Olympiad problems

2012 BMT Spring, 6

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.

2014 Rioplatense Mathematical Olympiad, Level 3, 3

Tags: game strategy , game , combinatorics

Kiko and Ñoño play with a rod of length $2n$ where $n \le 3$ is an integer. Kiko cuts the rod in $ k \le 2n$ pieces of integer lengths. Then Ñoño has to arrange these pieces so that they form a hexagon of equal opposite sides and equal angles. The pieces can not be split and they all have to be used. If Ñoño achieves his goal, he wins, in any other case, Kiko wins. Determine which victory can be secured based on $k$.

2019 Thailand TST, 3

Tags: polynomial , algebra

Determine all polynomials $P (x, y), Q(x, y)$ and $R(x, y)$ with real coefficients satisfying $$P (ux + vy, uy + vx) = Q(x, y)R(u, v)$$ for all real numbers $u, v, x$ and $y$.

2022 Durer Math Competition Finals, 7

Tags: number theory , digit

The [i]fragments [/i] of a positive integer are the numbers seen when reading one or more of its digits in order. The [i]fragment sum[/i] equals the sum of all the fragments, including the number itself. For example, the fragment sum of $2022$ is $2022+202+022+20+02+22+2+0+2+2 = 2296$. There is another four-digit number with the same fragment sum. What is it? As the example shows, if a fragment occurs multiple times, then all its occurrences are added, and the fragments beginning with $0$ also count (for instance, $022$ is worth $22$).

2022 Belarusian National Olympiad, 8.8

Tags: algebra

Vitya and Masha are playing a game. At first, Vitya thinks of three different integers. In one move Masha can ask one of the following three numbers: the sum of the numbers, the product of the numbers or the sum of pairwise products of the numbers. Masha asks questions and Vitya immediately answers before Masha asks the next question. a) Prove that Masha can always guess Vitya's numbers. b) What is the least amount of questions Masha needs to ask to guaranteely guess them?