Olympiad problems

2016 Saudi Arabia BMO TST, 2

Let $ABC$ be a triangle and $I$ its incenter. The point $D$ is on segment $BC$ and the circle $\omega$ is tangent to the circumcirle of triangle $ABC$ but is also tangent to $DC, DA$ at $E, F$, respectively. Prove that $E, F$ and $I$ are collinear.

2020 Israel Olympic Revenge, P3

Tags: number theory , algebra , polynomial

For each positive integer $n$, define $f(n)$ to be the least positive integer for which the following holds: For any partition of $\{1,2,\dots, n\}$ into $k>1$ disjoint subsets $A_1, \dots, A_k$, [u]all of the same size[/u], let $P_i(x)=\prod_{a\in A_i}(x-a)$. Then there exist $i\neq j$ for which \[\deg(P_i(x)-P_j(x))\geq \frac{n}{k}-f(n)\] a) Prove that there is a constant $c$ so that $f(n)\le c\cdot \sqrt{n}$ for all $n$. b) Prove that for infinitely many $n$, one has $f(n)\ge \ln(n)$.

2000 Bundeswettbewerb Mathematik, 1

Tags: combinatorics

We are given $n \geq 3$ weights of masses $1, 2, 3, \ldots , n$ grams. Find all $n$ for which it is possible to divide these weights into three groups with the same mass.

2000 Bundeswettbewerb Mathematik, 2

Tags: number theory , sum , divisible

A $5$-tuple $(1,1,1,1,2)$ has the property that the sum of any three of them is divisible by the sum of the remaining two. Is there a $5$-tuple with this property whose all terms are distinct?

1997 Pre-Preparation Course Examination, 5

Tags: circumcircle , parallelogram , rotation , geometry

Let $H$ be the orthocenter of the triangle $ABC$ and $P$ an arbitrary point on circumcircle of triangle. $BH$ meets $AC$ at $E$. $PAQB$ and $PARC$ are two parallelograms and $AQ$ meets $HR$ at $X$. Show that $EX \parallel AP$.

2023 Assara - South Russian Girl's MO, 3

Tags: number theory , combinatorics

In equality $$1 * 2 * 3 * 4 * 5 * ... * 60 * 61 * 62 = 2023$$ Instead of each asterisk, you need to put one of the signs “+” (plus), “-” (minus), “•” (multiply) so that the equality becomes true. What is the smallest number of "•" characters that can be used?

2010 Poland - Second Round, 1

Tags: incenter , perpendicular bisector , angle bisector , geometry

In the convex pentagon $ABCDE$ all interior angles have the same measure. Prove that the perpendicular bisector of segment $EA$, the perpendicular bisector of segment $BC$ and the angle bisector of $\angle CDE$ intersect in one point.

2014 Sharygin Geometry Olympiad, 18

Tags: geometry , circumcircle , incenter

Let $I$ be the incenter of a circumscribed quadrilateral $ABCD$. The tangents to circle $AIC$ at points $A, C$ meet at point $X$. The tangents to circle $BID$ at points $B, D$ meet at point $Y$ . Prove that $X, I, Y$ are collinear.

PEN A Problems, 104

Tags: divisibility theory

A wobbly number is a positive integer whose $digits$ in base $10$ are alternatively non-zero and zero the units digit being non-zero. Determine all positive integers which do not divide any wobbly number.

1985 IMO Longlists, 93

Tags: 3d geometry , sphere , tetrahedron , geometry

The sphere inscribed in tetrahedron $ABCD$ touches the sides $ABD$ and $DBC$ at points $K$ and $M$, respectively. Prove that $\angle AKB = \angle DMC$.

2021 Durer Math Competition Finals, 10

Tags: combinatorics

Billy owns some really energetic horses. They are hopping around on points of the plane having two integer coordinates. Each horse can make the following types of jumps. They can hop to a point obtained from their current position via a translation by vector $(15, 9)$, $(-9, 15)$, $(-15,-9)$ or $(9,-15)$. The horses are now standing on lattice points such that no two can meet by making jumps as above. What is the maximal possible number of horses Billy can own?

Russian TST 2016, P1

Tags: number theory , divisibility

Let $a{}$ and $b{}$ be natural numbers greater than one. Let $n{}$ be a natural number for which $a\mid 2^n-1$ and $b\mid 2^n+1$. Prove that there is no natural $k{}$ such that $a\mid 2^k+1$ and $b\mid 2^k-1$.

2012 Turkmenistan National Math Olympiad, 7

Tags: algebra

If $a,b,c$ are positive real numbers and satisfy: $\frac{a_1}{b_1}=\frac{a_2}{b_2}=...=\frac{a_n}{b_n}$ then prove that :$ \sum_{i=1}^{n} a^{2}_i \cdot \sum_{i=1}^{n} b^{2}_i =(\sum_{i=1}^{n} a_{i}b_{i})^2$

Ukraine Correspondence MO - geometry, 2012.7

Tags: geometry , bisects sements

Let $O$ and $H$ be the center of the circumcircle and the point of intersection of the altitudes of the acute triangle $ABC$ respectively, $D$ be the foot of the altitude drawn to $BC$, and $E$ be the midpoint of $AO$. Prove that the circumcircle of the triangle $ADE$ passes through the midpoint of the segment $OH$.

2016 District Olympiad, 1

Tags: geometry , 3d geometry , pyramid

Let be a pyramid having a square as its base and the projection of the top vertex to the base is the center of the square. Prove that two opposite faces are perpendicular if and only if the angle between two adjacent faces is $ 120^{\circ } . $

2018 Saudi Arabia GMO TST, 3

Tags: tangent , geometry , perpendicular , arc midpoint

Let $C$ be a point lies outside the circle $(O)$ and $CS, CT$ are tangent lines of $(O)$. Take two points $A, B$ on $(O)$ with $M$ is the midpoint of the minor arc $AB$ such that $A, B, M$ differ from $S, T$. Suppose that $MS, MT$ cut line $AB$ at $E, F$. Take $X \in OS$ and $Y \in OT$ such that $EX, FY$ are perpendicular to $AB$. Prove that $X Y$ and $C M$ are perpendicular.

2015 Junior Regional Olympiad - FBH, 1

Tags: counting years

Father is $42$ years old, and son has $14$ years. In how many years father will be twice as old as his son?

2000 Bosnia and Herzegovina Team Selection Test, 4

Tags: inequality , inequalities , algebra

Prove that for all positive real $a$, $b$ and $c$ holds: $$ \frac{bc}{a^2+2bc}+\frac{ac}{b^2+2ac}+\frac{ab}{c^2+2ab} \leq 1 \leq \frac{a^2}{a^2+2bc}+\frac{b^2}{b^2+2ac}+\frac{c^2}{c^2+2ab}$$

2013 Dutch IMO TST, 4

Tags: modular arithmetic , number theory

Determine all positive integers $n\ge 2$ satisfying $i+j\equiv\binom ni +\binom nj \pmod{2}$ for all $i$ and $j$ with $0\le i\le j\le n$.

2021 Azerbaijan IMO TST, 3

Tags: algebra , polynomial , algorithm , lagrange s interpolation

A magician intends to perform the following trick. She announces a positive integer $n$, along with $2n$ real numbers $x_1 < \dots < x_{2n}$, to the audience. A member of the audience then secretly chooses a polynomial $P(x)$ of degree $n$ with real coefficients, computes the $2n$ values $P(x_1), \dots , P(x_{2n})$, and writes down these $2n$ values on the blackboard in non-decreasing order. After that the magician announces the secret polynomial to the audience. Can the magician find a strategy to perform such a trick?

2013 Brazil Team Selection Test, 1

Tags: quadratic , number theory

Call admissible a set $A$ of integers that has the following property: If $x,y \in A$ (possibly $x=y$) then $x^2+kxy+y^2 \in A$ for every integer $k$. Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. [i]Proposed by Warut Suksompong, Thailand[/i]

1985 IMO Longlists, 88

Tags: polynomial , calculus , binomial theorem , algebra

Determine the range of $w(w + x)(w + y)(w + z)$, where $x, y, z$, and $w$ are real numbers such that \[x + y + z + w = x^7 + y^7 + z^7 + w^7 = 0.\]

2013 Romania National Olympiad, 1

Tags: geometry , angle bisector , median , parallel

In the triangle $ABC$, the angle - bisector $AD$ ($D \in BC$) and the median $BE$ ($E \in AC$) intersect at point $P$. Lines $AB$ and $CP$ intesect at point $F$. The parallel through $B$ to $CF$ intersects $DF$ at point $M$. Prove that $DM = BF$

2021 Korea National Olympiad, P5

Tags: sequence , inequalities , algebra , polynomial

A real number sequence $a_1, \cdots ,a_{2021}$ satisfies the below conditions. $$a_1=1, a_2=2, a_{n+2}=\frac{2a_{n+1}^2}{a_n+a_{n+1}} (1\leq n \leq 2019)$$ Let the minimum of $a_1, \cdots ,a_{2021}$ be $m$, and the maximum of $a_1, \cdots ,a_{2021}$ be $M$. Let a 2021 degree polynomial $$P(x):=(x-a_1)(x-a_2) \cdots (x-a_{2021})$$ $|P(x)|$ is maximum in $[m, M]$ when $x=\alpha$. Show that $1<\alpha <2$.

2011 Greece Junior Math Olympiad, 2

Tags: number theory

We consider the set of four-digit positive integers $x =\overline{abcd}$ with digits different than zero and pairwise different. We also consider the integers $y = \overline{dcba}$ and we suppose that $x > y$. Find the greatest and the lowest value of the difference $x-y$, as well as the corresponding four-digit integers $x,y$ for which these values are obtained.