Olympiad problems

2025 Harvard-MIT Mathematics Tournament, 1

Tags: combinatorics

Compute the number of ways to arrange the numbers $1, 2, 3, 4, 5, 6,$ and $7$ around a circle such that the product of every pair of adjacent numbers on the circle is at most $20.$ (Rotations and reflections count as different arrangements.)

2015 Purple Comet Problems, 18

Tags:

Deﬁne the determinant $D_1$ = $|1|$, the determinant $D_2$ = $|1 1|$ $|1 3|$ , and the determinant $D_3=$ |1 1 1| |1 3 3| |1 3 5| . In general, for positive integer n, let the determinant $D_n$ have 1s in every position of its ﬁrst row and ﬁrst column, 3s in the remaining positions of the second row and second column, 5s in the remaining positions of the third row and third column, and so forth. Find the least n so that $D_n$ $\geq$ 2015.

1999 Yugoslav Team Selection Test, Problem 2

Tags: right triangle , harmonic division , inversion , geometry

Let $ABC$ be a triangle such that $\angle A=90^{\circ }$ and $\angle B<\angle C$. The tangent at $A$ to the circumcircle $\omega$ of triangle $ABC$ meets the line $BC$ at $D$. Let $E$ be the reflection of $A$ in the line $BC$, let $X$ be the foot of the perpendicular from $A$ to $BE$, and let $Y$ be the midpoint of the segment $AX$. Let the line $BY$ intersect the circle $\omega$ again at $Z$. Prove that the line $BD$ is tangent to the circumcircle of triangle $ADZ$. [hide="comment"] [i]Edited by Orl.[/i] [/hide]

2001 AMC 10, 7

Tags:

When the decimal point of a certain positive decimal number is moved four places to the right, the new number is four times the reciprocal of the original number. What is the original number? $ \textbf{(A) }0.0002\qquad\textbf{(B) }0.002\qquad\textbf{(C) }0.02\qquad\textbf{(D) }0.2\qquad\textbf{(E) }2$

2005 Grigore Moisil Urziceni, 2

Tags: primitive , real analysis , calculus , function

Let be a function $ f:\mathbb{R}\longrightarrow\mathbb{R}_{\ge 0} $ that admits primitives and such that $ \lim_{x\to 0 } \frac{f(x)}{x} =0. $ Prove that the function $ g:\mathbb{R}\longrightarrow\mathbb{R} , $ defined as $$ g(x)=\left\{ \begin{matrix} f(x)/x ,&\quad x\neq 0\\ 0,& \quad x=0 \end{matrix} \right. , $$ is primitivable.

2023 SEEMOUS, P3

Tags: matrix , linear algebra

Prove that if $A{}$ is an $n\times n$ matrix with complex entries such that $A+A^*=A^2A^*$ then $A=A^*$. (Here, we denote by $M^*$ the conjugate transpose $\overline{M}^t$ of the matrix $M{}$).

2020 BAMO, 4

Tags:

Consider $\triangle ABC$. Choose a point $M$ on side $BC$ and let $O$ be the center of the circle passing through the vertices of $\triangle ABM$. Let $k$ be the circle that passes through $A$ and $M$ and whose center lies on $BC$. Let line $MO$ intersect $K$ again in point $K$. Prove that the line $BK$ is the same for any point $M$ on segment $BC$, so long as all of these constructions are well-defined. [i]Proposed by Evan Chen[/i]

2013 Sharygin Geometry Olympiad, 3

Tags: convex polygon , projection , geometry

Each vertex of a convex polygon is projected to all nonadjacent sidelines. Can it happen that each of these projections lies outside the corresponding side?

1961 Putnam, B3

Tags: circumcircle , geometry

Consider four points in the plane, no three of which are collinear, and such that the circle through three of them does not pass through the fourth. Prove that one of the four points can be selected having the property that it lies inside the circle determined by the other three.

MIPT student olimpiad autumn 2024, 3

Tags: function , real analysis

$\exists ? f: R\to R$ continuos function that: $\forall x_0\in R \lim\limits_{x \to x_0} \frac{|f(x)-f(x_0)|}{|x-x_0|}=+\infty$

2007 ITest, 29

Tags: modular arithmetic

Let $S$ be equal to the sum \[1+2+3+\cdots+2007.\] Find the remainder when $S$ is divided by $1000$.

2019 India IMO Training Camp, P1

Tags: number theory , polynomial

Determine all non-constant monic polynomials $f(x)$ with integer coefficients for which there exists a natural number $M$ such that for all $n \geq M$, $f(n)$ divides $f(2^n) - 2^{f(n)}$ [i] Proposed by Anant Mudgal [/i]

1972 IMO Longlists, 6

Tags: trigonometry , inequalities

Prove the inequality \[(n + 1)\cos\frac{\pi}{n + 1}- n\cos\frac{\pi}{n}> 1\] for all natural numbers $n \ge 2.$

2022 District Olympiad, P2

Tags: group theory , abelian group

Let $(G,\cdot)$ be a group and $H\neq G$ be a subgroup so that $x^2=y^2$ for all $x,y\in G\setminus H.$ Show that $(H,\cdot)$ is an Abelian group.

2009 Purple Comet Problems, 12

Tags: geometry

In isosceles triangle $ABC$ sides $AB$ and $BC$ have length $125$ while side $AC$ has length $150$. Point $D$ is the midpoint of side $AC$. $E$ is on side $BC$ so that $BC$ and $DE$ are perpendicular. Similarly, $F$ is on side $AB$ so that $AB$ and $DF$ are perpendicular. Find the area of triangle $DEF$.

2019 Stanford Mathematics Tournament, 1

Tags: geometry

Let $ABCD$ be a quadrilateral with $\angle DAB = \angle ABC = 120^o$. If $AB = 3$, $BC = 2$, and $AD = 4$, what is the length of $CD$?

2017 Regional Olympiad of Mexico West, 5

Tags: quadratic , algebra

Laura and Daniel play with quadratic polynomials. First Laura says a nonzero real number $r$. Then Daniel says a nonzero real number $s$, and then again Laura says another nonzero real number $t$. Finally. Daniel writes the polynomial $P(x) = ax^2 + bx + c$ where $a,b$, and $c$ are $r,s$, and $t$ in some order Daniel chooses. Laura wins if the equation $P(x) = 0$ has two different real solutions, and Daniel wins otherwise. Determine who has a winning strategy and describe that strategy.

2015 Latvia Baltic Way TST, 12

Tags: algebra , inequalities

For real positive numbers $a, b, c$, the equality $abc = 1$ holds. Prove that $$\frac{a^{2014}}{1 + 2 bc}+\frac{b^{2014}}{1 + 2ac}+\frac{c^{2014}}{1 + 2ab} \ge \frac{3}{ab+bc+ca}.$$

2020 Cono Sur Olympiad, 3

Tags: circumcircle , butterfly theorem , lines meeting at circmucircle , geometry

Let $ABC$ be an acute triangle such that $AC<BC$ and $\omega$ its circumcircle. $M$ is the midpoint of $BC$. Points $F$ and $E$ are chosen in $AB$ and $BC$, respectively, such that $AC=CF$ and $EB=EF$. The line $AM$ intersects $\omega$ in $D\neq A$. The line $DE$ intersects the line $FM$ in $G$. Prove that $G$ lies on $\omega$.

2024 Israel TST, P2

Tags: combinatorics , function , tree , average

A positive integer $N$ is given. Panda builds a tree on $N$ vertices, and writes a real number on each vertex, so that $1$ plus the number written on each vertex is greater or equal to the average of the numbers written on the neighboring vertices. Let the maximum number written be $M$ and the minimal number written $m$. Mink then gives Panda $M-m$ kilograms of bamboo. What is the maximum amount of bamboo Panda can get?

2024 Malaysian IMO Training Camp, 4

Tags: algebra

Fix a real polynomial $P$ with degree at least $1$, and a real number $c$. Prove that there exist a real number $k$ such that for all reals $a$ and $b$, $$P(a)+P(b)=c \quad \Rightarrow \quad |a+b|<k$$ [i]Proposed by Wong Jer Ren[/i]

2018 AIME Problems, 2

Tags: abstract algebra

Let $a_0 = 2$, $a_1 = 5$, and $a_2 = 8$, and for $n>2$ define $a_n$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$. Find $a_{2018}\cdot a_{2020}\cdot a_{2022}$.

2011 Junior Balkan MO, 3

Tags: symmetry , geometry , rhombus , ratio , combinatorics

Let $n>3$ be a positive integer. Equilateral triangle ABC is divided into $n^2$ smaller congruent equilateral triangles (with sides parallel to its sides). Let $m$ be the number of rhombuses that contain two small equilateral triangles and $d$ the number of rhombuses that contain eight small equilateral triangles. Find the difference $m-d$ in terms of $n$.

2009 IMO Shortlist, 2

Tags: geometry , circumcircle , reflection

Let $ ABC$ be a triangle with circumcentre $ O$. The points $ P$ and $ Q$ are interior points of the sides $ CA$ and $ AB$ respectively. Let $ K,L$ and $ M$ be the midpoints of the segments $ BP,CQ$ and $ PQ$. respectively, and let $ \Gamma$ be the circle passing through $ K,L$ and $ M$. Suppose that the line $ PQ$ is tangent to the circle $ \Gamma$. Prove that $ OP \equal{} OQ.$ [i]Proposed by Sergei Berlov, Russia [/i]

2012 ELMO Shortlist, 5

Tags: polynomial , relatively prime , number theory , pigeonhole principle , algebra

Prove that if $m,n$ are relatively prime positive integers, $x^m-y^n$ is irreducible in the complex numbers. (A polynomial $P(x,y)$ is irreducible if there do not exist nonconstant polynomials $f(x,y)$ and $g(x,y)$ such that $P(x,y) = f(x,y)g(x,y)$ for all $x,y$.) [i]David Yang.[/i]