Olympiad problems

2000 Brazil Team Selection Test, Problem 1

Tags: geometry

Show that if the sides $a, b, c$ of a triangle satisfy the equation \[2(ab^2 + bc^2 + ca^2) = a^2b + b^2c + c^2a + 3abc,\] then the triangle is equilateral. Show also that the equation can be satisfied by positive real numbers that are not the sides of a triangle.

2012 Indonesia TST, 3

Tags: circumcircle , incenter , cyclic quadrilateral , projective geometry , geometry

Given a cyclic quadrilateral $ABCD$ with the circumcenter $O$, with $BC$ and $AD$ not parallel. Let $P$ be the intersection of $AC$ and $BD$. Let $E$ be the intersection of the rays $AB$ and $DC$. Let $I$ be the incenter of $EBC$ and the incircle of $EBC$ touches $BC$ at $T_1$. Let $J$ be the excenter of $EAD$ that touches $AD$ and the excircle of $EAD$ that touches $AD$ touches $AD$ at $T_2$. Let $Q$ be the intersection between $IT_1$ and $JT_2$. Prove that $O,P,Q$ are collinear.

2004 All-Russian Olympiad, 1

Tags: geometry , geometric transformation , rotation , analytic geometry , combinatorics , combinatorial geometry

Each grid point of a cartesian plane is colored with one of three colors, whereby all three colors are used. Show that one can always find a right-angled triangle, whose three vertices have pairwise different colors.

2014 Purple Comet Problems, 4

Tags: vector , algorithm

Find the least positive integer $n$ such that the prime factorizations of $n$, $n + 1$, and $n + 2$ each have exactly two factors (as $4$ and $6$ do, but $12$ does not).

2015 India Regional MathematicaI Olympiad, 1

Tags: rearrangement inequality , am-gm , geometry , inequalities

Let $ABCD$ be a convex quadrilateral with $AB=a$, $BC=b$, $CD=c$ and $DA=d$. Suppose \[a^2+b^2+c^2+d^2=ab+bc+cd+da,\] and the area of $ABCD$ is $60$ sq. units. If the length of one of the diagonals is $30$ units, determine the length of the other diagonal.

2022 Thailand TST, 3

Tags: number theory , sequence

Let $a_1,a_2,a_3,\ldots$ be an infinite sequence of positive integers such that $a_{n+2m}$ divides $a_{n}+a_{n+m}$ for all positive integers $n$ and $m.$ Prove that this sequence is eventually periodic, i.e. there exist positive integers $N$ and $d$ such that $a_n=a_{n+d}$ for all $n>N.$

2025 China Team Selection Test, 5

Tags: combinatorics

There are $2025$ people and $66$ colors, where each person has one ball of each color. For each person, their $66$ balls have positive mass summing to one. Find the smallest constant $C$ such that regardless of the mass distribution, each person can choose one ball such that the sum of the chosen balls of each color does not exceed $C$.

1968 AMC 12/AHSME, 20

Tags:

The measures of the interior angles of a convex polygon of $n$ sides are in arithmetic progression. If the common difference is $5^\circ$ and the largest angle is $160^\circ$, then $n$ equals: $\textbf{(A)}\ 9\qquad \textbf{(B)}\ 10\qquad \textbf{(C)}\ 12\qquad \textbf{(D)}\ 16\qquad \textbf{(E)}\ 32 $

2020 AMC 10, 17

Tags: counting

There are 10 people standing equally spaced around a circle. Each person knows exactly 3 of the other 9 people: the 2 people standing next to her or him, as well as the person directly across the circle. How many ways are there for the 10 people to split up into 5 pairs so that the members of each pair know each other? $\textbf{(A) } 11 \qquad \textbf{(B) } 12 \qquad \textbf{(C) } 13 \qquad \textbf{(D) } 14 \qquad \textbf{(E) } 15$

1967 Putnam, B2

Tags: inequalities

Let $0\leq p,r\leq 1$ and consider the identities $$a)\; (px+(1-p)y)^{2}=a x^2 +bxy +c y^2, \;\;\;\, b)\; (px+(1-p)y)(rx+(1-r)y) =\alpha x^2 + \beta xy + \gamma y^2.$$ Show that $$ a)\; \max(a,b,c) \geq \frac{4}{9}, \;\;\;\; b)\; \max( \alpha, \beta , \gamma) \geq \frac{4}{9}.$$

1998 Mexico National Olympiad, 2

Tags: geometry , locus , circles

Rays $l$ and $m$ forming an angle of $a$ are drawn from the same point. Let $P$ be a fixed point on $l$. For each circle $C$ tangent to $l$ at $P$ and intersecting $m$ at $Q$ and $R$, let $T$ be the intersection point of the bisector of angle $QPR$ with $C$. Describe the locus of $T$ and justify your answer.

2023 Malaysian IMO Training Camp, 3

Tags: geometry

Let triangle $ABC$ with $AB<AC$ has orthocenter $H$, and let the midpoint of $BC$ be $M$. The internal angle bisector of $\angle BAC$ meet $CH$ at $X$, and the external angle bisector of $\angle BAC$ meet $BH$ at $Y$. The circles $(BHX)$ and $(CHY)$ meet again at $Z$. Prove that $\angle HZM=90^{\circ}$. [i]Proposed by Ivan Chan Kai Chin[/i]

1979 Bulgaria National Olympiad, Problem 5

Tags: triangle , geometry , pentagon

A convex pentagon $ABCDE$ satisfies $AB=BC=CA$ and $CD=DE=EC$. Let $S$ be the center of the equilateral triangle $ABC$ and $M$ and $N$ be the midpoints of $BD$ and $AE$, respectively. Prove that the triangles $SME$ and $SND$ are similar.

2010 Dutch IMO TST, 4

Tags: geometry , square , circles , equal segments

Let $ABCD$ be a square with circumcircle $\Gamma_1$. Let $P$ be a point on the arc $AC$ that also contains $B$. A circle $\Gamma_2$ touches $\Gamma_1$ in $P$ and also touches the diagonal $AC$ in $Q$. Let $R$ be a point on $\Gamma_2$ such that the line $DR$ touches $\Gamma_2$. Proof that $|DR| = |DA|$.

2019 Czech-Polish-Slovak Junior Match, 5

Tags: combinatorics

Given is a group in which everyone has exactly $d$ friends and every two strangers have exactly one common friend. Prove that there are at most $d^2 + 1$ people in this group.

2023 AMC 10, 6

Tags: parity , sequence

Let $L_1=1$, $L_2=3$, and $L_{n+2}=L_{n+1}+L_n$ for $n\geq1$. How many terms in the sequence $L_1, L_2, L_3, \dots, L_{2023}$ are even? $\textbf{(A) }673\qquad\textbf{(B) }1011\qquad\textbf{(C) }675\qquad\textbf{(D) }1010\qquad\textbf{(E) }674$

2005 Turkey Team Selection Test, 1

Tags: function , algebra

Find all functions $ f :\mathbb{R}_{0}^{+}\mapsto\mathbb{R}_{0}^{+} $ satisfying the conditions $4f(x)\geq 3x$ and $f(4f(x)-3x)=x$ for all $x\geq 0$ .

VI Soros Olympiad 1999 - 2000 (Russia), 8.3

Tags: number theory , digit

$72$ was added to the natural number $n$ and in the sum we got a number written in the same digits as the number $n$, but in the reverse order. Find all numbers $n$ that satisfy the given condition.

STEMS 2023 Math Cat A, 1

Tags: combinatorics

The following $100$ numbers are written on the board: $$2^1 - 1, 2^2 - 1, 2^3 - 1, \dots, 2^{100} - 1.$$ Alice chooses two numbers $a,b,$ erases them and writes the number $\dfrac{ab - 1}{a+b+2}$ on the board. She keeps doing this until a single number remains on the board. If the sum of all possible numbers she can end up with is $\dfrac{p}{q}$ where $p, q$ are coprime, then what is the value of $\log_{2}(p+q)$?

2020 MBMT, 40

Tags:

Wu starts out with exactly one coin. Wu flips every coin he has [i]at once[/i] after each year. For each heads he flips, Wu receives a coin, and for every tails he flips, Wu loses a coin. He will keep repeating this process each year until he has $0$ coins, at which point he will stop. The probability that Wu will stop after exactly five years can be expressed as $\frac{a}{2^b}$, where $a, b$ are positive integers such that $a$ is odd. Find $a+b$. [i]Proposed by Bradley Guo[/i]

2019 District Olympiad, 4

Tags: inequality , algebra

Find the smallest positive real number $\lambda$ such that for every numbers $a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right]$ and $b_1,b_2,b_3 \in (0, \infty)$ with $\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1,$ we have $$b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3).$$

2023 Turkey MO (2nd round), 6

Tags: geometry , circumcircle , symmedian

On a triangle $ABC$, points $D$, $E$, $F$ are given on the segments $BC$, $AC$, $AB$ respectively such that $DE \parallel AB$, $DF \parallel AC$ and $\frac{BD}{DC}=\frac{AB^2}{AC^2}$ holds. Let the circumcircle of $AEF$ meet $AD$ at $R$ and the line that is tangent to the circumcircle of $ABC$ at $A$ at $S$ again. Let the line $EF$ intersect $BC$ at $L$ and $SR$ at $T$. Prove that $SR$ bisects $AB$ if and only if $BS$ bisects $TL$.

2023 Math Prize for Girls Problems, 16

Tags:

Let $f(x) = x^2 - 3/4$. Let $f^{(n)}(x)$ denote the composition of $f$ with itself $n$ times. For example, $f^{(3)}(x) = f(f(f(x)))$. Let $R$ be the set of complex numbers that is the union of the roots of the polynomials $f^{(n)}(x^2 + 3/4)$ over positive integers $n$. Let $B$ be the smallest rectangle in the complex plane with sides parallel to the real and imaginary axes that contains $R$. What is the square of the area of $B$?

2011 AMC 12/AHSME, 5

Tags:

Let $N$ be the second smallest positive integer that is divisible by every positive integer less than 7. What is the sum of the digits of $N$? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 9 $

2024 HMIC, 1

Tags: combinatorics

In an empty $100 \times 100$ grid, $300$ cells are colored blue, $3$ in each row and each column. Compute the largest positive integer $k$ such that you can always recolor $k$ of these blue cells red so that no contiguous $2 \times 2$ square has four red cells. [i]Arul Kolla[/i]