Olympiad problems

2024 MMATHS, 12

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Let $ABC$ be a triangle with $\angle{A}=60^\circ$ and orthocenter $H.$ Let $B'$ be the reflection of $B$ over $AC,$ $C'$ be the reflection of $C$ over $AB,$ and $A'$ be the intersection of $BC'$ and $B'C.$ Let $D$ be the intersection of $A'H$ and $BC.$ If $BC=5$ and $A'D=4,$ then the area of $\triangle{ABC}$ can be expressed as $a\sqrt{b}+\sqrt{c},$ where $a,b,$ and $c$ are positive integers, and $b$ and $c$ are not divisible by the square of any prime. Find $a+b+c.$

1997 Junior Balkan MO, 2

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Let $\frac{x^2+y^2}{x^2-y^2} + \frac{x^2-y^2}{x^2+y^2} = k$. Compute the following expression in terms of $k$: \[ E(x,y) = \frac{x^8 + y^8}{x^8-y^8} - \frac{ x^8-y^8}{x^8+y^8}. \] [i]Ciprus[/i]

2022 Stanford Mathematics Tournament, 1

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An ant starts at the point $(1,1)$. It can travel along the integer lattice, only moving in the positive $x$ and $y$ directions. What is the number of ways it can reach $(5,5)$ without passing through $(3,3)$?

1988 Tournament Of Towns, (181) 4

There is a set of cards with numbers from $1$ to $30$ (which may be repeated) . Each student takes one such card. The teacher can perform the following operation: He reads a list of such numbers (possibly only one) and then asks the students to raise an arm if their number was in this list. How many times must he perform such an operation in order to determine the number on each student 's card? (Indicate the number of operations and prove that it is minimal . Note that there are not necessarily 30 students.)

2002 Federal Competition For Advanced Students, Part 1, 1

Tags: number theory

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

2013 AMC 12/AHSME, 5

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The average age of $33$ fifth-graders is $11$. The average age of $55$ of their parents is $33$. What is the average age of all of these parents and fifth-graders? $\textbf{(A) }22\qquad\textbf{(B) }23.25\qquad\textbf{(C) }24.75\qquad\textbf{(D) }26.25\qquad\textbf{(E) }28$

2018 Moscow Mathematical Olympiad, 11

Tags: geometry

Ivan want to paint ball. Ivan can put ball in the glass with some paint, and then one half of ball will be painted. Ivan use $5$ glasses to paint glass competely. Prove, that one glass was not needed, and Ivan can paint ball with $4$ glasses, putting ball in it by same way.

2014 Contests, 2

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Paul owes Paula $35$ cents and has a pocket full of $5$-cent coins, $10$-cent coins, and $25$-cent coins that he can use to pay her. What is the difference between the largest and the smallest number of coins he can use to pay her? $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) }5$

2002 Estonia National Olympiad, 5

Tags: combinatorics

There is a lottery at Juku’s birthday party with a number of identical prizes, where each guest can win at most one prize. It is known that if there was one prize less, then the number of possible distributions of the prizes among the guests would be $50\%$ less than it actually is, while if there was one prize more, then the number of possible distributions of the prizes would be $50\%$ more than it actually is. Find the number of possible distributions of the prizes.

2022 Indonesia TST, N

Tags: number theory , infinity , product , prime , polynomial

Let $n$ be a natural number, with the prime factorisation \[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define \[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.

2022 USA TSTST, 7

Tags: parallelogram , angle chasing , trapezoid , geometry

Let $ABCD$ be a parallelogram. Point $E$ lies on segment $CD$ such that \[2\angle AEB=\angle ADB+\angle ACB,\] and point $F$ lies on segment $BC$ such that \[2\angle DFA=\angle DCA+\angle DBA.\] Let $K$ be the circumcenter of triangle $ABD$. Prove that $KE=KF$. [i]Merlijn Staps[/i]

1996 Spain Mathematical Olympiad, 4

Tags: equation , radical , algebra

For each real value of $p$, find all real solutions of the equation $\sqrt{x^2 - p}+2\sqrt{x^2-1} = x$.

1976 AMC 12/AHSME, 9

Tags: geometry , ratio

In triangle $ABC$, $D$ is the midpoint of $AB$; $E$ is the midpoint of $DB$; and $F$ is the midpoint of $BC$. If the area of $\triangle ABC$ is $96$, then the area of $\triangle AEF$ is $\textbf{(A) }16\qquad\textbf{(B) }24\qquad\textbf{(C) }32\qquad\textbf{(D) }36\qquad \textbf{(E) }48$

2019 Romanian Master of Mathematics Shortlist, C3

Tags: divisible , permutation , combinatorics

Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$. Croatia

MOAA Gunga Bowls, 2023.10

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A number is called [i]winning[/i] if it can be expressed in the form $\frac{a}{20}+\frac{b}{23}$ where $a$ and $b$ are positive integers. How many [i]winning[/i] numbers are less than 1? [i]Proposed by Andy Xu[/i]

2009 All-Russian Olympiad, 1

Tags: polynomial , combinatorics , algebra

Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax \plus{} b)^{100} \minus{} (cx \plus{} d)^{100}$ has exactly $ n$ nonzero coefficients.

2012 Brazil Team Selection Test, 4

Tags: parallelogram , circumcircle , power of a point , perpendicular bisector , geometry

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

2013 Sharygin Geometry Olympiad, 2

Tags: circumcircle , geometry

Let $ABC$ be an isosceles triangle ($AC = BC$) with $\angle C = 20^\circ$. The bisectors of angles $A$ and $B$ meet the opposite sides at points $A_1$ and $B_1$ respectively. Prove that the triangle $A_1OB_1$ (where $O$ is the circumcenter of $ABC$) is regular.

2012 Math Prize For Girls Problems, 15

Tags: expected value , probability

Kate has two bags $X$ and $Y$. Bag $X$ contains $5$ red marbles (and nothing else). Bag $Y$ contains $4$ red marbles and $1$ blue marble (and nothing else). Kate chooses one of her bags at random (each with probability $\frac{1}{2}$) and removes a random marble from that bag (each marble in that bag being equally likely). She repeats the previous step until one of the bags becomes empty. At that point, what is the probability that the blue marble is still in bag $Y$?

1982 Bundeswettbewerb Mathematik, 3

Tags: inequalities , algebra , minimum

Given that $a_1, a_2, . . . , a_n$ are nonnegative real numbers with $a_1 + \cdots + a_n = 1$, prove that the expression $$ \frac{a_1}{1+a_2 +a_3 +\cdots +a_n }\; +\; \frac{a_2}{1+a_1 +a_3 +\cdots +a_n }\; +\; \cdots \; +\, \frac{a_n }{1+a_1 +a_2+\cdots +a_{n-1} }$$ attains its minimum, and determine this minimum.

2016 Mathematical Talent Reward Programme, SAQ: P 1

Tags: algebra , polynomial

Show that there exist a polynomial $P(x)$ whose one cofficient is $\frac{1}{2016}$ and remaining cofficients are rational numbers, such that $P(x)$ is an integer for any integer $x$ .

1994 All-Russian Olympiad, 8

Tags: combinatorics

There are $30$ students in a class. In an examination, their results were all different from each other. It is given that everyone has the same number of friends. Find the maximum number of students such that each one of them has a better result than the majority of his friends. PS. Here majority means larger than half.

2024 Czech and Slovak Olympiad III A, 3

Tags: tiling , tiles , combinatorics

Find the largest natural number $n$ such that any set of $n$ tetraminoes, each of which is one of the four shapes in the picture, can be placed without overlapping in a $20 \times 20$ table (no tetramino extends beyond the borders of the table), such that each tetramino covers exactly 4 cells of the 20x20 table. An individual tetramino is allowed to turn and flip at will. [img]https://cdn.artofproblemsolving.com/attachments/b/9/0dddb25c2aa07536b711ded8363679e47972d6.png[/img]

1977 Putnam, B3

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An (ordered) triple $(x_1,x_2,x_3)$ of positive [i]irrational[/i] numbers with $x_1+x_2+x_3=1$ is called [b]balanced[/b] if each $x_i< 1/2.$ If a triple is not balanced, say if $x_j>1/2$, one performs the following [b] balancing act[/b] $$B(x_1,x_2,x_3)=(x'_1,x'_2,x'_3),$$ where $x'_i=2x_i$ if $i\neq j$ and $x'_j=2x_j-1.$ If the new triple is not balanced, one performs the balancing act on it. Does the continuation of this process always lead to a balanced triple after a finite number of performances of the balancing act?

2004 Regional Olympiad - Republic of Srpska, 3

Tags: modular arithmetic , function , arithmetic sequence , number theory

Given a sequence $(a_n)$ of real numbers such that the set $\{a_n\}$ is finite. If for every $k>1$ subsequence $(a_{kn})$ is periodic, is it true that the sequence $(a_n)$ must be periodic?