Olympiad problems

2012 District Olympiad, 1

Let $a,b,c$ three positive distinct real numbers. Evaluate: \[\lim_{t\to \infty} \int_0^t \frac{1}{(x^2+a^2)(x^2+b^2)(x^2+c^2)}dx\]

2013 Harvard-MIT Mathematics Tournament, 2

Tags: geometry

Let $ABCD$ be an isosceles trapezoid such that $AD = BC$, $AB = 3$, and $CD = 8$. Let $E$ be a point in the plane such that $BC = EC$ and $AE \perp EC$. Compute $AE$.

2006 All-Russian Olympiad Regional Round, 11.4

Tags: geometry , angle bisector , parallel

The bisectors of angles $A$ and $C$ of triangle $ABC$ intersect its sides at points $A_1$ and $C_1$, and the circumcircle of this triangle is at points $A_0$ and $C_0$, respectively. Lines $A_1C_1$ and $A_0C_0$ intersect at point P. Prove that the segment connecting $P$ to the center of the incircle of triangle $ABC$ is parallel to $AC$.

2021 Argentina National Olympiad, 4

Tags: combinatorics

Martu wants to build a set of cards with the following properties: • Each card has a positive integer on it. • The number on each card is equal to one of $5$ possible numbers. • If any two cards are taken and added together, it is always possible to find two other cards in the set such that the sum is the same. Determine the fewest number of cards Martu's set can have and give an example for that number.

2013 May Olympiad, 4

Tags:

Is it possible to write $100$ odd numbers on a line such that the sum of any $5$ consecutive numbers is a square number and the sum of any $9$ consecutive numbers is also a square number?

2018 HMNT, 1

Tags: hmmt , number

What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?

1992 ITAMO, 2

Tags: area , combinatorial geometry , area of a triangle , area of triangle , geometry

A convex quadrilateral of area $1$ is given. Prove that there exist four points in the interior or on the sides of the quadrilateral such that each triangle with the vertices in three of these four points has an area greater than or equal to $1/4$.

2012 Estonia Team Selection Test, 1

Tags: perfect square , perfect cube , number theory

Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.

2020 Iran MO (2nd Round), P2

Tags: inequality , floor function , inequalities , algebra

let $x,y,z$ be positive reals , such that $x+y+z=1399$ find the $$\max( [x]y + [y]z + [z]x ) $$ ( $[a]$ is the biggest integer not exceeding $a$)

1979 IMO, 3

Tags: geometry , parallelogram , circles , fixed point , conic

Two circles in a plane intersect. $A$ is one of the points of intersection. Starting simultaneously from $A$ two points move with constant speed, each travelling along its own circle in the same sense. The two points return to $A$ simultaneously after one revolution. Prove that there is a fixed point $P$ in the plane such that the two points are always equidistant from $P.$

1973 Chisinau City MO, 65

Tags: combinatorial geometry , chord , geometric inequality , geometry , combinatorics

A finite number of chords is drawn in a circle $1$ cm in diameter so that any diameter of the circle intersects at most $N$ of these chords. Prove that the sum of the lengths of all chords is less than $3.15 \cdot N$ cm.

2006 Kyiv Mathematical Festival, 1

Tags: combinatorics

See all the problems from 5-th Kyiv math festival [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=506789#p506789]here[/url] The number $123456789$ is written on the blackboard. At each step it is allowed to choose its digits $a$ and $b$ of the same parity and to replace each of them by $\frac{a+b}{2}.$ Is it possible to obtain a number larger then a)$800000000$; b)$880000000$ by such replacements?

2021 Putnam, B5

Tags:

Say that an $n$-by-$n$ matrix $A=(a_{ij})_{1\le i,j \le n}$ with integer entries is very odd if, for every nonempty subset $S$ of $\{1,2,\dots,n \}$, the $|S|$-by-$|S|$ submatrix $(a_{ij})_{i,j \in S}$ has odd determinant. Prove that if $A$ is very odd, then $A^k$ is very odd for every $k \ge 1$.

2016 Sharygin Geometry Olympiad, P24

Tags: geometry , 3d geometry , sphere , prism

A sphere is inscribed into a prism $ABCA'B'C'$ and touches its lateral faces $BCC'B', CAA'C', ABB'A' $ at points $A_o, B_o, C_o$ respectively. It is known that $\angle A_oBB' = \angle B_oCC' =\angle C_oAA'$. a) Find all possible values of these angles. b) Prove that segments $AA_o, BB_o, CC_o$ concur. c) Prove that the projections of the incenter to $A'B', B'C', C'A'$ are the vertices of a regular triangle.

2014 Contests, 2

Tags: hmmt

Find the integer closest to \[\frac{1}{\sqrt[4]{5^4+1}-\sqrt[4]{5^4-1}}\]

2003 AMC 12-AHSME, 8

Tags: modular arithmetic

Let $ \clubsuit(x)$ denote the sum of the digits of the positive integer $ x$. For example, $ \clubsuit(8)\equal{}8$ and $ \clubsuit(123)\equal{}1\plus{}2\plus{}3\equal{}6$. For how many two-digit values of $ x$ is $ \clubsuit(\clubsuit(x))\equal{}3$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 10$

2015 Oral Moscow Geometry Olympiad, 4

Tags: geometry , trapezoid , angle bisector , incircle , equal segments

In trapezoid $ABCD$, the bisectors of angles $A$ and $D$ intersect at point $E$ lying on the side of $BC$. These bisectors divide the trapezoid into three triangles into which the circles are inscribed. One of these circles touches the base $AB$ at the point $K$, and two others touch the bisector $DE$ at points $M$ and $N$. Prove that $BK = MN$.

2016 Postal Coaching, 3

Tags: function , functional equation , algebra

Find all real numbers $a$ such that there exists a function $f:\mathbb R\to \mathbb R$ such that the following conditions are simultaneously satisfied: (a) $f(f(x))=xf(x)-ax,\;\forall x\in\mathbb{R};$ (b) $f$ is not a constant function; (c) $f$ takes the value $a$.

2016 Sharygin Geometry Olympiad, P18

Tags: geometry

Let $ABC$ be a triangle with $\angle C=90^{\circ}$, and $K, L $ be the midpoints of the minor arcs AC and BC of its circumcircle. Segment $KL$ meets $AC$,at point $N$. Find angle $NIC$ where $I$is the incenter of $ABC$.

2018 USA TSTST, 1

Tags: algebra , functional equation

As usual, let ${\mathbb Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : {\mathbb Z}[x] \to {\mathbb Z}$ such that for any polynomials $p,q \in {\mathbb Z}[x]$, [list] [*]$\theta(p+1) = \theta(p)+1$, and [*]if $\theta(p) \neq 0$ then $\theta(p)$ divides $\theta(p \cdot q)$. [/list] [i]Evan Chen and Yang Liu[/i]

1982 AMC 12/AHSME, 26

Tags: quadratic

If the base $8$ representation of a perfect square is $ab3c$, where $a\ne 0$, then $c$ equals $\textbf{(A) } 0\qquad \textbf{(B) }1 \qquad \textbf{(C) } 3\qquad \textbf{(D) } 4\qquad \textbf{(E) } \text{not uniquely determined}$

2006 Croatia Team Selection Test, 4

Tags: inequalities , number theory

Find all natural solutions of $3^{x}= 2^{x}y+1.$

2001 Croatia National Olympiad, Problem 1

Tags: diophantine equation , number theory , preschool

Find all integers $x$ for which $2x^2-x-36$ is the square of a prime number.

2011 Federal Competition For Advanced Students, Part 2, 2

Tags: combinatorics

We consider permutations $f$ of the set $\mathbb{N}$ of non-negative integers, i.e. bijective maps $f$ from $\mathbb{N}$ to $\mathbb{N}$, with the following additional properties: \[f(f(x)) = x \quad \mbox{and}\quad \left| f(x)-x\right| \leqslant 3\quad\mbox{for all } x \in\mathbb{N}\mbox{.}\] Further, for all integers $n > 42$, \[\left.M(n)=\frac{1}{n+1}\sum_{j=0}^n \left|f(j)-j\right|<2,011\mbox{.}\right.\] Show that there are infinitely many natural numbers $K$ such that $f$ maps the set \[\left\{ n\mid 0\leqslant n\leqslant K\right\}\] onto itself.

2017 District Olympiad, 2

Tags: morphism , group theory , abstract algebra , superior algebra

Let be a group and two coprime natural numbers $ m,n. $ Show that if the applications $ G\ni x\mapsto x^{m+1},x^{n+1} $ are surjective endomorphisms, then the group is commutative.