Olympiad problems

1991 Hungary-Israel Binational, 4

Find all the real values of $ \lambda$ for which the system of equations $ x\plus{}y\plus{}z\plus{}v\equal{}0$ and $ \left(xy\plus{}yz\plus{}zv\right)\plus{}\lambda\left(xz\plus{}xv\plus{}yv\right)\equal{}0$, has a unique real solution.

Estonia Open Senior - geometry, 2018.1.1

Tags: geometry , lattice points , equilateral

Is there an equilateral triangle in the coordinate plane, both coordinates of each vertex of which are integers?

2001 Mexico National Olympiad, 3

Tags: geometry , cyclic quadrilateral , equal segments

$ABCD$ is a cyclic quadrilateral. $M$ is the midpoint of $CD$. The diagonals meet at $P$. The circle through $P$ which touches $CD$ at $M$ meets $AC$ again at $R$ and $BD$ again at $Q$. The point $S$ on $BD$ is such that $BS = DQ$. The line through $S$ parallel to $AB$ meets $AC$ at $T$. Show that $AT = RC$.

2019 Durer Math Competition Finals, 1

Tags: algebra , sequence , sum , inequalities

Let $a_o,a_1,a_2,..,a_ n$ be a non-decreasing sequence of $n+1$ real numbers where $a_0 = 0$ and for every $j > i $ we have $a_j - a_i \le j - i$. Show that $$\left (\sum_{i=0}^n a_i \right )^2 \ge \sum_{i=0}^n a_i^3$$

1967 IMO Shortlist, 6

Tags: inequality , polynomial , algebra , n-variable inequality

Prove the following inequality: \[\prod^k_{i=1} x_i \cdot \sum^k_{i=1} x^{n-1}_i \leq \sum^k_{i=1} x^{n+k-1}_i,\] where $x_i > 0,$ $k \in \mathbb{N}, n \in \mathbb{N}.$

2020 Turkey Junior National Olympiad, 1

Tags: quadratic equation , number theory

Determine all real number $(x,y)$ pairs that satisfy the equation. $$2x^2+y^2+7=2(x+1)(y+1)$$

1985 IMO Longlists, 46

Tags: analytic geometry , geometry

Let $C$ be the curve determined by the equation $y = x^3$ in the rectangular coordinate system. Let $t$ be the tangent to $C$ at a point $P$ of $C$; t intersects $C$ at another point $Q$. Find the equation of the set $L$ of the midpoints $M$ of $PQ$ as $P$ describes $C$. Is the correspondence associating $P$ and $M$ a bijection of $C$ on $L$ ? Find a similarity that transforms $C$ into $L.$

2007 Kyiv Mathematical Festival, 4

Tags: geometric transformation , reflection , trapezoid , geometry

The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$

2020 China Northern MO, BP5

Tags: number theory , relatively prime , combinatorics , set

It is known that subsets $A_1,A_2, \cdots , A_n$ of set $I=\{1,2,\cdots ,101\}$ satisfy the following condition $$\text{For any } i,j \text{ } (1 \leq i < j \leq n) \text{, there exists } a,b \in A_i \cap A_j \text{ so that } (a,b)=1$$ Determine the maximum positive integer $n$. *$(a,b)$ means $\gcd (a,b)$

2022 JHMT HS, 4

Tags: combinatorics

For a nonempty set $A$ of integers, let $\mathrm{range} \, A=\max A-\min A$. Find the number of subsets $S$ of \[ \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \] such that $\mathrm{range} \, S$ is an element of $S$.

2022 Bangladesh Mathematical Olympiad, 8

Tags: number theory

Solve the following problems - A) Find any $158$ consecutive integers such that the sum of digits for any of the numbers is not divisible by $17.$ B) Prove that, among any $159$ consecutive integers there will always be at least one integer whose sum of digits is divisible by $17.$

2006 Pre-Preparation Course Examination, 4

Tags: abstract algebra , algebra

If $d\in \mathbb{Q}$, is there always an $\omega \in \mathbb{C}$ such that $\omega ^n=1$ for some $n\in \mathbb{N}$ and $\mathbb{Q}(\sqrt{d})\subseteq \mathbb{Q}(\omega)$?

2012 Portugal MO, 1

Tags: floor function , number theory

Find the number of positive integers $n$ such that $1\leq n\leq 1000$ and $n$ is divisible by $\lfloor \sqrt[3]{n} \rfloor$.

2020 Ukraine Team Selection Test, 2

Tags: algebra

Let $n\geqslant 2$ be a positive integer and $a_1,a_2, \ldots ,a_n$ be real numbers such that \[a_1+a_2+\dots+a_n=0.\] Define the set $A$ by \[A=\left\{(i, j)\,|\,1 \leqslant i<j \leqslant n,\left|a_{i}-a_{j}\right| \geqslant 1\right\}\] Prove that, if $A$ is not empty, then \[\sum_{(i, j) \in A} a_{i} a_{j}<0.\]

2025 USA IMO Team Selection Test, 6

Tags: number theory

Prove that there exists a real number $\varepsilon>0$ such that there are infinitely many sequences of integers $0<a_1<a_2<\hdots<a_{2025}$ satisfying \[\gcd(a_1^2+1, a_2^2+1,\hdots, a_{2025}^2+1) > a_{2025}^{1+\varepsilon}.\] [i]Pitchayut Saengrungkongka[/i]

2020 USMCA, 6

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Alex is thinking of a number that is divisible by all of the positive integers 1 through 200 inclusive except for two consecutive numbers. What is the smaller of these numbers?

2000 Harvard-MIT Mathematics Tournament, 27

Tags:

What is the smallest number that can be written as a sum of $2$ squares in $3$ ways?

1996 Tournament Of Towns, (517) 4

Tags: combinatorics

For what integers $n > 1$ can it happen that in a group of $n +1$ girls and $n$ boys, all the girls know a different number of boys while all the boys know the same number of girls? (NB Vassiliev)

2021 Alibaba Global Math Competition, 15

Tags: calculus , integration , topology , analysis , manifold , differential geometry

Let $(M,g)$ be an $n$-dimensional complete Riemannian manifold with $n \ge 2$. Suppose $M$ is connected and $\text{Ric} \ge (n-1)g$, where $\text{Ric}$ is the Ricci tensor of $(M,g)$. Denote by $\text{d}g$ the Riemannian measure of $(M,g)$ and by $d(x,y)$ the geodesic distance between $x$ and $y$. Prove that \[\int_{M \times M} \cos d(x,y) \text{d}g(x)\text{d}g(y) \ge 0.\] Moreover, equality holds if and only if $(M,g)$ is isometric to the unit round sphere $S^n$.

PEN J Problems, 13

Tags: divisor function

Determine all positive integers $k$ such that \[\frac{d(n^{2})}{d(n)}= k\] for some $n \in \mathbb{N}$.

2019 Junior Balkan MO, 1

Tags: number theory , prime number

Find all prime numbers $p$ for which there exist positive integers $x$, $y$, and $z$ such that the number $x^p + y^p + z^p - x - y - z$ is a product of exactly three distinct prime numbers.

Novosibirsk Oral Geo Oly VIII, 2020.5

Tags: perpendicular bisector , midpoint , geometry

Line $\ell$ is perpendicular to one of the medians of the triangle. The median perpendiculars to the sides of this triangle intersect the line $\ell$ at three points. Prove that one of them is the midpoint of the segment formed by the other two.

2010 LMT, 35

Tags:

Consider a set of $6$ fixed points in the plane, with no three collinear. Between some pairs of these points, we may draw one arrow from one point to the other. How many possible configurations of arrows are there such that if there is an arrow from point $A$ to point $B$ and an arrow from $B$ to $C,$ then there is an arrow from $A$ to $C?$ Your score will be $16-\frac{1}{800}|\textbf{Your Answer}-\textbf{Actual Answer}|$ rounded to the nearest integer or zero, whichever is higher.

1969 IMO Longlists, 27

Tags: geometry , 3d geometry , ratio , sphere

$(GBR 4)$ The segment $AB$ perpendicularly bisects $CD$ at $X$. Show that, subject to restrictions, there is a right circular cone whose axis passes through $X$ and on whose surface lie the points $A,B,C,D.$ What are the restrictions?

1991 Putnam, B3

Tags: geometry , rectangle , combinatorics

Can we find $N$ such that all $m\times n$ rectangles with $m,n>N$ can be tiled with $4\times6$ and $5\times7$ rectangles?