Olympiad problems

2000 Belarusian National Olympiad, 4

Tags: geometry , trapezoid

The lateral sides and diagonals of a trapezoid intersect a line $l$, determining three equal segments on it. Must $l$ be parallel to the bases of the trapezoid?

1981 Austrian-Polish Competition, 8

Tags: combinatorial geometry , combinatorics , Plane , lines , parallel

The plane has been partitioned into $N$ regions by three bunches of parallel lines. What is the least number of lines needed in order that $N > 1981$?

2025 Malaysian IMO Training Camp, 6

Tags: number theory

Let $a_1, a_2, \ldots, a_{2024}$ be positive integers such that $a_{i+1}+1$ is a multiple of $a_i$ for all $i = 1, 2, \ldots , 2024$, with indices taken modulo $2024$. Find the maximum possible value of $a_1 + a_2 + \ldots + a_{2024}$. [i](Proposed by Ivan Chan Guan Yu)[/i]

1968 IMO Shortlist, 7

Tags: trigonometry , geometric inequality , Triangle , IMO Shortlist

Prove that the product of the radii of three circles exscribed to a given triangle does not exceed $A=\frac{3\sqrt 3}{8}$ times the product of the side lengths of the triangle. When does equality hold?

1986 China National Olympiad, 1

Tags: inequalities

We are given $n$ reals $a_1,a_2,\cdots , a_n$ such that the sum of any two of them is non-negative. Prove that the following statement and its converse are both true: if $n$ non-negative reals $x_1,x_2,\cdots ,x_n$ satisfy $x_1+x_2+\cdots +x_n=1$, then the inequality $a_1x_1+a_2x_2+\cdots +a_nx_n\ge a_1x^2_1+ a_2x^2_2+\cdots + a_nx^2_n$ holds.

2022 BMT, 5

Tags: geometry

Steve has a tricycle which has a front wheel with a radius of $30$ cm and back wheels with radii of $10$ cm and $9$ cm. The axle passing through the centers of the back wheels has a length of $40$ cm and is perpendicular to both planes containing the wheels. Since the tricycle is tilted, it goes in a circle as Steve pedals. Steve rides the tricycle until it reaches its original position, so that all of the wheels do not slip or leave the ground. The tires trace out concentric circles on the ground, and the radius of the circle the front wheel traces is the average of the radii of the other two traced circles. Compute the total number of degrees the front wheel rotates. (Express your answer in simplest radical form.)

2014 Contests, 2

Tags: calculus , integration , arithmetic sequence , algebra proposed , algebra

The roots of the equation \[ x^3-3ax^2+bx+18c=0 \] form a non-constant arithmetic progression and the roots of the equation \[ x^3+bx^2+x-c^3=0 \] form a non-constant geometric progression. Given that $a,b,c$ are real numbers, find all positive integral values $a$ and $b$.

2002 Federal Competition For Advanced Students, Part 1, 1

Tags: number theory proposed , number theory

Determine all integers $a$ and $b$ such that \[(19a + b)^{18} + (a + b)^{18} + (a + 19b)^{18}\] is a perfect square.

2002 District Olympiad, 1

Tags: combinatorics , number theory

Find the number of representations of the number $180$ in the form $180 =x+y+z$, where $x, y, z$ are positive integers that are proportional with some three consecutive positive integers

ICMC 5, 2

Tags: table , combinatorics , number theory , college contests , ICMC

Find all integers $n$ for which there exists a table with $n$ rows, $2022$ columns, and integer entries, such that subtracting any two rows entry-wise leaves every remainder modulo $2022$. [i]Proposed by Tony Wang[/i]

1988 AMC 12/AHSME, 2

Tags: ratio

Triangles $ABC$ and $XYZ$ are similar, with $A$ corresponding to $X$ and $B$ to $Y$. If $AB=3$, $BC=4$, and $XY=5$, then $YZ$ is: $ \textbf{(A)}\ 3\frac 3 4 \qquad \textbf{(B)}\ 6 \qquad \textbf{(C)}\ 6\frac 1 4 \qquad \textbf{(D)}\ 6\frac 2 3 \qquad \textbf{(E)}\ 8$

2003 USAMO, 5

Tags: AMC , USA(J)MO , USAMO , inequalities , function , calculus

Let $ a$, $ b$, $ c$ be positive real numbers. Prove that \[ \dfrac{(2a \plus{} b \plus{} c)^2}{2a^2 \plus{} (b \plus{} c)^2} \plus{} \dfrac{(2b \plus{} c \plus{} a)^2}{2b^2 \plus{} (c \plus{} a)^2} \plus{} \dfrac{(2c \plus{} a \plus{} b)^2}{2c^2 \plus{} (a \plus{} b)^2} \le 8. \]

1974 Putnam, A6

Tags: Putnam , algebra , polynomial

Given $n$, let $k = k(n)$ be the minimal degree of any monic integral polynomial $$f(x)=x^k + a_{k-1}x^{k-1}+\ldots+a_0$$ such that the value of $f(x)$ is exactly divisible by $n$ for every integer $x.$ Find the relationship between $n$ and $k(n)$. In particular, find the value of $k(n)$ corresponding to $n = 10^6.$

V Soros Olympiad 1998 - 99 (Russia), 11.10

Tags: geometry , concurrency

Consider a circle tangent to sides $AB$ and $AC$ (these sides are not equal) of triangle $ABC$ and the circumscribed circle around it. Let $K$, $M$ and $P$ be the touchpoints of this circle with the sides of the triangle and with the circle circumscribed around it, respectively, and let $L$ be the midpoint of the arc $BC$ (not containing $A$). Prove that the lines $KM$, $PL$ and $BC$ intersect at one point.

2017 Dutch IMO TST, 3

Tags: algebra

let $x,y$ be non-zero reals such that : $x^3+y^3+3x^2y^2=x^3y^3$ find all values of $\frac{1}{x}+\frac{1}{y}$

1983 IMO Longlists, 37

Tags: combinatorics unsolved , combinatorics

The points $A_1,A_2, \ldots , A_{1983}$ are set on the circumference of a circle and each is given one of the values $\pm 1$. Show that if the number of points with the value $+1$ is greater than $1789$, then at least $1207$ of the points will have the property that the partial sums that can be formed by taking the numbers from them to any other point, in either direction, are strictly positive.

2023/2024 Tournament of Towns, 5

Tags: combinatorics

5. Tom has 13 weight pieces that look equal, however 12 of them weigh the same and the 13th piece is fake and weighs more than the others. He also has two balances: one shows correctly which pan is heavier or that their weights are equal, the other one gives the correct result when the weights on the pans differ, and gives a random result when the weights are equal. (Tom does not know which balance is which). Tom can choose the balance before each weighting. Prove that he can surely determine the fake weight piece in three weighings. Andrey Arzhantsev

2013 Purple Comet Problems, 5

Tags:

How many four-digit positive integers have exactly one digit equal to $1$ and exactly one digit equal to $3$?

2010 F = Ma, 12

Tags: 2010 , Problem 12

A ball with mass $m$ projected horizontally off the end of a table with an initial kinetic energy $K$. At a time $t$ after it leaves the end of the table it has kinetic energy $3K$. What is $t$? Neglect air resistance. (A) $(3/g)\sqrt{K/m}$ (B) $(2/g)\sqrt{K/m}$ (C) $(1/g)\sqrt{8K/m}$ (D) $(K/g)\sqrt{6/m}$ (E) $(2K/g)\sqrt{1/m}$

2024 Myanmar IMO Training, 1

Tags: geometry

Let $ABC$ be a triangle with $AC > BC,$ let $\omega$ be the circumcircle of $\triangle ABC,$ and let $r$ be its radius. Point $P$ is chosen on $\overline{AC}$ such taht $BC=CP,$ and point $S$ is the foot of the perpendicular from $P$ to $\overline{AB}$. Ray $BP$ mets $\omega$ again at $D$. Point $Q$ is chosen on line $SP$ such that $PQ = r$ and $S,P,Q$ lie on a line in that order. Finally, let $E$ be a point satisfying $\overline{AE} \perp \overline{CQ}$ and $\overline{BE} \perp \overline{DQ}$. Prove that $E$ lies on $\omega$.

2022 MMATHS, 11

Tags: MMATHS , complex numbers , algebra

Denote by $Re(z)$ and $Im(z)$ the real part and imaginary part, respectively, of a complex number $z$; that is, if $z = a + bi$, then $Re(z) = a$ and $Im(z) = b$. Suppose that there exists some real number $k$ such that $Im \left( \frac{1}{w} \right) = Im \left( \frac{k}{w^2} \right) = Im \left( \frac{k}{w^3} \right) $ for some complex number $w$ with $||w||=\frac{\sqrt3}{2}$ , $Re(w) > 0$, and $Im(w) \ne 0$. If $k$ can be expressed as $\frac{\sqrt{a}-b}{c}$ for integers $a$, $b$, $c$ with $a$ squarefree, find $a + b + c$.

CVM 2020, Problem 2+

Tags: algebra

Find all the real solutions to $$n=\sum_{i=1}^n x_i=\sum_{1\le i<j\le n} x_ix_j$$ [i]Proposed by Carlos Dominguez, Valle[/i]

2008 JBMO Shortlist, 9

Tags: JBMO , geometry

Let $O$ be a point inside the parallelogram $ABCD$ such that $\angle AOB + \angle COD = \angle BOC + \angle AOD$. Prove that there exists a circle $k$ tangent to the circumscribed circles of the triangles $\vartriangle AOB, \vartriangle BOC, \vartriangle COD$ and $\vartriangle DOA$.

2009 District Olympiad, 3

Tags: geometry , 3D geometry , prism , planes , angle

Consider the regular quadrilateral prism $ABCDA'B'C 'D'$, in which $AB = a,AA' = \frac{a \sqrt {2}}{2}$, and $M$ is the midpoint of $B' C'$. Let $F$ be the foot of the perpendicular from $B$ on line $MC$, Let determine the measure of the angle between the planes $(BDF)$ and $(HBS)$.

2022 Germany Team Selection Test, 3

Tags: number theory , IMO Shortlist , AZE IMO TST

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]