Olympiad problems

2025 Malaysian APMO Camp Selection Test, 3

Tags: geometry

A fixed triangle $ABC$ is right angled at $A$, and $M$ is a fixed point inside the triangle such that $BM=BA$. Let $O$ be a point on line $BC$, and suppose the ray $OM$ beyond $M$ intersects the interior and exterior angle bisector of $\angle ACM$ at $S$ and $T$ respectively. Prove that there exist a fixed point $J$ such that circumcircles of triangles $JOM$ and $CST$ are always tangent, regardless of the choice of $O$. [i]Proposed by Ivan Chan Kai Chin[/i]

2016 ISI Entrance Examination, 7

Tags: integration , calculus , real analysis

$f$ is a differentiable function such that $f(f(x))=x$ where $x \in [0,1]$.Also $f(0)=1$.Find the value of $$\int_0^1(x-f(x))^{2016}dx$$

2001 National High School Mathematics League, 15

Tags:

The schematic wiring diagram below is made of six electric resistances $a_1,a_2,a_3,a_4,a_5,a_6(a_1>a_2>a_3>a_4>a_5>a_6)$. How can we choose the electric resistances, so that the all-in resistance takes its minimum value? [center][img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvNy81LzBiZGVjZjczN2Y4MWY0YmNhMTg1YmQxNGEzZWMwZDc2OTE1NzUwLnBuZw==&rn=MjExMTExMTExMTExMTFnaGoucG5n[/img][/center]

2021 Princeton University Math Competition, A1 / B3

Tags: algebra

Compute the sum of all real numbers x which satisfy the following equation $$\frac {8^x - 19 \cdot 4^x}{16 - 25 \cdot 2^x}= 2$$

ICMC 7, 5

Tags: geometry

Is it possible to dissect an equilateral triangle into three congruent polygonal pieces (not necessarily convex), one of which contains the triangle’s centre in its interior? [i]Note:[/i] The interior of a polygon is the polygon without its boundary. [i]Proposed by Dylan Toh[/i]

1999 Iran MO (2nd round), 3

Tags: combinatorics

Let $A_1,A_2,\cdots,A_n$ be $n$ distinct points on the plane ($n>1$). We consider all the segments $A_iA_j$ where $i<j\leq{n}$ and color the midpoints of them. What's the minimum number of colored points? (In fact, if $k$ colored points coincide, we count them $1$.)

2016 Mediterranean Mathematics Olympiad, 2

Tags: inequalities

Let $a,b,c$ be positive real numbers with $a+b+c=3$. Prove that \[ \sqrt{\frac{b}{a^2+3}}+ \sqrt{\frac{c}{b^2+3}}+ \sqrt{\frac{a}{c^2+3}} ~\le~ \frac32\sqrt[4]{\frac{1}{abc}}\]

2010 South East Mathematical Olympiad, 3

Tags: inequalities

Let $n$ be a positive integer. The real numbers $a_1,a_2,\cdots,a_n$ and $r_1,r_2,\cdots,r_n$ are such that $a_1\leq a_2\leq \cdots \leq a_n$ and $0\leq r_1\leq r_2\leq \cdots \leq r_n$. Prove that $\sum_{i=1}^n\sum_{j=1}^n a_i a_j \min (r_i,r_j)\geq 0$

2011 Nordic, 3

Tags: function , algebra

Find all functions $f$ such that \[f(f(x) + y) = f(x^2-y) + 4yf(x)\] for all real numbers $x$ and $y$.

2021 Middle European Mathematical Olympiad, 1

Tags: sequence , algebra , constant

Determine all real numbers A such that every sequence of non-zero real numbers $x_1, x_2, \ldots$ satisfying \[ x_{n+1}=A-\frac{1}{x_n} \] for every integer $n \ge 1$, has only finitely many negative terms.

2019 Regional Olympiad of Mexico Southeast, 6

Tags: number theory , prime number

Let $p\geq 3$ a prime number, $a$ and $b$ integers such that $\gcd(a, b)=1$. Let $n$ a natural number such that $p$ divides $a^{2^n}+b^{2^n}$, prove that $2^{n+1}$ divides $p-1$.

2008 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: sequence , function , algebra , functional equation

Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that: $i)$ $a(0)=0$ $ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$. If exists prove: $a)$ $a(k)\geq a(k-1)$ $b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.

2014 ITAMO, 5

Tags: algebra

Prove that there exists a positive integer that can be written, in at least two ways, as a sum of $2014$-th powers of $2015$ distinct positive integers $x_1 <x_2 <\cdots <x_{2015}$.

2014 China Northern MO, 6

Tags: inequalities

Let $x,y,z,w $ be real numbers such that $x+2y+3z+4w=1$. Find the minimum of $x^2+y^2+z^2+w^2+(x+y+z+w)^2$.

2007 Mathematics for Its Sake, 1

Tags: floor function , real analysis , sequence , parity , number theory

Prove that the parity of each term of the sequence $ \left( \left\lfloor \left( \lfloor \sqrt q \rfloor +\sqrt{q} \right)^n \right\rfloor \right)_{n\ge 1} $ is opposite to the parity of its index, where $ q $ is a squarefree natural number.

2005 China Team Selection Test, 1

Tags: inequalities , algebra

Prove that for any $n$ ($n \geq 2$) pairwise distinct fractions in the interval $(0,1)$, the sum of their denominators is no less than $\frac{1}{3} n^{\frac{3}{2}}$.

2018 India PRMO, 30

Tags: polynomial , algebra

Let $P(x)$ = $a_0+a_1x+a_2x^2+\cdots +a_nx^n$ be a polynomial in which $a_i$ is non-negative integer for each $i \in$ {$0,1,2,3,....,n$} . If $P(1) = 4$ and $P(5) = 136$, what is the value of $P(3)$?

1999 National High School Mathematics League, 10

Tags: hyperbola , conic

$P$ is a point on hyperbola $\frac{x^2}{16}-\frac{y^2}{9}=1$, if the distance from $P$ to right directrix is the arithmetic mean of the distance from $P$ to two focal points, then the $x$-axis of $P$ is________.

2007 Peru Iberoamerican Team Selection Test, P1

Tags: algebra

Solve in the set of real numbers, the system: $$x(3y^2+1)=y(y^2+3)$$ $$y(3z^2+1)=z(z^2+3)$$ $$z(3x^2+1)=x(x^2+3)$$

1974 IMO, 1

Tags: combinatorics , game , algebra , system of equations

Three players $A,B$ and $C$ play a game with three cards and on each of these $3$ cards it is written a positive integer, all $3$ numbers are different. A game consists of shuffling the cards, giving each player a card and each player is attributed a number of points equal to the number written on the card and then they give the cards back. After a number $(\geq 2)$ of games we find out that A has $20$ points, $B$ has $10$ points and $C$ has $9$ points. We also know that in the last game B had the card with the biggest number. Who had in the first game the card with the second value (this means the middle card concerning its value).

2017 F = ma, 22

Tags: kinetic energy

22) A particle of mass m moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$. The fractional energy transfer is the absolute value of the final kinetic energy of $M$ divided by the initial kinetic energy of $m$. If the collision is perfectly elastic, under what condition will the fractional energy transfer between the two objects be a maximum? A) $\frac{m}{M} \ll 1$ B) $0.5 < \frac{m}{M} < 1$ C) $m = M$ D) $1 < \frac{m}{M} < 2$ E) $\frac{m}{M} \gg 1$

2016 Junior Balkan Team Selection Tests - Moldova, 6

Tags: diophantine , diophantine equation , number theory

Determine all pairs $(x, y)$ of natural numbers satisfying the equation $5^x=y^4+4y+1$.

2001 Slovenia National Olympiad, Problem 4

Tags: combinatorics

Cross-shaped tiles are to be placed on a $8\times8$ square grid without overlapping. Find the largest possible number of tiles that can be placed. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvMy8zL2EyY2Q4MDcyMWZjM2FmZGFhODkxYTk5ZmFiMmMwNzk0MzZmYmVjLnBuZw==&rn=U2NyZWVuIFNob3QgMjAyMS0wNC0wNyBhdCA2LjIzLjU4IEFNLnBuZw[/img]

2007 Iran MO (3rd Round), 1

Tags: rotation , geometric transformation , reflection , rectangle , geometry

Consider two polygons $ P$ and $ Q$. We want to cut $ P$ into some smaller polygons and put them together in such a way to obtain $ Q$. We can translate the pieces but we can not rotate them or reflect them. We call $ P,Q$ equivalent if and only if we can obtain $ Q$ from $ P$(which is obviously an equivalence relation). [img]http://i3.tinypic.com/4lrb43k.png[/img] a) Let $ P,Q$ be two rectangles with the same area(their sides are not necessarily parallel). Prove that $ P$ and $ Q$ are equivalent. b) Prove that if two triangles are not translation of each other, they are not equivalent. c) Find a necessary and sufficient condition for polygons $ P,Q$ to be equivalent.

1970 Regional Competition For Advanced Students, 1

Tags: inequalities

Let $x,y,z$ be positive real numbers such that $x+y+z=1$ Prove that always $\left( 1+\frac1x\right)\times\left(1+\frac1y\right)\times\left(1 +\frac1z\right)\ge 64$ When does equality hold?