Olympiad problems

2002 Moldova National Olympiad, 1

Tags: modular arithmetic

Find all triplets of primes in the form $ (p, 2p\plus{}1, 4p\plus{}1)$.

2003 USA Team Selection Test, 3

Tags: modular arithmetic , number theory , order

Find all ordered triples of primes $(p, q, r)$ such that \[ p \mid q^r + 1, \quad q \mid r^p + 1, \quad r \mid p^q + 1. \] [i]Reid Barton[/i]

2003 All-Russian Olympiad, 4

Tags: 3d geometry , sphere , tetrahedron , circumcircle , radical axis , geometry

The inscribed sphere of a tetrahedron $ABCD$ touches $ABC,ABD,ACD$ and $BCD$ at $D_1,C_1,B_1$ and $A_1$ respectively. Consider the plane equidistant from $A$ and plane $B_1C_1D_1$ (parallel to $B_1C_1D_1$) and the three planes defined analogously for the vertices $B,C,D$. Prove that the circumcenter of the tetrahedron formed by these four planes coincides with the circumcenter of tetrahedron of $ABCD$.

1993 Brazil National Olympiad, 4

Tags: trigonometry , geometry , circumcircle , law of sines , perpendicular bisector

$ABCD$ is a convex quadrilateral with \[\angle BAC = 30^\circ \]\[\angle CAD = 20^\circ\]\[\angle ABD = 50^\circ\]\[\angle DBC = 30^\circ\] If the diagonals intersect at $P$, show that $PC = PD$.

2000 Harvard-MIT Mathematics Tournament, 5

Tags: geometry

Find the interior angle between two sides of a regular octagon (degrees).

2016 Hanoi Open Mathematics Competitions, 14

Tags: algebra , radical , natural

Given natural numbers $a,b$ such that $2015a^2+a = 2016b^2+b$. Prove that $\sqrt{a-b}$ is a natural number.

2020 BMT Fall, 2

Tags: algebra

Let $m$ be the answer to this question. What is the value of $2m - 5$?

2023 Assam Mathematics Olympiad, 10

Tags:

If $a,b,c \neq 0$, prove that $\frac{a^2+b^2}{c^2} +\frac{b^2+c^2}{a^2} +\frac{c^2+a^2}{b^2} \geq 6$.

2008 Kyiv Mathematical Festival, 1

Tags:

Find all positive integers $ k$ for which equation $ n^m\minus{}m^n\equal{}k$ has solution in positive integers.

2016 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: trapezoid , ratio , geometry

Nine lines are given such that every one of them intersects given square $ABCD$ on two trapezoids, which area ratio is $2 : 3$. Prove that at least $3$ of those $9$ lines pass through the same point

2018 Serbia JBMO TST, 4

Tags: combinatorics

Two players are playing the following game. They are alternatively putting blue and red coins on the board $2018$ by $2018$. If first player creates $n$ blue coins in a row or column, he wins. Second player wins if he can prevent it. Who will win if: $a)n=4$; $b)n=5$? Note: first player puts only blue coins, and second only red.

2022 USAJMO, 2

Tags: combinatorics , grid

Let $a$ and $b$ be positive integers. The cells of an $(a+b+1)\times (a+b+1)$ grid are colored amber and bronze such that there are at least $a^2+ab-b$ amber cells and at least $b^2+ab-a$ bronze cells. Prove that it is possible to choose $a$ amber cells and $b$ bronze cells such that no two of the $a+b$ chosen cells lie in the same row or column.

1981 Miklós Schweitzer, 2

Tags: combinatorics

Consider the lattice $ L$ of the contradictions of a simple graph $ G$ (as sets of vertex pairs) with respect to inclusion. Let $ n \geq 1$ be an arbitrary integer. Show that the identity \[ x \bigwedge \left( \bigvee_{i\equal{}0}^n y_i \right) \equal{} \bigvee_{j\equal{}0}^n \left( x \bigwedge \left( \bigvee_{0 \leq i \leq n, \;i\not\equal{}j\ } y_i \right)\right)\] holds if and only if $ G$ has no cycle of size at least $ n\plus{}2$. [i]A. Huhn[/i]

1977 IMO Shortlist, 8

Tags: geometry , convex quadrilateral , perpendicular

Let $S$ be a convex quadrilateral $ABCD$ and $O$ a point inside it. The feet of the perpendiculars from $O$ to $AB, BC, CD, DA$ are $A_1, B_1, C_1, D_1$ respectively. The feet of the perpendiculars from $O$ to the sides of $S_i$, the quadrilateral $A_iB_iC_iD_i$, are $A_{i+1}B_{i+1}C_{i+1}D_{i+1}$, where $i = 1, 2, 3.$ Prove that $S_4$ is similar to S.

2024 AMC 10, 20

Tags: set

Let $S$ be a subset of $\{1, 2, 3, \dots, 2024\}$ such that the following two conditions hold: - If $x$ and $y$ are distinct elements of $S$, then $|x-y| > 2$ - If $x$ and $y$ are distinct odd elements of $S$, then $|x-y| > 6$. What is the maximum possible number of elements in $S$? $ \textbf{(A) }436 \qquad \textbf{(B) }506 \qquad \textbf{(C) }608 \qquad \textbf{(D) }654 \qquad \textbf{(E) }675 \qquad $

2001 IMO Shortlist, 2

Tags: geometry , circumcenter , triangle , geometric inequality

Consider an acute-angled triangle $ABC$. Let $P$ be the foot of the altitude of triangle $ABC$ issuing from the vertex $A$, and let $O$ be the circumcenter of triangle $ABC$. Assume that $\angle C \geq \angle B+30^{\circ}$. Prove that $\angle A+\angle COP < 90^{\circ}$.

1971 AMC 12/AHSME, 10

Tags:

Each of a group of $50$ girls is blonde or brunette and is blue eyed of brown eyed. If $14$ are blue-eyed blondes, $31$ are brunettes, and $18$ are brown-eyed, then the number of brown-eyed brunettes is $\textbf{(A) }5\qquad\textbf{(B) }7\qquad\textbf{(C) }9\qquad\textbf{(D) }11\qquad \textbf{(E) }13$

2023 AMC 10, 12

Tags: polynomial

When the roots of the polynomial \[P(x)=\prod_{i=1}^{10}(x-i)^{i}\] are removed from the real number line, what remains is the union of $11$ disjoint open intervals. On how many of those intervals is $P(x)$ positive? $\textbf{(A)}~3\qquad\textbf{(B)}~4\qquad\textbf{(C)}~5\qquad\textbf{(D)}~6\qquad\textbf{(E)}~7$

2022 JHMT HS, 5

Tags: geometry

Suppose $\triangle JHU$ satisfies $JH = JU = 44$ and $HU = 32$. There is a unique circle passing through $U$ that is tangent to $\overline{JH}$ at its midpoint; let this circle intersect $\overline{JU}$ and $\overline{HU}$ again at points $X \neq U$ and $Y \neq U$, respectively. Let $Z$ be the unique point on $\overline{JH}$ such that $JZ = XU$. Compute the perimeter of quadrilateral $UXZY$.

2002 District Olympiad, 1

Tags: algebra

Let $x, y, z$ be positive real numbers such that $xyz(x+y+z) = 1$. Show that the following equality holds: $$\sqrt{\left( x^2+\frac{1}{y^2}\right)\left( y^2+\frac{1}{z^2}\right)\left( z^2+\frac{1}{x^2}\right)}=(x+y)(y+z)(z+x)$$ Find some numbers $x ,y ,z$ which satisfy the given property.

1935 Eotvos Mathematical Competition, 3

Tags: geometry , combinatorics , combinatorial geometry , 3d geometry , prism

A real number is assigned to each vertex of a triangular prism so that the number on any vertex is the arithmetic mean of the numbers on the three adjacent vertices. Prove that all six numbers are equal.

2007 IMC, 5

Tags: function , modular arithmetic , algebra , polynomial , induction

Let $ n$ be a positive integer and $ a_{1}, \ldots, a_{n}$ be arbitrary integers. Suppose that a function $ f: \mathbb{Z}\to \mathbb{R}$ satisfies $ \sum_{i=1}^{n}f(k+a_{i}l) = 0$ whenever $ k$ and $ l$ are integers and $ l \ne 0$. Prove that $ f = 0$.

1958 Czech and Slovak Olympiad III A, 4

Tags: locus , geometry , 3d geometry

Consider positive numbers $d,v$ such that $d>v$. Moreover, consider two perpendicular skew lines $p,q$ of distance $v$ (that is direction vectors of both lines are orthogonal and $\min_{X\in p,Y\in q}XY = v$). Finally, consider all line segments $PQ$ such that $P\in p, Q\in q, PQ=d$. a) Find the locus of all points $P$. b) Find the locus of all midpoints of segments $PQ$.

2015 German National Olympiad, 4

Tags: divisibility , number theory

Let $k$ be a positive integer. Define $n_k$ to be the number with decimal representation $70...01$ where there are exactly $k$ zeroes. Prove the following assertions: a) None of the numbers $n_k$ is divisible by $13$. b) Infinitely many of the numbers $n_k$ are divisible by $17$.

2013 Bangladesh Mathematical Olympiad, 4

Tags: algebra

Higher Secondary P4 If the fraction $\dfrac{a}{b}$ is greater than $\dfrac{31}{17}$ in the least amount while $b<17$, find $\dfrac{a}{b}$.