Olympiad problems

2019 Peru IMO TST, 1

In each cell of a chessboard with $2$ rows and $2019$ columns a real number is written so that: [LIST] [*] There are no two numbers written in the first row that are equal to each other.[/*] [*] The numbers written in the second row coincide with (in some another order) the numbers written in the first row.[/*] [*] The two numbers written in each column are different and they add up to a rational number.[/*] [/LIST] Determine the maximum quantity of irrational numbers that can be in the chessboard.

PEN I Problems, 18

Tags: floor function

Do there exist irrational numbers $a, b>1$ and $\lfloor a^{m}\rfloor \not=\lfloor b^{n}\rfloor $ for any positive integers $m$ and $n$?

2000 AIME Problems, 12

Tags: geometry , 3D geometry , sphere , circumcircle , trigonometry , tetrahedron , number theory

The points $A, B$ and $C$ lie on the surface of a sphere with center $O$ and radius 20. It is given that $AB=13, BC=14, CA=15,$ and that the distance from $O$ to triangle $ABC$ is $\frac{m\sqrt{n}}k,$ where $m, n,$ and $k$ are positive integers, $m$ and $k$ are relatively prime, and $n$ is not divisible by the square of any prime. Find $m+n+k.$

2011 Indonesia TST, 4

Tags: modular arithmetic , induction , number theory unsolved , number theory

A prime number $p$ is a [b]moderate[/b] number if for every $2$ positive integers $k > 1$ and $m$, there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \] If $q$ is the smallest [b]moderate[/b] number, then determine the smallest prime $r$ which is not moderate and $q < r$.

1966 IMO Shortlist, 7

Tags: geometry , 3D geometry , IMO Shortlist , IMO Longlist

For which arrangements of two infinite circular cylinders does their intersection lie in a plane?

2003 Austrian-Polish Competition, 4

Tags: divides , divisible , Product , number theory

A positive integer $m$ is alpine if $m$ divides $2^{2n+1} + 1$ for some positive integer $n$. Show that the product of two alpine numbers is alpine.

2002 National High School Mathematics League, 9

Tags: geometry , 3D geometry , pyramid

Points $P_1,P_2,P_3,P_4$ are vertexes of a regular triangular pyramid, and $P_5,P_6,P_7,P_8,P_9,P_{10}$ midpoints of edges. The number of groups $(P_1,P_i,P_j,P_k)(1<i<j<k\leq10)$ that $P_1,P_i,P_j,P_k$ are coplane is________.

2024 LMT Fall, 21

Tags: guts

Let $ABC$ be a triangle such that $AB=2$, $BC=3$, and $AC=4$. A circle passing through $A$ intersects $AB$ at $D$, $AC$ at $E$, and $BC$ at $M$ and $N$ such that $BM=MN=NC$. Find $DE$.

2001 Tuymaada Olympiad, 6

Tags: geometry , geometric inequality , isosceles , Isosceles Triangle

On the side $AB$ of an isosceles triangle $AB$ ($AC=BC$) lie points $P$ and $Q$ such that $\angle PCQ \le \frac{1}{2} \angle ACB$. Prove that $PQ \le \frac{1}{2} AB$.

2010 Princeton University Math Competition, 8

Tags: modular arithmetic

Let $N$ be the number of (positive) divisors of $2010^{2010}$ ending in the digit $2$. What is the remainder when $N$ is divided by 2010?

2013 Kazakhstan National Olympiad, 1

Tags: number theory unsolved , number theory , Kazakhstan

Find all triples of positive integer $(m,n,k)$ such that $ k^m|m^n-1$ and $ k^n|n^m-1$

2002 Romania National Olympiad, 1

Tags: conics , hyperbola , analytic geometry , geometry proposed , geometry

In the Cartesian plane consider the hyperbola \[\Gamma=\{M(x,y)\in\mathbb{R}^2 \vert \frac{x^2}{4}-y^2=1\} \] and a conic $\Gamma '$, disjoint from $\Gamma$. Let $n(\Gamma ,\Gamma ')$ be the maximal number of pairs of points $(A,A')\in\Gamma\times\Gamma '$ such that $AA'\le BB'$, for any $(B,B')$ For each $p\in\{0,1,2,4\}$, find the equation of $\Gamma'$ for which $n(\Gamma ,\Gamma ')=p$. Justify the answer.

1988 Putnam, B3

Tags: Putnam , real analysis

For every $n$ in the set $\mathrm{N} = \{1,2,\dots \}$ of positive integers, let $r_n$ be the minimum value of $|c-d\sqrt{3}|$ for all nonnegative integers $c$ and $d$ with $c+d=n$. Find, with proof, the smallest positive real number $g$ with $r_n \leq g$ for all $n \in \mathbb{N}$.

2023 IFYM, Sozopol, 3

Tags: geometry

A positive real number $k$, a triangle $ABC$ with circumcircle $\omega$, and a point $M$ on its side $AB$ are fixed. The point $P$ moves along $\omega$, and $Q$ on segment $CP$ is such that $CQ : QP = k$. The line through $P$, parallel to $CM$, intersects the line $MQ$ at point $N$. Prove that $N$ lies on a constant circle, independent of the choice of $P$.

1971 Putnam, B1

Tags: college contests

Let $S$ be a set and let $\circ$ be a binary operation on $S$ satisfying two laws $$x\circ x=x \text{ for all } x \text{ in } S, \text{ and}$$ $$(x \circ y) \circ z= (y\circ z) \circ x \text{ for all } x,y,z \text{ in } S.$$ Show that $\circ$ is associative and commutative.

1995 AMC 12/AHSME, 11

Tags:

How many base 10 four-digit numbers, $N = \underline{a} \underline{b} \underline{c} \underline{d}$, satisfy all three of the following conditions? (i) $4,000 \leq N < 6,000;$ (ii) $N$ is a multiple of 5; (iii) $3 \leq b < c \leq 6$. $ \mathbf{(A)}\; 10\qquad \mathbf{(B)}\; 18\qquad \mathbf{(C)}\; 24\qquad \mathbf{(D)}\; 36\qquad \mathbf{(E)}\; 48$

2004 India IMO Training Camp, 3

Tags: trigonometry , function , algebra , functional equation , system of equations , algebra unsolved

Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that \[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.

1984 All Soviet Union Mathematical Olympiad, 394

Tags: combinatorial geometry , geometry , area , 3D geometry , cube

Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.

2006 Tuymaada Olympiad, 4

Tags: function , induction , algebra unsolved , algebra

Find all functions $f: (0,\infty)\rightarrow(0,\infty)$ with the following properties: $f(x+1)=f(x)+1$ and $f\left(\frac{1}{f(x)}\right)=\frac{1}{x}$. [i]Proposed by P. Volkmann[/i]

2023 District Olympiad, P1

Tags: functional equation , continuity , algebra

Determine all continuous functions $f:\mathbb{R}\to\mathbb{R}$ for which $f(1)=e$ and \[f(x+y)=e^{3xy}\cdot f(x)f(y),\]for all real numbers $x{}$ and $y{}$.

1994 Greece National Olympiad, 2

Tags: algebra , polynomial

Fow which real values of $m$ does the polynomial $x^3+1995x^2-1994x+m$ have all three roots integers?

2007 Germany Team Selection Test, 3

Tags: number theory , IMO Shortlist

Find all integer solutions of the equation \[\frac {x^{7} \minus{} 1}{x \minus{} 1} \equal{} y^{5} \minus{} 1.\]

1986 Tournament Of Towns, (121) 3

Tags: combinatorics , game , game strategy , combinatorial geometry

A game has two players. In the game there is a rectangular chocolate bar, with $60$ pieces, arranged in a $6 \times 1 0$ formation , which can be broken only along the lines dividing the pieces. The first player breaks the bar along one line, discarding one section . The second player then breaks the remaining section, discarding one section. The first player repeats this process with the remaining section , and so on. The game is won by the player who leaves a single piece. In a perfect game which player wins? {S. Fomin , Leningrad)

2011 VTRMC, Problem 2

Tags: recurrence relation , algebra , Sequences

A sequence $(a_n)$ is defined by $a_0=-1,a_1=0$, and $a_{n+1}=a_n^2-(n+1)^2a_{n-1}-1$ for all positive integers $n$. Find $a_{100}$.

1989 Kurschak Competition, 2

Tags: number theory unsolved , number theory

For any positive integer $n$ denote $S(n)$ the digital sum of $n$ when represented in the decimal system. Find every positive integer $M$ for which $S(Mk)=S(M)$ holds for all integers $1\le k\le M$.