lines $OA$ and $BC$ intersect each other at $M$ ; the points $N$ and $P$ are defined in an analogous way.The tangent line in $A$ at the circumcircle of triangle $ABC$ intersect $NP$ in the point $X$ ; the points $Y$ and $Z$ are defined in an analogous way.Prove that the points $X$ , $Y$ and $Z$ are collinear.

2010 Contests, 1

Tags: combinatorics

There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors. P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.

2015 South East Mathematical Olympiad, 8

Tags: number theory

For any integers $m,n$, we have the set $A(m,n) = \{ x^2+mx+n \mid x \in \mathbb{Z} \}$, where $\mathbb{Z}$ is the integer set. Does there exist three distinct elements $a,b,c$ which belong to $A(m,n)$ and satisfy the equality $a=bc$?

2008 Grigore Moisil Intercounty, 4

Tags: number theory

Let $ n$ be a positive integer, and $ k\leq n\minus{}1$, $ k\in \mathbb{N}$. Denote $ a_k\equal{}k!(1\plus{}\frac12\plus{}\frac13\plus{}\cdots\plus{}\frac1k)$. Prove that the number $ k! \cdot\left[\binom{n\minus{}1}{k}\minus{}(\minus{}1)^k\right]\plus{}(\minus{}1)^k\cdot a_k \cdot n$ is divisible by $ n^2$.

2002 India IMO Training Camp, 2

Tags: combinatorics

Show that there is a set of $2002$ consecutive positive integers containing exactly $150$ primes. (You may use the fact that there are $168$ primes less than $1000$)

1974 Miklós Schweitzer, 4

Tags: superior algebra , ring theory , function , induction , number theory , prime number

Let $ R$ be an infinite ring such that every subring of $ R$ different from $ \{0 \}$ has a finite index in $ R$. (By the index of a subring, we mean the index of its additive group in the additive group of $ R$.) Prove that the additive group of $ R$ is cyclic. [i]L. Lovasz, J. Pelikan[/i]

1996 National High School Mathematics League, 10

Tags: geometry , 3d geometry , pyramid

Give two congruent regular triangular pyramids, stick their bottom surfaces together. Then ,it becomes a hexahedron with all dihedral angles equal. The length of the shortest edge of the hexahedron is $2$. Then, the furthest distance between two vertexes is________.

2023 Hong Kong Team Selection Test, Problem 2

Tags: coordinate geometry , geometry

Giiven $\Delta ABC$, $\angle CAB=75^{\circ}$ and $\angle ACB=45^{\circ}$. $BC$ is extended to $T$ so that $BC=CT$. Let $M$ be the midpoint of the segment $AT$. Find $\angle BMC$.

1992 All Soviet Union Mathematical Olympiad, 578

Tags: combinatorics , tiling , equilateral , combinatorial geometry

An equilateral triangle side $10$ is divided into $100$ equilateral triangles of side $1$ by lines parallel to its sides. There are m equilateral tiles of $4$ unit triangles and $25 - m$ straight tiles of $4$ unit triangles (as shown below). For which values of $m$ can they be used to tile the original triangle. [The straight tiles may be turned over.]