Olympiad problems

2024 Iran MO (3rd Round), 4

Tags: number theory

For a given positive integer number $n$ find all subsets $\{r_0,r_1,\cdots,r_n\}\subset \mathbb{N}$ such that $$ n^n+n^{n-1}+\cdots+1 | n^{r_n}+\cdots+ n^{r_0}. $$ Proposed by [i]Shayan Tayefeh[/i]

2009 Croatia Team Selection Test, 4

Tags: modular arithmetic , number theory

Determine all triplets off positive integers $ (a,b,c)$ for which $ \mid2^a\minus{}b^c\mid\equal{}1$

1978 All Soviet Union Mathematical Olympiad, 252

Tags: sum , algebra

Let $a_n$ be the closest to $\sqrt n$ integer. Find the sum $$1/a_1 + 1/a_2 + ... + 1/a_{1980}$$

2010 Sharygin Geometry Olympiad, 4

Tags: projection , geometry , parallelogram , concentric circles , concentric , inscribed

Projections of two points to the sidelines of a quadrilateral lie on two concentric circles (projections of each point form a cyclic quadrilateral and the radii of circles are different). Prove that this quadrilateral is a parallelogram.

2012 BMT Spring, 6

Tags: cyclic quadrilateral , geometry

Let $ \text{ABCD} $ be a cyclic quadrilateral, with $ \text{AB} = 7 $, $ \text{BC} = 11 $, $ \text{CD} = 13 $, and $ \text{DA} = 17 $. Let the incircle of $ \text{ABD} $ hit $ \text{BD} $ at $ \text{R} $ and the incircle of $ \text{CBD} $ hit $ \text{BD} $ at $ \text{S} $. What is $ \text{RS} $?

2004 Germany Team Selection Test, 2

Tags: fixed point , geometry

Three distinct points $A$, $B$, and $C$ are fixed on a line in this order. Let $\Gamma$ be a circle passing through $A$ and $C$ whose center does not lie on the line $AC$. Denote by $P$ the intersection of the tangents to $\Gamma$ at $A$ and $C$. Suppose $\Gamma$ meets the segment $PB$ at $Q$. Prove that the intersection of the bisector of $\angle AQC$ and the line $AC$ does not depend on the choice of $\Gamma$.

2023 AMC 10, 9

Tags:

A digital display shows the current date as an $8$-digit integer consisting of a $4$-digit year, followed by a $2$-digit month, followed by a $2$-digit date within the month. For example, Arbor Day this year is displayed as 20230428. For how many dates in $2023$ will each digit appear an even number of times in the 8-digital display for that date? $\textbf{(A)}~5\qquad\textbf{(B)}~6\qquad\textbf{(C)}~7\qquad\textbf{(D)}~8\qquad\textbf{(E)}~9$

1977 All Soviet Union Mathematical Olympiad, 235

Tags: combinatorial geometry , geometry

Given a closed broken line without self-intersections in a plane. Not a triple of its vertices belongs to one straight line. Let us call "special" a couple of line's segments if the one's extension intersects another. Prove that there is even number of special pairs.

2013 USA TSTST, 6

Tags: function , number theory , algebra , functional equation

Let $\mathbb N$ be the set of positive integers. Find all functions $f: \mathbb N \to \mathbb N$ that satisfy the equation \[ f^{abc-a}(abc) + f^{abc-b}(abc) + f^{abc-c}(abc) = a + b + c \] for all $a,b,c \ge 2$. (Here $f^1(n) = f(n)$ and $f^k(n) = f(f^{k-1}(n))$ for every integer $k$ greater than $1$.)

2025 Caucasus Mathematical Olympiad, 7

Tags: geometry

From a point $O$ lying outside the circle $\omega$, two tangents are drawn touching $\omega$ at points $M$ and $N$. A point $K$ is chosen on the segment $MN$. Let points $P$ and $Q$ be the midpoints of segments $KM$ and $OM$ respectively. The circumcircle of triangle $MPQ$ intersects $\omega$ again at point $L$ ($L \neq M$). Prove that the line $LN$ passes through the centroid of triangle $KMO$.

2005 Postal Coaching, 12

Tags: ratio , analytic geometry , geometry

Let $ABC$ be a triangle with vertices at lattice points. Suppose one of its sides in $\sqrt{n}$, where $n$ is square-free. Prove that $\frac{R}{r}$ is irraational . The symbols have usual meanings.

2009 Hong Kong TST, 3

Tags: circumcircle , geometry

Let $ ABCDE$ be an arbitrary convex pentagon. Suppose that $ BD\cap CE \equal{} A'$, $ CE\cap DA \equal{} B'$, $ DA\cap EB \equal{} C'$, $ EB\cap AC \equal{} D'$ and $ AC\cap BD \equal{} E'$. Suppose also that $ eABD'\cap eAC'E \equal{} A''$, $ eBCE'\cap eBD'A \equal{} B''$, $ eCDA'\cap eCE'B \equal{} C''$, $ eDEB'\cap eDA'C \equal{} D''$, $ eEAC'\cap eEB'D \equal{} E''$. Prove that $ AA'', BB'', CC'', DD'', EE''$ are concurrent. (Here $ l_1\cap l_2 \equal{} P$ means that $ P$ is the intersection of lines $ l_1$ and $ l_2$. Also $ eA_1A_2A_3\cap eB_1B_2B_3 \equal{} Q$ means that $ Q$ is the intersection of the circumcircles of $ \Delta A_1A_2A_3$ and $ \Delta B_1B_2B_3$.)

2023 Harvard-MIT Mathematics Tournament, 22

Tags: guts

Let $a_0, a_1, a_2, \ldots$ be an infinite sequence where each term is independently and uniformly at random in the set $\{1, 2, 3, 4\}.$ Define an infinite sequence $b_0, b_1, b_2, \ldots$ recursively by $b_0=1$ and $b_{i+1}=a_i^{b_i}.$ Compute the expected value of the smallest positive integer $k$ such that $b_k \equiv 1 \pmod{5}.$

2010 Contests, 2

Tags: geometry

Given a triangle $ABC$, let $D$ be the point where the incircle of the triangle $ABC$ touches the side $BC$. A circle through the vertices $B$ and $C$ is tangent to the incircle of triangle $ABC$ at the point $E$. Show that the line $DE$ passes through the excentre of triangle $ABC$ corresponding to vertex $A$.

Kyiv City MO Juniors 2003+ geometry, 2014.851

Tags: geometry , ratio , median , equal segments

On the side $AB$ of the triangle $ABC$ mark the point $K$. The segment $CK$ intersects the median $AM$ at the point $F$. It is known that $AK = AF$. Find the ratio $MF: BK$.

Novosibirsk Oral Geo Oly IX, 2017.5

Tags: geometry , rectangle , equal segments

Point $K$ is marked on the diagonal $AC$ in rectangle $ABCD$ so that $CK = BC$. On the side $BC$, point $M$ is marked so that $KM = CM$. Prove that $AK + BM = CM$.

2013 Hanoi Open Mathematics Competitions, 13

Tags: algebra , system of equations

Solve the system of equations $\begin{cases} xy=1 \\ \frac{x}{x^4+y^2}+\frac{y}{x^2+y^4}=1\end{cases}$

2023 JBMO Shortlist, A3

Tags: inequality

Prove that for all non-negative real numbers $x,y,z$, not all equal to $0$, the following inequality holds $\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geq 3.$ Determine all the triples $(x,y,z)$ for which the equality holds. [i]Milan Mitreski, Serbia[/i]

1969 Spain Mathematical Olympiad, 2

Tags: complex number , locus , geometry

Find the locus of the affix $M$, of the complex number $z$, so that it is aligned with the affixes of $i$ and $iz$ .

2006 Taiwan TST Round 1, 1

Tags: number theory

Find the largest integer that is a factor of $(a-b)(b-c)(c-d)(d-a)(a-c)(b-d)$ for all integers $a,b,c,d$.

2004 Brazil Team Selection Test, Problem 4

Tags: number theory , algebra

Let $b$ be a number greater than $5$. For each positive integer $n$, consider the number $$x_n=\underbrace{11\ldots1}_{n-1}\underbrace{22\ldots2}_n5,$$ written in base $b$. Prove that the following condition holds if and only if $b=10$: There exists a positive integer $M$ such that for every integer $n$ greater than $M$, the number $x_n$ is a perfect square.

2001 Estonia Team Selection Test, 6

Tags: incircle , circumcircle , geometry

Let $C_1$ and $C_2$ be the incircle and the circumcircle of the triangle $ABC$, respectively. Prove that, for any point $A'$ on $C_2$, there exist points $B'$ and $C'$ such that $C_1$ and $C_2$ are the incircle and the circumcircle of triangle $A'B'C'$, respectively.

2014 District Olympiad, 1

Tags: linear algebra

[list=a] [*]Give an example of matrices $A$ and $B$ from $\mathcal{M}_{2}(\mathbb{R})$, such that $ A^{2}+B^{2}=\left( \begin{array} [c]{cc} 2 & 3\\ 3 & 2 \end{array} \right) . $ [*]Let $A$ and $B$ be matrices from $\mathcal{M}_{2}(\mathbb{R})$, such that $\displaystyle A^{2}+B^{2}=\left( \begin{array} [c]{cc} 2 & 3\\ 3 & 2 \end{array} \right) $. Prove that $AB\neq BA$.[/list]

Kyiv City MO Seniors 2003+ geometry, 2015.10.5

Tags: circles , equal segments , geometry

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers at points ${{O} _ {1}}$ and ${{ O} _ {2}}$ intersect at points $A$ and $B$, respectively. Around the triangle ${{O} _ {1}} {{O} _ {2}} B$ circumscribe a circle $w$ centered at the point $O$, which intersects the circles ${{w } _ {1}}$ and ${{w} _ {2}}$ for the second time at points $K$ and $L$, respectively. The line $OA$ intersects the circles ${{w} _ {1}}$ and ${{w} _ {2}}$ at the points $M$ and $N$, respectively. The lines $MK$ and $NL$ intersect at the point $P$. Prove that the point $P$ lies on the circle $w$ and $PM = PN$. (Vadym Mitrofanov)

2003 Brazil National Olympiad, 2

Tags: floor function , combinatorics , logarithm

Let $S$ be a set with $n$ elements. Take a positive integer $k$. Let $A_1, A_2, \ldots, A_k$ be any distinct subsets of $S$. For each $i$ take $B_i = A_i$ or $B_i = S - A_i$. Find the smallest $k$ such that we can always choose $B_i$ so that $\bigcup_{i=1}^k B_i = S$, no matter what the subsets $A_i$ are.