Olympiad problems

2009 IMO, 6

Let $ a_1, a_2, \ldots , a_n$ be distinct positive integers and let $ M$ be a set of $ n \minus{} 1$ positive integers not containing $ s \equal{} a_1 \plus{} a_2 \plus{} \ldots \plus{} a_n.$ A grasshopper is to jump along the real axis, starting at the point $ 0$ and making $ n$ jumps to the right with lengths $ a_1, a_2, \ldots , a_n$ in some order. Prove that the order can be chosen in such a way that the grasshopper never lands on any point in $ M.$ [i]Proposed by Dmitry Khramtsov, Russia[/i]

2024 Serbia Team Selection Test, 1

Tags: algebra

Does there exist a positive integer $n$ and a) complex numbers $a_0, a_1, \ldots, a_n;$ b) reals $a_0, a_1, \ldots, a_n, $ such that $P(x) Q(x)=x^{2024}+1$ where $P(x)=a_nx^n+\ldots +a_1x+a_0$ and $Q(x)=a_0x^n+a_1x^{n-1}+\ldots+a_n?$

2015 Thailand TSTST, 1

Tags: circumcircle , bisects segment , right angle , geometry

Let $O$ be the circumcenter of an acute $\vartriangle ABC$ which has altitude $AD$. Let $AO$ intersect the circumcircle of $\vartriangle BOC$ again at $X$. If $E$ and $F$ are points on lines $AB$ and $AC$ such that $\angle XEA = \angle XFA = 90^o$ , then prove that the line $DX$ bisects the segment $EF$.

1988 Austrian-Polish Competition, 2

Tags: inequalities , sum , algebra

If $a_1 \le a_2 \le .. \le a_n$ are natural numbers ($n \ge 2$), show that the inequality $$\sum_{i=1}^n a_ix_i^2 +2\sum_{i=1}^{n-1} x_ix_{i+1} >0$$ holds for all $n$-tuples $(x_1,...,x_n) \ne (0,..., 0)$ of real numbers if and only if $a_2 \ge 2$.

2001 China Team Selection Test, 1

Tags: algebra , polynomial

Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer

2022 Brazil EGMO TST, 6

Tags: geometry , angle bisector , italy

The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC \plus{}MA \cdot CD \equal{} MB \cdot MD$, prove that $ \angle BKC \equal{} \angle CDB$.

2023 IMC, 9

Tags: 3d geometry , convexity , geometry , analytic geometry

We say that a real number $V$ is [i]good[/i] if there exist two closed convex subsets $X$, $Y$ of the unit cube in $\mathbb{R}^3$, with volume $V$ each, such that for each of the three coordinate planes (that is, the planes spanned by any two of the three coordinate axes), the projections of $X$ and $Y$ onto that plane are disjoint. Find $\sup \{V\mid V\ \text{is good}\}$.

1998 ITAMO, 1

Tags: sum , algebra , radical

Calculate the sum $\sum_{n=1}^{1.000.000}[ \sqrt{n} ]$ . You may use the formula $\sum_{i=1}^{k} i^2=\frac{k(k +1)(2k +1)}{6}$ without a proof.

2010 AMC 12/AHSME, 1

Tags:

What is $ (20\minus{}(2010\minus{}201)) \plus{} (2010\minus{}(201\minus{}20))$? $ \textbf{(A)}\ \minus{}4020\qquad \textbf{(B)}\ 0\qquad \textbf{(C)}\ 40\qquad \textbf{(D)}\ 401\qquad \textbf{(E)}\ 4020$

1967 IMO Longlists, 29

Tags: geometry , triangle , similar triangles , area of a triangle

$A_0B_0C_0$ and $A_1B_1C_1$ are acute-angled triangles. Describe, and prove, how to construct the triangle $ABC$ with the largest possible area which is circumscribed about $A_0B_0C_0$ (so $BC$ contains $B_0, CA$ contains $B_0$, and $AB$ contains $C_0$) and similar to $A_1B_1C_1.$

2016 Saudi Arabia BMO TST, 2

Tags: incenter , collinear , circumcircle , tangent circle , geometry

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 } . $