Found problems: 85335
1993 AMC 8, 6
A can of soup can feed $3$ adults or $5$ children. If there are $5$ cans of soup and $15$ children are fed, then how many adults would the remaining soup feed?
$\text{(A)}\ 5 \qquad \text{(B)}\ 6 \qquad \text{(C)}\ 7 \qquad \text{(D)}\ 8 \qquad \text{(E)}\ 10$
the 11th XMO, 9
$x,y\in\mathbb{R},(4x^3-3x)^2+(4y^3-3y)^2=1.\text { Find the maximum of } x+y.$
EGMO 2017, 3
There are $2017$ lines in the plane such that no three of them go through the same point. Turbo the snail sits on a point on exactly one of the lines and starts sliding along the lines in the following fashion: she moves on a given line until she reaches an intersection of two lines. At the intersection, she follows her journey on the other line turning left or right, alternating her choice at each intersection point she reaches. She can only change direction at an intersection point. Can there exist a line segment through which she passes in both directions during her journey?
1970 IMO Longlists, 16
Show that the equation $\sqrt{2-x^2}+\sqrt[3]{3-x^3}=0$ has no real roots.
2010 Malaysia National Olympiad, 1
Triangles $OAB,OBC,OCD$ are isoceles triangles with $\angle OAB=\angle OBC=\angle OCD=\angle 90^o$. Find the area of the triangle $OAB$ if the area of the triangle $OCD$ is 12.
Novosibirsk Oral Geo Oly IX, 2017.3
Medians $AA_1, BB_1, CC_1$ and altitudes $AA_2, BB_2, CC_2$ are drawn in triangle $ABC$ . Prove that the length of the broken line $A_1B_2C_1A_2B_1C_2A_1$ is equal to the perimeter of triangle $ABC$.
2010 All-Russian Olympiad, 1
There are $24$ different pencils, $4$ different colors, and $6$ pencils of each color. They were given to $6$ children in such a way that each got $4$ pencils. What is the least number of children that you can randomly choose so that you can guarantee that you have pencils of all colors.
P.S. for 10 grade gives same problem with $40$ pencils, $10$ of each color and $10$ children.
2020 ASDAN Math Tournament, 6
Triangle $\vartriangle ABC$ has side lengths $AB = 26$, $BC = 34$, and $CA = 24\sqrt2$. A fourth point $D$ makes a right angle $\angle BDC$. What is the smallest possible length of $\overline{AD}$?
1970 AMC 12/AHSME, 25
For every real number $x$, let $[x]$ be the greatest integer less than or equal to $x$. If the postal rate for first class mail is six cents for every ounce or portion thereof, then the cost in cents of first-class postage on a letter weighing $W$ ounces is always
$\textbf{(A) }6W\qquad\textbf{(B) }6[W]\qquad\textbf{(C) }6([W]-1)\qquad\textbf{(D) }6([W]+1)\qquad \textbf{(E) }-6[-W]$
1971 IMO Longlists, 52
Prove the inequality
\[ \frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4, \]
where $a_i > 0, i = 1, 2, 3, 4.$
2014 Purple Comet Problems, 1
In the diagram below $ABCD$ is a square and both $\triangle CFD$ and $\triangle CBE$ are equilateral. Find the degree measure of $\angle CEF$.
[asy]
size(4cm);
pen dps = fontsize(10);
defaultpen(dps);
pair temp = (1,0);
pair B = (0,0);
pair C = rotate(45,B)*temp;
pair D = rotate(270,C)*B;
pair A = rotate(270,D)*C;
pair F = rotate(60 ,D)*C;
pair E = rotate(60 ,C)*B;
label("$B$",B,SW*.5);
label("$C$",C,W*2);
label("$D$",D,NW*.5);
label("$A$",A,W);
label("$F$",F,N*.5);
label("$E$",E,S*.5);
draw(A--B--C--D--cycle^^D--F--C--E--B^^F--E);
[/asy]
2014 IMO Shortlist, N8
For every real number $x$, let $||x||$ denote the distance between $x$ and the nearest integer.
Prove that for every pair $(a, b)$ of positive integers there exist an odd prime $p$ and a positive integer $k$ satisfying \[\displaystyle\left|\left|\frac{a}{p^k}\right|\right|+\left|\left|\frac{b}{p^k}\right|\right|+\left|\left|\frac{a+b}{p^k}\right|\right|=1.\]
[i]Proposed by Geza Kos, Hungary[/i]
2021 OMpD, 5
Let $ABC$ be a triangle with $\angle BAC > 90^o$ and with $AB < AC$. Let $r$ be the internal bisector of $\angle ACB$ and let $s$ be the perpendicular, through $A$, on $r$. Denote by $F$ the intersection of $r$ and $ s$, and denote by $E$ the intersection of $s$ with the segment $BC$. Let also $D$ be the symmetric of $A$ with respect to the line $BF$. Assuming that the circumcircle of triangle $EAC$ is tangent to line $AB$ and $ D$ lies on $r$, determine the value of $\angle CDB$.
2015 District Olympiad, 3
Find $ \#\left\{ (x,y)\in\mathbb{N}^2\bigg| \frac{1}{\sqrt{x}} -\frac{1}{\sqrt{y}} =\frac{1}{2016}\right\} , $ where $ \# A $ is the cardinal of $ A . $
2023 Indonesia Regional, 2
Let $K$ be a positive integer such that there exist a triple of positive integers $(x,y,z)$ such that
\[x^3+Ky , y^3 + Kz, \text{and } z^3 + Kx\]
are all perfect cubes.
(a) Prove that $K \ne 2$ and $K \ne 4$
(b) Find the minimum value of $K$ that satisfies.
[i]Proposed by Muhammad Afifurrahman[/i]
2015 Turkey MO (2nd round), 6
Find all positive integers $n$ such that for any positive integer $a$ relatively prime to $n$, $2n^2 \mid a^n - 1$.
1982 Miklós Schweitzer, 5
Find a perfect set $ H \subset [0,1]$ of positive measure and a continuous function $ f$ defined on $ [0,1]$ such that for any twice differentiable function $ g$ defined on $ [0,1]$, the set $ \{ x \in H : \;f(x)\equal{}g(x)\ \}$ is finite.
[i]M. Laczkovich[/i]
2017 Brazil Team Selection Test, 5
Let $n$ be a positive integer. A pair of $n$-tuples $(a_1,\cdots{}, a_n)$ and $(b_1,\cdots{}, b_n)$ with integer entries is called an [i]exquisite pair[/i] if
$$|a_1b_1+\cdots{}+a_nb_n|\le 1.$$
Determine the maximum number of distinct $n$-tuples with integer entries such that any two of them form an exquisite pair.
[i]Pakawut Jiradilok and Warut Suksompong, Thailand[/i]
2014 India Regional Mathematical Olympiad, 6
In the adjacent fi gure, can the numbers $1,2,3, 4,..., 18$ be placed, one on each line segment, such that the sum of
the numbers on the three line segments meeting at each point is divisible by $3$?
2023 Belarus Team Selection Test, 4.1
A $\pm 1$-[i]sequence[/i] is a sequence of $2022$ numbers $a_1, \ldots, a_{2022},$ each equal to either $+1$ or $-1$. Determine the largest $C$ so that, for any $\pm 1$-sequence, there exists an integer $k$ and indices $1 \le t_1 < \ldots < t_k \le 2022$ so that $t_{i+1} - t_i \le 2$ for all $i$, and $$\left| \sum_{i = 1}^{k} a_{t_i} \right| \ge C.$$
2012 Baltic Way, 8
A directed graph does not contain directed cycles. The number of edges in any directed path does not exceed 99. Prove that it is possible to colour the edges of the graph in 2 colours so that the number of edges in any single-coloured directed path in the graph will not exceed 9.
2017 Middle European Mathematical Olympiad, 5
Let $ABC$ be an acute-angled triangle with $AB > AC$ and circumcircle $\Gamma$. Let $M$ be the midpoint of the shorter arc $BC$ of $\Gamma$, and let $D$ be the intersection of the rays $AC$ and $BM$. Let $E \neq C$ be the intersection of the internal bisector of the angle $ACB$ and the circumcircle of the triangle $BDC$. Let us assume that $E$ is inside the triangle $ABC$ and there is an intersection $N$ of the line $DE$ and the circle $\Gamma$ such that $E$ is the midpoint of the segment $DN$.
Show that $N$ is the midpoint of the segment $I_B I_C$, where $I_B$ and $I_C$ are the excentres of $ABC$ opposite to $B$ and $C$, respectively.
2024 Assara - South Russian Girl's MO, 1
There is a set of $2024$ cards. Each card on both sides is colored in one of three colors — red, blue or white, and for each card its two sides are colored in different colors. The cards were laid out on the table. The card [i]lies beautifully[/i] if at least one of two conditions is met: its upper side — red; its underside is blue. It turned out that exactly $150$ cards are lying beautifully. Then all the cards were turned over. Now some of the cards are lying beautifully on the table. How many of them can there be?
[i]K.A.Sukhov[/i]
2001 Tournament Of Towns, 3
Let $n\ge3$ be an integer. Each row in an $(n-2)\times n$ array consists of the numbers 1,2,...,$n$ in some order, and the numbers in each column are all different. Prove that this array can be expanded into an $n\times n$ array such that each row and each column consists of the numbers 1,2,...,$n$.
1996 Baltic Way, 2
In the figure below, you see three half-circles. The circle $C$ is tangent to two of the half-circles and to the line $PQ$ perpendicular to the diameter $AB$. The area of the shaded region is $39\pi$, and the area of the circle $C$ is $9\pi$. Find the length of the diameter $AB$.