Olympiad problems

2018 Stanford Mathematics Tournament, 2

Tags: geometry , trapezoid

Let $ABCD$ be a trapezoid with $AB$ parallel to $CD$ and perpendicular to $BC$. Let $M$ be a point on $BC$ such that $\angle AMB = \angle DMC$. If $AB = 3$, $BC = 24$, and $CD = 4$, what is the value of $AM + MD$?

2004 Oral Moscow Geometry Olympiad, 1

Tags: midpoint , geometry , ratio

$E$ and $F$ are the midpoints of the sides $BC$ and $AD$ of the convex quadrilateral $ABCD$. Prove that the segment $EF$ divides the diagonals $AC$ and $BD$ in the same ratio.

BIMO 2022, 4

Tags: algebra , polynomial

Given a polynomial $P\in \mathbb{Z}[X]$ of degree $k$, show that there always exist $2d$ distinct integers $x_1, x_2, \cdots x_{2d}$ such that $$P(x_1)+P(x_2)+\cdots P(x_{d})=P(x_{d+1})+P(x_{d+2})+\cdots + P(x_{2d})$$ for some $d\le k+1$. [Extra: Is this still true if $d\le k$? (Of course false for linear polynomials, but what about higher degree?)]

2016 Taiwan TST Round 3, 2

Tags: geometry

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

2023 Macedonian Team Selection Test, Problem 1

Tags: number theory

Let $s(n)$ denote the smallest prime divisor and $d(n)$ denote the number of positive divisors of a positive integer $n>1$. Is it possible to choose $2023$ positive integers $a_{1},a_{2},...,a_{2023}$ with $a_{1}<a_{2}-1<...<a_{2023}-2022$ such that for all $k=1,...,2022$ we have $d(a_{k+1}-a_{k}-1)>2023^{k}$ and $s(a_{k+1}-a_{k}) > 2023^{k}$? [i]Authored by Nikola Velov[/i]

2021 Benelux, 2

Tags: combinatorics

Pebbles are placed on the squares of a $2021\times 2021$ board in such a way that each square contains at most one pebble. The pebble set of a square of the board is the collection of all pebbles which are in the same row or column as this square. (A pebble belongs to the pebble set of the square in which it is placed.) What is the least possible number of pebbles on the board if no two squares have the same pebble set?

2013 Estonia Team Selection Test, 5

Tags: number theory , power

Call a tuple $(b_m, b_{m+1},..., b_n)$ of integers perfect if both following conditions are fulfilled: 1. There exists an integer $a > 1$ such that $b_k = a^k + 1$ for all $k = m, m + 1,..., n$ 2. For all $k = m, m + 1,..., n,$ there exists a prime number $q$ and a non-negative integer $t$ such that $b_k = q^t$. Prove that if $n - m$ is large enough then there is no perfect tuples, and find all perfect tuples with the maximal number of components.

2022 Bosnia and Herzegovina BMO TST, 2

Tags: number theory

Determine all positive integers $A= \overline{a_n a_{n-1} \ldots a_1 a_0}$ such that not all of its digits are equal and no digit is $0$, and $A$ divides all numbers of the following form: $A_1 = \overline{a_0 a_n a_{n-1} \ldots a_2 a_1}, A_2 = \overline{a_1 a_0 a_{n} \ldots a_3 a_2}, \ldots ,$ $ A_{n-1} = \overline{a_{n-2} a_{n-3} \ldots a_0 a_n a_{n-1}}, A_n = \overline{a_{n-1} a_{n-2} \ldots a_1 a_0 a_n}$.

1988 Brazil National Olympiad, 5

Tags: combinatorics , game strategy , coloring

A figure on a computer screen shows $n$ points on a sphere, no four coplanar. Some pairs of points are joined by segments. Each segment is colored red or blue. For each point there is a key that switches the colors of all segments with that point as endpoint. For every three points there is a sequence of key presses that makes the three segments between them red. Show that it is possible to make all the segments on the screen red. Find the smallest number of key presses that can turn all the segments red, starting from the worst case.

2013 Tournament of Towns, 2

Tags: digit , number theory , composite

Does there exist a ten-digit number such that all its digits are different and after removing any six digits we get a composite four-digit number?

1997 Slovenia Team Selection Test, 4

Tags: trigonometry , trig identities , law of cosines

Let $ABC$ be an equilateral triangle and let $P$ be a point in its interior. Let the lines $AP$, $BP$, $CP$ meet the sides $BC$, $CA$, $AB$ at the points $A_1$, $B_1$, $C_1$, respectively. Prove that $A_1B_1 \cdot B_1C_1 \cdot C_1A_1 \ge A_1B \cdot B_1C \cdot C_1A$.

2015 CCA Math Bonanza, TB1

Tags:

Compute the greatest $4$-digit number $\underline{ABCD}$ such that $(A^3+B^2)(C^3+D^2)=2015$. [i]2015 CCA Math Bonanza Tiebreaker Round #1[/i]

2024 CMIMC Integration Bee, 2

Tags: calculus , integration

\[\int_0^2 |\sin(\pi x)|+|\cos(\pi x)|\mathrm dx\] [i]Proposed by Anagh Sangavarapu[/i]

2013 CIIM, Problem 6

Tags: topology

Let $(X,d)$ be a metric space with $d:X\times X \to \mathbb{R}_{\geq 0}$. Suppose that $X$ is connected and compact. Prove that there exists an $\alpha \in \mathbb{R}_{\geq 0}$ with the following property: for any integer $n > 0$ and any $x_1,\dots,x_n \in X$, there exists $x\in X$ such that the average of the distances from $x_1,\dots,x_n$ to $x$ is $\alpha$ i.e. $$\frac{d(x,x_1)+d(x,x_2)+\cdots+d(x,x_n)}{n} = \alpha.$$

2011 Danube Mathematical Competition, 1

Tags: geometry , parallelogram , equal angles , quadrilateral

Let $ABCM$ be a quadrilateral and $D$ be an interior point such that $ABCD$ is a parallelogram. It is known that $\angle AMB =\angle CMD$. Prove that $\angle MAD =\angle MCD$.

2013 Korea National Olympiad, 7

Tags: vieta , algebra

For positive integer $k$, define integer sequence $\{ b_n \}, \{ c_n \} $ as follows: \[ b_1 = c_1 = 1 \] \[ b_{2n} = kb_{2n-1} + (k-1)c_{2n-1}, c_{2n} = b_{2n-1} + c_{2n-1} \] \[ b_{2n+1} = b_{2n} + (k-1)c_{2n}, c_{2n+1} = b_{2n} + kc_{2n} \] Let $a_k = b_{2014} $. Find the value of \[ \sum_{k=1}^{100} { (a_k - \sqrt{{a_k}^2-1} )^{ \frac{1}{2014}} }\]

2003 AMC 8, 7

Tags:

Blake and Jenny each took four $100$ point tests. Blake averaged $78$ on the four tests. Jenny scored $10$ points higher than Blake on the first test, $10$ points lower on the second test, and $20$ points higher on both the third and fourth test. What is the difference between Blake's average on the four tests and Jenny's average on the four tests? $\textbf{(A)}\ 10 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 20 \qquad \textbf{(D)}\ 25 \qquad \textbf{(E)}\ 40$

LMT Team Rounds 2010-20, 2020.S15

Tags:

Let $\phi(k)$ denote the number of positive integers less than or equal to $k$ that are relatively prime to $k$. For example, $\phi(2)=1$ and $\phi(10)=4$. Compute the number of positive integers $n \leq 2020$ such that $\phi(n^2)=2\phi(n)^2$.

2004 Baltic Way, 14

Tags: modular arithmetic , induction , geometry , combinatorics

We say that a pile is a set of four or more nuts. Two persons play the following game. They start with one pile of $n \geq 4$ nuts. During a move a player takes one of the piles that they have and split it into two nonempty sets (these sets are not necessarily piles, they can contain arbitrary number of nuts). If the player cannot move, he loses. For which values of $n$ does the first player have a winning strategy?

1994 National High School Mathematics League, 1

Tags: complex number

In the equation $x^2+z_1x+z_2+m=0$, $z_1,z_2,m$ are complex numbers, and $z_1^2-4z_2=16+20\text{i}$. Two roots of the equations are $\alpha,\beta$. If $|\alpha-\beta|=2\sqrt7$, find the maximum and minumum value of $|m|$.

2017 HMNT, 8

Tags: number theory

Find the number of quadruples $(a, b, c, d)$ of integers with absolute value at most $5$ such that $$(a^2 + b^2 + c^2 + d^2)^2 = (a + b + c + d)(a - b + c -d)((a - c)^2 + (b - d)^2).$$

2002 AMC 10, 7

Tags:

The dimensions of a rectangular box in inches are all positive integers and the volume of the box is $2002\text{ in}^3$. Find the minimum possible sum in inches of the three dimensions. $\textbf{(A) }36\qquad\textbf{(B) }38\qquad\textbf{(C) }42\qquad\textbf{(D) }44\qquad\textbf{(E) }92$

2018 AIME Problems, 12

Tags:

For every subset $T$ of $U = \{ 1,2,3,\ldots,18 \}$, let $s(T)$ be the sum of the elements of $T$, with $s(\emptyset)$ defined to be $0$. If $T$ is chosen at random among all subsets of $U$, the probability that $s(T)$ is divisible by $3$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m$.

2013 EGMO, 6

Tags: pigeonhole principle , combinatorics

Snow White and the Seven Dwarves are living in their house in the forest. On each of $16$ consecutive days, some of the dwarves worked in the diamond mine while the remaining dwarves collected berries in the forest. No dwarf performed both types of work on the same day. On any two different (not necessarily consecutive) days, at least three dwarves each performed both types of work. Further, on the first day, all seven dwarves worked in the diamond mine. Prove that, on one of these $16$ days, all seven dwarves were collecting berries.

2006 IberoAmerican Olympiad For University Students, 3

Tags: polynomial , induction , trigonometry , algebra

Let $p_1(x)=p(x)=4x^3-3x$ and $p_{n+1}(x)=p(p_n(x))$ for each positive integer $n$. Also, let $A(n)$ be the set of all the real roots of the equation $p_n(x)=x$. Prove that $A(n)\subseteq A(2n)$ and that the product of the elements of $A(n)$ is the average of the elements of $A(2n)$.