Olympiad problems

2023 China Team Selection Test, P5

Tags: geometry , combinatorics

Let $\triangle ABC$ be a triangle, and let $P_1,\cdots,P_n$ be points inside where no three given points are collinear. Prove that we can partition $\triangle ABC$ into $2n+1$ triangles such that their vertices are among $A,B,C,P_1,\cdots,P_n$, and at least $n+\sqrt{n}+1$ of them contain at least one of $A,B,C$.

2021 Bundeswettbewerb Mathematik, 1

Tags: number theory

Let $Q(n)$ denote the sum of the digits of $n$ in its decimal representation. Prove that for every positive integer $k$, there exists a multiple $n$ of $k$ such that $Q(n)=Q(n^2)$.

2023 CMIMC Combo/CS, 9

Tags: combinatorics

A grid is called $k$-special if in each cell is written a distinct integer such that the set of integers in the grid is precisely the set of positive divisors of $k$. A grid is called $k$-awesome if it is $k$-special and for each positive divisor $m$ of $k$, there exists an $m$-special grid within this $k$-special grid (within meaning you could draw a box in this grid to obtain the new grid). Find the sum of the $4$ smallest integers $k$ for which no $k$-awesome grid exists. [i]Proposed by Oliver Hayman[/i]

2009 Mathcenter Contest, 4

Tags: inequalities , algebra

Let $x,y,z\in \mathbb{R}^+_0$ such that $xy+yz+zx=1$. Prove that $$\frac{1}{\sqrt{x+y}}+\frac{1}{\sqrt{y+z}}+\frac{1}{\sqrt{z+x}}\ge 2+\frac{1}{\sqrt{2}}.$$ [i](Anonymous314)[/i]

1995 Balkan MO, 1

Tags: induction , algebra

For all real numbers $x,y$ define $x\star y = \frac{ x+y}{ 1+xy}$. Evaluate the expression \[ ( \cdots (((2 \star 3) \star 4) \star 5) \star \cdots ) \star 1995. \] [i]Macedonia[/i]

2014 India Regional Mathematical Olympiad, 6

Tags: inequalities

Let $x_1,x_2,x_3 \ldots x_{2014}$ be positive real numbers such that $\sum_{j=1}^{2014} x_j=1$. Determine with proof the smallest constant $K$ such that \[K\sum_{j=1}^{2014}\frac{x_j^2}{1-x_j} \ge 1\]

2024 Nordic, 3

Tags: algebra

Find all functions $f: \mathbb{R} \to \mathbb{R}$ $f(f(x)f(y)+y)=f(x)y+f(y-x+1)$ For all $x,y \in \mathbb{R}$

2003 China Team Selection Test, 2

Tags: function , induction , algebra

Let $S$ be a finite set. $f$ is a function defined on the subset-group $2^S$ of set $S$. $f$ is called $\textsl{monotonic decreasing}$ if when $X \subseteq Y\subseteq S$, then $f(X) \geq f(Y)$ holds. Prove that: $f(X \cup Y)+f(X \cap Y ) \leq f(X)+ f(Y)$ for $X, Y \subseteq S$ if and only if $g(X)=f(X \cup \{ a \}) - f(X)$ is a $\textsl{monotonic decreasing}$ funnction on the subset-group $2^{S \setminus \{a\}}$ of set $S \setminus \{a\}$ for any $a \in S$.

2023 Thailand October Camp, 5

Tags: geometry

Let $ABC$ be an acute-angled triangle with $AC > AB$, let $O$ be its circumcentre, and let $D$ be a point on the segment $BC$. The line through $D$ perpendicular to $BC$ intersects the lines $AO, AC,$ and $AB$ at $W, X,$ and $Y,$ respectively. The circumcircles of triangles $AXY$ and $ABC$ intersect again at $Z \ne A$. Prove that if $W \ne D$ and $OW = OD,$ then $DZ$ is tangent to the circle $AXY.$

2013 Turkey Junior National Olympiad, 4

Tags: combinatorics

Player $A$ places an odd number of boxes around a circle and distributes $2013$ balls into some of these boxes. Then the player $B$ chooses one of these boxes and takes the balls in it. After that the player $A$ chooses half of the remaining boxes such that none of two are consecutive and take the balls in them. If player $A$ guarantees to take $k$ balls, find the maximum possible value of $k$.

2002 IMO, 3

Tags: algebra , polynomial , laurentiu panaitopol

Find all pairs of positive integers $m,n\geq3$ for which there exist infinitely many positive integers $a$ such that \[ \frac{a^m+a-1}{a^n+a^2-1} \] is itself an integer. [i]Laurentiu Panaitopol, Romania[/i]

2017 Hanoi Open Mathematics Competitions, 12

Tags: number theory , consecutive integers , prime

Does there exist a sequence of $2017$ consecutive integers which contains exactly $17$ primes?

2009 Junior Balkan MO, 1

Tags: trapezoid , pentagon , geometry

Let $ ABCDE$ be a convex pentagon such that $ AB\plus{}CD\equal{}BC\plus{}DE$ and $ k$ a circle with center on side $ AE$ that touches the sides $ AB$, $ BC$, $ CD$ and $ DE$ at points $ P$, $ Q$, $ R$ and $ S$ (different from vertices of the pentagon) respectively. Prove that lines $ PS$ and $ AE$ are parallel.

2005 Germany Team Selection Test, 2

Tags: circumcircle , homothety , triangle , geometry

Let $O$ be the circumcenter of an acute-angled triangle $ABC$ with ${\angle B<\angle C}$. The line $AO$ meets the side $BC$ at $D$. The circumcenters of the triangles $ABD$ and $ACD$ are $E$ and $F$, respectively. Extend the sides $BA$ and $CA$ beyond $A$, and choose on the respective extensions points $G$ and $H$ such that ${AG=AC}$ and ${AH=AB}$. Prove that the quadrilateral $EFGH$ is a rectangle if and only if ${\angle ACB-\angle ABC=60^{\circ }}$. [i]Proposed by Hojoo Lee, Korea[/i]

2020-21 KVS IOQM India, 30

Tags: combinatorics

Ari chooses $7$ balls at random from $n$ balls numbered $1$ to$ n$. If the probability that no two of the drawn balls have consecutive numbers equals the probability of exactly one pair of consecutive numbers in the chosen balls, find $n$.

2020 LMT Spring, 18

Tags:

Compute the maximum integer value of $k$ such that $2^k$ divides $3^{2n+3}+40n-27$ for any positive integer $n$.

2020 LMT Fall, 4

Tags:

At the Lexington High School, each student is given a unique five-character ID consisting of uppercase letters. Compute the number of possible IDs that contain the string "LMT". [i]Proposed by Alex Li[/i]

Indonesia Regional MO OSP SMA - geometry, 2004.2

Tags: geometry , ratio , cevian

Triangle $ABC$ is given. The points $D, E$, and $F$ are located on the sides $BC, CA$ and $AB$ respectively so that the lines $AD, BE$ and $CF$ intersect at point $O$. Prove that $\frac{AO}{AD} + \frac{BO}{BE} + \frac{CO}{ CF}=2$

2009 Indonesia MO, 1

Tags: modular arithmetic , quadratic , algebra , polynomial , divisibility tests , number theory

Find all positive integers $ n\in\{1,2,3,\ldots,2009\}$ such that \[ 4n^6 \plus{} n^3 \plus{} 5\] is divisible by $ 7$.

2011 Dutch IMO TST, 4

Tags: polynomial , integer polynomial , algebra

Determine all integers $n$ for which the polynomial $P(x) = 3x^3-nx-n-2$ can be written as the product of two non-constant polynomials with integer coeffcients.

2011 Thailand Mathematical Olympiad, 1

Tags:

Given a natural number $n$ $\geq 3$. If $p,q$ are primes, such that, $p \mid n!$ and $q \mid (n-1)!-1$. Prove that, $p<q$

2018 Ecuador NMO (OMEC), 3

Tags: geometry , trapezoid

Let $ABCD$ be a convex quadrilateral with $AB\le CD$. Points $E ,F$ are chosen on segment $AB$ and points $G ,H$ are chosen on the segment $CD$, are chosen such that $AE = BF = CG = DH <\frac{AB}{2}$. Let $P, Q$, and $R$ be the midpoints of $EG$, $FH$, and $CD$, respectively. It is known that $PR$ is parallel to $AD$ and $QR$ is parallel to $BC$. a) Show that $ABCD$ is a trapezoid. b) Let $d$ be the difference of the lengths of the parallel sides. Show that $2PQ\le d$.

2023 AIME, 7

Tags:

Call a positive integer $n$ [i]extra-distinct[/i] if the remainders when $n$ is divided by $2, 3, 4, 5,$ and $6$ are distinct. Find the number of extra-distinct positive integers less than $1000$.

2025 Turkey Team Selection Test, 2

Tags: number theory , arithmetic sequence

For all positive integers $n$, the function $\gamma: \mathbb{Z}^+ \to \mathbb{Z}_{\geq 0}$ is defined as, $\gamma(1) = 0$ and for all $n > 1$, if the prime factorization of $n$ is $n = p_1^{\alpha_1} p_2^{\alpha_2} \dots p_k^{\alpha_k},$ then $\gamma(n) = \alpha_1 + \alpha_2 + \dots + \alpha_k$. We have an arithmetic sequence $X = \{x_i\}_{i=1}^{\infty}$. If for a positive integer $a > 1$, the sequence $\{ \gamma(a^{x_i} -1) \}$ is also an arithmetic sequence, show that the sequence $X$ has to be constant.

2015 CCA Math Bonanza, T5

Tags:

Emily Thorne is throwing a Memorial Day Party to kick off the Summer in the Hamptons, and she is trying to figure out the seating arrangment for all of her guests. Emily saw that if she seated $4$ guests to a table, there would be $1$ guest left over (how sad); if she seated $5$ to a table, there would be $3$ guests left over; and if she seated $6$ to a table, there would again be $1$ guest left over. If there are at least $100$ but no more than $200$ guests (because she’s rich and her house is $20000$ square feet), what is the greatest possible number of guests? [i]2015 CCA Math Bonanza Team Round #5[/i]