This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 85335

2015 South East Mathematical Olympiad, 7

Tags: geometry
In $\triangle ABC$, we have $AB>AC>BC$. $D,E,F$ are the tangent points of the inscribed circle of $\triangle ABC$ with the line segments $AB,BC,AC$ respectively. The points $L,M,N$ are the midpoints of the line segments $DE,EF,FD$. The straight line $NL$ intersects with ray $AB$ at $P$, straight line $LM$ intersects ray $BC$ at $Q$ and the straight line $NM$ intersects ray $AC$ at $R$. Prove that $PA \cdot QB \cdot RC = PD \cdot QE \cdot RF$.

1991 All Soviet Union Mathematical Olympiad, 556

$X$ is a set with $100$ members. What is the smallest number of subsets of $X$ such that every pair of elements belongs to at least one subset and no subset has more than $50$ members? What is the smallest number if we also require that the union of any two subsets has at most $80$ members?

Kyiv City MO Seniors 2003+ geometry, 2016.11.4.1

In the triangle $ABC$ the angle bisector $AD$ is drawn, $E$ is the point of tangency of the inscribed circle to the side $BC$, $I$ is the center of the inscribed circle $\Delta ABC$. The point ${{A} _ {1}}$ on the circumscribed circle $\Delta ABC$ is such that $A {{A} _ {1}} || BC$. Denote by $T$ - the second point of intersection of the line $E {{A} _ {1}}$ and the circumscribed circle $\Delta AED$. Prove that $IT = IA$.

1994 IMO Shortlist, 4

Let $ ABC$ be an isosceles triangle with $ AB \equal{} AC$. $ M$ is the midpoint of $ BC$ and $ O$ is the point on the line $ AM$ such that $ OB$ is perpendicular to $ AB$. $ Q$ is an arbitrary point on $ BC$ different from $ B$ and $ C$. $ E$ lies on the line $ AB$ and $ F$ lies on the line $ AC$ such that $ E, Q, F$ are distinct and collinear. Prove that $ OQ$ is perpendicular to $ EF$ if and only if $ QE \equal{} QF$.

2015 Caucasus Mathematical Olympiad, 5

Let's call a natural number a palindrome, the decimal notation of which is equally readable from left to right and right to left (decimal notation cannot start from zero; for example, the number $1221$ is a palindrome, but the numbers $1231, 1212$ and $1010$ are not). Which palindromes among the numbers from $10,000$ to $999,999$ have an odd sum of digits, which have an one even, and how many times are the ones with odd sum more than the ones with the even sum?

2016 Harvard-MIT Mathematics Tournament, 10

Tags:
Kristoff is planning to transport a number of indivisible ice blocks with positive integer weights from the north mountain to Arendelle. He knows that when he reaches Arendelle, Princess Anna and Queen Elsa will name an ordered pair $(p,q)$ of nonnegative integers satisfying $p + q \le 2016$. Kristoff must then give Princess Anna \emph{exactly} $p$ kilograms of ice. Afterward, he must give Queen Elsa $\emph{exactly}$ $q$ kilograms of ice. What is the minimum number of blocks of ice Kristoff must carry to guarantee that he can always meet Anna and Elsa's demands, regardless of which $p$ and $q$ are chosen?

2016 CMIMC, 7

There are eight people, each with their own horse. The horses are arbitrarily arranged in a line from left to right, while the people are lined up in random order to the left of all the horses. One at a time, each person moves rightwards in an attempt to reach their horse. If they encounter a mounted horse on their way to their horse, the mounted horse shouts angrily at the person, who then scurries home immediately. Otherwise, they get to their horse safely and mount it. The expected number of people who have scurried home after all eight people have attempted to reach their horse can be expressed as simplified fraction $\tfrac{m}{n}$. Find $m+n$.

2017 SDMO (High School), 1

Tags:
As shown in the diagram, three circles of radius $1$ are all externally tangent to each other, and there are two circles that are tangent to all three of these circles. Find the area of the shaded region. [asy] size(5cm); real r=2/sqrt(3)-1, R=2/sqrt(3)+1; pair O=(0,0), C_1=O+(0,2/sqrt(3)), C_2=O+(-1,-1/sqrt(3)), C_3=O+(1,-1/sqrt(3)); fill(circle(O,R),rgb(0.5,0.5,0.5)); draw(circle(O,R)); fill(circle(C_1,1),rgb(1,1,1)); draw(circle(C_1,1)); fill(circle(C_2,1),rgb(1,1,1)); draw(circle(C_2,1)); fill(circle(C_3,1),rgb(1,1,1)); draw(circle(C_3,1)); fill(circle(O,r),rgb(1,1,1)); draw(circle(O,r)); [/asy]

2019 PUMaC Geometry B, 5

Tags: geometry
Let $BC=6$, $BX=3$, $CX=5$, and let $F$ be the midpoint of $\overline{BC}$. Let $\overline{AX}\perp\overline{BC}$ and $AF=\sqrt{247}$. If $AC$ is of the form $\sqrt{b}$ and $AB$ is of the form $\sqrt{c}$ where $b$ and $c$ are nonnegative integers, find $2c+3b$.

2015 AMC 10, 21

Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. However, instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose the Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$? $\textbf{(A) } 9 \qquad\textbf{(B) } 11 \qquad\textbf{(C) } 12 \qquad\textbf{(D) } 13 \qquad\textbf{(E) } 15 $

2000 Turkey Team Selection Test, 1

Show that any triangular prism of infinite length can be cut by a plane such that the resulting intersection is an equilateral triangle.

1998 Akdeniz University MO, 2

$100$ points at a circle with radius $1$ $cm$. Show that, we find an another point such that, this point's distance to other $100$ points is greater than $100$ $cm$.

1998 Gauss, 23

Tags: gauss
A cube measures $10 \text{cm} \times 10 \text{cm} \times10 \text{cm}$ . Three cuts are made parallel to the faces of the cube as shown creating eight separate solids which are then separated. What is the increase in the total surface area? $\textbf{(A)}\ 300 \text{cm}^2 \qquad \textbf{(B)}\ 800 \text{cm}^2 \qquad \textbf{(C)}\ 1200 \text{cm}^2 \qquad \textbf{(D)}\ 600 \text{cm}^2 \qquad \textbf{(E)}\ 0 \text{cm}^2$

Indonesia MO Shortlist - geometry, g1.1

$ABCD$ is a parallelogram. $g$ is a line passing $A$. Prove that the distance from $C$ to $g$ is either the sum or the difference of the distance from $B$ to $g$, and the distance from $D$ to $g$.

2016 Korea Summer Program Practice Test, 2

Tags: geometry
Let the incircle of triangle $ABC$ meet the sides $BC$, $CA$, $AB$ at $D$, $E$, $F$, and let the $A$-excircle meet the lines $BC$, $CA$, $AB$ at $P$, $Q$, $R$. Let the line passing through $A$ and perpendicular to $BC$ meet the lines $EF$, $QR$ at $K$, $L$. Let the intersection of $LD$ and $EF$ be $S$, and the intersection of $KP$ and $QR$ be $T$. Prove that $A$, $S$, $T$ are collinear.

2019 Saudi Arabia Pre-TST + Training Tests, 5.2

Let the bisector of the outside angle of $A$ of triangle $ABC$ and the circumcircle of triangle $ABC$ meet at point $P$. The circle passing through points $A$ and $P$ intersects segments $BP$ and $CP$ at points $E$ and $F$ respectively. Let $AD$ is the angle bisector of triangle $ABC$. Prove that $\angle PED = \angle PFD$. [img]https://cdn.artofproblemsolving.com/attachments/0/3/0638429a220f07227703a682479ed150302aae.png[/img]

1995 Korea National Olympiad, Problem 3

Let $ABC$ be an equilateral triangle of side $1$, $D$ be a point on $BC$, and $r_1, r_2$ be the inradii of triangles $ABD$ and $ADC$. Express $r_1r_2$ in terms of $p = BD$ and find the maximum of $r_1r_2$.

1997 AIME Problems, 13

Let $ S$ be the set of points in the Cartesian plane that satisfy \[ \Big|\big|{|x| \minus{} 2}\big| \minus{} 1\Big| \plus{} \Big|\big|{|y| \minus{} 2}\big| \minus{} 1\Big| \equal{} 1. \] If a model of $ S$ were built from wire of negligible thickness, then the total length of wire required would be $ a\sqrt {b},$ where $ a$ and $ b$ are positive integers and $ b$ is not divisible by the square of any prime number. Find $ a \plus{} b.$

PEN A Problems, 36

Let $n$ and $q$ be integers with $n \ge 5$, $2 \le q \le n$. Prove that $q-1$ divides $\left\lfloor \frac{(n-1)!}{q}\right\rfloor $.

2007 Mexico National Olympiad, 3

Tags: geometry
Let $ABC$ be a triangle with $AB>BC>CA$. Let $D$ be a point on $AB$ such that $CD=BC$, and let $M$ be the midpoint of $AC$. Show that $BD=AC$ and that $\angle BAC=2\angle ABM.$

1997 All-Russian Olympiad Regional Round, 11.5

Members of the State Duma formed factions in such a way that for any two fractions $A $ and $B$ (not necessarily different), $\overline{A \cup B}$ is also faction ($\overline{C}$ denotes the set of all members of the Duma, not including in $C$). Prove that for any two factions $A$ and $B$, $A \cup B$ is also a faction.

2012 Middle European Mathematical Olympiad, 3

In a given trapezium $ ABCD $ with $ AB$ parallel to $ CD $ and $ AB > CD $, the line $ BD $ bisects the angle $ \angle ADC $. The line through $ C $ parallel to $ AD $ meets the segments $ BD $ and $ AB $ in $ E $ and $ F $, respectively. Let $ O $ be the circumcenter of the triangle $ BEF $. Suppose that $ \angle ACO = 60^{\circ} $. Prove the equality \[ CF = AF + FO .\]

2017 IFYM, Sozopol, 1

Find all prime numbers $p$, for which there exist $x, y \in \mathbb{Q}^+$ and $n \in \mathbb{N}$, satisfying $x+y+\frac{p}{x}+\frac{p}{y}=3n$.

2007 Alexandru Myller, 4

Let be a number $ n\ge 2, $ a binary funcion $ b:\mathbb{Z}\rightarrow\mathbb{Z}_2, $ and $ \frac{n^3+5n}{6} $ consecutive integers. Show that among these consecutive integers there are $ n $ of them, namely, $ b_1,b_2,\ldots ,b_n, $ that have the properties: $ \text{(i)} b\left( b_1\right) =b\left( b_2\right) =\cdots =b\left( b_n\right) $ $ \text{(ii)} 1\le b_2-b_1\le b_3-b_2\le \cdots\le b_n-b_{n-1} $

2018 CMIMC CS, 3

You are given the existence of an unsorted sequence $a_1,\ldots, a_5$ of five distinct real numbers. The Erdos-Szekeres theorem states that there exists a subsequence of length $3$ which is either strictly increasing or strictly decreasing. You do not have access to the $a_i$, but you do have an oracle which, when given two indexes $1\leq i < j\leq 5$, will tell you whether $a_i < a_j$ or $a_i > a_j$. What is the minimum number of calls to the oracle needed in order to identify an ordered triple of integers $(r,s,t)$ such that $a_r,a_s,a_t$ is one such sequence?