Olympiad problems

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 $O$.

2023 LMT Fall, 21

Tags: algebra , number theory , combinatorics

Let $(a_1,a_2,a_3,a_4,a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of $$\sum^5_{i=1} |a_i -i | = |a_1 -1|+|a_2 -2|+|a_3 -3|+|a_4 -4|+|a_5 -5|.$$ [i]Proposed by Muztaba Syed[/i]

2021 CCA Math Bonanza, T2

Tags:

Given that real numbers $a$, $b$, and $c$ satisfy $ab=3$, $ac=4$, and $b+c=5$, the value of $bc$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$. [i]2021 CCA Math Bonanza Team Round #2[/i]

2014 PUMaC Number Theory A, 4

Tags:

Find the sum of all positive integers $x$ such that $3 \times 2^x = n^2 -1$ for some positive integer $n$.

2012 Today's Calculation Of Integral, 798

Tags: calculus , integration , function , derivative , quadratic , analytic geometry , geometry

Denote by $C,\ l$ the graphs of the cubic function $C: y=x^3-3x^2+2x$, the line $l: y=ax$. (1) Find the range of $a$ such that $C$ and $l$ have intersection point other than the origin. (2) Denote $S(a)$ by the area bounded by $C$ and $l$. If $a$ move in the range found in (1), then find the value of $a$ for which $S(a)$ is minimized. 50 points

2011 Switzerland - Final Round, 9

Tags: ceiling function , floor function , induction , strong induction , number theory

For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$. Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 9)[/i]

2021 Saint Petersburg Mathematical Olympiad, 5

Tags: geometry

Given is an isosceles trapezoid $ABCD$, such that $AD$ and $BC$ are bases and $AD=2AB$, and it is inscribed in a circle $c$. Points $E$ and $F$ are selected on a circle $c$ so that $AC$ || $DE$ and $BD$ || $AF$. The line $BE$ intersects lines $AC$ and $AF$ at points $X$ and $Y$, respectively. Prove that the circumcircles of triangles $BCX$ and $EFY$ are tangent to each other.

2020 Federal Competition For Advanced Students, P2, 5

Tags: geometry , mixtilinear , fixed , fixed angle , semicircle

Let $h$ be a semicircle with diameter $AB$. Let $P$ be an arbitrary point inside the diameter $AB$. The perpendicular through $P$ on $AB$ intersects $h$ at point $C$. The line $PC$ divides the semicircular area into two parts. A circle will be inscribed in each of them that touches $AB, PC$ and $h$. The points of contact of the two circles with $AB$ are denoted by $D$ and $E$, where $D$ lies between $A$ and $P$. Prove that the size of the angle $DCE$ does not depend on the choice of $P$. (Walther Janous)

2005 Peru MO (ONEM), 2

Tags: trigonometry , algebra

The measures, in degrees, of the angles , $\alpha, \beta$ and $\theta$ are greater than $0$ less than $60$. Find the value of $\theta$ knowing, also, that $\alpha + \beta = 2\theta$ and that $$\sin \alpha \sin \beta \sin \theta = \sin(60 - \alpha ) \sin(60 - \beta) \sin(60 - \theta ).$$

2021 Kazakhstan National Olympiad, 1

Tags: inequalities

Given $a,b,c>0$ such that $$a+b+c+\frac{1}{abc}=\frac{19}{2}$$ What is the greatest value for $a$?