Olympiad problems

2018 Bundeswettbewerb Mathematik, 3

Tags: parallel , orthocenter , circles , geometry , perpendicular

Let $H$ be the orthocenter of the acute triangle $ABC$. Let $H_a$ be the foot of the perpendicular from $A$ to $BC$ and let the line through $H$ parallel to $BC$ intersect the circle with diameter $AH_a$ in the points $P_a$ and $Q_a$. Similarly, we define the points $P_b, Q_b$ and $P_c,Q_c$. Show that the six points $P_a,Q_a,P_b,Q_b,P_c,Q_c$ lie on a common circle.

2021 Balkan MO Shortlist, A6

Tags: functional equation , algebra

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(xy) = f(x)f(y) + f(f(x + y))$$ holds for all $x, y \in \mathbb{R}$.

2012 EGMO, 8

Tags: complex number , modular arithmetic , combinatorics

A [i]word[/i] is a finite sequence of letters from some alphabet. A word is [i]repetitive[/i] if it is a concatenation of at least two identical subwords (for example, $ababab$ and $abcabc$ are repetitive, but $ababa$ and $aabb$ are not). Prove that if a word has the property that swapping any two adjacent letters makes the word repetitive, then all its letters are identical. (Note that one may swap two adjacent identical letters, leaving a word unchanged.) [i]Romania (Dan Schwarz)[/i]

2014 Math Prize For Girls Problems, 7

Tags:

If $x$ is a real number and $k$ is a nonnegative integer, recall that the binomial coefficient $\binom{x}{k}$ is defined by the formula \[ \binom{x}{k} = \frac{x(x - 1)(x - 2) \dots (x - k + 1)}{k!} \, . \] Compute the value of \[ \frac{\binom{1/2}{2014} \cdot 4^{2014}}{\binom{4028}{2014}} \, . \]

1999 Estonia National Olympiad, 3

Tags: square , ratio , geometry

Let $E$ and $F$ be the midpoints of the lines $AB$ and $DA$ of a square $ABCD$, respectively and let $G$ be the intersection of $DE$ with $CF$. Find the aspect ratio of sidelengths of the triangle $EGC$, $| EG | : | GC | : | CE |$.

Russian TST 2020, P3

Tags: polynomial , algebra

A polynomial $P(x, y, z)$ in three variables with real coefficients satisfies the identities $$P(x, y, z)=P(x, y, xy-z)=P(x, zx-y, z)=P(yz-x, y, z).$$ Prove that there exists a polynomial $F(t)$ in one variable such that $$P(x,y,z)=F(x^2+y^2+z^2-xyz).$$

1968 IMO, 1

Tags: triangle , triangle inequality , algebra , trigonometry

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

2006 Singapore Junior Math Olympiad, 1

Tags: number theory , diophantine , diophantine equation

Find all integers $x,y$ that satisfy the equation $x+y=x^2-xy+y^2$

2017-IMOC, C6

Tags: geometry , combinatorics

Consider a convex polygon in a plane such that the length of all edges and diagonals are rational. After connecting all diagonals, prove that any length of a segment is rational.

2004 China Team Selection Test, 2

Tags: ramsey theory , extremal combinatorics , matrix , combinatorics

Twenty-one girls and twenty-one boys took part in a mathematical competition. It turned out that each contestant solved at most six problems, and for each pair of a girl and a boy, there was at least one problem that was solved by both the girl and the boy. Show that there is a problem that was solved by at least three girls and at least three boys.

1997 AMC 12/AHSME, 21

Tags: function , matrix , logarithm , integration , linear algebra , calculus

For any positive integer $ n$, let \[f(n) \equal{} \begin{cases} \log_8{n}, & \text{if }\log_8{n}\text{ is rational,} \\ 0, & \text{otherwise.} \end{cases}\] What is $ \sum_{n \equal{} 1}^{1997}{f(n)}$? $ \textbf{(A)}\ \log_8{2047}\qquad \textbf{(B)}\ 6\qquad \textbf{(C)}\ \frac {55}{3}\qquad \textbf{(D)}\ \frac {58}{3}\qquad \textbf{(E)}\ 585$

2017 AMC 12/AHSME, 18

Tags:

Let $S(n)$ equal the sum of the digits of positive integer $n$. For example, $S(1507) = 13$. For a particular positive integer $n$, $S(n) = 1274$. Which of the following could be the value of $S(n+1)$? $\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 1239\qquad\textbf{(E)}\ 1265$

2023 Federal Competition For Advanced Students, P2, 4

Tags: algebra

The number $2023$ is written $2023$ times on a blackboard. On one move, you can choose two numbers $x, y$ on the blackboard, delete them and write $\frac{x+y} {4}$ instead. Prove that when one number remains, it is greater than $1$.

2016 Japan Mathematical Olympiad Preliminary, 4

Tags: geometry , rectangle , combinatorics

There is a $11\times 11$ square grid. We divided this in $5$ rectangles along unit squares. How many ways that one of the rectangles doesn’t have a edge on basic circumference. Note that we count as different ways that one way coincides with another way by rotating or reversing.

1962 AMC 12/AHSME, 6

Tags: perimeter , geometry

A square and an equilateral triangle have equal perimeters. The area of the triangle is $ 9 \sqrt{3}$ square inches. Expressed in inches the diagonal of the square is: $ \textbf{(A)}\ \frac{9}{2} \qquad \textbf{(B)}\ 2 \sqrt{5} \qquad \textbf{(C)}\ 4 \sqrt{2} \qquad \textbf{(D)}\ \frac{9 \sqrt{2}}{2} \qquad \textbf{(E)}\ \text{none of these}$

1982 Poland - Second Round, 1

Tags: algebra , polynomial

Prove that if $ c, d $ are integers with $ c \neq d $, $ d > 0 $ then the equation $$ x^3 - 3cx^2 - dx + c = 0$$ has no more than one rational root.

2020 LMT Fall, B28

Tags: algebra

There are $2500$ people in Lexington High School, who all start out healthy. After $1$ day, $1$ person becomes infected with coronavirus. Each subsequent day, there are twice as many newly infected people as on the previous day. How many days will it be until over half the school is infected?

2014 IMAR Test, 3

Tags: algebra , irreducible , polynomial

Let $f$ be a primitive polynomial with integral coefficients (their highest common factor is $1$ ) such that $f$ is irreducible in $\mathbb{Q}[X]$ , and $f(X^2)$ is reducible in $\mathbb{Q}[X]$ . Show that $f= \pm(u^2-Xv^2)$ for some polynomials $u$ and $v$ with integral coefficients.

2021 MMATHS, 7

Tags:

Let $P_k(x) = (x-k)(x-(k+1))$. Kara picks four distinct polynomials from the set $\{P_1(x), P_2(x), P_3(x), \ldots ,$ $P_{12}(x)\}$ and discovers that when she computes the six sums of pairs of chosen polynomials, exactly two of the sums have two (not necessarily distinct) integer roots! How many possible combinations of four polynomials could Kara have picked? [i]Proposed by Andrew Wu[/i]

2018 Dutch IMO TST, 2

Tags: tangent , perpendicular , right triangle , geometry , circumcircle

Suppose a triangle $\vartriangle ABC$ with $\angle C = 90^o$ is given. Let $D$ be the midpoint of $AC$, and let $E$ be the foot of the altitude through $C$ on $BD$. Show that the tangent in $C$ of the circumcircle of $\vartriangle AEC$ is perpendicular to $AB$.

2024 CCA Math Bonanza, L1.1

Tags:

Find the sum of the squares of all solutions to $$(x^2 + 7x + 9)^2 = 9.$$ [i]Lightning 1.1[/i]

MOAA Accuracy Rounds, 2023.3

Tags:

Ms. Raina's math class has 6 students, including the troublemakers Andy and Harry. For a group project, Ms. Raina randomly divides the students into three groups containing 1, 2, and 3 people. The probability that Andy and Harry unfortunately end up in the same group can be expressed in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Proposed by Andy Xu[/i]

2022 Princeton University Math Competition, 8

Tags: number theory

Ryan Alweiss storms into the Fine Hall common room with a gigantic eraser and erases all integers $n$ in the interval $[2, 728]$ such that $3^t$ doesn’t divide $n!$, where $t = \left\lceil \frac{n-3}{2} \right\rceil$. Find the sum of the leftover integers in that interval modulo $1000$.

MOAA Gunga Bowls, 2023.11

Tags:

Let $s(n)$ denote the sum of the digits of $n$ and let $p(n)$ be the product of the digits of $n$. Find the smallest integer $k$ such that $s(k)+p(k)=49$ and $s(k+1)+p(k+1)=68$. [i]Proposed by Anthony Yang[/i]

1985 Austrian-Polish Competition, 2

Tags: graph theory , vertex degree , combinatorics

Suppose that $n\ge 8$ persons $P_1,P_2,\dots,P_n$ meet at a party. Assume that $P_k$ knows $k+3$ persons for $k=1,2,\dots,n-6$. Further assume that each of $P_{n-5},P_{n-4},P_{n-3}$ knows $n-2$ persons, and each of $P_{n-2},P_{n-1},P_n$ knows $n-1$ persons. Find all integers $n\ge 8$ for which this is possible. (It is understood that "to know" is a symmetric nonreflexive relation: if $P_i$ knows $P_j$ then $P_j$ knows $P_i$; to say that $P_i$ knows $p$ persons means: knows $p$ persons other than herself/himself.)