Olympiad problems

1993 Kurschak Competition, 1

Let $a$ and $b$ be positive integers. Prove that the numbers $an^2+b$ and $a(n+1)^2+b$ are both perfect squares only for finitely many integers $n$.

2013 China Team Selection Test, 3

Tags: analytic geometry , geometry , combinatorics

A point $(x,y)$ is a [i]lattice point[/i] if $x,y\in\Bbb Z$. Let $E=\{(x,y):x,y\in\Bbb Z\}$. In the coordinate plane, $P$ and $Q$ are both sets of points in and on the boundary of a convex polygon with vertices on lattice points. Let $T=P\cap Q$. Prove that if $T\ne\emptyset$ and $T\cap E=\emptyset$, then $T$ is a non-degenerate convex quadrilateral region.

1980 Vietnam National Olympiad, 3

Tags: inequalities

Let be given an integer $n\ge 2$ and a positive real number $p$. Find the maximum of \[\displaystyle\sum_{i=1}^{n-1} x_ix_{i+1},\] where $x_i$ are non-negative real numbers with sum $p$.

Kharkiv City MO Seniors - geometry, 2021.10.5

Tags: geometry , collinear , incircle

The inscribed circle $\Omega$ of triangle $ABC$ touches the sides $AB$ and $AC$ at points $K$ and $ L$, respectively. The line $BL$ intersects the circle $\Omega$ for the second time at the point $M$. The circle $\omega$ passes through the point $M$ and is tangent to the lines $AB$ and $BC$ at the points $P$ and $Q$, respectively. Let $N$ be the second intersection point of circles $\omega$ and $\Omega$, which is different from $M$. Prove that if $KM \parallel AC$ then the points $P, N$ and $L$ lie on one line.

2022 BMT, 27

Tags: combinatorics

Submit a positive integer $n$ less than $10^5$. Let the sum of the valid submissions from all teams to this question be $S$. If you submit an invalid answer, you will receive $0$ points. Otherwise, your score will be $ \max \left(0,\lfloor 25 - \frac{|S'-n|}{10} \rfloor \right)$ , where $S'$ is the sum of the squares of the digits of $S$.

2020 LMT Fall, B15

Tags: geometry

Let $\vartriangle AMO$ be an equilateral triangle. Let $U$ and $G$ lie on side $AM$, and let $S$ and $N$ lie on side $AO$ such that $AU =UG = GM$ and $AS = SN = NO$. Find the value of $\frac{[MONG]}{[U S A]}$

1966 AMC 12/AHSME, 17

Tags: ellipse , conic

The number of distinct points common to the curves $x^2+4y^2=1$ and $4x^2+y^2=5$ is: $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ 3 \qquad \text{(E)} \ 4$

2015 Iran Team Selection Test, 5

Tags: number theory , combinatorics

We call a permutation $(a_1, a_2,\cdots , a_n)$ of the set $\{ 1,2,\cdots, n\}$ "good" if for any three natural numbers $i <j <k$, $n\nmid a_i+a_k-2a_j$ find all natural numbers $n\ge 3$ such that there exist a "good" permutation of a set $\{1,2,\cdots, n\}$.

2019 Purple Comet Problems, 18

Tags: system of equations , algebra

Suppose that $a, b, c$, and $d$ are real numbers simultaneously satisfying $a + b - c - d = 3$ $ab - 3bc + cd - 3da = 4$ $3ab - bc + 3cd - da = 5$ Find $11(a - c)^2 + 17(b -d)^2$.

2022 AMC 12/AHSME, 23

Tags:

Let $x_{0}$, $x_{1}$, $x_{2}$, $\cdots$ be a sequence of numbers, where each $x_{k}$ is either $0$ or $1$. For each positive integer $n$, define \[S_{n} = \displaystyle\sum^{n-1}_{k=0}{x_{k}2^{k}}\] Suppose $7S_{n} \equiv 1\pmod {2^{n}}$ for all $n\geq 1$. What is the value of the sum \[x_{2019}+2x_{2020}+4x_{2021}+8x_{2022}?\] $ \textbf{(A)}\ 6 \qquad \textbf{(B)}\ 7 \qquad \textbf{(C)}\ 12 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

2023 Middle European Mathematical Olympiad, 4

Tags: number theory

Let $n, m$ be positive integers. A set $S$ of positive integers is called $(n, m)$-good, if: (1) $m \in S$; (2) for all $a\in S$, all divisors of $a$ are also in $S$; (3) for all distinct $a, b \in S$, $a^n+b^n \in S$. For which $(n, m)$, the only $(n, m)$-good set is $\mathbb{N}$?

2005 Tournament of Towns, 1

Tags:

A palindrome is a positive integer which reads in the same way in both directions (for example, $1$, $343$ and $2002$ are palindromes, while $2005$ is not). Is it possible to find $2005$ pairs in the form of $(n, n + 110)$ where both numbers are palindromes? [i](3 points)[/i]

2010 Finnish National High School Mathematics Competition, 5

Tags: geometry

Let $S$ be a non-empty subset of a plane. We say that the point $P$ can be seen from $A$ if every point from the line segment $AP$ belongs to $S$. Further, the set $S$ can be seen from $A$ if every point of $S$ can be seen from $A$. Suppose that $S$ can be seen from $A$, $B$ and $C$ where $ABC$ is a triangle. Prove that $S$ can also be seen from any other point of the triangle $ABC$.

1992 Brazil National Olympiad, 1

Tags: calculus , derivative , analytic geometry , graphing lines , slope , algebra

The equation $x^3+px+q=0$ has three distinct real roots. Show that $p<0$

2010 China National Olympiad, 3

Tags: inequalities , trigonometry , calculus , function , complex number , algebra

Given complex numbers $a,b,c$, we have that $|az^2 + bz +c| \leq 1$ holds true for any complex number $z, |z| \leq 1$. Find the maximum value of $|bc|$.

2021 Iran Team Selection Test, 2

Tags: function , number theory

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have : $$f(n)+1400m^2|n^2+f(f(m))$$

2014 Czech-Polish-Slovak Junior Match, 2

Tags: equation , algebra

Solve the equation $a + b + 4 = 4\sqrt{a\sqrt{b}}$ in real numbers

2023/2024 Tournament of Towns, 1

Tags: polymonial , algebra

For every polynomial of degree 45 with coefficients $1,2,3, \ldots, 46$ (in some order) Tom has listed all its distinct real roots. Then he increased each number in the list by 1 . What is now greater: the amount of positive numbers or the amount of negative numbers? Alexey Glebov

2020 MBMT, 20

Tags: geometry , geometric transformation , rotation

Sam colors each tile in a 4 by 4 grid white or black. A coloring is called [i]rotationally symmetric[/i] if the grid can be rotated 90, 180, or 270 degrees to achieve the same pattern. Two colorings are called [i]rotationally distinct[/i] if neither can be rotated to match the other. How many rotationally distinct ways are there for Sam to color the grid such that the colorings are [i]not[/i] rotationally symmetric? [i]Proposed by Gabriel Wu[/i]

2016 Indonesia TST, 3

Tags: incenter , circumcircle , geometric transformation , homothety , power of a point , geometry

Circles $\Omega $ and $\omega $ are tangent at a point $P$ ($\omega $ lies inside $\Omega $). A chord $AB$ of $\Omega $ is tangent to $\omega $ at $C;$ the line $PC$ meets $\Omega $ again at $Q.$ Chords $QR$ and $QS$ of $ \Omega $ are tangent to $\omega .$ Let $I,X,$ and $Y$ be the incenters of the triangles $APB,$ $ARB,$ and $ASB,$ respectively. Prove that $\angle PXI+\angle PYI=90^{\circ }.$

1994 Austrian-Polish Competition, 6

Tags: system of equations , algebra

Let $n > 1$ be an odd positive integer. Assume that positive integers $x_1, x_2,..., x_n \ge 0$ satisfy: $$\begin{cases} (x_2 - x_1)^2 + 2(x_2 +x_1) + 1 = n^2 \\ (x_3 -x_2)^2 + 2(x_3 +x_2) + 1 = n^2 \\ ...\\ (x_1 - x_n)^2 + 2(x_1 + x_n)+ 1 = n^2 \end {cases}$$ Show that there exists $j, 1 \le j \le n$, such that $x_j = x_{j+1}$. Here $x_{n+1} = x_1$.

2024 Korea Summer Program Practice Test, 3

Tags: combinatorics

Find all pairs of positive integers $n$ such that one can partition a $n\times (n+1)$ board with $1\times 2$ or $2\times 1$ dominoes and draw one of the diagonals on each of the dominos so that none of the diagonals share endpoints.

2010 Romania National Olympiad, 3

Tags: floor function , algebra

For any integer $n\ge 2$ denote by $A_n$ the set of solutions of the equation \[x=\left\lfloor\frac{x}{2}\right\rfloor+\left\lfloor\frac{x}{3}\right\rfloor+\cdots+\left\lfloor\frac{x}{n}\right\rfloor .\] a) Determine the set $A_2\cup A_3$. b) Prove that the set $A=\bigcup_{n\ge 2}A_n$ is finite and find $\max A$. [i]Dan Nedeianu & Mihai Baluna[/i]

2010 Princeton University Math Competition, 5

Tags:

We say that a rook is "attacking" another rook on a chessboard if the two rooks are in the same row or column of the chessboard and there is no piece directly between them. Let $n$ be the maximum number of rooks that can be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other. How many ways can $n$ rooks be placed on a $6\times 6$ chessboard such that each rook is attacking at most one other?

2019 India IMO Training Camp, P3

Tags: function , number theory

Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.