Olympiad problems

1953 Moscow Mathematical Olympiad, 258

A knight stands on an infinite chess board. Find all places it can reach in exactly $2n$ moves.

1997 Tournament Of Towns, (558) 3

Tags: number theory , diophantine , diophantine equation

Prove that the equation $$xy(x -y) + yz(y-z) + zx(z-x) = 6$$ has infinitely many solutions in integers $x, y$ and $z$. (N Vassiliev)

2005 iTest, 36

Tags: linear algebra , matrix

Find the determinant of this matrix: $\begin{bmatrix} 2 & 2 & 2 & 2 & 2 & 2 \\ 4 & 2 & 2 & 2 & 2 & 2 \\ 4 & 4 & 2 & 2 & 2 & 2 \\ 4 & 4 & 4 & 2 & 2 & 2 \\ 4 & 4 & 4 & 4 & 2 & 2 \\ 4 & 4 & 4& 4 & 4 & 2 \end{bmatrix} $

2008 Balkan MO Shortlist, G8

Tags: geometric inequality , geometry , max , distance

Let $P$ be a point in the interior of a triangle $ABC$ and let $d_a,d_b,d_c$ be its distances to $BC,CA,AB$ respectively. Prove that max $(AP, BP, CP) \ge \sqrt{d_a^2+d_b^2+d_c^2}$

2022 JBMO Shortlist, A4

Tags: inequality , shortlist , algebra

Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$ Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

2020 Ukrainian Geometry Olympiad - April, 1

Tags: geometry , isosceles , angle bisector , perpendicular

In triangle $ABC$, bisectors are drawn $AA_1$ and $CC_1$. Prove that if the length of the perpendiculars drawn from the vertex $B$ on lines $AA1$ and $CC_1$ are equal, then $\vartriangle ABC$ is isosceles.

2009 Purple Comet Problems, 8

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Find the least positive integer that has exactly $20$ positive integer divisors.

2024 Mongolian Mathematical Olympiad, 3

Tags: combinatorics , number theory , complete residue system

A set $X$ consisting of $n$ positive integers is called $\textit{good}$ if the following condition holds: For any two different subsets of $X$, say $A$ and $B$, the number $s(A) - s(B)$ is not divisible by $2^n$. (Here, for a set $A$, $s(A)$ denotes the sum of the elements of $A$) Given $n$, find the number of good sets of size $n$, all of whose elements is strictly less than $2^n$.

2005 Bundeswettbewerb Mathematik, 4

Tags: invariant , induction , combinatorics

Prove that each finite set of integers can be arranged without intersection.

1978 IMO Longlists, 2

Tags: algebra , generating functions

If \[f(x) = (x + 2x^2 +\cdots+ nx^n)^2 = a_2x^2 + a_3x^3 + \cdots+ a_{2n}x^{2n},\] prove that \[a_{n+1} + a_{n+2} + \cdots + a_{2n} =\dbinom{n + 1}{2}\frac{5n^2 + 5n + 2}{12}\]

2023 Harvard-MIT Mathematics Tournament, 5

Tags: guts

If $a$ and $b$ are positive real numbers such that $a \cdot 2^b=8$ and $a^b=2,$ compute $a^{\log_2 a} 2^{b^2}.$

2015 India IMO Training Camp, 2

Tags: combinatorics

A $10$-digit number is called a $\textit{cute}$ number if its digits belong to the set $\{1,2,3\}$ and the difference of every pair of consecutive digits is $1$. a) Find the total number of cute numbers. b) Prove that the sum of all cute numbers is divisibel by $1408$.

1977 Putnam, A3

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Let $u,f$ and $g$ be functions, defined for all real numbers $x$, such that $$\frac{u(x+1)+u(x-1)}{2}=f(x) \text{ and } \frac{u(x+4)+u(x-4)}{2}=g(x).$$ Determine $u(x)$ in terms of $f$ and $g$.

2021-2022 OMMC, 3

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Parabolas $P_1, P_2$ share a focus at $(20,22)$ and their directrices are the $x$ and $y$ axes respectively. They intersect at two points $X,Y.$ Find $XY^2.$ [i]Proposed by Evan Chang[/i]

PEN A Problems, 46

Tags: divisibility theory

Let $a$ and $b$ be integers. Show that $a$ and $b$ have the same parity if and only if there exist integers $c$ and $d$ such that $a^2 +b^2 +c^2 +1 = d^2$.

1989 AMC 12/AHSME, 3

Tags: geometry , rectangle , perimeter

A square is cut into three rectangles along two lines parallel to a side, as shown. If the perimeter of each of the three rectangles is 24, then the area of the original square is [asy] draw((0,0)--(9,0)--(9,9)--(0,9)--cycle); draw((3,0)--(3,9), dashed); draw((6,0)--(6,9), dashed);[/asy] $\text{(A)} \ 24 \qquad \text{(B)} \ 36 \qquad \text{(C)} \ 64 \qquad \text{(D)} \ 81 \qquad \text{(E)} \ 96$

1981 Austrian-Polish Competition, 4

Tags: combinatorics

Let $n \ge 3$ cells be arranged into a circle. Each cell can be occupied by $0$ or $1$. The following operation is admissible: Choose any cell $C$ occupied by a $1$, change it into a $0$ and simultaneously reverse the entries in the two cells adjacent to $C$ (so that $x,y$ become $1 - x$, $1 - y$). Initially, there is a $1$ in one cell and zeros elsewhere. For which values of $n$ is it possible to obtain zeros in all cells in a finite number of admissible steps?

2016 China Team Selection Test, 5

Tags: number theory , algebra

Does there exist two infinite positive integer sets $S,T$, such that any positive integer $n$ can be uniquely expressed in the form $$n=s_1t_1+s_2t_2+\ldots+s_kt_k$$ ,where $k$ is a positive integer dependent on $n$, $s_1<\ldots<s_k$ are elements of $S$, $t_1,\ldots, t_k$ are elements of $T$?

1984 All Soviet Union Mathematical Olympiad, 378

Tags: incircle , concurrency , geometry

The circle with the centre $O$ is inscribed in the triangle $ABC$ . The circumference touches its sides $[BC], [CA], [AB]$ in $A_1, B_1, C_1$ points respectively. The $[AO], [BO], [CO]$ segments cross the circumference in $A_2, B_2, C_2$ points respectively. Prove that lines $(A_1A_2),(B_1B_2)$ and $(C_1C_2)$ intersect in one point.

2000 AMC 10, 18

Tags: geometry , rectangle

Charlyn walks completely around the boundary of a square whose sides are each $5$ km long. From any point on her path she can see exactly $1$ km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number? $\text{(A)}\ 24 \qquad\text{(B)}\ 27\qquad\text{(C)}\ 39\qquad\text{(D)}\ 40 \qquad\text{(E)}\ 42$

1998 Putnam, 3

Tags: function , integration

Let $f$ be a real function on the real line with continuous third derivative. Prove that there exists a point $a$ such that \[f(a)\cdot f^\prime(a)\cdot f^{\prime\prime}(a)\cdot f^{\prime\prime\prime}(a)\geq 0.\]

2023 Azerbaijan BMO TST, 3

Tags: algebra , functional equation

Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and \[f(f(x)) + f(f(y)) = f(x + y)f(xy),\] for all $x, y \in\mathbb{R}$.

2007 Pan African, 1

Tags: system of equations , algebra

Solve the following system of equations for real $x,y$ and $z$: \begin{eqnarray*} x &=& \sqrt{2y+3}\\ y &=& \sqrt{2z+3}\\ z &=& \sqrt{2x+3}. \end{eqnarray*}

2020 Azerbaijan IMO TST, 1

Tags: combinatorics

A finite number of stones are [i]good[/i] when the weight of each of these stones is less than the total weight of the rest. It is known that arbitrary $n-1$ of the given $n$ stones is [i]good[/i]. Prove that it is possible to choose a [i]good[/i] triple from these stones.

2017 ASDAN Math Tournament, 7

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Alice and Bob play a game where on each turn, Alice rolls a die and Bob flips a coin. Bob wins the game if he flips $3$ heads before Alice rolls a $6$. What is the probability that Bob wins? Note that Bob does not in if he flips his third head the same turn Alice rolls her first $6$.