Olympiad problems

2024 All-Russian Olympiad, 8

Tags: combinatorics

$1000$ children, no two of the same height, lined up. Let us call a pair of different children $(a,b)$ good if between them there is no child whose height is greater than the height of one of $a$ and $b$, but less than the height of the other. What is the greatest number of good pairs that could be formed? (Here, $(a,b)$ and $(b,a)$ are considered the same pair.) [i]Proposed by I. Bogdanov[/i]

2006 Finnish National High School Mathematics Competition, 5

Tags: combinatorics

The game of Nelipe is played on a $16\times16$-grid as follows: The two players write in turn numbers $1, 2,..., 16$ in different squares. The numbers on each row, column, and in every one of the 16 smaller squares have to be different. The loser is the one who is not able to write a number. Which one of the players wins, if both play with an optimal strategy?

1971 IMO Longlists, 30

Tags: parameterization , system of equations , algebra

Prove that the system of equations \[2yz+x-y-z=a,\\ 2xz-x+y-z=a,\\ 2xy-x-y+z=a, \] $a$ being a parameter, cannot have five distinct solutions. For what values of $a$ does this system have four distinct integer solutions?

2008 ITest, 84

Tags: algebra , polynomial

Let $S$ be the sum of all integers $b$ for which the polynomial $x^2+bx+2008b$ can be factored over the integers. Compute $|S|$.

2020 BMT Fall, 8

Tags: algebra , inequalities

Compute the smallest value $C$ such that the inequality $$x^2(1+y)+y^2(1+x)\le \sqrt{(x^4+4)(y^4+4)}+C$$ holds for all real $x$ and $y$.

2012 Tournament of Towns, 5

Tags: reflection , angle bisector , congruent triangles , incircle , tangent , geometry

Let $\ell$ be a tangent to the incircle of triangle $ABC$. Let $\ell_a,\ell_b$ and $\ell_c$ be the respective images of $\ell$ under reflection across the exterior bisector of $\angle A,\angle B$ and $\angle C$. Prove that the triangle formed by these lines is congruent to $ABC$.

2024 HMNT, 13

Tags: guts

Let $f$ and $g$ be two quadratic polynomials with real coefficients such that the equation $f(g(x)) = 0$ has four distinct real solutions: $112, 131, 146,$ and $a.$ Compute the sum of all possible values of $a.$

2018 AMC 10, 4

Tags:

How many ways can a student schedule 3 mathematics courses -- algebra, geometry, and number theory -- in a 6-period day if no two mathematics courses can be taken in consecutive periods? (What courses the student takes during the other 3 periods is of no concern here.) $\textbf{(A) }3\qquad\textbf{(B) }6\qquad\textbf{(C) }12\qquad\textbf{(D) }18\qquad\textbf{(E) }24$

2010 Harvard-MIT Mathematics Tournament, 3

Tags:

Let $S_0=0$ and let $S_k$ equal $a_1+2a_2+\ldots+ka_k$ for $k\geq 1$. Define $a_i$ to be $1$ if $S_{i-1}<i$ and $-1$ if $S_{i-1}\geq i$. What is the largest $k\leq 2010$ such that $S_k=0$?

2018 Stanford Mathematics Tournament, 7

Tags: geometry , 3d geometry

Two equilateral triangles $ABC$ and $DEF$, each with side length $1$, are drawn in $2$ parallel planes such that when one plane is projected onto the other, the vertices of the triangles form a regular hexagon $AF BDCE$. Line segments $AE$, $AF$, $BF$, $BD$, $CD,$ and $CE$ are drawn, and suppose that each of these segments also has length $1$. Compute the volume of the resulting solid that is formed.

2023 Romania National Olympiad, 1

Tags: number theory , divisibility

The non-zero natural number n is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223- \frac{3}{2} \cdot n$. Find the quotient of the division.

2005 Junior Balkan Team Selection Tests - Romania, 18

Tags: algebra

Consider two distinct positive integers $a$ and $b$ having integer arithmetic, geometric and harmonic means. Find the minimal value of $|a-b|$. [i]Mircea Fianu[/i]

2023 VN Math Olympiad For High School Students, Problem 4

Tags: algebra , polynomial

Prove that: a polynomial is irreducible in $\mathbb{Z}[x]$ if and only if it is irreducible in $\mathbb{Q}[x].$

2008 Mexico National Olympiad, 2

Tags: geometry , 3d geometry , greatest common divisor , number theory

We place $8$ distinct integers in the vertices of a cube and then write the greatest common divisor of each pair of adjacent vertices on the edge connecting them. Let $E$ be the sum of the numbers on the edges and $V$ the sum of the numbers on the vertices. a) Prove that $\frac23E\le V$. b) Can $E=V$?

2021-IMOC qualification, A3

Tags: function , functional inequality , functional , inequalities , algebra

Find all injective function $f: N \to N$ satisfying that for all positive integers $m,n$, we have: $f(n(f(m)) \le nm$

2006 Harvard-MIT Mathematics Tournament, 1

Tags: geometry , perimeter

Octagon $ABCDEFGH$ is equiangular. Given that $AB=1$, $BC=2$, $CD=3$, $DE=4$, and $EF=FG=2$, compute the perimeter of the octagon.

2018 HMNT, 1

Tags: geometry

Square $CASH$ and regular pentagon $MONEY$ are both inscribed in a circle. Given that they do not share a vertex, how many intersections do these two polygons have?

2002 IMC, 7

Tags: linear algebra , matrix

Compute the determinant of the $n\times n$ matrix $A=(a_{ij})_{ij}$, $$a_{ij}=\begin{cases} (-1)^{|i-j|} & \text{if}\, i\ne j,\\ 2 & \text{if}\, i= j. \end{cases}$$

2006 China Second Round Olympiad, 1

Tags:

Let $\triangle ABC$ be a given triangle. If $|BA-tBC| \ge |AC|$ for any $t \in \mathbb{R}$, then $\triangle ABC$ is $ \textbf{(A)}\ \text{an acute triangle}\qquad\textbf{(B)}\ \text{an obtuse triangle}\qquad\textbf{(C)}\ \text{a right triangle}\qquad\textbf{(D)}\ \text{not known}\qquad$

1964 All Russian Mathematical Olympiad, 050

Tags: geometry , tangential

The quadrangle $ABCD$ is circumscribed around the circle with the centre $O$. Prove that $$\angle AOB+ \angle COD=180^o. $$

2003 All-Russian Olympiad Regional Round, 10.3

Tags: combinatorics

$45$ people came to the alumni meeting. It turned out that any two of them, having the same number of acquaintances among those who came, don't know each other. What is the greatest number of pairs of acquaintances that could to be among those participating in the meeting?

1960 Miklós Schweitzer, 3

Tags:

[b]3.[/b] Let $f(z)$ with $f(0)=1$ be regular in the unit disk and let $\left [\frac{\partial^2 \mid f(z)\mid}{\partial x\partial y} \right ] _{z=0} =1$. Show thatthe area of the image of the unit disk by $w= f(z)$ (taken with multiplicity) is not less than $\frac {1} {2}$ .[b](f. 6)[/b]

2009 All-Russian Olympiad, 8

Tags: number theory

Let $ x$, $ y$ be two integers with $ 2\le x, y\le 100$. Prove that $ x^{2^n} \plus{} y^{2^n}$ is not a prime for some positive integer $ n$.

2023 Kyiv City MO Round 1, Problem 4

Tags: combinatorics

For $n \ge 2$ consider $n \times n$ board and mark all $n^2$ centres of all unit squares. What is the maximal possible number of marked points that we can take such that there don't exist three taken points which form right triangle? [i]Proposed by Mykhailo Shtandenko[/i]

1981 IMO, 2

Tags: geometry , incenter , circumcircle , similar triangles

Three circles of equal radius have a common point $O$ and lie inside a given triangle. Each circle touches a pair of sides of the triangle. Prove that the incenter and the circumcenter of the triangle are collinear with the point $O$.